Penrose Roger - The Emperor's New Mind_ Concerning Computers, Minds, and the Laws of Physics

Introduction to AI and Consciousness

The text discusses the question of whether computers can have minds or consciousness
It explores the concept of artificial intelligence (AI) and its implications
The Turing test is introduced as a way to evaluate machine intelligence

The Turing Test

Proposed by Alan Turing in 1950
Involves an interrogator trying to distinguish between a human and a computer through text-based conversation
If the interrogator cannot consistently identify the computer, it passes the test
The test is meant to evaluate whether a machine can exhibit intelligent behavior

Strong AI

A philosophical position that claims a sufficiently advanced computer program can have genuine consciousness and understanding
Argues that mental states are essentially computational processes that can be replicated in any suitable hardware
Contrasts with weak AI, which only claims computers can simulate intelligent behavior

Searle's Chinese Room Argument

Thought experiment proposed by philosopher John Searle to challenge strong AI
Imagines a person who doesn't understand Chinese following instructions to respond to Chinese messages
Argues that syntax (following rules) is not sufficient for semantics (understanding)
Aims to show that running a program is not equivalent to understanding

Hardware vs. Software

Hardware refers to physical components of a computer
Software refers to programs and algorithms run on the hardware
Strong AI view emphasizes software/algorithms as the essence of mind, downplaying hardware

Philosophical Issues

Questions the nature of consciousness, understanding, and personal identity
Explores whether consciousness can be reduced to information processing
Considers thought experiments like teleportation and copying minds

Criticisms of Strong AI

Searle and others argue genuine understanding requires more than just information processing
Questions whether simulating intelligence is the same as having intelligence
Raises issues about the importance of biological brains vs. artificial hardware

Unresolved Questions

The text leaves open many philosophical and empirical questions about machine consciousness
Suggests the need for further developments in neuroscience and physics to address these issues
Implies that current AI may be far from genuine machine consciousness or understanding

Prologue
1 Can A Computer Have A Mind?
2 Algorithms And Turing Machines
3 Mathematics And Reality
4 Truth, Proof, And Insight
5 The Classical World
6 Quantum Magic And Quantum Mystery
7 Cosmology And The Arrow Of Time
8 In Search Of Quantum Gravity
9 Real Brains And Model Brains
10 Where Lies The Physics Of Mind?

Prologue

The Great Gathering and the Ultronic Computer

President Pollo's Speech:

Marked the initiation of the new Ultronic computer
President was not fond of such occasions, but knew it would save him time
He had spent a great deal of treasury gold on the computer, hoping it would take over "awkward decisions"

Adam's Perspective:

Sat in the third row
His mother, a chief technocrat, was two rows in front of him
His father was uninvited and surrounded by security guards because he had tried to bomb the computer
Adam had little feeling for either parent and had been raised almost entirely by computers since birth

The Chief Designer's Speech:

The Ultronic computer has over 10^17 logical units, more than the number of neurons in everyone in the country
It will be "unimaginable" in its intelligence, but they did not need to imagine it
The First Lady, Madame Isabella Pollo, threw the switch to turn on the computer
The Chief Designer asked if anyone had a question for the new Ultronic system

Adam's Question:

Adam raised his hand and was the rst to ask a question
He was curious about what it might be like to "be a computer" since he had grown up with computers

1 Can A Computer Have A Mind?

Computers and Minds: Understanding the Relation Between Physical Laws, Mathematics, Conscious Thinking, and the Nature of Minds

Introduction:

Discusses the impact of computer technology on human thinking capabilities
Questions whether a mechanical device can have a mind or experience feelings
Examines deep philosophical issues: meaning of thinking/feeling, nature of minds, existence of minds, and their relationship with physical structures.

Modern Computer Technology:

Rapid advancements in computer technology
Machines perform tasks previously exclusive to human thinking
Questions raised about the implications for human superiority if machines can outperform us mentally

Historical Context:

Question of whether a mechanical device can think or have a mind is not new
Receives renewed urgency with modern computer technology development
Issues touched upon include: nature of mathematics, conscious thinking, and the relationship between physical structures and minds.

Philosophical Questions:

What does it mean to think or feel?
What is a mind?
Do minds exist?
Are minds functionally dependent on physical structures?
Can minds exist independently of such structures?
Must relevant structures be biological (brains) or electronic?
Are minds subject to the laws of physics?
What are the fundamental laws of physics?
What is the relationship between thought and consciousness, mathematics, and physical laws?
Can we come to grips with the concept of mind in physical or logical terms?

Unconventional Viewpoint:

The author's point of view is unconventional among physicists
Argues that our current lack of understanding of fundamental physics prevents us from fully comprehending the nature of minds.

Challenges in Physics:

Understanding the laws governing subatomic particles and their interactions
Reconciling quantum theory with special relativity
Constructing a consistent quantum gravity theory
Understanding physics at extremely small scales (less than 1/100000000000000000000 of the size of known fundamental particles)
Understanding black hole physics and the nature of space at their core
Determining whether the universe is finite or infinite in extent.

Author's Purpose:

To raise new issues concerning the relation between physical laws, mathematics, conscious thinking, and mind operation
To present a viewpoint that has not been widely expressed before
To attempt to stimulate future research and make specific suggestions about the potential role of mind within physics development.

Turing Test Operationalizing Consciousness and Intelligence

The Turing Test and Artificial Intelligence

Background:

Imagining a new, intelligent computer with human attributes
Manufacturers claim it thinks, feels, understands, etc.
Operational view: machine passes if acts indistinguishably from human being
Originated in Alan Turing's article "Computing Machinery and Intelligence" (1950)

The Test:

Computer and human hidden, interrogator asks questions
Answers are impersonal, no other info about subjects
Human answers truthfully while trying to persuade as human
Computer lies convincingly to appear human
Passes if interrogator cannot identify real human consistently

Problems:

Unfair test for computer?
- If roles reversed, human could outperform with complex calculations.

However, making a computer appear stupider might not be a serious problem for its programmers.

Artificial Intelligence and the Turing Test Ethical Implications

Challenges in Assessing Computer Understanding:

Simples Questions: Computers struggle with questions requiring real understanding or original thought.
Examples of Questions: Rhinoceros questions, nonsensical queries can reveal a computer's limitations.
Turing Test: Debate over whether passing this test indicates consciousness or just imitation.
- Manufacturers' claims: Moral responsibilities and implications if true.
- Personal judgments based on conversation vs. objective test results.
- Perceptive interrogator's role in deciding computer's consciousness.
- Original Turing test's advantages for objectivity despite biases against computers.
Future Expectations: Supporters believe computers will pass the test by 2010 or sooner.

Artificial Intelligence Imitating Human Mental Capacities

Turing Test and Mental Qualities in Computers

Importance of Imitation vs. Real Thing

Imitation (e.g., Turing test) may not be equivalent to the real thing
Personal position: imitation is a rough guide, but ultimately requires further evidence

Criteria for Judging Consciousness in Computers

Successful theory of consciousness required for making judgments
Hypothetical consciousness detector could provide contrary evidence
Approach based on evidence and faith in science/optimism

Artificial Intelligence (AI)

Interest in AI:

Robotics: industrial applications, mechanical devices performing intelligent tasks
Expert systems: replacing human expertise with factual information and comprehensive cross-referencing
Psychological insights: studying brain function through electronic devices or failures to imitate intelligence
Philosophical questions: potential contributions to understanding the concept of mind

Progress in AI:

Early achievements: W. Grey Walters tortoise, Terry Winograd's program
Difficulty in designing control systems for simple tasks (e.g., robot arm)
Impressive early successes but genuine intelligence still elusive

AI Simulation of Pleasure and Pain in Machines Tortoise Example

AI and Human-Like Behavior: Chess and Pleasure/Pain

Computer Programs Simulating Intelligence:

Colby's psychotherapist program: successful in deceiving patients, but has no understanding (mid-1960s)
Chess computers: achieving respectable levels of performance (2110 Elo rating for Fidelity Excel, 2500 Elo rating for Deep Thought)
Computers excel at solving chess problems and outstrip humans in calculation
Human judgment plays a crucial role in achieving greater depth during analysis

AI Approach to Pleasure and Pain:

Grey Walters tortoise: behavior change based on battery charge level (analogous to human hunger)
AI's potential understanding of mental qualities like happiness, pain, or hunger
Simplified example with a device registering its pp-score and maximizing it through various factors influencing the score.

Computer Programs and Pleasure/Pain:

Device has means to register its own pp-score (pleasure-pain)
Behavior geared towards maximizing the pp-score
Factors influencing pp-score: battery charge, solar panels, moving towards light.

AIs Approach to Mimicking Human Emotions and Desires

Artificial Intelligence and Pleasure-Pain Scale (pp-score)

Designing an AI Device:

Introduce probability weightings based on data reliability
Provide goals beyond maintaining energy supply: companionship, learning for own sake, performing services
Identifying pleasure and pain with pp-score as a reasonable approximation of human behavior

Operationalist Perspective:

Claim that device feels pleasure when its score is positive and pain when it's negative based on its actions
Justification comes from the fact that humans react similarly to pain and pleasure, but actions are influenced by more complex criteria

Understanding Human Behavior:

Influences driving behavior: conscious (pain, pleasure) and unconscious (involuntary actions)
Analogies between AI device's pp-score and human behavior helpful for understanding its behavior
More to the understanding of mental qualities than can be directly obtained from AI

AI Philosophy:

Simulation of actual intelligence: a serious case that must be respected and reckoned with
Improvements in technology, design, and programming techniques will lead to better results
Potential for creating truly intelligent devices or new principles needed

Chinese Room Argument and Strong AI Controversy

Strong AI and Searle's Chinese Room Argument

Strong AI:

Extreme position on artificial intelligence (AI)
Mental activity is simply the carrying out of a well-defined sequence of operations, an algorithm
All mental qualities (thinking, feeling, intelligence, understanding, consciousness) are aspects of this complicated functioning

Arguments against Strong AI:

Chinese Room Argument:
- A computer program can pass Turing tests by mimicking human responses to stories
- However, it does not demonstrate genuine understanding or consciousness on the part of the computer or algorithm
- Searle's example: a computer program designed by Roger Schank can correctly answer questions about simple stories in Chinese without actually understanding them. The program is just manipulating symbols according to its instructions (algorithm) given in English.
Critique:
- A successful algorithm does not necessarily imply genuine understanding or consciousness.

Searles Chinese Room Argument and Objections against Strong AI

Imagined Searle in Chinese Room

Couldn't understand single word of stories in Chinese

Objections to Searle's Argument:

Pattern Perception: Understanding involves patterns as much as individual words. One might perceive story patterns without understanding individual symbols.
Algorithm Length and Complexity: Executing a simple computer program is lengthy and tedious for humans, but not a serious objection to the principle argument.
Critical Amount of Complication: There might be a critical level of algorithm complexity required to exhibit mental qualities, which no human could carry out.
Team of Symbol Manipulators: Replacing single person with a team or even entire India does not alter the fact that individuals do not understand Chinese stories.
Country vs. Brain vs. Thermostat: Country, like thermostat, doesn't understand stories; strong AI argues only algorithm matters, not physical embodiment.
Algorithm versus Understanding: Strong AI asserts understanding comes from algorithms alone, which Searle argues is a form of dualism and lacks evidence.

Searle's Reply:

Individual neurons don't understand thoughts, same for symbol manipulators in Chinese room.
India as a whole can't understand stories no one does.
Countries, thermostats, etc., are not in the business of understanding; only individuals can.

Strong AI's Position:

Algorithm logic structure is significant for mental state representation regardless of physical embodiment.

Arguments Against Strong AI:

No evidence provided that algorithms alone can embody genuine understanding or consciousness.
Dualism raises questions about how mind and matter interact if they are separate substances.

The Philosophical Dilemma of Strong AI

The Strong AI Debate: Mind vs Algorithm

Background:

Mind is not composed of matter and can exist independently
Strong AI believes mind is a logical structure of an algorithm
Algorithms have a disembodied existence, apart from physical realization

Irony in Strong AI's Position:

Supporters of strong AI take the reality of algorithms seriously
This leads to extreme dualism, contradicting their desired perspective

Hofstadter's Dialogue with Einstein's Brain:

Book containing a complete description of Einstein's brain can answer questions like him
Questions answered through detailed instructions in book
Not a mere misnomer but equivalent to Einstein in operational sense

Issues Arising from Strong AI:

Activating an algorithm vs embodying it in physical form
Changing algorithms and mental events association
Awareness of the book-Einstein when unopened or disturbed
Multiple instances of the same state of awareness
Role of changes in algorithms in mental events
Self-awareness of the book-Einstein if never examined or disturbed

Response from Searle:

Brains are digital computers, but they can have intentionality and semantics
Distinction between biological objects (brains) and electronic ones lies in material construction
Reasons for this distinction not well explained by Searle
Suspicion that this assertion is dogmatic and lacks scientific theory of mind.

Computers Identity and Quantum Physics Material Replacement and Unique Arrangement

Background

Widespread belief that everything is a digital computer (physicists led astray)
Two concepts: hardware and software
- Hardware: machinery involved in a computer, including specification for connections
- Software: programs run on the machine
Alan Turing's discovery of universal Turing machines
- Any two such machines are equivalent through specific pieces of software
- Modern computers are universal Turing machines
- Differences between them are only in software (speed, storage)

Importance of Software vs Hardware

Strong-AI philosophy: hardware is unimportant, software is vital
Underlying factors from physics

Identity and Material Composition

Identity not dependent on specific atoms or particles
- Continual turnover in living body's material
- Quantum mechanics: all particles must be identical

Pattern of Arrangement vs Individuality of Constituents

Person's identity depends on pattern of how constituents are arranged, not individuality itself
Analogy with everyday technology and electronic word processor.

Quantum Identity and Strong AI A Discussion on Individuality and Consciousness

Concepts Related to Individuality and Quantum Mechanics:

Classical vs. quantum physics: atoms as individuals in classical physics, but not in quantum mechanics (distinction between same and indistinguishable)
Persons individuality: configuration of material constituents, not limited to individual objects or atoms
Strong AI argument for preserving individuality: information content can be translated into another form and recovered without loss
Analogy with sequences of letters on screen or magnetic storage media
Claims of strong AI supporters: consciousness as software running on hardware (brain and body)
Discussion of teleportation machine in science fiction, debated as transportation or creation of a duplicate.

Classical vs. Quantum Physics:

In classical physics: atoms can be treated as individual objects when reasonably well separated
Atoms maintain individual identities by being continuously tracked and tabulated
From quantum mechanics perspective, the distinction is not trivial but convenience of speech at this level

Individuality and Configuration:

Individuality not tied to any specific configuration of material constituents
In quantum mechanics, replacing one particle by an identical one does not change the situation

Strong AI and Information Preservation:

If information content can be translated into another form and recovered, then a person's individuality remains intact
Analogy with sequences of letters on display screen or magnetic storage media

Consciousness as Software:

Supporters of strong AI claim that consciousness is to be taken as software running on hardware (brain and body)
Human manifestation seen as the operation of this software by the hardware.

Teleportation Machine Debate:

Teleportation machine in science fiction: intended for transportation from one planet to another
Questions raised about its distinction from creating a duplicate and destroying the original copy
Discussion on reliability, preparation, and ethical considerations if shown to be completely reliable.

Exploring Quantum Consciousness in Teleportation

Teleportation and Strong AI

Identity of Atoms:

Preservation of identity not meaningful for atoms
Moving pattern of atoms constitutes a wave of information

Paradoxical Situation:

Traveller's awareness in two places at once after teleportation?
Laws of physics do not make teleportation impossible in principle
Question is whether preserving two viable copies is impossible

Role of Quantum Mechanics:

Something gained about the physical nature of consciousness and individuality
More to be discussed after examining quantum theory in Chapter 6

Strong AI and Teleportation:

Strong AI supporters believe awareness would be present in magnetic or electronic device
All relevant physics can be modeled by digital calculations according to strong AI
Question rests on equivalence of universal Turing machines and the ability of brain to act according to algorithmic action.

Notes:

Gardner (1958), Gregory (1981) for information on calculating prodigies
Resniko and Wells (1984), Rouse Ball (1892), Smith (1983) for more details about computer chess
OConnell (1988), Keene (1988) for articles on computer chess
Levy (1984) for more information about computer chess problems
Searle's criticism of strong AI from an operational viewpoint
Turing test and the equivalence of universal Turing machines and algorithms
Strong AI terminology vs functionalism, some proponents include Minsky (1968), Fodor (1983), Hofstadter (1979), Moravec (1989)
Searle's original paper criticized by Hofstadter for internalizing another human being's mind in its entirety
Human beings cannot internalize the entire description of another human being's mind, but the point is to execute that part of an algorithm which embodies a mental event
Searle's criticism of strong AI and the distinction between human and machine intelligence
Rotating one electron completely through 360 degrees when making interchange eliminates this signi cant difference for strong AI proponents.

2 Algorithms And Turing Machines

Algorithm Concept Background

Algorithm: procedure or set of instructions for solving a problem, often with a finite number of steps
Origins: derived from Al-Khowrizmi's mathematical textbook "Kitab al-jabr wal-muqabala" (c. 825 AD)
Named after Muhammad ibn Musa al-Khowarzmi, Persian mathematician
Ancient examples: Euclid's algorithm for finding the highest common factor of two numbers

Euclid's Algorithm

Procedure for finding the highest common factor (HCF) of two numbers A and B
Divide one number by another, find remainder R
Replace original numbers with new pair (R, A) and repeat until a clear answer is obtained
Last number divided is HCF
Clear cut at each step what operation to perform and decision on termination
Description can be presented in finite terms despite unlimited size of natural numbers
Can be represented by a flowchart

Remainder Operation Algorithm

Obtaining remainder when A is divided by B
Keep removing successions of marks representing B from A until no more operations can be performed
Last remaining succession provides answer
Example: finding the remainder of 17 divided by 5 (two)

Subroutines in Algorithm Programming

Subroutine: a previously known algorithm called upon and used as part of another algorithm's operation
Euclid's algorithm includes a subroutine for finding a remainder.

Notes:

The text discusses the concept of algorithms, with a focus on Euclid's algorithm for finding the highest common factor of two numbers
Algorithms are procedures or sets of instructions for solving problems
Origins in ancient Greek mathematics (Euclid's algorithm) and named after Al-Khowrizmi, a Persian mathematician
Euclid's algorithm is an example of a clear cut procedure with finite steps that can be presented in finite terms
The text also discusses the concept of subroutines, which are algorithms called upon as part of another algorithm's operation.

Turing Machines Idealized Calculation Devices

The Concept of a Mechanical Procedure

Ancient Algorithms vs. Modern Formulation:

Euclid's algorithm is just one among many classical algorithms
The precise formulation of the concept of a general algorithm dates from the 1930s
The most direct and persuasive description was in terms of Turing machines

Turing Machines:

Introduced by Alan Turing in 1935-6 as part of the solution to Hilbert's Entscheidungsproblem
A piece of abstract mathematics, not a physical object
Designed to tackle the question of whether there exists a general mechanical procedure for solving all mathematical problems

Hilbert's Program:

Sought to place mathematics on a firm foundation with axioms and rules
Crushed by Gdel's incompleteness theorem, considered in Chapter 4

Turing's Approach:

Tried to formalize the concept of a machine by breaking down its operations into elementary terms
Regarded human thinking as an example of a machine, but this is not necessary

The Turing Machine Concept:

A device with a finite number of internal states that can deal with inputs of unlimited size
Can call upon unlimited external storage space and produce outputs of unlimited size
Must examine only the relevant parts of the data or previous calculations, noting down the results and proceeding to the next stage

Turing Machine Design and Operation for Computation

Turing Machines and Computation

Idealization of Turing's Machine: approximated by modern electronic computers
Tape: in nite, linear sequence of squares; can be moved back and forth
Each square is either blank or contains a single mark (0 or 1)
Tape represents external environment, used for input/output
Device reads tape one square at a time and moves accordingly

Behavior of the Machine

Determined by internal state and input
Operations include changing internal state, modifying tape symbols, moving along the tape
Terminates when calculation is complete

Definition of Machine's Operation

Specified through an explicit list of replacements (rules)
Large figure on left is symbol being read; replacement is on the right
R: move one square to the right, L: move one square left
STOP: terminate calculation when complete

Numbering Internal States

Labelled using binary numbers based on powers of 2 (0, 1, 2^1, 2^2, ...)
Alternative notation: succession of 0s and Is.

Binary Coding and Turing Machine Operations

Turing Machines and Numbering Systems:

Binary Notation:

Using binary notation for internal states, Turing machine specification abbreviated: R.STOP STOP
Machine in state 11010010 during calculation with tape as p.49
Read digit on tape (0) is replaced by a 1 and internal state changes to 11
Device moves one step left, reads next digit (1), changes internal state to 100101, and moves back right
Further instructions replace digits read and move device along tape until STOP

Primitive Numbering Systems:

Unary system: represent numbers by a string of 1s, with natural number 0 represented as blank space
Important to separate numbers from one another for algorithms that operate on sets

Example Turing Machines:

UN + 1 adds one to a unary number: simple exercise using the provided text
EUC eects Euclid's algorithm: more complex exercise, check if it correctly determines highest common factor of two unary numbers

Binary Coding of Numerical Data:

Unary system is inefficient for large numbers
Binary number system used instead with a terminator to denote end of number representation.

Expanded Binary Notation for Sequences of Natural Numbers

Expanded Binary Notation for Representing Numbers

Problems with Original Approach:

Difficulty distinguishing spaces between numbers from 0s or strings of 0s in binary representation
Cannot code complicated instructions and numbers together

Contraction Procedure:

Replace string of 0s and 1s (with finite number of Is) by a string of 0s, 1s, 2s, 3s, etc.
Each digit in second sequence is the number of Is between successive 0s in first sequence

Example:

Sequence "5, 13, 0, 1, 1, 4," becomes "9, 3, 4 (instruction 3), 3 (instruction 4), 0"
In binary notation: 1011, 11011, 0, 1, 1, 100
Coded on tape as ". . . 00001001011010100101101101011010110100011000 . . ."

Benefits of Expanded Binary Notation:

Terminates description of number and distinguishes it from blank tape on right
Enables coding any finite sequence of natural numbers in binary notation as a single string of 0s and 1s, separated by commas

Redundancy in Binary Representation:

In denary representation, 0s on far left are usually omitted
Extends to number zero (which can be written as 00 or 0)
Does not cause confusion when using comma notation and expanded binary form

Euclid's Algorithm in Expanded Binary Notation:

Can be implemented using a Turing machine that operates on pairs of numbers written in expanded binary notation
More eficient than unary numbering system, which would show up in device efficiency and external paper usage.

Expanded Binary Operations and the Church-Turing Thesis

Turing Machines and their Operations

Basic Arithmetic Operations:

Adding two numbers: can be achieved by a Turing machine
Multiplying numbers: can be achieved by XN x 2 Turing machine
Division with remainder: also achievable on Turing machines

Turing Machines vs. Unary Notation:

Adding one in expanded binary notation is more complicated than unary notation (UN + 1)
Multiplying by two is simpler in expanded binary notation than unary notation (XN x 2)

Complexity of Turing Machines:

Turing machines can perform more complex algorithmic tasks once basic arithmetic operations are established
The ultimate scope of these devices is to perform any mechanical operation, as defined by the Church-Turing thesis

Expanded Notations and Tape Configurations:

Allowing multiple tapes or symbols does not expand the class of achievable operations
A single tape can be extended to accommodate more data if needed

Parallelism in Turing Machines:

Having multiple separate devices doesn't gain anything in principle, as they can be treated as a single device if necessary.

Turing Machines and Computability Exploration

Turing Machines and Computability

Background:

Concept of a Turing machine proposed by Alan Turing to address Hilbert's Entscheidungsproblem (question of algorithmic procedure)
Equivalent proposals from Church, Kleene, and Post
Church-Turing Thesis: Turing machine concept defines what is computationally effective or mechanical

Natural Numbers:

Single Turing machines can handle natural numbers of arbitrarily large size
No need to question original mathematical form of the Church-Turing Thesis

More Complicated Types of Numbers:

Negative numbers, fractions, and infinite decimals handled as special cases of natural numbers
Digit generation procedure exists for generating decimal expansions of irrational numbers
Some irrationals cannot be generated by any Turing machine (non-computable)

Computability in Mathematics:

Turing machines can operate on mathematical formulae and formal calculus manipulations with proper coding
Important concept in mathematics, not just limited to numbers.

Universal Turing Machine Coding for Numbering Turing Machines

Turing Machines Coding

Coding Scheme:

Code list of instructions for a Turing machine into a string of 0s and 1s using:
- Symbols R, L, STOP, arrow (), and comma represented as contractions: 2, 3, 4, 5, and 6
- Digits 0 and 1 coded as 0 and 10, respectively
Economize by leaving out arrow symbols, digits preceding them, commas, initial 110 or final 110 (with infinite blank tape following), and using the numerical ordering of instructions to specify what those symbols must be

Example: Turing machine XN + 1

Leave out arrows, digits preceding them, and commas
Code sequence: 10101101101001011010100111010010110101111 01000011101001010111010001011101010001101
Delete initial 110 and final 110 (and infinite sequence of 0s following it)
Binary number: 450813704461563958982113775643437908

Other Examples:

UN + 1: binary number 177642, denary notation
XN 2: binary number 10389728107, denary notation

Not all Natural Numbers Have Working Turing Machines:

Majority do not give working Turing machines at all.

First 13 Turing Machines:

T 0: moves on to the right obliterating everything
T 1: ultimately achieves the same eect, but clumsily
T 2: leaves everything on the tape as it was before (never stops)
T 3: first respectable machine, changes rst (leftmost) 1 into a 0
T 4: encounters problem with no listing for an internal state
T 8, T 9, and T 10: encounter the same problem as T 4
T 7: gets stuck when it finds its rst 1 on the tape due to a sequence of five consecutive 1s
T 5, T 6, and T 12: encounter similar problems as T 0, T 1, and T 2 (never stop)

Universal Turing Machines Construction and Implications

Turing Machines

List of Non-Functioning Machines:

T 0, T 1, T 2, T 4, T 5, T 6 (twice), T 8, T 9, T 10, and T 12 are duds
Only T 3 and T 11 are working Turing machines, but not very interesting

Redundancy in List:

T 6 is identical with T 0 and T 12
This redundancy cannot be fully removed without complicating the universal Turing machine

Interpreting the Tape Representation:

The tape with infinite 0s at both ends, ending with a non-zero number of 1s, represents the binary representation of some natural number
Zero is also represented as an infinitely repeating sequence of 0s
Reading the string between the first and last 1 as the binary description gives all natural numbers

Turing Machine Behavior on Tape Input:

A Turing machine's behavior on a finite input string can be interpreted as producing another binary number, which represents the answer to a calculation
This operation is an algorithmic procedure that can be carried out by a Universal Turing Machine (U) on the pair of numbers (n, m)

Universal Turing Machine:

The U machine reads the binary representation of n and m from its input tape
At each step, it examines the digits in n to determine the appropriate replacement for m's digits
This procedure is tedious but possible to construct, leading to a universal Turing machine that can mimic any Turing machine

Size of Universal Turing Machine:

The number encoding the U machine is large but unavoidable for a universal Turing machine design

Comparison with Modern Computers:

Modern general-purpose computers are essentially universal Turing machines, despite not resembling the U machine design

Turing Machine Halting Problem and Mathematical Undecidability

Turing Machines and the Halting Problem

Purpose: Turing originally proposed his ideas to resolve Hilbert's broad-ranging Entscheidungsproblem (decision problem)
Halting Problem: Phrase of whether a given Turing machine will halt or not when applied to a particular input number m
Importance: Deciding whether a Turing machine halts is crucial as an algorithm that runs forever without stopping is essentially useless
Example: Fermat's Last Theorem
- Unsolved problem in mathematics about whether there exist natural numbers w, x, y, z satisfying an equation
- Could be phrased as a Turing machine halting problem
Example: Goldbach Conjecture
- Assertion that every even number greater than 2 can be expressed as the sum of two prime numbers
- Can be tested algorithmically by checking divisibility of the given even number by primes up to its square root
- Turing machine devised to decide whether an even number is a sum of two primes or not
Deciding Halting Problem
- Difficulty lies in determining whether a particular Turing machine halts for all possible inputs
Turing's Argument against Solving the Halting Problem Algorithmically:
- Assumes existence of algorithm H that decides whether a given Turing machine halts or not when applied to m
- The universal Turing machine (U) can be constructed to simulate H and hence solve the halting problem for any Turing machine
Conclusion: There is no algorithmic procedure to decide the halting problem of all possible Turing machines.

Diagonalization Argument Proving Non-Existence of Turing Machine Halting Problem Solver

The Undecidability of Turing Machines

Imagining an Infinite Array:

Lists all the outputs of all possible Turing machines acting on various inputs
Each row displays the output of a specific Turing machine, as applied to different inputs

Cheating for a More Interesting List:

The author has "cheated a little" by not listing Turing machines in order
This makes the list look more interesting than just having rows of 's or 0's

Defining Computable Sequences:

A computable sequence is an infinite sequence whose successive values can be generated by an algorithm
There is some Turing machine that can generate such a computable sequence when given the natural numbers as inputs

The Diagonal Slash Argument:

Consider the elements of the main diagonal, marked with bold 's
Add 1 to each term in this sequence: 1, 1, 2, 3, 2, 1, 4, 8, 2, ...
This is a computable procedure that generates a new computable sequence: 1 + Tn(nn) * H(nn)
The table contains every computable sequence, so this new sequence must be somewhere in the list
However, this new sequence differs from the first term in each row of the table, which is a contradiction
This establishes that there is no algorithm to decide whether a Turing machine stops or not (Hilbert's Entscheidungsproblem)

Deciding Individual Cases:

For some Turing machines and inputs, the question of whether it stops can be decided by intuition or common sense
But there is no single algorithm that works for all mathematical questions or Turing machines

The Limits of Algorithms Churchs Lambda Calculus

The Undecidability of Algorithms and Computability

Turings Argument:

Shows that there is no algorithm to determine if all Turing machines will halt or not
Provides a method for constructing a Turing machine that defeats a given algorithm
This contradiction shows the algorithm cannot be correct

Finding the Contradictory Turing Machine:

To find the contradictory Turing machine, we need to examine the construction of the algorithm (H) and the Turing machine (Tn)
By looking at the diagonal terms, we can identify a specific Turing machine (k) that defeats the algorithm
This process is complicated but well-defined, and could potentially be automated into an algorithm itself

The Meaning of "Knowing":

When discussing algorithms, it is important to consider what "knowing" means
Is it the algorithm that knows, or are we the ones doing the knowing?
This raises questions about mathematical truth and the limitations of algorithms

Computability and Mathematical Puzzles:

The concept of computability is a fundamental idea in mathematics
It is interesting because some well-defined operations are non-computable, which challenges mathematicians
The general solution to determining computability is also non-computable, making it an intriguing puzzle

Abstract Mathematics Lambda Calculus for Computability

Computability as a Mathematical Concept

Computability:

Genuine absolute mathematical concept
Abstract idea that lies beyond any particular realization in terms of Turing machines
Turing machines are one way to express the idea, but not the only way
Other approaches include Church's lambda calculus and Kleene's work

Church's Lambda Calculus:

Designed to illustrate computability as a mathematical concept
Abstract in its essence, focusing on mathematical operations or functions
Functions can take other functions as arguments, with the result also being a function
The operation of abstraction allows for simplifying complex expressions into abstracted functions
Examples: sine function, power operations (cubing), iterating functions like multiplication and exponentiation

Expressing Arithmetic Operations:

Church's scheme can express arithmetical operations like addition and multiplication using elementary functional operations
For example, S is the function that adds a given number to another number
Other basic arithmetical operations (subtraction, division) are more difficult to define in this framework

Universal Turing Machine Coding and Algorithm Improvements

Notes on Computability and Universal Turing Machines

Background:

Discovery of computability in the early 1930s by Kleene, Church, and Turing
Church's notion of computability has fundamental relations to practical computing
LISP language incorporates Church's calculus structure
Various ways to define computability (Post's machine, recursiveness)

Universal Turing Machines:

Concept remains the same despite differing approaches
Turings original one vs. modern ones (more practical but less straightforward)
Platonic reality of mathematical concepts

Computability and Practical Computing:

Church's notion of computability has fundamental relations to practical computing
LISP language incorporates Church's calculus in an essential way

Defining Computability:

Post's concept of computing machine was very close to Turing's, produced independently at almost the same time
Currently more usable definition of computability (recursiveness) due to Herbrand and Gdel
Curry and Schnfelds approach somewhat earlier than Church's calculus

Universal Turing Machines:

Concept remains constant despite various approaches
Can be defeated by applying the same procedure iteratively, leading into complex considerations discussed in connection with Gdel's theorem.

3 Mathematics And Reality

TorBled-Nam: A Mathematician's Discovery

Background:

Imaginary journey to an unknown world, TorBled-Nam
Remote sensing device reveals strange landscape (Fig. 3.1)
Island with various tributaries and warts

Further Examination:

Enlarging the image:
- Revealed larger object resembling the island but not identical
- Complex structure with numerous protuberances, laments, and spiral formations
Analyzing a wart:
- Resembles creature as a whole but different at attachment point
- Various places exhibit numerical patterns (veness, sevenness, nineness)
Entering crevices:
- Discovering smaller warts and swirling activity
- Sea anemones or regions with oral appearance in seahorse valley
Examining a spiral tail:
- Complex structure made up of tiny spirals and other forms (Fig. 3.5)
- Revealing a baby creature (Fig. 3.6 and 3.7) almost identical to parent
Conclusion:
- TorBled-Nam is not an actual land or landscape but a mathematical concept, the Mandelbrot set.

The Mandelbrot Set:

Complex number system including real and imaginary parts
Generated by simple rule yet displays infinite complexity.

Real Numbers and Their Extensions Infinite Decimal Expansions

Complex Numbers and Real Numbers

Real Numbers:

Natural numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
Quantified by the use of natural numbers
Can be added or multiplied to produce new natural numbers
Need negative numbers for systematic subtraction
Extended to include rational numbers (fractions) and real numbers (infinite decimal expansions)
Includes square roots, cube roots, etc. of positive real numbers

Complex Numbers:

Absolutely fundamental to the structure of quantum mechanics and the workings of the world
A Great Miracle of Mathematics
Can be represented as imaginary (purely imaginary) or complex (includes both real and imaginary components) numbers

Complex Number Representation:

In Cartesian form: a + bi, where a is the real part and b is the imaginary part
Magnitude (modulus): |a + bi| = sqrt(a^2 + b^2)
Argument (phase): arg(a + bi) = atan2(b, a)

Uncountable Real Numbers Cantors Diagonal Argument

The Complexity of Real Numbers

Computable Numbers vs. Real Numbers:

Computable numbers are "awkward" to work with, as one cannot keep all operations computable
Deciding whether two computable numbers are equal is not a computable task
Prefer to work with all real numbers, which can have any decimal expansion

Counting Real Numbers:

The number of integers and natural numbers is the same, denoted as 0 (aleph nought)
Fractions, integers, and natural numbers can be put into one-to-one correspondence
These sets are called countable

Uncountable Sets:

There exist sets that are not countable, such as the real numbers
The argument uses a diagonal slash to show there is no one-to-one correspondence between real numbers and natural numbers
This establishes the real numbers are greater than the rational numbers and uncountable

The Continuum Hypothesis:

The symbol C stands for the system of real numbers, also called the continuum
It is an unsolved problem whether C = 1 (the next infinite number) is true or not

Complex Number System A Mathematical Extension of Real Numbers

Diagonal Procedure and Real Numbers

Halting Problem:

Some Turing machines may produce digits of a real number but get stuck, unable to produce another digit
There is no computable method to determine which Turing machines will get stuck

Computability vs. Real Numbers:

Real numbers are called "real" because they seem to provide magnitudes for measurement
However, the relationship between abstract real numbers and physical quantities is not as clear-cut as it may seem
At very small scales (e.g., quantum gravity), the concept of distance or time may have limited meaning

Real Number System in Physics:

Chosen for mathematical utility, simplicity, elegance, and agreement with physical concepts over a wide range
However, it is not clear that this agreement extends to very tiny scales or vast distances (e.g., quasars)
Familiar concept of real-number distance may hold up to 10^60 orders of magnitude, though extrapolation from limited experience

Complex Numbers:

Real number system has limitations, such as the inability to take square roots of negative numbers
Postulate the existence of i, such that i^2 = 1, and extend this to all real numbers
Complex numbers are formed by combining real and imaginary parts (a + ib)
Rules for adding and multiplying complex numbers follow ordinary algebraic rules with the added condition i^2 = 1

Geometrical Representation of Complex Numbers and the Mandelbrot Set

Understanding Complex Numbers and the Argand Plane

The Importance of Complex Numbers:

Provide the ability to take square roots, cube roots, etc.
Has applications in various mathematical fields
Euler proved their usefulness in simplifying trigonometric formulas

Representing Complex Numbers: The Argand Plane

Complex number z = x + iy represented by a point (x, y) on the Argand plane
- 0 is at the origin, 1 is located on the real axis
Allows for visual organization of complex numbers in a geometrically useful way

Basic Operations with Complex Numbers

Addition: Represented as vector sum of two points (parallelogram law)
Multiplication: Distance from origin and angle relative to real axis determined by the angles between 0, 1, u, and v

The Mandelbrot Set

Complex numbers can be visualized using the Argand plane
The Mandelbrot set is a deformation of the complex number plane that has been used to generate intricate patterns through iterations.

Iterated Function Systems and the Realness of Mathematical Structures

The Mandelbrot Set: Bounded vs Unbounded Complex Numbers

Concept:

Arbitrary complex number z represented as a point on Argand plane
Mapping: z replaced by new complex number z + c, where c is given

Bounded Sequence:

Replacement rule: replace 0 with c, then apply sequence iteratively
- Example: if c = 1.63i4.2, 3 becomes 9+1.63i4.2; 2.7+0.3 becomes (2.7)+(0.3)+c
Sequence remains within some fixed circle for certain choices of c (Mandelbrot set)
- Examples: c=0, c=1, c=i
Sequences can be visualized on Argand plane as a series of points
Bounded sequences do not wander far from origin

Unbounded Sequence:

Some choices of c result in unbounded sequences that cannot be contained within any fixed circle
- Example: c=1, sequence: 0, 1, 2, 5, 26, ...; c=3, sequence: 0, 3, 6, 33, ...; c=i1, sequence: 0, i, i1, i1, ...
Unbounded sequences wander farther and farther from origin to infinite distance

The Mandelbrot Set:

Region in Argand plane consisting of points c for which the sequence remains bounded (black region)
White region consists of unbounded sequences
Detailed pictures drawn from outputs of computers, based on criterion whether sequence is bounded or not.

Philosophical Question:

What reality do mathematical concepts have?
- Seem arbitrary mental idealizations but may reveal some external truth with its own reality.
The Mandelbrot set as an example: complex structure not invented by a particular person, revealed partially to individuals through computation.

The Eternal Nature of Mathematics and Complex Numbers Invention or Discovery

The Mandelbrot Set and Complex Numbers

The Mandelbrot set is a fundamental mathematical structure discovered through computer analysis of complex numbers
It was not an invention but a discovery, like Mount Everest
The system of complex numbers has a profound reality beyond mental constructions
Cardano's work on algebra in the 16th century led to the understanding of complex numbers
- He noticed that taking square roots of negative numbers allowed for solving equations
- Raphael Bombelli extended Cardano's work and began the study of complex numbers
Complex numbers have properties beyond their original purpose, such as the Cauchy integral formula, Riemann mapping theorem, Lewy extension property
Mathematics can be seen as both discovery and invention
- Discoveries are more compelling and universal than inventions
- Inventions are works of man with limited scope compared to discoveries
Plato's philosophical views on mathematics support the idea that mathematical concepts have a timeless, ethereal existence.

Exploring Non-Algorithmic Mathematics Beyond Mandelbrot Sets

Chapter 8: The Role of Algorithms in Understanding Mental Phenomena

Algorithmic versus Non-algorithmic Mathematics

Discussion of strong AI perspective and its connection to algorithmic ideas (Chapter 1)
Importance of mathematical notions, including algorithms (Chapter 2)
Arguments for a timeless existence for mathematical concepts
Possible credence given to strong AI viewpoint
Belief that mental phenomena may not reside within the concept of an algorithm

Mandelbrot Set

Discussion of Mandelbrot set and its complexities (Chapters 1, 8)
Uncertainty regarding discovery vs. invention of the Mandelbrot set

Mathematical Perspective

Demand for a rule determining digits in arbitrary real numbers
Consistency with unconventional point of view

Notes

Mandelbrot (1986) and Peitgen & Richter (1986) provided further information on the Mandelbrot set, including coloured illustrations. For more striking illustrations, see Peitgen & Saupe (1988).
Debate over who first discovered the Mandelbrot set; supporters of discovery viewpoint believe it was a natural finding rather than an invention within a formal system.
Consistent point of view: there should always be some rule determining digits in arbitrary real numbers, even if not effective or definable within a preassigned formal system.

4 Truth, Proof, And Insight

Mathematical Truth

Background:

Old question: What is mathematical truth?
Simplified by focusing on mathematical judgements
Confidence in powerful methods of proof shattered with Russell's paradox (1902)
Previous advances from Greek philosophers and mathematicians like Hilbert, Cantor, Poincar

Sets and Properties:

Concept of sets as completed wholes
- Characterized by properties (e.g., redness = set of all red things)
Freges definition of numbers: totalities of equivalent sets
- Number 3: set of all sets with the property of having exactly three members

Russell's Paradox:

Set R defined as the set of all sets not belonging to themselves
Contradiction arises when asking if R is a member of itself or not
Same type of reasoning used in earlier mathematical proofs led to paradox
Need for more precise reasoning and avoiding contradictions

Conclusion:

Mathematical truth is an essential question with significant implications
Past developments provided insights but also challenges, as seen through Russell's paradox.

Formalization of Mathematical Systems Hilberts Program and Gdels Theorem

Formal Mathematical Systems

Overview:

Russell and Whitehead developed a formal system to translate mathematical reasoning into their scheme, preventing paradoxes
Hilbert aimed for a comprehensive scheme that would prove consistency of mathematical area
Kurt Gdel's theorem destroyed Hilbert's program due to undecidable statements

Formal Mathematical System:

Alphabet of symbols including natural number notation and logical operators
Need space symbol, variables (t, u, v...), arithmetic operations, brackets, and logical symbols
Propositions: Fermat's theorem example using capital letters P, Q, R...
Axioms: self-evident propositions like truth of conjunction or negation
Rules of procedure for deriving new propositions from established ones

Building Up Theorems:

Starting with axioms and applying rules of procedure to build up long list of propositions called theorems
Prove specific proposition P by finding a correctly assembled list terminating in P.

Godels Incompleteness Theorem and Formal Systems

Hilbert's Programme and Godel's Theorem

Background:

Hilbert's programme: find a comprehensive system of axioms and rules for arithmetic to incorporate all correct mathematical reasoning
Goal: enable decision on truth or falsity of any statement within the system
Hope: achieve completeness (proof either P or ~P) and consistency (no string both true and false)
Problem: Godel's theorem

Godel's Argument:

Central idea: create a propositional function that asserts there is no proof of a specific proposition
Propositional function: ~x[x proves Pw(w)]
- Part in square brackets: well-defined arithmetical statement about two natural numbers x and w
- Negated existential quantifier removes one variable, leaving an arithmetical proposition dependent on w only
- Statement asserts there is no proof for Pw(w)

Godel's Theorem:

Showing that such statements can be coded into arithmetic is the major effort in Godel's argument
Complicated part: coding individual rules of procedure and axioms into arithmetical operations
No proof of Pw(w) if Pw(w) is not syntactically correct, but statement remains true.

Incompleteness of Formal Systems The Self-Referential Proof

Gdel's Theorem and Unprovable Statements

Introduction:

Dependency on numbering systems and formal system details
Assignment of numbers to propositional functions (k)
Discussion of Pk(k), a specific proposition

Properties of Pk(k):

True statement: proven through argument
No proof within the formal system
Implications for formalists' notion of truth a. Incomplete according to strict formalists b. Pragmatic viewpoint questionable

Alternative Perspectives:

Simple mathematical statements equivalent to Gdel propositions
Formalists' lack of interest in mathematical truth strange

Procedure Suggestion:

Accept Pk(k) as valid proposition
Adjoin it to the system as an additional axiom
Repeat process with resulting system and its Gdel proposition
Continue indefinitely, forming a list of unlimited axioms.

The Limits of Formal Systems and Mathematical Intuition

The Systematic Expansion of Formal Systems

Gdel Propositions:

Any true, universally quantified proposition in arithmetic can be obtained through a repeated Gdelization procedure
Leads to a new system and a new Gdel proposition (e.g., G2, G2+1, G2+2, etc.)

The Procedure:

Entirely systematic and has its own Gdel proposition (G2)
However, it requires systematizing the adjoining of an infinite family of Gdel propositions into a single additional axiom

Referential Principles:

Provide insights that go beyond the rigid confines of formal systems
Involve reasoning about infinite sets, which must be done carefully to avoid paradoxes
Enable obtaining new mathematical insights not deducible from existing axioms and rules

Comparison to Russell's Paradox:

Similarities between the arguments establishing unprovability of Pk(k) and the Russell paradox
Historical connection between Turing's argument, Gdel's theorem, and Russells paradox

The Limits of Formalism:

Strict formalist viewpoint does not hold together
Attempts to avoid reasoning with enormous sets still have limitations
Mathematical truth goes beyond mere formalism

Intuition in Mathematics:

Deciding on axioms and rules requires intuitive understanding of self-evident truths
Concepts like self-consistency are not adequate for this purpose

Mathematical Platonism and Intuitionism Philosophical Perspectives on Set Existence

Gdel's Theorem and Platonism in Mathematics

Platonism vs Formalism:

Gdel's theorem shows that the formalist perspective on mathematical truth is not tenable
Platonistic view: Mathematical truth is absolute and God-given, beyond the concept of formalism

Distinctions within Platonism:

Distinction between the existence of mathematical objects and the absolute nature of mathematical truth
Whether mathematical concepts have actual existence or just an abstract, conceptual existence

Platonic Existence vs Absolute Truth:

The Mandelbrot set's existence is tied to its absolute nature, independent of specific mathematicians or computers
Controversy around wildly enormous and nebulously constructed sets, which may seem like "matters of opinion" rather than absolute truth

Platonistic Viewpoints:

Gdel was a strong Platonist, accepting the absolute truth/falsity of even controversial mathematical statements
Other Platonists may limit their Platonism to reasonably constructive sets

Intuitionism as an Alternative:

Initiated by L. E. J. Brouwer in 1924 as a response to the paradoxes of infinite sets
Rejects the law of the excluded middle and constructive existence, only accepting mathematical objects that can be explicitly constructed

Brouwerian Intuitionism vs Ordinary Mathematics:

Rejects non-constructive proofs by contradiction, as they don't provide a means to actually construct the object in question
Sees this kind of existence as "meaningless" and refuses to accept it as true

Turing-Inspired Proofs of Gdels Incompleteness Theorems

Brouwer's Ideas (1924)

Put forward ideas more than ten years before Church and Turing's work
Intuitionism: denies use of powerful arguments within mathematics, makes subjective truth claims
- Example: existence of a succession of twenty consecutive sevens in the decimal expansion of pi
  - Intuitionists would not accept this as true or false until constructively proven
  - Direct calculation not feasible due to computational complexity
  - More likely to be established mathematically, but not intuitionistically acceptable
- Similar issues with Fermat's Last Theorem
Extreme form of intuitionism: mathematical truth is subjective and time-dependent
- Not accepted by most present-day mathematicians due to its limitations in reasoning

Brouwer vs. Platonism (Present-Day Mathematical Philosophy)

Brouwer's intuitionism: denies absolute, external mathematical truth; focuses on human society and physical objects
Present-day Platonists: believe mathematical truth is absolute, eternal, not dependent on man-made criteria or physical objects
- My sympathies lie with the Platonistic view
- Important for understanding later concepts

Gdel's Theorem and Its Derivatives (Turing's Argument)

Three main streams of mathematical philosophy: formalism, Platonism, intuitionism
Emphasized the undecidability of axiom consistency as historically important but not for understanding truth proposition
Turing derived key aspects of Gdel's result through his own argument to establish insolubility of the halting problem
- A formal system must have an algorithm for checking proofs, but not necessarily finding them.
- An algorithm exists to find a proof when one does exist within any formal system.

Recursively Enumerable Sets and Gdels Incompleteness Theorems Set-theoretic Interpretation of Turing Machines and Formal Systems

Hilbert's Program and Gdel's Theorem

Symbolic Language:

System formulated in terms of some symbolic language, expressible in a finite alphabet of symbols
Ordered lexicographically (alphabetically for each fixed length of string)

Listing Proofs and Theorems:

List of all correctly constructed proofs numerically ordered according to the lexicographical scheme
Corresponding list of all theorems of the formal system, as the last lines of those proofs
Computable listing by testing each string in the list using a proof-testing algorithm

Hilbert's Program and Completeness:

If Hilbert's scheme was successful, it would provide a general algorithmic method to decide truth or falsity of any proposition within the system
This would contradict Turing's result that there is no general algorithm for deciding mathematical propositions
Gdel's theorem establishes that no formal system of Hilbert's intended type can be complete in the sense discussed

Recursively Enumerable Sets:

Concerned with sets (finite or infinite) of natural numbers 0, 1, 2, ...
Includes sets like {4, 5, 8}, {0, 57, 100003}, {6}, {0}, {1, 2, 3, ..., 9999}, {1, 2, 3, ...}, and the entire set of natural numbers {0, 1, 2, ...} or the empty set
Focuses on computability issues: Which sets of natural numbers can be generated by algorithms and which cannot?
Identifying natural numbers with propositions in a formal system, using known algorithms to convert between them.

Choosing a Formal System:

The chosen formal system is consistent and broad enough to include all actions of all Turing machines
Its axioms and rules of procedure are self-evidently true

Undecidable Propositions and Incomplete Formals Systems

Properties of Recursively Enumerable Sets:

A set, such as P, that can be generated by an algorithm is called recursively enumerable.
The set of propositions disprovable within a formal system (i.e., their negations are provable) is also recursively enumerable.
Simple examples of recursively enumerable sets include: even numbers, squares, primes, and their complements.
A set that has both the set and its complementary set recursively enumerated is called a recursive set.

Properties of Recursive Sets:

Both the set and its complementary set are recursively enumerable in a recursive set.
The complement of a recursive set is also a recursive set.

Recursively Enumerable but Not Recursive Sets:

A set that is recursively enumerable but not recursive would mean that there exists an algorithm to generate its elements, but no algorithm for generating the complementary set.
The question of whether such a situation arises is whether or not there are recursively enumerated sets that are not recursive.

Turing Machines and the Set P:

The set P consists of propositions provable within a formal system.
Turings fundamental result states that there is no algorithm to determine when a Turing machine will stop for all natural numbers n (S(n)).
The false S(n) are not recursively enumerable, so the complement of set P cannot be recursively enumerated either.
This establishes that the set P is not recursive and demonstrates that the formal system cannot be complete, as there must be undecidable propositions neither provable nor disprovable within it.

Algorithmic Boundaries of Complicated Sets Mandelbrot Set and Gdels Theorem

Gdel's Theorem: Recursive vs. Recursively Enumerable Sets

Recursive Sets:

Represent the true propositions of a formal system
Are not recursive or recursively enumerable
True propositions are more complicated than provable within the system
Simple classes of arithmetical propositions form recursively enumerable sets (e.g., those involving functions constructed from basic operations)

The Mandelbrot Set:

Exhibits an endless variety of highly elaborate structure, defying algorithmic systematization
Computer generates the Mandelbrot set by applying a simple map repeatedly to a complex number c
Determining if a point in the Argand plane belongs to the Mandelbrot set involves knowing what happens after an infinite number of terms
The complement of the Mandelbrot set (white region) is recursively enumerable, but whether the Mandelbrot set itself (black region) has an algorithm for belonging is unknown.

Complicated Set Theory in Mandelbrot Set Discussion Recursive Enumerability and Local Connectedness

Recursively Enumerable Sets and Complex Numbers

Issues to Address:

The term "recursively enumerable" and "recursive" should only be used for countable sets like natural numbers, not complex numbers.
The real numbers are uncountable, and the complex numbers cannot be countable due to their relationship with real numbers.

Problem of Equality of Computable Complex Numbers:

There is no general algorithm for deciding whether two given computable complex numbers are equal or not, unlike with computable real numbers.
This problem arises because the decimal expansions of the real and imaginary parts of a complex number may not converge to 0 or 1, making it impossible to determine if they are on the unit circle or not.

Unit Disc as Recursive Set:

The unit disc should count as recursive but requires an appropriate viewpoint.
One way around this problem is to ignore the boundary and only consider points in the interior or exterior, where algorithms exist to determine their position.

Alternative Perspective: Recursively Enumerable Sets:

Another approach is to design an algorithm that decides if a given complex number lies within the set or its complement, without needing to generate digits of the real and imaginary parts.
This perspective also ignores boundaries, considering only the interior and exterior of the unit disc as recursively enumerable sets.

Mandelbrot Set Complexity:

When applied to the Mandelbrot set, ignoring boundaries may miss its complexity, particularly the tendrils which do not lie in the interior of the set.
The Mandelbrot set's recursive enumerability depends on whether it is locally connected, a term whose relevance and meaning need clarification.

Non-Algorithms in Mathematics Diophantine Equations and Word Problems

Non-Recursive Mathematics

Complex Numbers:

Complex numbers are not countable, but subsets can be defined that are computable
Examples: Rational complex numbers with rational real and imaginary parts, algebraic numbers (solutions to algebraic equations)
Algebraic numbers are countable and computable, but may not fully resolve the algorithmic nature of the Mandelbrot set boundary

Examples of Non-Recursive Mathematics:

Diophantine equations: Systems of algebraic equations with integer coefficients
- The problem is to find integer solutions, if any
- No general algorithm exists for this
Topological equivalence of manifolds (3D and above)
- Equivalence not known for dimensions 3 and higher
- Could be relevant to quantum gravity in 4D space-time
Word problem: Given a set of word equations, can two words be derived from each other using those equations?
- Difficulty lies in the complexity of possible substitutions

Word Problem:

Given an initial list of word equalities (E.g., EAT = AT)
Derive further equalities by making substitutions
Determine if two words can be derived from each other using these rules
Examples: Caterpillar to Man, Carpet to Meat

Non-recursive Mathematics Word Problem and Tiling of Plane

Caterpillar Transformation

Changes:

Caterpillar: C A R P I L L A R
Carpet: C A R P E T
Man: M A N
Meat: M E A T
Low: C A R P I L O W

Comparison:

Number of As, Ws, and Ms is the same on both sides in every equality.
Total number of As, Ws, and Ms differs between CARPET (1) and MEAT (2).
No way to get from CARPET to MEAT by allowed substitutions.

General Word Problem:

Algorithm for establishing equality when words are equal: lexicographical listing of all possible sequences, strike non-consecutive unequal pairs.
Deciding when two words are not equal is more complex and may require intelligence.
No single algorithm can be used universally for all initial lists (non-recursive mathematics).

Specific Word Problem:

Initial list: C A R P E T, M A T E N, M A N, C A R P I L O W, L O W.
There is no algorithm for deciding when two words are unequal using this initial list (non-recursive mathematics).

Periodic Tilings:

Examples: squares, equilateral triangles, regular hexagons, irregular pentagons, versatile tile shapes.
Periodic tilings have a period parallelogram that reproduces the pattern when repeated in two directions.
Non-periodic tilings do not reproduce the pattern periodically but still cover the entire plane.

Non-Periodic Tilings:

Raphael Robinson's set of six tiles (Figure 4.10) will tile the entire plane only in a non-periodic way.

History:

Hao Wang addressed the question of whether there is a decision procedure for the tiling problem.
If every finite set of distinct tiles that tile the plane also tiles it periodically, then there would be an algorithm to decide whether a given set tiles the plane or not (decision procedure).

Non-Recursive Geometry Exploring the Mandelbrot Sets Complexity

Non-Periodic Tiling Problem

History:

Discovered non-periodic tilings in the late 1960s, following Hao Wang's leads
Robert Berger exhibited the first aperiodic set of tiles with 20,426 di erent ones; later reduced to 104 by Raphael Robinson
In 1971, Robinson further reduced the number to six tiles (Fig. 4.10)
Another set of six non-periodic tiles was produced independently in about 1973 (Fig. 4.11)
Reduced the number of tiles to two sets by slicing and re-gluing (Fig. 4.12)

Properties:

Aperiodic patterns exhibit crystallographically impossible quasi-periodic structures with fivefold symmetry
Non-recursive mathematics, as Wang, Berger, and Robinson's work was part of this area
Many unsolved problems in this domain, including whether there exists an aperiodic single tile

Mandelbrot Set

Assumption:

If Mandelbrot set is non-recursive, its complement would not be recursively enumerable

Analysis:

Major parts of the interior can be defined by simple algorithmic equations (Fig. 4.13)
Most evident regions and overwhelming percentage of the area can be dealt with algorithmically
If complete set is non-recursive, hidden delicate regions would be hard to find and improve algorithms to reach them
Process similar to how mathematics progresses in di cult, non-recursive areas.

Complexity of Algorithms Polynomial Time Classification

Complexity Theory

Arguments about algorithms are typically at the "in principle" level
Discussion of practicality and efficiency is complex and technical
Complexity theory: concerned with limitations of algorithms for solving problems
Complexity theory focuses on infinite families of problems with general algorithms
Size of a problem measured by some natural number n
- Greater n, greater number of steps required by algorithm
N (number of steps) can increase more rapidly than n
- Depends on type of computing machine used
Broad categorization of ways N increases:
- P: polynomial time, includes all rates that are at most fixed multiples of n, n2, n3, etc.
  - Example: multiplying two numbers belongs to P category
- Other categories exist with even faster growth rates for N.

Complexity Theory and Algorithms:

No single problem's solution time is discussed in complexity theory
Instead, focus on families of problems with general algorithms
Different sizes (n) have varying numbers of steps required (N)
- Greater n, greater N
P category includes all rates that are at most fixed multiples of n, n2, n3, etc.
Algorithms can be developed and improved to handle specific cases better.

Polynomial Time Algorithms and Intractable Problems in Graph Theory

Binary Multiplication and Addition Steps

Direct way to carry out multiplication: write it out in binary system
0 0 = 0, 0 1 = 0, 1 0 = 0, 1 1 = 1
Number of individual binary multiplications: (n/2) * (n/2) = n/4
Total number of steps: N = n/2 (ignoring lower order terms)
Classes of problems with size measure n: total number of binary digits
Polynomial time algorithm for uniformly coping with all instances within that problem class

NP Problems

Problems where answers can be written down and checked in polynomial time
Example: Finding a Hamiltonian circuit in a graph
- Collection of vertices, connected by edges
- Problem: Decide existence, display if exists
- Checking procedure: Polynomial time
- Graph representation: List pairs as binary sequence
  - 1: edge between two vertices
  - 0: no edge
- Hamiltonian circuit checking: Verify it's a closed loop and each vertex used exactly twice.
NP problems are intractable for large n but soluble in principle, unlike P problems which are tractable.
NP-complete problems have polynomial time solutions that can be used to solve any other NP problem.
Examples of NP-complete problems: Hamiltonian circuit problem and traveling salesman problem.

Complexity and Quantum Computing in Physical Things Potential Quantum Advancements for Problem Solving

Complexity Theory and Computability

Belief that it's impossible to solve NP-complete problems in polynomial time with any Turing machine-like device
This is the most important unsolved problem in complexity theory
Complexity theory raises questions about algorithmic solutions being useful for mental phenomena
In later chapters, less focus on complexity theory than computability
Potential differences between complexity theory for theoretical objects vs. physical things like brains
Possibility of quantum computers solving problems beyond P class
Unclear how to construct reliable physical device for a quantum computer
Human brain may use quantum effects in problem solving and judgment formation
May need to go beyond present-day quantum theory for potential benefits
Difficulty understanding how actual physical devices could improve upon complexity theory for Turing machines.

Exploring Undecidable Problems in Set Theory and Computability

Mathematical Concepts Discussed:

Continuum hypothesis: A statement that the real numbers form a continuum without finite subsets. It is independent of standard set theory and requires new reasoning to establish its truth or falsity (Rucker, 1984; Cohen, 1966).
Formalism vs. Platonism: Formalists believe that the continuum hypothesis cannot be determined using standard formal systems because it is undecidable, while Platonists argue that it holds true or false values but requires new reasoning to ascertain (Rucker, 1984; Cohen, 1966).
Brouwer's fixed point theorem: A topological theorem stating that a continuous function from a disc to itself has at least one fixed point. The problem of finding this fixed point may not be constructive in Brouwer's sense (Brouwer, undated).
Computability theory for real functions: A new computational theory developed by Blum, Shub, and Smale to deal with real-valued functions, which could potentially shed light on the issues discussed.

Mathematical Problems:

Word problem for semigroups: Finding a specific element in a semigroup that satisfies certain conditions (Hanf, 1974; Myers, 1974).
Tile problem: Determining if a given set of tiles can cover the entire plane without overlapping or gaps. The solution to this problem is non-computable for some sets.

Notes:

Brouwer's concerns about non-constructiveness in his proofs led him to develop new methods for solving problems (Brouwer, undated).
Knuth (1981) provides further information on efficient algorithms for addressing these types of mathematical problems.

5 The Classical World

Chapter 5: The Classical World and Physical Theory

The Relevance of Understanding Physical Laws

Consciousness may be part of Nature
Understanding physical laws important for appreciating consciousness
Detailed ways we are constituted relevant, as are precise physical laws governing matter's behavior

Classical Physics Overview

Includes Newtons mechanics and Einstein's relativity
Precedes quantum theory (unpredictable, uncertain)
Deterministic: future completely fixed by the past
Dramatic achievements in understanding nature with remarkable accuracy
Amenable to precise mathematical treatment

Newton's Principia (1687)

Major work demonstrating how basic physical principles govern object behavior
Underpinned technology through impressive power and practical methods
Built upon earlier thinkers like Galileo, Descartes, Kepler, Plato, Euclid, Archimedes, Apollonios

Departures from Newton's Dynamics

Electromagnetic theory by James Clerk Maxwell (mid-19th century)
- Encompassed classical behavior of electric and magnetic fields, as well as light
- Relevance to technology and brain workings less clear than electromagnetic phenomena
Special relativity by Henri Poincar, Lorentz, Einstein, Minkowski (early 20th century)
- Explained puzzling behavior of moving bodies near the speed of light
- Famous equation E = mc part of this theory
  - Impact on technology minimal so far except nuclear physics
  - May have significance for understanding quantum theory and perceived flow of time.

Comparison of Physics Theories Superb Useful Tentative

Categories of Basic Physical Theories

SUPERB: This category includes theories that apply without refutation to a phenomenal range and accuracy of phenomena in the physical world. Examples: Euclidean geometry, Archimedes' theory of statics, Pappos and Stevin's development of this theory (now subsumed by Newtonian mechanics), Galileo's dynamics, Newtonian mechanics, Maxwell's electromagnetism, Einstein's special relativity, and general relativity. Quantum mechanics is also considered SUPERB due to its accurate predictions and explanation of various inexplicable phenomena like chemical laws, atomic stability, sharpness of spectral lines, superconductivity, and laser behavior.
USEFUL: This category includes theories that have a significant impact on technology or provide new insights into certain areas. Examples: Some quantum physics theories such as those related to chemistry and metallurgy.
TENTATIVE: This category includes relatively new physical theories that are not yet proven or lack enough evidence for widespread acceptance. Examples: Quarks, GUT (Grand Unified Theories), the inflationary scenario, supersymmetry, string theory, etc.

Classification of Modern Physics Theories

Quantum Electrodynamics (QED)

Emerged from work of Jordan, Heisenberg, Pauli, and Dirac in the 1920s and 1930s
Formulated by Dirac in 1926-1934, made workable by Bethe, Feynman, Schwinger, and Tomonaga in 1947-1948
Combines principles of quantum mechanics with special relativity, incorporating Maxwell's equations and a fundamental equation governing electron motion and spin
Does not have the same level of elegance or consistency as earlier SUPERB theories but qualifies due to its phenomenal accuracy
Implication: magnetic moment value of an electron (1.001159 65246 vs. experimental value 1.001159652193)

Current Theories in the USEFUL Category

GellMann-Zweig quark model and quantum chromodynamics (QCD)
- Explain hadrons (protons, neutrons, mesons, etc.) as made up of constituents called quarks
- Interactions between quarks explained by generalization of Maxwell's theory (Yang-Mills theory)
- Some experimental support but more untidy than SUPERB theories
Glashow, Salam, Ward, and Weinberg's theory combining electromagnetic forces with weak interactions
- Describes leptons (electrons, muons, neutrinos; W- and Z-particles)
- Incorporates a description of the weakly interacting particles
- Good experimental support but less consistent than SUPERB theories

Theory of the Big Bang Origin of the Universe

Plays an important role in discussions of Chapters 7 and 8
Lacks significant experimental support compared to SUPERB theories

Tentative Theories

Kaluza-Klein theories, supersymmetry/supergravity, string or superstring theories, GUT theories, inflationary scenario
Contain original ideas with potential for advancement but lack experimental support

Ptolemaic System

Ancient Greek theory of planetary motion governed by complicated circular motions
Effective for making predictions, but became increasingly over-complicated as greater accuracy was needed
Faded as a physical theory though it played an organizational role of historical importance

Mendeleev's Periodic Table of Chemical Elements

Did not provide predictive schemes of phenomenal character on their own
Later became correct deductions within SUPERB theories (Newtonian dynamics, quantum theory)

Euclidean Geometry as Physical Theory and Platos Ideal World

Euclidean Geometry

Euclidean geometry is the mathematics learned in school, describing the physical space of our world with near exact accuracy
Not a logical necessity, but an observed feature of the physical world
Lobachevskian (hyperbolic) geometry exists, similar to Euclidean but with certain differences: sum of angles in a triangle is less than 180 degrees proportional to its area
Dutch artist Maurits C. Escher's depiction shows the boundary as an infinite pattern with straight lines being circles meeting the boundary circle at right angles
Lobachevskian geometry might be true on a cosmological scale, but Euclidean is an excellent approximation for ordinary scales
Deviations from Euclidean geometry occur in Einstein's general relativity on smaller scales than cosmological ones

Plato and Euclidean Geometry

Plato's view: objects of pure Euclidean geometry inhabit a distinct, more perfect world (Platonic world) accessible through intellect, not necessarily applicable to the physical world
Mathematics should be studied for its own sake without demanding complete applicability to physical experiences
Understanding the external world requires precise mathematics based on Plato's ideal world.

Eudoxos of Cnidus (mathematician and astronomer)

A member of Plato's Academy, considered a great thinker in antiquity.

Real Numbers and Geometry A Historical Perspective on their Influence on Mathematics and Physics

Euclidean Geometry: Understanding Lengths and Angles through Real Numbers

Introduction:

Euclidean geometry introduced real numbers as a crucial concept (fourth century BC)
Eudoxos' idea for describing ratios of lengths in terms of integers
- Criteria to determine if one length ratio is greater or equal to another
Modern development: Precise mathematical theory by Dedekind and Weierstrass

Eudoxos vs Modern Theorists:

Ancient Greeks viewed real numbers as properties of physical space (geometrical magnitudes)
- Necessary for rigorous argument about geometrical measures, sums, and products
Nineteenth century mathematicians prefer to think of real numbers as logically primitive
- Construct various types of geometry based on number concepts

Significance of Euclidean Geometry:

Important for scientific and mathematical thinking after its publication (300 BC)
- Influenced Albert Einstein's theory of relativity
Pioneering work by Archimedes using Eudoxos' theory of proportion
- Calculated areas and volumes of various shapes before calculus was introduced.

The Ancient Greek Approach:

Real numbers extracted from geometry of physical space
Euclid's Elements (Book V) established theory of proportion
- Set stage for scientific and mathematical thinking thereafter

Eudoxos vs Modern Theorists:

Eudoxos viewed real numbers as things given in terms of geometrical magnitudes
- Describing geometrical magnitudes using arithmetic was necessary to argue rigorously about them
Mathematicians today prefer to think of real numbers as logically primitive
- Construct various types of geometry starting from the concept of number.

Eudoxos' Contribution:

Crisis in Greek geometry due to Pythagorean discovery of irrational numbers (2, not expressible as a fraction)
Solution: Criteria for comparing length ratios using integers (M, N)
- Algebraic operation on lengths: a + N > b + M and d + M > c + N
Enabled rigorous computation of areas and volumes.

Eudoxos' Legacy:

Essential ingredient in modern physical theories since Euclid's work
- Remains unchanged from Eudoxos to Einstein.

Galileos Dynamics and Relativity in Motion

The Dynamics of Galileo and Newton

Apollonios (c. 262-200 BC):

A contemporary of Archimedes
A great geometer with profound insights and ingenuity
Studied the theory of conic sections (ellipses, parabolas, hyperbolas), which had important influence on Kepler and Newton

The Dynamics of Motion:

The ancient Greeks had a good understanding of static objects but lacked a good theory of dynamics, the laws governing how bodies move
Lacked sufficient timekeeping (clocks) to accurately measure changes in position, speed, and acceleration
Galileo's observation that a pendulum could be used as a reliable means of keeping time was a key breakthrough

Galileo's Contributions:

Velocity: A rate of change of position with respect to time, a vector quantity (direction matters)
Acceleration: The rate of change of velocity with respect to time, also a vector quantity
Force on a body determines its acceleration, not its velocity
In the absence of force, velocity is constant and bodies move uniformly in straight lines
Galileo's principle of relativity: Uniform motion and rest are physically indistinguishable

Implications for Copernicanism:

Before Galilean relativity, the earth's motion around the sun was a puzzle because it wasn't directly observable
With Galilean relativity, there is no local physical meaning to the concept of "at rest"
This has implications for how space and time should be viewed

Newtons Laws and Celestial Mechanics

Galileo's Insights and Newtons Laws

Relativity of Space:

Galilean space-time: four-dimensional picture of physical reality
Three-dimensional Euclidean spaces joined together for complete understanding
Particles in uniform motion depicted as straight lines (world-lines)

Understanding Gravity and Conservation of Energy:

Galileo's insights into conservation of energy and falling bodies Objects released from rest have the same speed depending only on height Energy is neither lost nor gained, just converted between forms
Law of energy conservation: important physical principle derived from Newton's laws
Descartes, Huygens, Leibniz, Euler, Kelvin made comprehensive formulations

Additional Conservation Laws:

Mass, momentum, angular momentum concepts have importance later on Mass and momentum are related: product of mass with velocity (Familiar examples: rocket propulsion, gun recoil) Angular momentum conservation: Earth's spin, tennis balls spin

Newtons Laws:

Three laws governing behavior of material objects in Euclidean space First and second laws: uniform motion in a straight line; force equals mass times acceleration Third law: every action has an equal and opposite reaction

Universe Structure:

Consists of particles moving around under the influence of forces
Accelerations determined by acting forces, which can be calculated using vector addition law
Forces between particles obtained from summing individual contributions (direct line rule) Gravitational force: attractive inverse square law

Kepler's Laws:

Elliptical orbits around the sun with sun at focus
Three laws governing planetary motion Newton showed that Keplers laws follow from his own scheme of things
Detailed corrections to elliptical orbits and other effects obtained using mathematical techniques.

Newtonian Determinism in Billiard-Ball Model of Reality

Newtonian Dynamics

Background:

Success of Newton's eorts due to mathematical skills and physical insights
Based on inverse square law of gravitation
Deterministic system with precise dynamical equations

Newtonian World

Constituent particles: points without spatial extent or rigid spheres
Laws of force known, like gravity, electric, magnetic, nuclear forces
Forces act along the line joining centers, magnitude depends on length
Rigid spherical balls collide elastically with no energy loss
Problem with triple collisions: outcome depends on which particles collide first

Determinism and Free Will

Deterministic model of physical behavior
No room for free will in this billiard-ball world
Question of free will arises from the lack of influence on material things by a mind.

Modeling Forces:

Electric forces: inverse square law, repulsion between similar particles, strength governed by electric charges
Magnetic forces: inverse square law, different dependence on distance than electric forces
Nuclear forces: bind particles together, extremely large at close separations but negligible at greater distances.

Computability in Physics Determinism vs Algorithmic Predictability

Issue of Determinism vs Computability

Deterministic but non-calculable world possible
Free will linked to non-computable ingredient in laws of universe
Distinction between computability and determinism
- Computability: algorithmic resolution of questions
- Determinism: future is determined by present state

Illustrative Example of Non-Calculable, Deterministic Universe:

Model universe described as pair of natural numbers (m, n)
State at next instant based on whether Turing machine (Tu) halts or not
No algorithm for halting problem of Tu
Model not computable despite being deterministic

Question of Computability in Newtonian Billiard Ball World:

Deterministic but possibly non-calculable physical theory
Question: Does ball A ever collide with ball B?
Complexity: continuous variables and infinite precision required for initial data
Approach using rational numbers to approximate initial data
Guess: With rational initial data, there might be no algorithm to decide if A and B will collide
Full Newtonian problem more complicated than Fredkin-Toole model
Problem persists even with rational initial data
Assertion: Newtonian billiard ball world may not be computable is not straightforward.

Hamiltonian Mechanics A Precursor to Quantum Theory

Possible Bulleted Notes:

Human Brain vs Turing Machine

Discussing whether human brain can outperform a Turing machine using non-computable physical laws
Newtonian billiard ball world is computable with approximations and discrete time steps
- Space grid and time intervals allow for velocity calculations
- Accelerations computed using force law
Unpredictability in Newtonian world: instability from limited initial data accuracy
- Large changes result from small differences in initial conditions (chaos or chaotic behavior)
Weather prediction example of unpredictability in deterministic physical laws

Hamiltonian Mechanics

Superb applicability and mathematical richness of Newtonian mechanics
Classical mechanics: theories of Nature that yield valuable mathematical ideas
- Calculus originated from Newtonian mechanics
- Developed by great mathematicians like Euler, Lagrange, Laplace, etc.
Hamiltonian theory summarizes classical mechanics
William Rowan Hamilton's contribution to Hamiltonian theory
- Emphasized analogy between waves and particles in the Hamilton equations
- Important for later development of quantum mechanics.

Visualizing Classical Systems with Phase Space

Hamiltonian Formulation of Classical Mechanics

Ingredients of Hamiltonian Scheme:

Novel use of variables: momenta instead of velocities
Position and momentum treated as independent quantities

Changes in Description:

Two sets of equations: momentum change with time and position change with time
Rates of change determined by positions and momenta at a given time

Equations:

Momentum Equation: pi/t = -H/xi, i = 1, ..., n
Position Equation: xi/t = H/pi, i = 1, ..., n

Total Energy:

Hamiltonian function H: expression for total energy in terms of all position and momentum variables

Advantages:

Elegant and symmetrical description of mechanics
Applicable to various systems including Newtonian equations, Maxwell theory, special relativity, general relativity, and quantum mechanics.

Phase Space:

A large number of dimensions: one for each coordinate (x1, x2, ..., pi, pj)
Visualize as a space that represents the entire state of a physical system
Six dimensions for n unconstrained particles (three position coordinates and three momentum coordinates per particle)
Complexity can't be pictured accurately but understanding the concept is essential.

Hamiltonian Systems and Computational Limitations

Physics and Phase Space

Hamilton's Equations:

Define a vector field on phase space
Determine the evolution of a system with time through this vector field
Provides deterministic solution for entire system based on initial conditions

Computability in Physics:

Assuming computable physical constants, if we start from a computable point in phase space and wait for a computable time t, does the resulting point remain computable?
- Depends on the choice of Hamiltonian function H
- Question is interesting but requires examination
The concept of computable number as applied to physical measurements may be limited by measurement accuracy

Non-Algorithmic Physical Devices:

Some devices may imitate non-algorithmic mathematical processes
Such devices could potentially yield correct answers to mathematically interesting questions without algorithms
However, the behavior of these devices might require ever-increasing accuracy in physical parameters, which is an unsatisfactory way of coding information
Preferable to acquire information in a discrete form using devices that can take up and evolve through discrete states

Hamiltonian Systems:

If Hamiltonian systems behave like this, they would require stability of behavior for clear determination of discrete states.

Liouvilles Theorem and Spreading Phase-Space Regions in Classical Dynamics

The Problem of Stability in Classical Mechanics

Goal: Maintain a system in a specific state without drifting into another and prevent errors from accumulating over time.

Approach: Use particles or sub-units with continuous parameters to represent discrete states, where each state corresponds to a region in phase space.

Concerns: Stability of the system as it evolves through time; potential spreading of regions in phase space leading to unpredictability.

Liouville's Theorem: Volume of any region in phase space remains constant under Hamiltonian evolution, but shape may change.

Behavior of Regions in Phase Space: Initial region R0 may distort and stretch, spreading out over an enormously larger area despite the volume remaining unchanged. This can be compared to a small drop of ink spreading throughout a large container of water.

Implications: Classical mechanics is essentially unpredictable due to the tendency for regions in phase space to spread outwards with time. However, despite this, Newtonian dynamics has been successful because initial states still provide some level of predictability within reasonable limits.

Maxwells Electromagnetic Field Theory Unifying Forces and Light

Newtonian Mechanics vs Celestial Mechanics

Comparatively small number of coherent bodies (sun, planets, moons) segregated by mass
Can ignore perturbing effects of less massive bodies and treat larger ones as acting under each other's influence
Dynamical laws applicable to individual particles also operate at level of celestial bodies
Treat sun, planets, and moons as particles, neglecting detailed motions of individual particles composing them
Main uses: overall property inference, conservation laws, statistical properties for predictive behavior
Phase space spreading effect has implications for the limitations of classical mechanics in explaining solid structures and quantum effects needed to understand them.

Maxwell's Electromagnetic Theory vs Newtonian Picture of the World

Electricity and magnetism act like gravitational forces with strength depending on electric charge/magnetic pole strength
Light can be accommodated within Newtonian framework through particle or wave concepts, but complications arise from moving electric charges giving rise to magnetic forces.
Challenge to Newtonian picture: Michael Faraday's research on physical fields (magnetic and electric) introduced the concept of vector elds in space.

Maxwells Electromagnetic Waves The New Physical Reality

Faraday's Discoveries and Maxwell's Equations

Prevailing Viewpoint on Electromagnetic Fields:

Electric and magnetic fields were not considered as real physical substances
They were viewed as convenient mathematical auxiliaries to the Newtonian point-particle action-at-a-distance picture

Faraday's Experimental Findings:

Led Faraday to believe that electric and magnetic fields are real physical entities
Varying electric and magnetic fields might be able to push each other through empty space, producing a disembodied wave
Light itself might consist of such waves

Maxwell's Equations:

Proposed a change in the equations for electric and magnetic fields
One implication was that electric and magnetic fields would push each other through empty space
Electromagnetic waves would propagate at the speed of light, exhibiting interference and polarization properties of light

Maxwell's Equations vs. Newtonian Mechanics:

Maxwell's equations were at variance with prevailing Newtonian wisdom where electric and magnetic fields were not considered real
Maxwell's equations required a radical change in understanding the nature of physical reality, taking electromagnetic fields seriously as entities in their own right

Hertz's Experimental Confirmation:

The existence of electromagnetic waves was experimentally confirmed by Heinrich Hertz in 1888
This provided firm basis for Faraday's inspired hopes

Lorentz Equations and Runaway Particles Resolving Singularities in Electromagnetic Fields

Computability and Wave Equation

Maxwell's equations imply the wave equation in regions with no charges or currents (j = 0, **= 0)
The wave equation is a simplified version of Maxwell's equations as it focuses on a single quantity
Solutions of the wave equation exhibit deterministic behavior but can be non-computable for certain initial data
Pour-El and Richards showed that this non-computability cannot arise if eld data is disallowed
This result does not contradict the expected computability of reasonable Hamiltonian systems
Non-computability may only be relevant when measurements of arbitrarily high precision are allowed

Lorentz Equation of Motion; Runaway Particles

Maxwell's equations do not provide rules for charged particle behavior, which is addressed by the Lorentz equations of motion
Lorentz equations describe how a charged particle's velocity changes due to electric and magnetic fields
When combined with Maxwell's equations, they form rules for time-evolution of both particles and electromagnetic fields
Issues arise when determining the eld at a particle's location due to its own field and potential problems with rotations or distortions.

Special Relativity and Light Contraction Lorentz-Einstein Theory

The Problem of Initial Data in Classical Physics

Dirac's Solution and Runaway Solutions:

In 1938, Dirac solved the problem of how to determine the behavior of particles and fields from their initial data
However, his solution led to "runaway solutions" - particles behaving in crazy ways with spontaneous acceleration towards the speed of light
These runaway solutions do not correspond to reality and must be ruled out by choosing the initial accelerations correctly
This is a departure from standard deterministic physical problems, where the initial data can be given without constraint

Determinism and Computability in Classical Equations:

The implications of Dirac's solution raised questions about determinism and computability in classical physical laws
Physicists argue that this issue is more relevant to quantum electrodynamics than classical electromagnetism
Poincar and Einstein were also concerned with the compatibility of Galilean relativity and Maxwell's equations

The Special Theory of Relativity:

Einstein and Poincar independently found that Maxwell's equations satisfy a relativity principle, preserving their form under changes in reference frames
This required modifying the laws of Galileo and Newton to be compatible with this relativistic principle
Lorentz had partially addressed these questions, suggesting matter contracts slightly when moving at speeds comparable to the speed of light

Minkowskis Four-Dimensional Space-Time Concept and Light Cones

Lorentz's Conclusion

Lorentz concluded that matter behaves exactly according to relativity principle underlying Maxwell equations, preventing uniform motion from being locally detected
Limited by specific theory of matter, no other forces were considered significant

Poincar's Contribution

In 1905, Poincar showed there is an exact way for matter to behave according to relativity principle under Maxwell equations
Understood physical implications including relativity of simultaneity
Regarded it as one possibility, did not share Einstein's conviction

Minkowski's Insight

In 1908, Minkowski introduced the concept of four-dimensional space-time
Space and time are considered together as a single entity
Each point in space-time diagram represents an event, whole history is represented by the diagram
Particle is not represented by a point but by a world-line; its entire existence is described

Special Relativity Basics with Minkowski's Spacetime

Difficulty: Four dimensions are hard to visualize
Space and time union preserves independent reality
Cheating allowed in terms of lower dimensions for better understanding

Light Cone in Minkowskian Spacetime

Each point in space-time has a future light cone, representing history of light ash from an explosion
Material particles cannot travel faster than light; their world-lines lie inside the light cone at each point
Convenient to describe photons as "classical" for better understanding, but they always travel in straight lines with fundamental speed c.

Minkowskian Distance in Special Relativity Time Measured by Moving Clocks

Minkowskian Geometry and Special Relativity:

In three-dimensional Euclidean geometry, distance from origin r is given by Pythagorean theorem: x + y + z.
Minkowskian geometry (also called spacetime) has similar expression with two minus signs: s = t - (x/c) - (y/c) - (z/c).
Essential difference is that we now have spacetime instead of just space.
Distance measure in Minkowskian geometry (s) represents time experienced by a particle between events O and P if P lies within light cone of O.
For uniformly moving observers, correct measure of time is s, not t.
Moving clocks run slower compared to stationary ones due to relativistic effects.
Photons don't experience any passage of time (s = 0).
Minkowskian distance applies to any pair of points within the light cone of each other.
World-line representing time measured by a clock moving uniformly from one point to another.
Twin paradox explanation using Minkowskian geometry: AC represents time experienced by stay-at-home twin, AB and BC represent traveller's time experiences during journey.
Inequality AC > AB + BC holds true for Minkowskian triangle inequality due to different sign conventions in distance calculation compared to Euclidean geometry.

Time Dilation and Relative Simultaneity

Minkowski Spacetime and Relativity Principle

Concept of time-measure in relativity: two twins at different velocities experience time differently
Experimental evidence for delayed decay times of subatomic particles traveling close to the speed of light
Nuclear clocks confirming Minkowskian distance measures
Implications of Einstein's equation E = mc and special relativity principle
Explanation of Poincar motion in Minkowski spacetime
- Rigid motion preserving all Minkowskian distances
- Uniformly moving observers have equivalent physics under these motions (stationary observer S vs. moving observer M)
Definition of simultaneous spaces for each observer: stationary spaces for S and tipped up simultaneous spaces for M
No unique concept of simultaneity in Minkowskian geometry but each uniformly moving observer carries their own idea of what simultaneous means.

Minkowski Spacetime and Relativity Principle (Continued)

Significant differences in time ordering for events at great distances even with slow relative velocities (example: people walking past each other)
The event that seems simultaneous to one observer may not be so for another, leading to di erent perceptions of reality depending on their motion.

Equivalence Principle and Gravity Distortion

Einstein's General Relativity: Galileo's Insight and Space-Time Curvature

Galileo's Insight

Galileo's great insight that all bodies fall equally fast in a gravitational field
This was an insight, not a direct observation (due to air resistance)
The idea of space-time curvature is seemingly different from Newton's scheme where particles accelerate under ordinary gravitational forces

Implications of Galileo's Insight

Newtons theory governs the acceleration of a body under gravity:
- Gravitational force (proportional to mass)
- Acceleration (inversely proportional to mass)
The mass that occurs in Newtons gravitational force law is the same as the mass in Newtons second law
Galileo's insight ensures that a body's acceleration under gravity is independent of its mass
This insight would not apply to electric forces, which depend on electric charge and do not follow the same principle as mass

Galileo's Experiment: Falling Objects

Galileo dropping two rocks (and a video camera) from the Leaning Tower of Pisa
The video camera would show a rock "hovering in space," seemingly unaffected by gravity, due to all objects falling at the same speed under gravity

Space Travel and Free Fall

In space, falling means following an appropriate orbit under gravity
An astronaut on a space walk views their vehicle as hovering, apparently unaffected by the gravitational force of the Earth
The eects of gravity can be cancelled by free fall, as the eects of a gravitational field are equivalent to those of acceleration
Even in a train or aeroplane, sensations may not coincide with visual evidence regarding the strength and direction of gravity

Principle of Equivalence

The local eects of gravity are equivalent to those of an accelerating reference frame
Allowing for more precise measurements could allow one to distinguish a true gravitational field from pure acceleration

Non-Uniformity of the Newtonian Gravitational Field

Particles nearer to the Earth's surface accelerate downwards faster, distorting the spherical shape of an initially stationary arrangement into an ellipsoid.

Gravitational Tidal Effects in Spacetime Curvature

Gravity: Tidal Effect and General Relativity

Tidal Effect of Gravity:

Distortion in gravitational field (non-uniformity)
Cannot be eliminated by free fall
Measured as inverse cube of distance from center of attraction, not inverse square

Newton's Inverse Square Law and Tidal Effect:

Volume of distorted sphere equals original sphere's volume in vacuum
Sphere surrounds matter with mass M results in additional inward acceleration
Proportional to M: force is proportional to attracting body's mass

Space-Time Picture of Gravity:

Worldlines of freely falling particles called geodesics
Geodesics on a surface are shortest routes, similar to strings stretched over curved surface
Positive curvature: particles move towards each other; negative curvature: they move away from each other
Space-time possesses curvature, with both positive and negative effects involved for different displacements

Newtonian Theory vs General Relativity:

Reformulation of Newtonian theory without requiring new physics (yet)
Combining this picture with Minkowski's description of special relativity results in Einstein's general relativity.

Gravitational Tidal Forces in General Relativity

Curved Spacetime in General Relativity:

Locally Flat Spacetime:

In the presence of gravity, spacetime is locally like Minkowski's geometry but allows some curviness on a larger scale.
Any point in space-time is vertex of a light cone, similar to Minkowski space.
Material particles have curves inside light cones; photons have world lines along them.
Concept of Minkowski distance for measuring time along world lines exists.

Geometry of Space-Time:

Geodesics in space-time are curves which locally maximize the distance (time) along the world line.
World lines of particles in free motion under gravity are geodesics according to this rule.
Rays of light in empty space are also geodesics, but with zero length.

Newtonian Gravity vs General Relativity:

In Newtonian theory, volume change of a sphere surrounded by vacuum does not occur.
When matter is surrounded, there's a volume reduction proportional to the mass M and pressure in the material.

Riemann Curvature Tensor:

Schematically written as RIEMANN.
Describes curvature of four-dimensional spacetime.
Can be split into Weyl tensor (WEYL) and Ricci tensor (RICCI).

Weyl and Ricci Tensors:

Weyl tensor measures tidal distortion of a sphere of freely falling particles.
Ricci tensor measures initial change in volume.

Equations of General Relativity:

Schematically written as RICCI = ENERGY.
Equates mass density (or energy density) to the Ricci tensor for consistency with Einsteins equations.

Tidal Effects:

Tidal effect experienced in empty space is entirely due to WEYL.
Weyl tensor measures gravitational field, analogous to electromagnetic eld in Maxwell's theory.

Relativistic Causality Constraints and Implications of Superluminary Signals

Relativistic Causality and Determinism

Principle of Relativity:

In Einstein's theory of relativity, material bodies cannot travel faster than light
World-lines of photons lie along the light cones, and for no particle is it permitted to have a world-line outside the cones

Implications of Superluminal Signals:

If superluminal signals were possible, an observer W could receive a message from a future event A before emitting a signal at event B
This would lead to contradictions and violations of free will and causality
Even if the observers are mechanical devices, the same essential contradiction is still obtained

Conclusion:

The rules of special relativity hold locally in general relativity, implying that all signals must be constrained by the light cones.

Computability in Classical Physics Determinism and Chaos

Determinism and Computability in Classical Physics

In Newtonian (or Hamiltonian) theory, determinism means initial data at one time completely determine behavior at all other times.
In relativity theory, there is no global concept of time to specify initial data for determinism.
- Special relativity: Use a flexible attitude, like a bounded region on a simultaneous space.
- General relativity: Use the more general notion of a spacelike surface instead.
Determinism in special relativity: Initial data on a bounded region of a simultaneous space xes behavior in whole space-time.
Determinism in general relativity is more complicated due to the need to consider geometry and light cone causality structure.
- Einstein equations give locally deterministic behavior for gravitational field assuming matter elds behave deterministically.
- Complications include determining which parts of spacelike surface are needed for future event.
Lack of determinism in extreme situations may lead to global indeterminacy and loss of determinism.
Computability issues may be as important as determinism when considering free will and mental phenomena.
Classical theories like special and general relativity do not seem to have significant non-computable elements, apart from the type encountered with Pour-El and Richards for the wave equation which does not occur for smoothly varying data.

Mass Matter and Reality Quantifying Substance in Classical Physics

Classical Physics and Reality

Spacetime: primary role as arena for physical activity
Physical objects: particles and fields

Particles: little known about their nature, have mass and charge
Fields: specifically defined (e.g., electromagnetic, gravitational)
Test particles: respond to elds without influencing them (point particles)
Source particles: interact with elds (spread-out sources for Maxwell's equations and Einstein's equations)
Space-time has a variable structure describing gravity

Mass: measure of total substance, conserved but interchangeable with energy due to E = mc
Energy and mass are interconnected in quantum theory
Classical physics learnings: great advancements but not final viewpoint.

Mass, Matter, and Reality (Continued)

Mass: measure of quantity of matter in a system
Einstein's formula E = mc: mass and energy are interchangeable
Distinction between rest mass and mass-energy of motion
Conserved mass but dependent on observer's viewpoint.

Mass, Matter, and Reality (Further Insights)

Classical physics presents a picture of physical reality with spacetime, objects, particles, and fields
Mass as the measure of total substance, conserved but interchangeable with energy due to E = mc
Viewpoint change: mass is an entire object called the energy-momentum four-vector.

Gravitational Energy and Mass in General Relativity Non-Locality and Empty Space Paradox

Mass and Energy Conservation

Arrow representing energy-momentum, with tip (height) determining mass or energy divided by speed of light squared
Massive objects decay into massless particles: energy-momentum four-vector is conserved using vector addition law
Rest-mass not additive for individual objects; entire arrow represents total mass/matter

Mass vs. Energy

Maxwell's electromagnetic field carries energy and mass (according to E=mc^2)
Gravitational waves carry energy but non-local, cannot be measured by standard tensor ENERGY or WEYL/RICCI
Energy and mass of gravitational eld are difficult to pinpoint and may exist in "empty" regions

Classical Theories and Reality

Established departures from Newtonian picture linked to behavior of light: electromagnetic fields, special relativity, general relativity, wave-particle duality, quantum electrodynamics.
Thermodynamics not considered a superb theory by author; may be reluctant to classify it as a physical theory due to applying to averages and being derived from other theories.

Quantum Field Theory Challenges and Solutions in Classification

Theory Classification:

The author is using a toy model to explore physical theories
Places theories in different categories: TENTATIVE, USEFUL, SUPERB
Discusses relationship between thermodynamics and big bang theory as potentially belonging to the SUPERB category
Mention of Galileo's use of water clocks for observations
Newtons views on physical world were less dogmatic than attached label of Newtonian mechanics

Twistor Theory:

An elaborate collection of ideas and procedures associated with the author
Not a separate theory but a mathematical transcription of earlier established physical theories
Placed in TENTATIVE category as different theory of physical world

Hamiltonian Mechanics:

Evolution of toy model can be computed similarly to Newtonian systems
Discusses calculational procedures and significance of Euler-Lagrange equations
Newtons third law important for understanding dynamical behavior of large bodies

Miraculous Facts for Science Development:

Inverse square law crucial for comprehension of Kepler's laws
Maxwell's equations presented in specific units, factors of c distributed differently with other choices
Charged particles can act as clocks due to oscillation frequency proportional to mass
Quantum particles behave like clocks on their own, accurate modern clocks based on this fact.

Einsteins General Relativity and Spacetime Geometry

Simultaneity and Relativity

Events judged as simultaneous according to Einstein's definition using light signals: M's perspective (Rindler, 1982)

Reformulation of Newtonian Theory

Cartan's mathematical description of the reformulated theory (1923)
Named after Bernhard Riemann, Gauss, Lorentzian manifolds or pseudo-Riemannian manifolds

Zero Length and Maximum Value of Length

Zero length geodesic: no other particle world lines connecting any pair of its points (locally)
In a vacuous sense, it represents the maximum value of length

Distortion Effects and Volume Change

Not clear cut division into distortion effects and volume change (Ricci tensor can give tidal distortion; Penrose and Rindler, 1986, Chapter 7)

Equations: Lorentzian Manifolds

Full Bianchi identities with Einstein's equations substituted into them
Correct form of the actual equations found by Hilbert in November 1915

Issues and Limitations

Lacking rigorous theorems concerning these issues
Unsatisfying answer on calculability: infinity (present theory)

Miscellaneous

Einstein's physical ideas solely due to him, but Hilbert found the correct form of equations in November 1915.
The term "hypersurface" is more appropriate than surface for a three-dimensional object.

6 Quantum Magic And Quantum Mystery

Classical Physics vs Quantum Theory

In classical physics:
- Objective world exists independently of perception
- Evolution is clear and deterministic
- Physical reality is not affected by observer's perspective
- Our bodies and brains are part of physical world, evolving deterministically
Classical philosophy arguments revolve around reality, perception, free will
Quantum theory emerged from discrepancies between actual behavior and classical descriptions
Quantum descriptions are precise but different from classical ones
Probabilities do not arise at quantum level; particles, atoms, or molecules evolve deterministically
Explanations for many familiar physical phenomena require quantum theory
Consciousness may not be understandable in entirely classical terms
Quantum world required for thinking, perceiving creatures like us to exist?
Science may provide more profound understanding than quantum theory in future
Two main views on the reality of quantum description: objective vs. calculational procedure only
Objective view: quantum state has deterministic time-evolution but odd relation to physical world behavior
Problem of deciding when a measurement is taken, as measuring apparatus is also made of quantum constituents.

Quantum Mechanics Resolving Classical Theorys Dilemmas

Classical vs. Quantum Theory

Reasons for questioning Classical Theory:

Experimental evidence: not in agreement with classical physics predictions
Inconsistencies between particles and fields
Stability issues: energy gets transferred from particles to elds, leaving no degrees of freedom for the particles

Problems with Classical Theory:

Coexistence of particles and elds is inconsistent
Orbiting electrons emit radiation in discreet bursts instead of continuously as predicted by classical theory
Black-body radiation does not follow Rayleigh-Jeans' predictions

Beginnings of Quantum Theory:

Max Planck proposed that electromagnetic oscillations occur only in quanta, with energy E = hv
This allowed for agreement with observed radiation law (Planck radiation law)
Einstein proposed that the electromagnetic field can exist only in discrete units
Light can exhibit particle and wave properties at the same time

Quantum Interference Light as Wave and Particle

The Two-Slit Experiment

Background:

Archetypical quantum mechanical experiment
Manifestation of wave and particle behavior in light or photons
Light behaves as particles (all-or-nothing phenomenon) and waves (interference pattern)

Particle Behavior:

Energy is received in discrete units related to frequency through Planck's formula: E = hv
Light arrives at screen with whole numbers of photons
Light reception is uniform when intensity is high
Photons appear to be detected randomly, with different probabilities for different angles of deflection

Wave Behavior:

Spreading out by the phenomenon called diffraction (wave propagation)
Smooth appearance of illumination is a statistical effect due to large number of photons involved

Problem for Particle Picture:

Opening the second slit: Intensity pattern is completely different from single slit, with points of zero intensity where photon could not reach before despite having an alternative route.

Quantum Interference Single Particles Acting as Waves

Interference Patterns and Wave-Particle Duality

Pattern of Intensity at the Screen:

When one slit is open, the pattern of intensity is a distribution of discrete tiny spots (Fig. 6.4)
When both slits are open, the pattern becomes a wavy distribution of discrete spots (Fig. 6.5)

Photon's Perspective:

The second slit is some 300 photon-sizes away from the first (each slit being about a couple of wavelengths wide)
Yet, the photon seems to know whether or not the other slit is open when passing through the first slit

Destructive Interference:

Cancellation or enhancement phenomenon occurs when light passes through the slits
This is similar to how waves can cancel each other out in ordinary situations (Fig. 6.7)
The bright places on the screen occur when the distances to the two slits differ by an integer number of wavelengths
Dark places occur when the differences are exactly midway between these values

Classical vs. Quantum Waves:

In classical waves, a disturbance can pass through both slits at once
However, each individual photon behaves like a wave entirely on its own and interferes with itself (Fig. 6.8)
Even though one route is open alone, the particle appears to be unable to travel through both when they are open together

Complex Number Combinations:

Quantum mechanics insists that alternatives can combine in complex number ways, such as: A + iB, where i = 1 (Chapter 3)
These combinations force themselves on physicists through surprising experimental results

Quantum Superposition and Probability Amplitudes

Quantum vs Classical Level

Two Different Levels of Physical Description:

Quantum level: molecules, atoms, subatomic particles
Classical level: macroscopic objects like cricket balls and elephants
Distinction not based on size but tiny energy differences

Probability Amplitudes (W & Z) at the Quantum Level:

Complex numbers instead of probabilities or ratios
Behave similarly to probabilities, referred to as amplitude or simply amplitudes
Normalized when sum equals 1 for actual probabilities: p + q = 1

Interpretation of Probability Amplitudes (W & Z)

Not actual probabilities but quantum level analogues
Pretend they are probabilities or ratios of probabilities for now

Classical Probability:

Describes uncertain situation with two alternatives, A and B
Probability of each alternative: p (for A) and q (for B)
Sum of probabilities must be 1 if there are no other alternatives: p + q = 1
Ratio of probabilities for actual occurrences: P(s,p) = P(s,t) P(t,p) or P(s,b) P(b,p) when only one hole is open
Total probability when both holes are open: P(s,p) = P(s,t) P(t,p) + P(s,b) P(b,p)

Quantum Level Rules:

Same as classical probabilities but with amplitudes (A(s,t), A(t,p), etc.) instead of probabilities.

Interpreting Amplitudes:

Unclear until we know how to interpret them at the classical level.
Need an actual probability for a click or observable event in order to make sense of the quantum amplitudes.

Quantum Amplitude Squared Modulus and Probability Transition

Passing from Quantum to Classical Level

Rule: Squared modulus of quantum complex amplitude = classical probability
Modulus |z| of a complex number is distance from origin (0) in Argand plane
Squared modulus |z| is square of this number
Point z in Argand plane must lie within unit circle for proper normalization as probability amplitude
Combination of alternatives w and z: require their squared moduli to sum to unity for normalization
Unnormalized amplitudes: w/(|w| + |z|) and z/(|w| + |z|)
Probability amplitude is not like a probability but more like a complex square root of a probability

Multiplication and Addition of Complex Numbers

Multiplication poses no problem in passing from quantum to classical rules due to: |zw| = |z| ||w|
Correction term 2|w||z|cos for sum of two complex amplitudes' squared modulus
Cosine of angle between points z and w in Argand plane () provides quantum interference between alternatives
- Reinforces: cos = 0, total probability greater than sum of individual probabilities
- Destructive: cos = 180, total probability less than sum of individual probabilities
- Intermediate: cos = 90, total probability between sum of individual probabilities
Large or complicated systems have correction terms average out to classical probability rules.

Two-Slit Experiment

Amplitude for particle reaching p is a sum of w (A(s,t) A(t,p)) and z (A(s,b) A(b,p))
Brightest points on screen: w = |z|, cos = 1, |w + z| = 4|w| = 4x probability and intensity
Darkest points on screen: w = z, cos = 1, |w + z| = 0, destructive interference
Intermediate points: w = iz or z = iw, cos = 0, |w + z| = 2|w| twice the intensity if a large number of photons.

Important Note: The squared modulus of an amplitude cannot be interpreted as the probability that the particle passed through a specific slit to reach p.

Visualizing Quantum State and Momentum in One-Dimensional Space

Understanding Quantum Theory: Interference and Wavefunctions

The Problem with Detecting Photons:

Detection at specific time obliterates interference pattern
At quantum level, individual routes have only amplitudes, not probabilities

Quantum State of a Particle:

Classically determined by position and velocity
Quantum mechanically: all positions are alternatives
Combined with complex-number weightings (amplitudes)
Collection of complex weightings is the quantum state

Picturing the Complex Function:

Difficult to imagine in three-dimensional space
Simplify by considering one-dimensional particle on x-axis
Use Argand plane for complex numbers: y (real axis), z (imaginary axis)
Plot wavefunction (psi-function) as a point in Argand plane at each position along x-axis
The resulting curve is the -curve of the particle
Probability of finding particle at a point is obtained by squaring modulus of amplitude

Understanding Momentum Amplitudes:

Classically, need velocity (or momentum) to determine future behavior
Quantum mechanically: wavefunction contains various momentum amplitudes
Apply harmonic analysis to function for determination of these amplitudes
Each pure tone corresponds to a different possible momentum value
Size of each pure-tone contribution provides the amplitude for that momentum value.

Momentum and Wave-Particle Duality in Quantum Mechanics

The Uncertainty Principle

Heisenberg's uncertainty principle: can't measure both position and momentum precisely at the same time
This is due to the wave nature of particles, as described by a wavefunction ()
The most wavelike waves are momentum states
In the two-slit experiment, photon wavefunctions spread out from source to screen:
- Each photon wavefunction is composed of momentum states in different directions
- These wave functions interfere with each other, causing bright or dark spots on the screen depending on the lengths of lines between slits and target point (p)
When lengths di er by an integer number of wavelengths: amplitudes add up, resulting in a bright spot
When lengths di er by an integer number plus half a wavelength: amplitudes cancel out, resulting in a dark spot
In all other cases: intermediate intensity due to some angle between displacements

Momentum States and Wavefunctions

Momentum states correspond to corkscrews in position space (Argand planes)
- Tightly wound = high momentum, energy, short wavelength, high frequency
- Loosely wound = low momentum, energy, long wavelength, low frequency
- Limiting case: no wind, straight line = zero momentum
Fourier transforms of position wavefunction (x) and momentum wavefunction (p) are symmetrical
- Position states correspond to delta functions in momentum space
- Momentum states give delta functions in position space
Useful for different types of measurements:
- Position space: measures particle position, uses position wavefunction (x)
- Momentum space: measures particle momentum, uses momentum wavefunction (p)

Quantum Wave Packet Evolution and Measurement Procedures

Heisenberg's Uncertainty Principle

Classical physics: can accurately measure both position and momentum of a particle at the same time
Heisenberg discovered an absolute limit on the product of accuracies (x and p) for measuring a particle: x * p
The more precisely one measures position, the less precisely one can measure momentum, and vice versa
Example: if position measured to nanometer level, momentum becomes uncertain enough to be located 100 km away within a second
Three interpretations of uncertainty principle:
1. Measurement process is clumsy, leading to random kicks in particle's motion
2. Particle has inherent randomness and unpredictability on quantum level
3. Concepts of classical position and momentum are not applicable to quantum particles
Description of wavefunction:
- Momentum state: corkscrew curve, equal squared moduli for all positions, completely uncertain position
- Position state: delta function, precisely located at one point, completely uncertain momentum
- Intermediate case: small region of appreciable distance from axis in both position and momentum space
Schrdinger's equation describes time development of wave packet as a deterministic process (U)
When measuring quantum eects at the classical level, we adopt a different procedure (R) to form squared moduli of quantum amplitudes to obtain probabilities. This introduces uncertainties and probabilities into quantum theory.

Quantum Spread and Particle Cohesion Two-Peaked Wavefunctions

Understanding Quantum Theory: Wavefunctions and Particles in Two Places at Once

Background:

Non-determinism in quantum theory comes from R (probabilistic rule) not U (deterministic Schrdinger equation)
Wavefunction describes objective reality of individual particle, spread out over space

Behavior of Momentum State:

Stays the same until interaction or measurement
Predictable and clear-cut as in classical theory
Probability comes from R when measuring position

Schrdinger Equation:

Determines particle's behavior precisely, no indeterminacy or probabilistic rule
Evolution of wavefunction (x) determined by this equation

Particle in Two Places at Once:

Spread-out view necessary for motion determination
Acceptance of reality: particle can be spread out over large regions until next measurement
Two-peaked state after passing through a pair of slits
- Sum of wavefunctions t and b, peaked at top and bottom slits respectively
- Particle in both places at once if accepting the reality of its state as represented by the wavefunction
- Standard objection: measurement reveals only one place, but various measurements can distinguish two-peaked states from others.

Quantum Superposition and Interference in Photons

Photon Wavefunction and Double-Peaked Distributions

Three Different Ways of Observing a Photon's Distribution:

Figure 6.16: Three different ways a photon wavefunction can be doubly peaked (see the -curves in each case).
Distinguishable through various measurements.
All possibilities are actual ways for the photon to exist.

Two-Slit Experiment vs Half-Silvered Mirror:

Two-slit experiment: photons pass through both slits at once, creating interference effects.
Wavefunction splits into two parts with different peak locations (closer together).
Separation between peaks increases over time without limit.

Experimental Set-up with Half-Silvered Mirror:

Re ects half of the light and transmits the other half through mirror.
Wavefunction splits into two separate parts, each describing a different location.
Can be much more widely separated than in two-slit experiment.
Photon may travel over vast distances while being in both places at once.

Interference Effects:

Two parts of the beam can interfere with each other upon meeting again, leading to unexpected results.
Cannot be explained by probability weighting for two alternatives.

Experiments and Results:

Experimental set-ups have been carried out with various path lengths (including long distances).
Results agree with quantum mechanical predictions.

Implications of Photon's State Between Half-Silvered Mirrors:

Niels Bohrs view on the unreality of a photon's state between measurements is too pessimistic.
Quantum mechanics provides a wave function to describe reality, and it can be doubly peaked in certain cases.
Distinction between different states (t+b vs t-b) determines whether or not a photon reaches specific detectors.

Superposition of States:

Linear superposition allows us to compose various quantum states by adding them using complex number weightings.
This feature is essential for understanding the reality of particles in multiple places at once.

Quantum State Vectors in Hilbert Space

Quantum Linear Superposition and Hilbert Space

Quantum Linear Superposition:

Any two states, regardless of how different they are, can coexist in a complex linear superposition
A physical object, made up of individual particles, could exist in such superpositions of spatially widely separated states, being in two places at once
The formalism of quantum mechanics does not distinguish between single particles and complicated systems of many particles in this respect

Question of Macroscopic Objects:

Why don't we experience macroscopic objects (e.g., cricket balls, people) having two completely different locations at once?
The explanation is that the system is at the classical level until an observation or measurement is made
This "begs the controversial question" of why the quantum rules can be changed from U to R in this manner

Hilbert Space:

In quantum theory, a Hilbert space represents the quantum state of an entire system
A single point in Hilbert space represents the quantum state of a system
The mathematical structure of a Hilbert space is a complex vector space, allowing addition and weighted sums of its elements
Elements of the Hilbert space are called state vectors and are denoted by symbols in angled brackets (e.g., |x)
Addition of two state vectors is written as | + |X, with weightings z and w
Multiplication of a single state by a complex number is equivalent to multiplying the length by the number while keeping the direction unchanged
Each dimension in a Hilbert space corresponds to an independent physical state of a quantum system

Orthogonal Complementation and Projections in Hilbert Space

Concepts in Quantum Mechanics: Hilbert Space

Hilbert space: mathematical structure used to describe quantum systems
Two-dimensional Hilbert space: elegant picture for location of a single particle with position and momentum states represented as axes
Infinite-dimensional Hilbert space: arises when considering multiple dimensions or aspects of a system (e.g., position, momentum)
- Each state corresponds to an independent direction in the Hilbert space
Importance of orthogonality between rays (states) for quantum mechanics
- Orthogonal rays: independent of one another
Measurements: selection of orthogonal basis vectors, representing a complete measurement as a set of orthogonal basis states
- After measurement, system jumps to selected axis based on probability amplitudes and their squared moduli.
Complete measurement: idealized measurement requiring infinite precision in locating or measuring properties (e.g., position) anywhere in the universe
Yes/no measurements: most fundamental type of measurement, can be used to narrow down other properties.

Quantum Spin Intrinsic Rotation of Particles

Quantum Mechanics: State Reduction upon Measurement

State vector reduction: A state jumps from one subspace Y or N upon measurement based on its projection into these orthogonal complements
The probability of jumping to Y is proportional to the ratio of the squared length reduction in the projection
Normalization ensures probabilities sum up to unity (Pythagorean theorem)
Spin: A measure of particle rotation, conserved like energy and momentum
- Amount of spin is always the same for a particular type of particle
- Direction can vary in odd way for fermions

Spin: Fermions vs Bosons

Fermions: Particles with odd-numbered spin (/2, 3/2, 5/2) that rotate differently under complete rotation
Bosons: Particles with even-numbered spin (0, 1, 2, etc.) whose state-vector returns to itself upon complete rotation
Electrons as an example of a spin-half particle with two-dimensional state space for up and down spins.

Geometrical Representation of Quantum States on Riemann Sphere

The Geometrical Relation Between Spin Direction and Complex Numbers

The ratio of z to w, denoted as q, is significant instead of the absolute complex numbers themselves
When q 0, it can be represented on the Argand plane, similar to Chapter 3's discussion
Stereographic projection maps the Argand plane onto a sphere (Riemann sphere)
The direction of spin given by | = w| + z| is provided by the point q on the Riemann sphere
Points on the sphere correspond to possible complex ratios for an electron's spin states

Calculating Probabilities with Angle Between Directions

Measuring spin in a particular direction (a) results in either YES or NO answer
Probability of YES for second measurement: (1 + cos), where is the angle between directions a and b
Probability of NO for second measurement: (1 - cos)
If measurements are at right angles, probability is 50% each way; acute angle makes YES more likely, obtuse angle makes NO more likely.

Objectivity and Measurability in Quantum States

Despite probabilities for outcomes, there's an objective aspect to quantum states represented by state vectors or wave functions
Some argue that state vectors describe a single system or provide probability information about an ensemble of similarly prepared systems.

Quantum State Objectivity and Impossibility of Copying

Quantum Mechanics and Physical Reality

Some doubt about physical reality of state vectors in quantum mechanics due to limited measurability
State vector is not measurable, but seems to be an objective property of the system
Objectivity vs. measurability:
- Spin state of single particle has a well-defined direction before measurement
- In some cases, the spin state cannot be considered on its own as it's entangled with other particles
Copying quantum states:
- Impossible to copy a state while leaving original intact
- Can be copied if the original is destroyed
Photon spin and polarization:
- Photons possess spin but cannot have a fixed point due to traveling at light speed
- Polarization is the phenomenon that determines how much light passes through a polarizing filter

Geometric Representation of Spin States in Quantum Physics

Electromagnetic Waves Polarization

Plane-Polarized Electromagnetic Wave:

Consists of electromagnetic waves oscillating in a single plane
Two polaroids filter waves based on polarization
- First polaroid: measures and allows through if polarized appropriately
- Second polaroid: asks same question but for another direction
Probability of passing through second polaroid given first one is cos angle between directions

Riemann Sphere:

Used to describe full complex number array of polarization states
Includes circular and elliptical polarization
Represents possible ways a photon can be polarized as points on the sphere

Objects with Large Spin:

More complicated space of physically distinguishable states than Riemann sphere for systems greater than two basis states
For spin n/2 particles:
- Every state uniquely characterized by an unordered set of n directions outwards from center (n points on Riemann sphere)
- Directions correspond to measurements and outcomes of spin values in those directions.

Remarks:

Massive particles or atoms have larger spin than half, resulting in more complex quantum systems with n + 1 possible outcomes
The quantum descriptions do not go over to classical Newtonian ones as previously believed for large and complicated systems.

Quantum Many-Particle States Non-Product Superpositions

Quantum Mechanics and Many-Particle Systems

Spin States:

Spin states of large angular momentum correspond to many points across the Riemann sphere
These spin states are a collection of individual spin states pointing in arbitrary directions

Classical Correspondence:

Most large-spin quantum states do not resemble classical ones
However, they are linear combinations of (orthogonal) states that do resemble classical states
Measurements can cause the system to "jump" to one of these classical-like states

Many-Particle Systems:

Quantum descriptions of many-particle systems are more complicated than single-particle states
Require specifying possible locations of all particles separately, leading to a vast space of possible states
The quantum state is a function of multiple positions, not just individual functions as in classical theory

Example:

For 10 possible positions, the state of a single particle requires 10 complex numbers
For two particles, it requires 100 complex numbers (one for each pair of positions)
The general two-particle state is a linear superposition, not a simple product of individual states

Identical Particles Fermions and Bosons in Quantum Mechanics

Quantum Mechanics and Distinction Between Meanings of 'And' and 'Plus'

Clear distinction must be maintained between meanings of words 'and' and 'plus' in quantum mechanics due to misuse of 'plus' like in insurance brochures
For three particles, general state requires 1,000 complex numbers for specification (as compared to ten alternative positions)
Rules are different for identical particles versus distinguishable ones: Pauli exclusion principle applies to fermions, allowing no two of the same type to be in the same state

Fermion and Boson States

Fermions (electrons, protons, neutrons): particles must have distinct states under interchange, leading to different number of basis states
- Two fermions: 45 complex numbers needed for specification
- Three fermions: 120 complex numbers required
- Pair of identical bosons: 55 complex numbers needed; three bosons: 220 complex numbers, etc.
More realistic description involves continuum of position states but essential ideas remain the same
Complication with spin: for a spin-half particle (fermion), two possible states per position (labeled by 'spin up' and 'spin down') leading to 20 basic states in simplified situation.

Identity of Particles According to Modern Theory:

No reference to individual electrons or photons as they cannot be specified uniquely due to fermion's Pauli exclusion principle. Instead, we can refer to pairs or conglomerate of particles.
Approximation allowed for individual electrons, protons, and photons based on total picture of all particles in a system.

Paradox of EPR and Non-locality in Quantum Mechanics

Einstein's Paradox and Quantum Theory

Introduction:

Einstein's contribution to quantum theory: photon concept, wave-particle duality, bosons
Rejection of probabilistic aspect and objective reality in quantum description
Argument against subjectivity and need for an objective physical world

Hidden Variables and Non-Locality

Hidden variables theory proposed by Einstein followers like Bohm
Objective reality but no direct access to hidden parameters
Successful example: de Broglie-Bohm model
Problem of non-locality and difficulties with relativity remain unresolved

Experiment:

Two subsystems A and B, e.g., particles
General combined state is a superposition (plus) of products instead of product (and)
Measuring A causes state of B to jump: | ||, NO ||
No need for localization between A and B

Objections:

Sudden change in second ball's uncertain state unrealistic
Instantaneous non-local influence from one particle to another problematic.

Implications:

Physical objectivity can be achieved without determinism but with some uncertainty or indeterminacy
Standard quantum theory provides a reality that usually evolves deterministically, but occasionally jumps due to measurements.

Quantum Entanglement Instantaneous Spin State Correlation

Non-Local Probabilities in Quantum Theory vs Classical Viewpoint

Experiment Scenario:

Particle A decays into electron E and positron P
Electron's spin determines posistron's spin, even when separated
Measurement choice on one particle seemingly influences the other instantaneously

Classical Hidden Variable Theory:

Each part of system already knows results of experiments beforehand
Probabilities arise due to experimenter's lack of knowledge

Quantum Formalism and Bell's Theorem:

Quantum formalism leads to conclusion that spins are a linear superposition of all possible directions, not just up/down or right/left
Measurement outcome determines which component of the superposition is observed
Repeating experiment with different measurement settings yields results:
- Same settings on both sides always disagree
- Randomly chosen settings have equal probability of agreement and disagreement

Limitations of Classical Model:

Cannot explain quantum phenomena where measurement choices instantaneously influence distant particles

Bell's Theorem:

Proves that no local realistic (hidden variable) description can give the correct quantum probabilities.

Quantum Non-Locality and Experiments with Photons

The EPR Paradox and Bell's Theorem

David Mermin's simple version of the EPR paradox: shows a contradiction between a local realistic view of nature and the results of quantum theory
The E-measurer and P-measurer each have three settings for measuring the spins of their respective particles
Properties (1) and (2) follow directly from the quantum probability rules

Property (1): Disagreement Between Measures

If P-measurer measures spin opposite to E-measurer, property (1) follows immediately

Property (2): Probability of Agreement vs. Disagreement

For measured directions at 120 degrees apart:
- If E-measurer says "YES", P-direction is at 60 degrees to spin state
- If E-measurer says "NO", P-direction is at 120 degrees to spin state
This leads to a probability of agreement = (1 + cos(60)) and disagreement = (1 + cos(120))
The averaged probability for the three possible P settings, if E gives "YES", is equally likely to agree or disagree

Local Realistic Models Are Ruled Out

If prepared answers were YES, YES, YES / NO, NO, NO, there are 9 cases of disagreement and 0 of agreement
No set of prepared answers can produce the quantum-mechanical probabilities

Experiments with Photons: A Problem for Relativity?

Similar experiments have been performed using photon polarizations
Experimental results are fully consistent with quantum theory predictions, but inconsistent with local realistic models
Aspect's experiment: decisions on which direction to measure were made after photons were in full light
- This "non-local influence" must travel faster than the speed of light, but cannot be used to send messages
The realistic view of the state-vector applies to this EPR-type experiment:
- Photon pair acts as a single unit until one is measured, then individual polarizations have definite states

Quantum-Mechanical Paradoxes and Field Theory

Non-Relativistic vs. Relativistic Quantum Mechanics

Spacelike-Separated Measurements:

Two measurements of polarization are spacelike separated, meaning each lies outside the others light cone (e.g., R and Q in Fig. 5.21)
The order in which these measurements occur is not physically meaningful, but depends on the observer's motion

Observer Relativity:

If an observer moves rapidly, they can judge the "rst" measurement to have occurred either on the right or left
This leads to mutually inconsistent pictures of physical reality (Fig. 6.32)

Schrdinger's Equation vs. Dirac's Equation:

Schrdinger's equation is a deterministic, linear equation that describes how a quantum state evolves over time
The Dirac equation (for fermions like electrons) was discovered in 1928 and is one of the "Great Field Equations" in physics

Quantum Electrodynamics:

Quantum electrodynamics (QED) is the most successful quantum field theory, describing the behavior of electrons and photons
QED allows for particle-antiparticle annihilation and creation from energy, with a varying number of particles

Quantum Superpositions and Schrdingers Cat Paradox

Quantum Electro-Magnetism (QED)

Precise value of magnetic moment of electron derived from Quantum Electro-Magnetism (QED)
Untidy and inconsistent theory initially gives nonsensical answers, requiring renormalization to remove
Not all quantum field theories amenable to renormalization and difficult to calculate with even when they are
Popular approach: path integrals involve quantum superpositions of entire space-time histories
Deeper understanding needed before confident in physical reality picture
Partially compatible with special relativity, but does not address relativistic interpretation of quantum jumps or gravity

Schrdinger's Cat Paradox

Thought experiment challenging the superposition principle and measurement problem in quantum mechanics
Closed system: sealed container with a cat and measuring apparatus
Quantum event triggers device that either kills or saves the cat
Inside observer sees R (probabilistic uncertainties), outside observer expects U (linear superposition)
Problem lies in maintaining linear superposition at large scales, which is not observed for classical objects like cats.

Quantum State Interference and the Observers Dilemma Challenges in Quantum Measurement

Physicist's Perspective on Quantum Superposition and Reality:

Subjective View of Physical Reality:

Some physicists argue that there is now so much experimental evidence for U (the linear combination of cat being alive and dead) that we have no right to abandon it.
This could lead to a subjective view of physical reality, where the outside observer sees the cat as in a superposition of life and death, but the inside observer has already collapsed the state.

Experimental Limitations:

Performing an experiment to distinguish U from anything orthogonal is practically impossible for the outside observer.
Quantum theory doesn't clearly distinguish between measurements that are possible and those that are not.

Environmental Considerations:

The cat itself and its environment involve a vast number of particles, making it difficult to distinguish complex linear combinations from probability mixtures.
Even if the reality is a complex superposition, how does it transform into one alternative or the other?

Density Matrices:

Some argue that complicated systems should be described by density matrices, but this doesn't fully resolve the issues around quantum measurement.
Classical probabilities and quantum amplitudes are involved in density matrices.

Quantum Superpositions Consciousness and Reality Dilemmas

Quantum Mechanics and Consciousness: Non-Linearity in Schrdinger's Cat Experiment

Chaos Effects and Linear Superpositions:

Linear U cannot resolve unwanted linear superpositions
Chaos eects cannot occur with U alone due to its linearity
A non-linear element is needed to resolve superpositions

Wigner's Model for Conscious Entities:

Linearity of Schrdinger equation may fail for conscious (or living) entities
Replaced by a non-linear procedure, resolving alternatives
Few and far between corners where consciousness exists

Participatory Universe:

Reality is lopsided towards areas with consciousness
Only in these regions do complex quantum linear superpositions get resolved into alternatives

Many Worlds Interpretation:

R never takes place, entire universe governed by deterministic U
Schrdinger cat exists in a superposition of life and death
Consciousness splits for each observer, creating multiple instances
Universes split at every measurement or macroscopic event

Summary of Standard Quantum Theory:

Applicable to quantum level where small energy differences exist between alternatives
Describes the world using complex number weighted superpositions called probability amplitudes
Quantum systems must be described by such a quantum state

Reviewing Standard Quantum Theory:

No clear answer on how to describe the actual world beyond quantum level
Puzzling issues persist regarding consciousness and measurement
Further ingredients needed to make many-worlds view work effectively.

Quantum Mechanics and the Quest for a More Comprehensive Theory

Quantum Mechanics and Classical Physics

Deterministic Evolution of Quantum States:

The quantum state evolves deterministically according to the Schrdinger equation (process U)
When quantum alternatives become large enough, they are no longer perceptible at the classical level
Superpositions of complex amplitudes appear as probabilities for the actual physical experience through the process R (reduction of state vector or collapse of wavefunction)

Non-Determinism in Quantum Theory:

The non-deterministic nature of quantum theory only enters at this point, not before U
This can lead to paradoxical and complicated quantum states for multiple particles, referred to as entanglements or correlations

Quantum States vs. Classical Physics:

Quantum states do not resemble classical physics for macroscopic bodies
Attempting to apply the linear laws of U to these bodies would violate common sense and relativity
A more comprehensive, non-linear law may be needed to replace U and R

Possible Changes in Quantum Theory:

The linearity of U may not extend into the macroscopic world
Some type of hidden variable viewpoint could potentially be acceptable but faces challenges due to non-locality from EPR experiments
A more radical change, possibly involving a new law, may be needed for quantum theory to merge with classical physics and explain the mind

Exploring Quantum Mechanics Probability Amplitudes and Fermion Exclusion Principle

Quantum Mechanics and Realism

Starting point: realistic perspective in philosophy
Opposition to subjective views of no real world (e.g., quantum mechanics' implications)
Einstein's unsuccessful attempts for a comprehensive theory beyond classical field theory (1955)

Background: Balmer's Spectral Lines and Quantum Mechanics

Discrete nature of particles revealed by spectral lines of hydrogen (Balmer, 1885)
Two evolution procedures in quantum mechanics: R reduction of state-vector and U unitary evolution (von Neumann, 1955)

Mathematical Concepts

Hilbert space and vectors
Scalar product/inner product
Unitarity, normalization, orthogonality

Measurement Processes

Measurement of a state involves jumping to either | or something orthogonal to | (Dirac notation)
Operators: || and their eigenvalues determine Yes or No answers
Stern-Gerlach apparatus for measuring spin (not applicable for electron spin due to technical reasons)

Fermions vs. Bosons

Fermi-Dirac statistics for fermions, Bose-Einstein statistics for bosons
Developed by Enrico Fermi and Paul Dirac in 1926
Distinguished from classical statistics (Boltzmann)

Experimental Indication of Quantum Nonlocality EPR Paradox

Experiment Results:

In 1924, Albert Einstein and Niraj Bose conducted an experiment with two measurement settings for E (up [] and right []) and P (45p to the right [] and 45 down to the right []).
The probability of agreement between measurements was (1 + cos 135) = 0.146 or approximately 15%.
If P measurements were not influenced by E settings, there would still be under 15% agreement between them.
However, this contradicts the expected over 85% probability of agreement based on the angle between settings.
This contradiction shows that the assumption that choice of measurement cannot influence results is false.
Mermin (1985) discussed this example, which was earlier addressed by Freedman and Clauser (1972).

Details of Experiment:

Settings: Two for E (up [] and right []) and P (45p to the right [] and 45 down to the right []). Actual settings are [] and [], respectively.
Probability of agreement without influence: Just under 15% between E and P measurements, or just over 85% if influenced.
Contradiction: Over 85% expected probability vs. actual under 15%.
Argument point: Possible non-computability in quantum field theory (cf. Komar 1964).

7 Cosmology And The Arrow Of Time

Cosmology and The Arrow of Time

The Flow of Time:

Central to our feelings of awareness is the sensation of time's progression
We seem to be moving ever forward, from a definite past into an uncertain future
The past is over and unchangeable, while the future seems undetermined with potentialities for reality
As we perceive time passing, the most immediate part of the future becomes realized and enters the fixed past

Physics vs Perception:

Physics tells a different story, with successful equations being symmetrical in time
The past and future seem physically on equal footing, contradicting our perception of time's arrow
Relativity challenges the concept of "now" with observer relativity
Discrepancies between conscious perception and physical theory suggest a time-asymmetrical ingredient is needed to reconcile the two

Time-Symmetry and Entropy Unraveling the Past vs Future Mystery

Understanding Time-Asymmetries in Physics through Thermodynamics:

Second Law of Thermodynamics

Important concept involving time-asymmetries
Inevitable increase of entropy (disorder)

Glass Falling from Table Example

In real life: glass falls, breaks into pieces, heat is generated from energy lost during fall
Time reversibility of laws: water could have flowed upwards and glass assembled itself back on the table
Unlikelihood of perfect coordination among atoms for assembly in reverse order
Normal parlance: causes precede effects; but be careful not to prejudice past vs future discussion
Deterministic equations (classical physics) have no preference for future or past
Future determines past equally as past determines future when evolving equations backwards in time
Imagining a universe where same symmetric classical equations apply but behavior is reversed: familiar kind of behavior, yet different interpretation of cause and effect.

Understanding Second Law of Thermodynamics:

Inevitable increase of entropy (disorder) in physical systems
Time asymmetry related to the fact that certain processes occur more frequently in one direction than the other.

Glass Falling from Table Example Reversed:

Heat energy is required to raise glass back onto table with coordinated motion
Such precise coordination is unlikely among atoms post event (unlike pre-event)
Energy conservation: heat energy comes from the loss of energy during fall, but reversing this would require absurdly precise atom movements to reassemble the system.
Coordinated motion after an event is acceptable and familiar, while coordinated motion before an event is viewed as magic or unlikely.
In our normal parlance, we assume cause precedes effect; however, in understanding time asymmetry in physics, this assumption may need to be reconsidered.
Deterministic equations have no preference for future or past direction, but the interpretation of cause and eect can differ depending on which direction is considered.
In a hypothetical universe where same symmetric classical equations apply but behavior is reversed: familiar kind of behavior but different interpretation of cause and eect (future as cause and past as ect or vice versa).

Time Reversal and Entropy A Counterfactual Scenario

The Coexistence of Perverse and Normal Experiences

Imagining a world where water glasses can self-assemble, eggs unscramble themselves, and sugar lumps form in coffee cups and jump into hands
In such a world, we would attribute the causes to teleological effects rather than random chance coincidences of atomic behavior
The glass on the table would be the cause, the atoms on the floor the effect, despite occurring earlier in time
Describing happenings in reverse of our normal conventions would not resolve the issue in this world

Understanding Our Actual World: Entropy and Special Ordering

In our world, causes precede effects; coordinated particle motions occur after large-scale changes
Introducing the concept of entropy as a measure of manifest disorder
Low entropy refers to special order or arrangement that is not manifest, such as precise coordination of individual particle motions
High entropy reflects manifest disorder and randomness in systems, like shattered glass and spilled water
The second law of thermodynamics states that the entropy of an isolated system increases over time or remains constant for a reversible system.

Coarse-Graining Phase Space for Entropy Measurement

Understanding Entropy: Second Law of Thermodynamics

Entropy Concept and Irreversible Systems:

Entropy is a measure of manifest disorder in a physical system, often associated with irreversibility (practical imprecision)
All systems are reversible when considering individual particle motions; irreversible refers to uncontrolled motions (heat)
Debate on subjectivity: different observers may perceive order differently but macroscopic changes dominate entropy calculations.

Entropy Concept Precision:

Utilitarian in precise scientific descriptions due to enormous contribution of random particle motions to entropy change
Phase space concept introduced for grouping identical physical systems based on manifest properties (coarse-graining)

Maxwellian Distribution and Thermal Equilibrium:

Most phase space corresponds to uniformly distributed gases in thermal equilibrium with characteristic random motion
Enormous volume of points representing these states occupies almost the entire phase space.

Entropy Discrepancies in Phase Space Gas Distribution Model

Gas Behavior in Thermal Equilibrium vs. Corner of Box

Macroscopically Indistinguishable States:

All describe gas being tucked up in the same way in corner of box
Constitute another compartment of phase space
Volume far tinier than thermal equilibrium states

Model for Gas in a Box:

Distribution of balls among cells represents different positions in box
Special subset of cells (1/10) represent gas molecule positions corresponding to region in corner

Probability of All Balls in Special Cells:

One ball: 1/10
Two balls: 1/190, then 1/110 with large number of cells
Three balls: 1/4060, then 1/1000 with very large number of cells
Four balls: 1/10000, then 1/100000 with very large number of cells
m balls: 1/10m

Special Compartment Volume:

For an m-atom gas, volume is 1/10 m of entire phase space volume

Entropy:

Measure of volume of compartment representing state in phase space
Logarithmic scale to make entropy additive for independent systems
Entropy difference between thermal equilibrium and corner region: about 1400 JK-1 (14 kJ)

Second Law in Action:

Gas starts in very special situation, then spreads and occupies larger volumes
Eventually settles into thermal equilibrium.

Time-Reversal Symmetry Breaking in Phase Space Dynamics

Second Law of Thermodynamics in Phase Space

Concept:

In a gas system, phase space represents the complete state of positions and motions of all particles
As time progresses, the point representing this information wanders about in phase space
Entropy increase is an exponential measure of the volume occupied by the point as it enters larger compartments over time

Explanation:

Initial state: Point starts off in a tiny region (low entropy) corresponding to all particles being in one corner of the box
Spreading gas: As gas spreads, the phase space point moves into larger volumes with each new volume dwarfing previous ones
Thermal equilibrium: The point eventually loses itself in the largest volume, representing thermal equilibrium (high entropy)
Entropy's role: Entropy increase provides a logarithmic measure of the volume occupied by the phase space point, which is inexorable and tends to increase over time
Time asymmetry: The argument deduces that entropy must decrease when moving backwards in time, but this is not consistent with actual behavior (entropy increases)
Alternative scenarios: If gas was previously in a small volume, more likely alternative scenarios involve removing a partition or heating to create space rather than starting from thermal equilibrium
Correcting the argument: The most likely way for the system to reach its current state might not be through reaching thermal equilibrium first and then collapsing into the corner; other scenarios could occur where entropy decreases before increasing.

Origins of Low Entropy in the Universe The Role of Photosynthesis and Sunlight

The Second Law of Thermodynamics and Low Entropy States

The Original Argument:

Attempting to apply the phase-space argument in reverse direction casts doubt on the original argument
The argument showed that, without other constraints, entropy would increase in both directions from a given low-entropy state
This is not in contradiction with the second law as the initial entropy was not constrained

The Puzzle of Low Entropy States:

The common presence of absurdly low entropy states in the universe is an "amazing fact" that we take for granted
For example, ourselves and other objects around us have a low entropy configuration
The puzzle lies in how this low entropy was created and maintained in the past, where there were constraining factors

The Origin of Low Entropy:

Our own low entropy comes from the food we eat and oxygen we breathe
Energy is conserved, but we must replace the heat lost as a high-entropy form to maintain internal organization
Plants play a crucial role in reducing entropy through photosynthesis, which uses sunlight to separate oxygen and carbon
The sunlight brings energy to earth in a low-entropy form, while re-radiated heat is in a high-entropy infra-red form

Entropy and Energy Flow in the Universe Sun-Earth Interactions

The Earth's Energy and Entropy

The earth (and its inhabitants) does not gain energy from the sun, but takes in low entropy energy from the sun and emits it back out as high entropy energy
The sun provides a source of low entropy through visible light photons that are absorbed by the earth and re-radiated as infra-red photons
This energy balance is maintained because there are fewer visible light photons reaching the earth than infra-green light photons leaving the earth

Plants and Entropy

Plants take in low entropy energy (fewer visible light photons) and re-radiate it as high entropy energy (more infra-red photons)
This allows plants to feed on this low entropy source and provide oxygen-carbon separation that is necessary

The Sun's High Temperature

The sun maintains a high temperature due to gravitational contraction from a previously uniform distribution of gas (mainly hydrogen)
Without gravity, the sun would not exist as a hot spot in the sky
Gravity holds the sun's material together and provides the necessary temperatures and pressures for thermonuclear reactions to occur

Fossil Fuels and Entropy

Fossil fuels come from prehistoric plant life that got their low entropy from the sun
The gravitational action in forming the sun out of a diffuse gas is crucial for this process
An alternative theory suggests oil and natural gas were trapped inside the earth when it formed and synthesized by sunlight before the earth existed

Nuclear Energy from Uranium-235

Nuclear energy in uranium-235 did not come originally from the sun, but from other stars that exploded thousands of millions of years ago
The material was thrown back into space during the supernova explosion, with a huge gain in entropy due to gravitational contraction
Gravity was ultimately responsible for the condensation of diffuse gas into a neutron star, which held the uranium-235 nucleus' low entropy store of energy

Cosmology and Expansion of the Universe Big Bang Theory and Friedmann-Robertson-Walker Models

Cosmology and The Big Bang

The Universe's Appearance:

Large scale, uniform appearance
Expanding universe: galaxies and quasars receding
Evidence for expansion: black-body background radiation (2.7K)

The Big Bang Theory:

Occurred approximately 10 billion years ago
Predicted by George Gamow in 1948 based on standard big bang picture
Observed by Penzias and Wilson in 1965, accidentally

Expansion of the Universe:

Uniform on a large scale, no special location
Comparison to expanding balloon: all points recede from each other
Three FRW models for universe expansion: spatially closed (positive curvature), zero/negative curvature (Euclidean)

Space-Time Description:

Euclidean planes for each moment of time, stacked together
Galaxy world-lines move away from each other into the future direction
No particular galaxy preferred in the description.

Origin of the Universe and the Big Bang Theory

The Expanding Universe: Positive vs Negative Curvature

Positively Closed Universe:

Cannot continue to expand forever according to Einstein's general relativity
Eventually reaches maximum expansion and collapses back in on itself (big crunch)
Only applies with a zero cosmological constant

Negatively Curved and Zero-Curved Universes:

Do not recollapse as they continue expanding forever
These models may have spatially infinite universe models or positively curved ones that expand infinitely

The Primordial Fireball: Origin of the Second Law of Thermodynamics

Big Bang Theory:

Universe was spewed out during creation, creating space itself
No pre-existing empty space into which material was ejected
Material was spread uniformly over the entire universe right from the start

The Hot Big Bang (Standard Model):

Universe moments after creation was in an extremely hot thermal state (primordial fireball)
Detailed calculations have been performed to understand the initial constituents and their changes during expansion and cooling
These calculations are based on non-controversial physics theories, though the events before 10^(-42) seconds remain uncertain.

Evolution of Stars and Black Holes Formation and Entropy

The Big Bang and Entropy: Understanding the Second Law

Material Composition of the Universe:

Mainly protons and electrons
Hydrogen gas formed at about 108 years after the big bang
Stars not formed instantaneously, gas expansion and cooling needed for clumping to occur
Unresolved issue: How galaxies form and initial irregularities required for galaxy formation

Distribution of Material in the Universe:

Stars collected in galaxies
Galaxies grouped into clusters and superclusters
Evidence shows early universe was very uniform

Entropy and the Second Law:

Initial entropy of the universe was low, not maximum as required by thermal equilibrium
Paradox: Big Bang initiated with high temperature (maximum entropy), yet second law demands minimum initial entropy
One explanation: Actual entropy lags behind permitted maximum due to expansion of the universe
However, this doesn't hold for a contracting universe model like a "big crunch"
Reasons for doubting reversal of entropy in collapsing universes: Black holes and violation of second law observed behavior

Black Holes:

Ultimate fate of stars like the sun (red giants)
Expanding to become red giants, eventually collapse into a black hole
Theories predict black holes have no entropy or temperature, challenging the second law.

Core Collapse and Formation of Compact Stars

White Dwarfs: Fate of Stars

Formation of White Dwarfs:

Core of red giant star, extremely dense and small concentration of matter
Eventually consumes entire red giant, leaving behind an actual white dwarf (about Earth size)
Sun's fate: red giant for a few thousand million years, then white dwarf for many more millions of years

Chandrasekhar Limit:

Calculation by Subrahmanyan Chandrasekhar in 1929 and Lev Landau around 1930
White dwarfs cannot exist if their masses are greater than approximately 1.4 solar masses (M)
Stars with mass exceeding this limit collapse catastrophically, forming a supernova

Neutron Stars:

Result of core collapse in some stars, above Chandrasekhar limit
Held apart by neutron degeneracy pressure
Densities much greater than white dwarfs and atomic nuclei
Modern value for maximum neutron star mass: about 2.5 M

Black Holes:

Result of core collapse in extremely massive stars (greater than Landau-Oppenheimer-Volkov limit)
Region of space where gravitational field is so strong that not even light can escape
Escape velocity exceeding the speed of light, impossible for objects to leave.

Space-Time Geometry of Black Holes and White Holes

Black Holes and White Holes

Formation of a Black Hole:

Collapse of body maintains spherical symmetry closely
Light cones indicate limitations on motion
Critical distance: Schwarzschild radius
- At this point, light's velocity is barely enough to counteract gravitational pull
Event horizon formed at the Schwarzschild radius
- Anything within cannot escape or communicate with outside world

Observations from Outside:

Portion inside event horizon not seen by observer A
Portion outside eventually visible, but image rapidly becomes too dim

Experience of Person B (inside black hole):

No noticeable change at horizon crossing
Local physics consistent with known laws
Entropy increases until final crunch

Theoretical Concepts:

White holes: time reverse of a black hole
- Probably don't exist in nature, but significant for theory

Space-Time Singularities:

Recall from Chapter 5 (p. 269)
Light cones indicate limitations on motion
Critical point where tips become vertical: singularity
- Extreme gravitational pull
Not expected to violate second law of thermodynamics inside a black hole.

Tidal Forces in Black Holes and Big Bang High and Low Entropy Singularities

The Tidal Effect of Space-Time Curvature

Tidal Distortion:

Spherical surface near a large body is stretched in one direction and squashed perpendicularly
The tidal distortion increases as the gravitating body is approached, varying inversely with the cube of the distance

Astronaut B's Experience:

Felt as a huge tidal effect, too large to survive close approach to a black hole
For larger black holes, the tidal eect at the horizon is smaller but still enough to make the astronaut uncomfortable
The tidal eect mounts rapidly towards infinity as the astronaut falls in

Ultimate Catastrophe:

The astronaut's body and molecules are torn apart
Even subatomic particles are destroyed, leading to a "crunch" that wreaks havoc on all matter and space-time
This is referred to as a "space-time singularity", the ultimate catastrophe where space-time ends

Classical Equations of General Relativity:

Show that such catastrophic behavior occurs when a black hole is formed
Challenged earlier hopes that this was an artifact of symmetries in the original model

Singularity Theorems:

Mathematical arguments, known as "singularity theorems", established the inevitability of space-time singularities
Provide evidence for the creation and destruction of all matter and space-time

Temporal Symmetry vs. Geometric Differences:

Appear to have exact temporal symmetry, but are not when examined in detail
Key to understanding the origin of the second law of thermodynamics

Astronaut B's Experiences:

Faces tidal forces that mount rapidly towards infinity
Experiences the "volume-preserving but distorting effects" provided by the space-time curvature tensor "WEYL"
The remaining part of the space-time curvature tensor, "RICCI", is zero in empty space

Singularity Behavior:

Close to a final singularity, the "distorting tidal eect" provided by "WEYL" goes to infinity, generally
With a big bang, the situation with the initial singularity is quite different:
- The distorting tidal eect of "WEYL" is entirely absent
- The symmetric inward acceleration of "RICCI" dominates near the initial singularity

Black Hole Formation and Entropy in Closed Universes

Reversal of Gravity's Effect on Clumping

Gravity causes reversal of clumping due to its universally attractive nature
As time goes on, clumping becomes more extreme leading to many black holes congealing and their singularities uniting in the final big crunch singularity

Comparison to Idealized Big Crunch Singularity

The final singularity does not resemble the idealized big crunch of the recollapsing FRW model with its constraint WEYL = 0
As more clumping occurs, there is a tendency for the Weyl tensor to get larger and larger, WEYL > 0 at any final singularity

Entropy in the Universe

For an ordinary gas, increasing entropy tends to make distribution more uniform
For gravitating bodies, high entropy is achieved by gravitational clumping and the highest by collapse into a black hole
This explains how a recollapsed universe does not need to have a small entropy as observed in the second law of thermodynamics

How Special Was the Big Bang?

Let's try to understand the constraint WEYL = 0 at the big bang
Imagining phase space: Each point represents a possible way the universe could start, Creator aims for a tiny volume (low entropy) to create our actual universe

Bekenstein-Hawking Formula for Black Hole Entropy

Proportional to surface area of black hole
For spherically symmetrical black hole, entropy is proportional to square mass
Greatest entropy achieved when all mass is concentrated in a black hole
Two black holes gain more entropy when they unite into one
Large black holes provide stupendous amounts of entropy compared to other situations.

Hawking's Analysis of Black Holes

Associated temperature with a black hole, not all mass-energy can be contained in the maximum entropy state
Hawking temperature is tiny for reasonable sized black holes and would become significant if mini-black holes or large numbers existed.

Quantum Gravity and the Origin of Entropy

Black Hole Entropy vs. Universe's Entropy

Background Radiation Entropy:

Previously thought to be the largest contributor to the universe's entropy
Approximately 10^8 for every baryon (photons per baryon)
With 10^8 baryons in the universe, total entropy would be 10^88

Black Hole Entropy:

Bekenstein-Hawking formula: Entropy per baryon = 10^20 (natural units) for a solar-mass black hole
If the universe consisted only of solar-mass black holes, total entropy would be 10^100
For realistic galaxy composition (10^11 stars and 10^6 solar-mass black hole per galaxy), entropy is ~10^21
After a long time, most galaxies' masses will be in black holes, leading to total entropy of 10^31

Creator's Precision:

To produce our observed universe, the Creator would have had to aim for an absurdly tiny volume (1/10^10123) of phase space

Singularities and the Second Law

Understanding the origin of the second law requires understanding the structure of space-time singularities
The "Weyl Curvature Hypothesis" suggests a time-asymmetric constraint at initial singularities that the Creator follows to create our universe

Exploring the Directionality and Flow of Time Insights from Relativity and Quantum Mechanics

Understanding the Directionality and Flow of Time: Theories and Insights from Journey to the Ends of Space and Time

Key Points:

Several prominent scientists have attempted to construct a theory explaining why time seems to flow in one direction, but with limited success.
Despite lack of an adequate theory, important lessons can be learned from journey through space and time concepts.
Theories on entropy play a role in understanding the directionality and flow of time:
- Entropy is gained through fusion of light nuclei (e.g., hydrogen to helium) in stars, but this is temporary as it requires gravity's concentrating effects.
- Black-body background radiation contains vastly greater entropy than matter in ordinary stars, which could potentially disintegrate heavier nuclei if concentrated back into material.

Speculative Return Trip:

Our return journey will be more speculative than the outward one but necessary to understand the mind.
Notes on some concepts discussed during our journey:
1. Relative vs. Absolute Space and Time: observers' light cones vs. simultaneous spaces make no difference in conclusions.
2. Controversial evidence for Gold's theory from ultra-deep well drillings in Sweden.
3. Entropy gain through fusion is temporary, made possible by gravity's concentrating effects.
4. Uranium production in supernovae unlikely to produce much entropy compared to gravity resources.
5. Infinite models vs. finite ones; no significant impact on discussion.
6. Confidence in unchanged parameters governing particle interactions since ancient times.
7. Pauli's principle and controversy around electrons occupying the same state.
8. Black hole theory development by Michell, Laplace, Oppenheimer, and Snyder using Newtonian theory with some debatable arguments.
9. Difficulty of ascertaining exact location of horizon in general non-stationary black holes.
10. Possible identification of gravitational contribution to entropy as Weyl curvature but no appropriate measure yet found.
11. Inflationary scenario and its limitations compared to the Weyl curvature hypothesis for explaining universe's uniformity on a large scale.

8 In Search Of Quantum Gravity

Why Quantum Gravity? What is New from Last Chapter?

New Insights:

We have not gained insights into why we perceive time to flow or why we perceive at all
More radical ideas are needed

Weyl Curvature Hypothesis:

Physics presentation has been less conventional than usual
Origin of the second law of thermodynamics can be traced to an enormous geometrical constraint on the big bang origin of the universe
Deductions from this hypothesis will be less conventional

Unconventional Viewpoint:

General relativity may have effects on the structure of quantum mechanics, rather than the other way around
The problems within quantum theory are fundamental and cannot be adequately resolved by interpretation alone
A radical new theory is needed to resolve the incompatibility between U (unitary evolution) and R (probabilistic state-vector reduction) procedures in quantum mechanics
Even if classical gravity has no direct influence, quantum corrections may still have relevance to mental phenomena

Conventional View:

Quantum effects on general relativity are insignificant at scales relevant to brains
Problems within quantum theory are best addressed by modifying Einstein's theory of space-time, not the standard structure of quantum mechanics

Quantum Gravity:

True quantum gravity theory will constitute a profound part of our understanding of Nature's universal laws
However, we are far from such an understanding
Putative quantum gravity theory would be remote from phenomena governing brain activity

Exploring Quantum Gravity Implications Weyl Curvature Hypothesis and Initial Singularities

The Weyl Curvature Hypothesis and Quantum Gravity

Background:

Conventional view: quantum gravity will resolve space-time singularities in general relativity
Surprising fact: quantum gravity's mark on singularities is time-asymmetrical (Weyl tensor = 0 at big bang, not restricted at future singularities)

Argument for Time-Asymmetric Quantized Theory:

Singularities require explanation from quantum gravity
Conventional procedures of quantization may not produce time-asymmetric theory
Historically, dynamical equations and boundary conditions have been separated in physics, but this distinction may dissolve in future comprehension of universe's laws
Personal viewpoint: we can explore implications of unknown theory (CQG) by accepting it will provide explanation for Weyl Curvature Hypothesis (WCH).

Implications:

Initial singularities have constraints based on CQG laws (WEYL = 0) if they exist, not just for big bang.

Time Asymmetry in Quantum State Reduction Probability Jump and Observation

Weyl Singularities and White Holes

Singularities in Black Holes vs. White Holes:

Singularities inside black holes satisfy Weyl conditions (Weyl = 0 for big bang singularity)
For white holes, the singularity is an initial singularity, requiring WEYL = 0 to comply with Weyl conformal gravity
This rules out the occurrence of white holes in our universe, as it contradicts the WCH requirement for white holes

Black Hole Explosions and Singularities:

The explosion of a black hole that has vanished after Hawking evaporation may not conflict with WCH
Such an explosion could be effectively instantaneous and symmetrical, allowing for WEYL = 0 hypothesis
It is likely that the first such explosion will not occur until after the universe has existed for much longer than its current age

Alternative Perspectives:

Some argue that CQG should allow two types of singularity structure, one with and one without WEYL
However, this does not explain why there are no other initial white hole singularities in the universe
The anthropic principle cannot account for the required level of specialness in the big bang

Time Asymmetry in State-Vector Reduction:

Quantum theory is time-symmetric in terms of unitary evolution, but not state-vector reduction (R)
The state-vector reduction procedure is time-asymmetrical as it involves a sudden jump to another state
This suggests that the R part of quantum mechanics may be inherently time-asymmetric, unlike the U part

Quantum Mechanics and Time Reversal A Simple Experiment

Reversed Quantum Mechanics Description

Background:

Illustrated in Fig. 8.2
State-vector taken as | immediately before O instead of after it
Unitary evolution applied backwards in time to the previous observation

State-Vector Evolution:

Reversed-time description: | represents state immediately past O
Normal forwards-evolved description: Some other state | just to the future of O (result of observation at O)

Calculating Probability p:

Same value as before, but what it refers to is different
Probability of finding result (|) at O given result (|) at O vs. probability of result at O given result at O

Experiment Example:

Lamp L emits photons randomly towards P with 50% chance of reflection or transmission by half-silvered mirror M
Reversed time description: When L registers a photon, wavefunction splits into two parts: one reaching photo-cell P and another reaching point A on the wall
Probability that photo-cell detects a photon given that source emits one is 50%, but probability that source emitted a photon given that photo-cell detected one is not 50% (time irreversibility)
Quantum mechanical calculation: Given L registers, probability of P registering = 1/2 (squared amplitudes for each path)
Experimentally observed result: One would obtain a 50% success rate in detecting the photon at both ends of the experiment.

Quantum Mechanics Time Reversal Limitations and Hawkings Box Thought Experiment

Quantum Mechanics and Time Symmetry: The Procedure R

Background:

Using eccentric reverse-time procedure to obtain answers in quantum mechanics
Two types of questions: given that L registers, what is the probability that P registers? vs. Given that the photon is ejected from the wall at B, what is the probability that P registers?
Standard R procedure works for future probabilities but not time-reversed ones
Discussion on second law of thermodynamics and its role in measurement process

The Box Experiment:

Thought experiment proposed by Stephen Hawking
Imaginary box with totally reflecting walls, no object or signal can pass through
Contains large amount of material substance
Question: what happens to the contents over time?
According to second law of thermodynamics, entropy should increase and reach maximum value
When thermal equilibrium is attained, most material collapses into a black hole with some remaining matter and radiation
Technological difficulties ignored for thought experiment purposes.

Evolution of Black Hole Formation and Disappearance in Hawkings Box Phase Space

Understanding Hawking's Box Equilibrium and Fluctuations

Hawking's Box:

Contains various states of the system, including those with a black hole (region ) or without a black hole (region )
Phase space schematic: Figure 8.4

Thermal Equilibrium States:

Major portion of region represents thermal equilibrium with a black hole present
Smaller major portion represents thermal equilibrium without a black hole

Hawking Evaporation and Arrows in Phase Space:

Classical theory of general relativity: Black holes can only swallow things, not emit them
Quantum mechanical effects (Hawking radiation): Black holes emit particles through Hawking radiation
Occasionally, a black hole may lose energy and mass due to Hawking evaporation, causing it to heat up and potentially disappear
Arrows in phase space represent the system's temporal evolution:
- Some arrows cross from region to region , representing black hole formation by gravitational collapse
- Other arrows cross from region to region , representing Hawking radiation emission when a black hole loses energy

Coarse-Graining and Entropy:

Coarse-graining compartments in phase space: Indistinguishable points within each compartment
Entropy increases as the system moves towards larger, more thermally equilibrated compartments (maximum entropy)
Occasionally, a large fluctuation can occur, causing the system to move into smaller, less thermally equilibrated compartments with lower entropy
This allows for transitions between regions and , representing Hawking evaporation or gravitational collapse events.

Quantum State Reduction Compensates for Black Hole-Induced Information Loss

Black Holes vs Hawking Radiation

Ease of Creating vs. Destroying Black Holes:

Important issue: same difficulty to build or destroy a black hole as Hawking radiation
Liouville's theorem: phase space volume is preserved by the flow, cannot accumulate on one side or another (incompressible fluid analogy)
Time symmetry argument: if we reverse clock, should get same answer as running forwards
- Black holes and white holes are physically identical? (Hawking suggestion but not believed)

Weyl Curvature Hypothesis (WCH):

Singularities absorb matter and destroy information
Implications for the discussion of Hawkings box: unknown physics governing behavior is involved
Time asymmetry introduced, different from Hawkings time-symmetric quantum gravity theory
WCH causes ow lines to merge in region 8 (violation of Liouville's theorem)

Quantum Mechanical Process of State Vector Reduction (R):

Compensates for merging of ow lines due to WCH
Bifurcation of ow lines required: time-asymmetric process
Example: photon emission from lamp has two possible outcomes, bifurcating phase space lines.

Conclusion: Black holes and white holes have different implications in the context of quantum mechanics and general relativity. While Hawking suggested that they might be physically identical due to time symmetry, this idea is not widely accepted. The Weyl curvature hypothesis introduces a time asymmetry into the discussion and offers an alternative explanation for the behavior of ow lines in relation to black holes. R, a quantum mechanical process, may play a role in compensating for the merging of ow lines caused by WCH.

Gravitational State-Vector Reduction Quantum-Gravity Interplay

Arguments for Linking WCH and R in Quantum Gravity Theory

Overview:

Suggestion: quantum mechanical state-vector reduction is linked to WCH (Wheeler's law of entropy conservation)
Both imply major implications for correct quantum gravity theory (CQG)
Discussion on when state-vector collapse might occur

Arguments Favoring the Link:

Heuristic Argument: State-vector reduction and WCH are intimately associated with second law of thermodynamics.
Classical vs Quantum Phase Space: Idea that correct theory requires a new type of mathematical space intermediate between Hilbert space and classical phase space.
Technical Difficulties in Applying General Relativity to Quantum Theory: Failure of rules of quantum theory to fit well with ideas of curved space-time geometry.
Significant Amount of Curvature: When this level is reached, linear superposition fails, and non-linear instability sets in, leading to a choice between alternatives and state-vector collapse.

State-Vector Reduction:

Suggestion: ultimate phenomenon may be gravitational
Connection to black holes: when not present, reduction can occur
Acceptance of this view faces difficulties with bringing general relativity principles into quantum theory due to technical complications.

Arguments Against the Link:

Uncertainty about building Hawking's box and idealizations involved therein.
Difering points of view: some physicists suggest that a small nugget of black hole would remain.
Postulating total volume of phase space as finite goes against basic ideas about black hole entropy and quantum systems.

State-Vector Collapse:

When might it occur?
- Acceptance of state-vector reduction as gravitational phenomenon leads to the question of when collapse should take place.
- Connection to significant amount of curvature: when this level is reached, rules of linear superposition fail and non-linear instability sets in.

Quantum Gravity in Cloud Chambers and Gas Boxes

One-Graviton Level and Reality Transition

Procedure R:

Occurs spontaneously in an objective way, independent of human intervention
Aims to bring reality between quantum level (atomic, molecular) and classical everyday experiences
Characterized by the Planck mass mp = 10^(5) grams

Observation with Wilson Cloud Chamber:

Particle emits spherical wavefunction on decay
Multiple possible particle tracks in superposition
Ionization of vapor leads to multiple strings of condensing droplets
Probability weighted collection of alternatives, only one realized in physical world

One-Graviton Stage:

Approx. 1/100th of mp = 10^(-8) g
Rough calculation suggests it occurs when a single droplet reaches this size
Uncertainties and complexities remain, including differences in calculations for strings of droplets

Alternative Manifestations of Quantum Effects:

Disturbance of gas atoms upon particle entry
Cascading effect leading to disturbance of virtually every atom in the gas
Two distinct spacetime geometries for superposition: particle entering and not entering

Implications:

Procedure R can occur even without a manifest magnification of quantum eects
The difference between two dierent spacetime geometries reaches the one-graviton level when it is unclear which geometry will be realized.

Quantum Gravity and State-Vector Reduction in Physics and Consciousness

Quantum Mechanics and State Vector Reduction

Description of Quantum Mechanically

Initial consideration only original particle's positions in superposition
After a short while, all atoms of gas must be included in complex linear superposition
Two possible states: one where the particle enters the box and another it doesn't
Extend this superposition to all the atoms in the gas
Difference in gravitational fields of individual atoms
When difference exceeds one-graviton level, state vector reduction occurs
Particle has either entered or not, complex linear superposition reduces to statistically weighted alternatives

Observation and State Vector Reduction

Cloud chamber used as a way for quantum mechanical observation
Other types of observations (photographic plates, spark chambers) can be treated using the one-graviton criterion by approaching them like the box of gas above
Work remains to be done in seeing how this procedure might apply in detail
New theory required involves radical ideas about nature of spacetime geometry and relativity

Notes:

Proposed modifications to Einstein's equations: changing RICCI = ENERGY via higher-order Lagrangians, changing number of dimensions of spacetime, introducing supersymmetry, string theory. All these proposals are tentative in the terminology of Chapter 5.
Violation of all four symmetries normally denoted by T, PT, CT, and CPT is required for a new physical action to be involved when thinking or perceiving consciously.
Proposed quantum gravity explanation of initial state from Hawking (1987, 1988) has theoretical substance but lacks time-asymmetric input.
Forward and backward-time descriptions have the same probability calculation due to unitary evolution.
Probability of past event given future event can be understood by imagining entire history of universe and counting fraction of occurrences accompanied by desired event.
Allowing for longitudinal gravitons (virtual gravitons that compose a static gravitational field) is problematic in terms of clear mathematical definition.
Precise mass value obtained from calculations by Abhay Ashtekar requires caution due to theoretical problems and assumptions used.
Other attempts at objective theory of state vector reduction include Krolyhzy (1974), Krolyhzy, Frenkel, and Lukcs (1986), Komar (1969), Pearle (1985, 1989), Ghirardi, Rimini, and Weber (1986).
Penrose has been involved in developing a non-local theory of space-time called twistor theory but it lacks essential ingredients for discussion here.

9 Real Brains And Model Brains

Anatomy and Structure of Brain

Brain: Large, convoluted structure controlling actions and consciousness
Cerebrum: Largest part, divided into left and right hemispheres (frontal lobe, parietal lobe, temporal lobe, occipital lobe)

Cerebrum has outer layers of grey matter (cortex) and inner regions of white matter

Cerebellum: Smaller portion at back, resembling two balls of wool
Other structures: pons, medulla, thalamus, hypothalamus, hippocampus, corpus callosum, etc.
Human brain has larger cerebrum and cerebellum than other animals

Functioning of Brain

Cerebral Cortex: Grey matter where computational tasks occur

Visual cortex: Interprets vision from both eyes (left side for left visual field, right side for right visual field)
Auditory cortex: Deals with sounds received on left and right sides
Olfactory cortex: Processes smells from nostrils
Somatosensory cortex: Handles touch sensations

Motor Cortex: Concerned with activating movement of different body parts

Right motor cortex controls left-hand side of the body, left motor cortex controls right-hand side.

Primary regions (visual, auditory, olfactory, somatosensory, and motor) are most directly concerned with input and output.
Secondary regions process sensory information in a subtle way and create more complex plans.
Tertiary or association cortex is where abstract and sophisticated brain activity occurs.

Cerebrums Role and Connected Brain Regions

The Motor Homunculus and Cerebrum's Role in Movement:

Motor Homunculus:

Illustrates portions of frontal lobes that most directly activate movement
Figures 9.4 and 9.5 provide an overview

Cerebrum's Role in Movement:

External Sensory Data: Enters at primary sensory regions, processed to varying degrees in secondary and tertiary regions
Output (Movement): Achieved by activating primary motor regions
Speech: Particularly interesting since it's thought of as a uniquely human cognitive function
- Most right-handed people and many left-handed ones have speech centers mainly on the left side of the brain
- Essential areas: Broca's area in frontal lobe for formulation, Wernicke's area in temporal lobe for comprehension
Processing: Movement of activity from primary to secondary and tertiary regions as input is analyzed
Action Implementation: Signals sent to primary motor cortex, ultimately reaching muscle groups
Internal Activity: Can also carry on independently without specific sensory input (thinking, calculating, reminiscing)
Cerebellum's Role: Responsible for precise coordination and control of body movements; timing, balance, delicate actions
Learning New Skills: Initially controlled by cerebrum when learning, but once mastered, cerebellum takes over
Other Brain Regions: Hippocampus (stores long-term memories), corpus callosum (connects right and left hemispheres), hypothalamus (seat of emotions), thalamus (processing center).

Brain Structure and Consciousness Reticular Formation and Upper Brain-stem

The Human Brain: Regions and Consciousness

Three Major Brain Regions:

Hindbrain (Rhombencephalon): Includes cerebellum, reticular formation (part in midbrain and hindbrain), and oldest evolutionarily.
Midbrain (Mesencephalon): Sits between hindbrain and forebrain.
Forebrain (Prosencephalon): Consists of newly developed parts like cerebrum, corpus callosum, thalamus, hypothalamus, hippocampus, etc.

Location of Consciousness:

No consensus on the relationship between brain state and consciousness.
All parts are not equally involved; cerebellum is more automaton-like than cerebrum.
Cerebrum likely more involved due to conscious decision making (e.g., walking) and complex plans.
Penfield's argument: Upper brainstem, including thalamus and midbrain (especially reticular formation), may be the seat of consciousness.
- Conscious awareness arises when this region is in communication with the appropriate cerebral cortex region.
Other neurophysiologists suggest that reticular formation could be the seat of consciousness due to its association with general brain alertness and state of unconsciousness.
Evolutionary argument against reticular formation: Ancient part, implying lower animals possess consciousness.
- Counterargument: Low-level forms of consciousness in other creatures cannot be ruled out based on current evidence.

Cerebellum vs. Cerebrum:

Controls detailed muscle movements during conscious actions (like walking) without requiring thought.
Unconscious reactions, such as removing hand from hot stove or blinking, are likely mediated by spinal cord instead of the brain.

Upper Brainstem and Consciousness:

Penfield's argument: Upper brainstem, including thalamus and midbrain (especially reticular formation), is involved in consciousness due to its role in communication with the appropriate cerebral cortex region for specific sensations, thoughts, memories or actions being consciously perceived or evoked at that moment.
Conscious awareness arises when this region is active and in direct communication with the cerebral cortex.

Duality of Consciousness in Split-Brain Subjects

Awareness and Consciousness in Different Species

Frogs, Lizards, Cod Shark:

Impression of conscious presence is weak
No demand for self-awareness or sophistication

Dogs, Cats, Apes, Monkeys:

Strong impression of conscious presence
No demand for complex feelings or self-awareness

Dreaming State:

Some form of awareness present at a low level
Reticular formation and hippocampus may play roles in manifesting consciousness

Viewpoints on Consciousness

Simple Presence: Awareness can be associated with the simple feeling of existence, as observed in animals.
Complex Algorithms: Consciousness may result from the complexity or depth of an algorithm, such as those carried out by the cerebral cortex.
Language and Humanity: Consciousness is linked to human language abilities, which distinguish humans from other animals and enable subtle thinking and self-expression.
Left vs. Right Hemisphere: Some neurophysiologists believe consciousness is associated with the left cerebral cortex but not the right, while others view this as an odd perspective.

Split Brain Experiments:

Subjects had corpus callosum severed for epilepsy treatment
Left and right hemispheres behave as virtually independent individuals
Communication between sides is primitive, primarily through emotional feelings
Controversy over whether each side should be considered a separate conscious individual.

Visual Cortex and Conscious Perception

Split Brain Research Findings:

Both hemispheres can be conscious as evidenced by P.S.'s ability to speak and have distinct desires
Multiple consciousnesses may result from bifurcation after operation
Blindsight experiments suggest some parts of cortex are more associated with consciousness than others
Visual Cortex processing: certain regions process visual information before reaching the visual cortex, like the lower temporal lobe in blindsight cases.

Split Brain Research: Consciousness and Hemispheres

Only left hemisphere can convincingly claim to be conscious due to language capabilities
Right hemisphere may struggle with verbal questions regarding consciousness
Two independent minds possible after bifurcation through split-brain procedure
No operational way of deciding which consciousness is original before operation.

Blindsight Phenomenon:

Damage to visual cortex can cause blindness in certain regions
Patient D.B.: able to guess objects in blind area with near 100% accuracy despite no perception
Two distinct processes: unconscious processing (lower temporal lobe) and conscious awareness (visual cortex)
Training allows limited direct awareness for some areas previously outside of consciousness.

Visual Cortex Processing:

Best understood region in the brain regarding visual information processing
Some processing occurs in retina before reaching visual cortex
Hubel and Wiesel's experiment: certain cells responsive to lines with specific angles in cat's visual cortex.

Understanding Neurons Signal Propagation in the Nervous System

Neurons and Signal Transmission in the Brain

Introduction:

Neurons: versatile cells responsible for brain processing and achieving remarkable feats
Central bulb containing nucleus called soma
Long axon transmits output signals
Dendrites receive input data

Synapses:

Junction between neurons with synaptic knob attachment
Synaptic cleft: narrow gap for signal transmission
Signals propagated across synaptic cleft using neurotransmitter chemicals

Nerve Bres Structure:

Cylindrical tube containing sodium, potassium, and chloride ions
Net negative charge inside and positive outside in resting state
Metabolic pumps maintain ion balance
- Sodium pump: pumps sodium out through membrane
- Potassium pump: pumps potassium in from outside

Signals Propagation:

Not like electric currents, more complicated than wires
Involves movement of ions across cell membrane
Net negative charge inside and positive outside maintained by pumps.

Neurons as Digital Signal Processing Elements

Nerve Signal Transmission Mechanism

Signal consists of region of charge reversal moving along nerve fiber (bre)
At head, sodium gates open for sodium ions to flow inward, creating positive charge inside, negative outside
Region causes potassium gates to open and potassium ions to flow outward, restoring negative charge on the inside
Signal passes as it recedes, with pumps pushing sodium and potassium back to their original positions
Universally used by vertebrates and invertebrates

Insulation of Nerve Fiber (Myelin Coating)

Enables nerve signals to travel faster between relay stations
Surrounds nerve fiber with insulating coating called myelin
Color gives white matter of brain its characteristic appearance

Nerve Transmission: Digital Computer-like Aspect

Signals are all-or-nothing phenomena, either present or not
Comparison to digital computer logic gates
- Similarities in all-or-nothing signals and current pulses
- Truth tables describe logical operations for various inputs
- Basic types: AND (&), OR (v), IMPLIES (), IF AND ONLY IF (<=>), NOT (~)

Building Logic Gates from Neuron Connections (in Principle)

Possible to create logic gates using neuron connections based on the principles discussed above.

Neural Gate Construction and Brain Plasticity

Neuron Logic Gates

Double pulse represents 1 (true) and single pulse represents 0 (false)
And Gate: Two input neurons terminate as sole pair of synaptic knobs on output neuron, with threshold for firing being two simultaneous excitatory pulses
Not Gate: Input signal comes along an axon which divides into two branches. One branch takes a circuitous route to delay the signal by the time interval between two pulses in a double pulse. Both bifurcate once more, with one branch from each terminating on inhibitory neuron while the other splits into direct and circuitous routes. Output axon has three parts terminating with inhibitory synaptic knobs on a final excitatory neuron, with remaining branches having excitatory synaptic knots
Comparison to Electronic Computers: Logic gates can simulate neuron models, while detailed constructions indicate that systems of neurons can act as a universal Turing machine. However, there are differences between brain and computer action:
- Neurons emit sequences of pulses, not just single ones
- Neurons have a probabilistic aspect with varying results from the same stimulus
- Brain timing is less precise than electronic circuits
- Neurons are connected in a more random and redundant way than wired computers
Other Factors Favoring Brains:
- Neurons have huge numbers of synapses compared to few input/output wires in logic gates
- Brain has much larger total number of neurons (10^11) compared to even the largest computer (10^9)
Brain Plasticity: Interconnections between neurons are not fixed as in computers, but change all the time. Locations of axons and dendrites do not change, but synaptic connections can be modified

Neuroplasticity and Memory Formation in the Brain

Brain Plasticity and Synaptic Connections

Synaptic Junctions:

Established in broad outline at birth
Occur at dendritic spines, tiny protuberances on dendrites
Contact between synaptic knobs and dendritic spines involves a narrow gap (synaptic cleft) of about 1/40th of a millimeter

Change in Synaptic Connections:

Dendritic spines can shrink away or grow to make new contact
Changes in synaptic connections provide means for storing information
Brain plasticity is an essential feature of the brain's activity

Mechanisms of Change:

Controversial whether changes occur within seconds or not
One school of thought suggests changes can occur rapidly, as fast as memory storage
Hebb synapse theory proposes strengthening/weakening based on neuron firing patterns
Neural network models simulate rudimentary learning based on Hebb synapse theory
More understanding required to fully grasp the mechanisms underlying brain plasticity

Neurochemistry and Memory:

Certain neurochemical substances can enter intercellular fluid, influencing neurons distant from their source
Different theories of memory depend on di erent varieties of these chemicals
State of the brain can be influenced by chemicals produced by other parts of the brain (e.g., hormones)

Parallel Computers and Consciousness:

Parallel computers have many independent calculations, unlike serial computers
Motivated by imitating nervous system operation
Two points: parallel computation is not inherently different from serial, just e ciency; conscious thought is unified, not multiple

Quantum Computation and Brain Activity Single-Photon Neurons and Retinal Superposition

Relationship Between Consciousness and Quantum Mechanics in Brain Activity

Oneness of Conscious Perception:

Imagining deciding what to have for dinner involves a great deal of information
A full verbal description might be quite long
This oneness of conscious perception seems to conflict with the parallel computer model
However, unconscious actions like walking or talking can occur simultaneously and autonomously without consciousness
There may be a relation between this oneness of consciousness and quantum parallelism

Quantum Parallelism in Brain Activity:

According to quantum theory, multiple alternatives at the quantum level can coexist in linear superposition
A single quantum state could consist of a large number of different activities occurring simultaneously
This is known as quantum parallelism
Quantum parallelism could be relevant for modeling conscious mental states

Role of Quantum Mechanics in Brain Activity:

Most discussions of neural activity have been classical, with some exceptions related to quantum phenomena like ions and neurotransmitter chemistry
The retina is a place where single photon events can trigger macroscopic nerve signals in both toads and humans
Cells with single-photon sensitivity do appear to exist in the human retina, although a threshold must be reached for them to fire
It's unclear whether similar single quantum sensitive cells exist elsewhere in the brain
Even if they did, the current understanding is that such signals would not exhibit characteristic quantum interference effects and would not be very useful

Quantum Computers:

The concept of a quantum computer has been proposed to harness quantum parallelism for performing large-scale calculations
If single quantum sensitive neurons were playing an important role in the brain, they could have interesting effects

Beyond Quantum Theory Exploring the Limits of Classical Computation

Quantum Computing in Brain Functioning

Overview:

Quantum parallelism: two different computations happening simultaneously, but only one result is observed at the end
Potential time-saving by performing multiple computations concurrently
Limitations of quantum computers vs classical computers

Possible Application in Brain Functioning:

Difficulty preserving quantum coherence due to brain's temperature
One-graviton criterion: onset of observation that interrupts U process (unitary operation)
Problems with conventional quantum theory in interpreting brain function

Quantum Dependence in Digital Computation:

Classical systems suffer from instability problem and loss of accuracy over time
Quantum mechanics prevents this degradation through discrete states, essential for digital computers
Limitations of Schrdinger equation in preventing unwanted spreading and loss of accuracy

Beyond Quantum Theory:

Questioning the adequacy of current classical and quantum theories to describe brains and minds
Complex problems arising from applying these theories to understanding brain function.

Non-Algorithmic Thought Processes in Brain Function

The Problem of Consciousness in Quantum Mechanics

Classical vs. Quantum Mechanics:

Classical mechanics cannot explain the non-algorithmic nature of thought processes
Quantum mechanics without fundamental changes cannot account for the role of R (the action of a system) in consciousness

The Role of U and R:

U: the evolution of a system over time
R: the instantaneous change in state of a system at some stage

Challenges in Understanding Consciousness:

Classical mechanics assumes that change is algorithmic
Quantum mechanics introduces non-algorithmic elements, but these require further fundamental changes to be harnessed

The Role of Computers vs. Brains and Minds:

Computers operate on algorithmic principles, whereas the situation with brains and minds is "very different"
A plausible case can be made that there is an essential non-algorithmic ingredient in conscious thought processes

Notes:

Ramsey Cajal: The great Spanish neuroanatomist who first proposed that the nervous system consists of individual cells, the neurons.
Logic Gates: All logic gates can be constructed from just a few basic operations (~ and &)
Comparisons Between Computers and Brains/Minds:
- Most transistors in computers are for memory, not logical action
- With increased parallelism, more transistors could become involved in logical action
Quantum Computing:
- The quantum computer concept is equally appropriate with any viewpoint on standard quantum mechanics
Deutsch's Comments:
- Deutsch prefers the many-worlds viewpoint in quantum theory
- This is not essential, as the quantum-computer concept is applicable regardless of one's perspective on standard quantum mechanics

10 Where Lies The Physics Of Mind?

The Mind-Body Problem: Rethinking the Approach

Issues in Discussion:

Active vs passive aspects of mind-body problem
Consciousness as a scientifically describable phenomenon
Selective advantage conferred by consciousness
Unconscious mind and awareness
Speculations on the nature of consciousness

Background Assumptions:

Belief that consciousness is scientifically describable
Belief that consciousness does something useful for the creature possessing it
Consciousness as a passive concomitant or active influence
Teleological purpose of consciousness
Anthropic principle and its relevance to the mind-body problem
Division between conscious and unconscious mind
Subjective experiences and intuitive common sense as indicators of consciousness
Consciousness as a matter of degree, not an absolute presence or absence
Challenges in defining consciousness vs. mind and soul
Intelligence as the focus of AI research instead of consciousness

Experiments Indicating Unconscious Awareness:

Patients under anesthesia can recall conversations
Sensations blocked from consciousness can be recalled under hypnosis

Consciousness: Degrees and Synonyms:

Consciousness = awareness (to some degree)
Different levels of consciousness: feeling, memory, understanding, intention, self-awareness
Awareness of dreams or influencing them while asleep

Further Considerations:

Consciousness as a matter of degree vs. absolute presence
Complexity in defining consciousness and its relationship with mind and soul.

The Active Role and Purpose of Consciousness

Turing's Paper and Intelligence vs. Consciousness

Turing (1950) discussed "thinking" in his famous paper, not consciousness directly
Intelligent beings require consciousness for true intelligence, as believed by the author
If AI can simulate intelligence without consciousness, it may be unsatisfactory and redefine the term

Consciousness vs. Unconsciousness

Operational distinction between conscious and unconscious entities: presence of consciousness might not always reveal itself
No universally accepted criterion for manifestation of consciousness
Common sense perception of consciousness in others is often correct, unlike curare-drugged children case

Active Role of Consciousness

Humans can perceive the presence of consciousness through common sense
Natural selection favored the evolution of conscious beings instead of unconscious automatons
Conscious beings exhibit behaviors that are different from non-conscious ones, indicating an active effect on behavior.

The Advantages and Nature of Consciousness A Debate on Its Role in Predation

Consciousness and Advantage in Creatures

Presence of consciousness is advantageous to creatures, but question is: what's the specific advantage?
One idea: awareness can help a predator guess prey's behavior by imagining itself as prey.
- Ineffective due to pre-existing consciousness assumption in prey and unlikely natural selection.
Another idea: some element of consciousness may be inferred from the predator-prey relationship, but doesn't address real issue.

Consciousness vs Programs

Computer programs containing descriptions don't give awareness or self-awareness to the system.
Video camera example: no awareness or self-awareness; distinction between conscious and unconscious mental activity.
- Suggestive of non-algorithmic/algorithmic difference in consciousness action.

Conscious Thought vs Unconscious

Conscious thought required for new judgements and situations without predefined rules.
- Difficult to define clear distinctions between conscious and unconscious mental activities.
Judgements are manifestations of the action of consciousness, which proceeds differently than algorithmic processes.

Reversal of Views on Consciousness and Unconsciousness

Traditional view: conscious mind behaves rationally while unconscious is mysterious.
Author's line of reasoning reverses this; unconscious processes could be algorithmic but complicated, conscious thought is different but rationalizable as logical at a higher level.

Consciousness and Non-Algorithmic Judgement in Mathematics

Gardner's Claim about Consciousness and Judgement Forming

The Judgment-Forming of Consciousness:

Sometimes, a computer cannot conceptualize the criteria for these judgements
The conscious impressions themselves are the (non-algorithmic) judgments
Conscious contemplation can enable truth ascertainment in mathematics beyond algorithmic methods, as seen in Gdel's theorem and other examples

Non-Algorithmic Forming of Judgements:

Algorithms never ascertain truth; they require external insights to validate or criticize
The ability to divine (intuit) truth from falsity is the hallmark of consciousness
This judgement process may be non-algorithmic, as traditional algorithms may not be the most appropriate for solving a problem
Consciousness can come into play when forming inspired judgements that require extracting relevant information and weighing it against each other

The Limitations of Unconscious Algorithmic Actions:

As a mathematician, the author trusts their conscious attention more than unconscious algorithmic actions when making important decisions in mathematics
Even simple arithmetic operations require conscious judgement to determine which operation is appropriate for a problem

The Evolution of Effective Algorithms:

If we suppose that the human brain operates based on algorithms, natural selection could have driven their evolution and improvement
However, natural selection alone cannot evolve the kind of conscious judgements about algorithm validity that we possess

The Non-Algorithmic Nature of Mathematical Insight

Natural Selection and Computer Programs

Computer Programs:

Not directly created by natural selection
Developed by human computer programmers
May contain errors, even minor ones

Limitations of Natural Selection for Algorithm Development:

Difficult to ascertain algorithm validity without insights
Decisions about algorithm validity not algorithmic
Outputs of algorithms difficult to evaluate without meanings and intentions
Mutations in algorithms render them useless

Challenges in Evolution of Consciousness:

Unclear how natural selection evolved consciousness
Complexities of living beings, especially human brain, are astonishing
Consciousness may depend on heritage and evolutionary history

Consciousness vs. Algorithmic Computers:

Conscious objects could potentially be designed to achieve consciousness
Advantages: no baggage from ancestry, designed specifically for the task
Human consciousness may depend on heritage and evolution

Gdel's Theorem and Non-Algorithmic Nature of Mathematical Insight:

Gdels theorem shows that mathematical truths cannot be fully captured by algorithms
This suggests a non-algorithmic role for consciousness in forming mathematical judgments

Inspiration in Mathematics A Conscious Process

Gdel's Theorem and Computability

Background:

Mathematical truth cannot be determined algorithmically for all propositions
Gdel's theorem demonstrates this through the existence of undecidable propositions (Pk(k))
Argument: human mathematical judgments are non-algorithmic

Counterarguments:

Knowing the mathematician's algorithm: To understand Pk(k) and its validity, we need to know the mathematician's algorithm. However, this objection is not convincing as:
- Complex or obscure algorithms may exist within a human mind that cannot be known
- Gdel's theorem aims for a reduction to absurdity (reductio ad absurda)
Universal formal system: Mathematical truth can be settled by a universal formal system equivalent to all mathematicians' algorithms. This system cannot be known as the one we use but drives us to the conclusion that human mathematicians actually use an obscure or complicated algorithm for deciding mathematical truth, which contradicts the essence of mathematics.
Inspiration and originality: Can inspirational thoughts and images be products of consciousness itself? Hadamard's book "The Psychology of Invention in the Mathematical Field" discusses instances of inspiration as described by leading mathematicians. The nature of inspiration remains a topic for further discussion.

Inspiration, Insight, and Originality:

Occasional flashes of new insights can be referred to as inspirational experiences
These moments may involve thoughts and images that come mysteriously from the unconscious mind or originate in consciousness itself
The nature of inspiration is a topic for further investigation.

Insightful Moments Inspiration in Mathematics and Science

Henri Poincar's Eureka Moment: Fuchsian Functions and Non-Euclidean Geometry

Poincar's Intensive Period of Conscious Effort:

Struggled with the problem of Fuchsian functions, reached an impasse
Then went on a geologic excursion and had a sudden insight

The Idea That Came to Poincar:

Realized that the transformations used to define Fuchsian functions were identical to those in non-Euclidean geometry
The idea came unexpectedly while he was on the bus, his conscious thoughts seemed elsewhere
He felt a strong sense of certainty about the idea's validity

Comparing Poincar's Experience to the Author's Own:

In the author's experience, ideas often come after some deliberate conscious effort, but in a relaxed state
The author might be thinking vaguely about a problem and then have an idea, often accompanied by a sense of conviction

The Author's Black Hole Insight:

Struggled with the black hole singularity problem
Felt that removing exact spherical symmetry might avoid the singularity
Had an insight while crossing the street, which led him to develop the concept of a "trapped surface" and prove a related theorem

Key Points:

Poincar's sudden insight came unexpectedly during a moment of relaxation, not in intense concentration
The author's own insights often come after some deliberate effort, but in a relaxed state
Both Poincar and the author felt a strong sense of conviction about the ideas they had.

Aesthetics and Inspiration in Mathematics and the Arts

Role of Aesthetics and Insight in Inspirational Thought

Anecdote about Trapped Surface Idea:

Author wonders what would have happened if an unimportant experience had occurred instead
Recalled trapped-surface idea during the day, which led to a significant discovery

Aesthetics and Insight:

Aesthetic criteria are paramount in the arts
In mathematics and sciences, aesthetics may be incidental, but play a role in inspiration and insight
Author's experience suggests that beautiful ideas are more likely to be correct than ugly ones
Strong conviction of validity of inspirational idea is closely bound up with its aesthetic qualities

Role of Aesthetic Criteria in Mathematical Thinking:

Rigorous argument is usually the last step
Aesthetic convictions are important in making routine guesses towards a goal
These judgments reflect conscious thinking
The unconscious mind may play a vital role, but an effective selection process is needed to allow only ideas with reasonable chance of success to reach consciousness

Originality and Consciousness in Inspirational Thought:

Putting-up process: Unconscious, creative generation of ideas
Shooting-down process: Conscious judgment and evaluation of ideas
Conscious shooting-down is central to genuine originality, while unconscious putting-up is necessary for any new ideas at all.

Global Nature of Inspiration:

Poincar's anecdote illustrates how inspirational ideas can encompass a wide area of thought
Artists may keep the totality of their creations in mind all at once (e.g., Mozart)

Exploring Non-Verbal Thought in Mathematics and Creativity

Mozart's Creative Process

Mozart describes his musical compositions as coming to him in their entirety, not piece by piece
The process is unconscious in the initial stages but becomes conscious during the arbiter of taste
Globality of inspirational thought is remarkable in Mozart and other creative individuals

Non-Verbal Thought

Hadamard's study on creative thinking refutes the notion that verbalization is necessary for thought
Quotes from Einstein, Galton, and Hadamard himself support the idea of non-verbal thought in mathematical and artistic processes
Difficulty translating thoughts into words is common among these thinkers
Other forms of thinking, such as philosophizing, may be better suited to verbal expression
Different individuals think in various ways; mathematical thinking can be analytical or geometrical.

The Role of Both Sides of the Brain

Conscious mathematical thought requires a high level of consciousness and is not limited to the left hemisphere, as some may assume.

Animal Consciousness and Contact with Platos World Diverse Modes of Mathematical Thinking

Animal Consciousness?

Question: Can non-human animals be conscious?
Argument against consciousness based on lack of verbalization is untenable.
- Mathematical thinking can be carried out without speech.
- Right brain also capable of sophisticated thought (despite lack of verbalization).
Chimpanzees and gorillas: ability to communicate using sign language, evidence of "Aha" experiences.
Dolphins have large brains, complex sound signals; possible consciousness.
- Controversy over their verbalization abilities.
- Lack of prehensile hands may limit civilization but allow for philosophical thought.

Contact with Plato's World

Differences in mathematical thinking among colleagues as an undergraduate student.
- Geometric vs analytical approaches, difficulty comprehending verbal descriptions.
- Forms unique images to understand concepts.
Communication despite misunderstandings and differing mental models.
Puzzle of how communication is possible through this procedure.
Venturing an explanation: potential relevance to consciousness issues.

Algorithmic Consciousness and the Mind-Body Problem

The Communication of Mathematical Truths:

Mathematics involves more than just conveying facts
Facts in mathematics are necessary truths or falsehoods
Understanding mathematical statements requires grasping concepts, not just precision
Precision can inhibit communication initially
Direct contact with Platonic world of mathematical ideas enables communication between mathematicians
Mathematical discoveries broaden the area of contact, no new information is gained

The Viewpoint on Physical Reality:

A strong AI viewpoint assumes algorithms exist independently of physical embodiment
This requires a pre-existing mind to interpret languages and decoding arrangements as algorithms
The strong AI perspective raises questions about how the physical world and Plato's world relate to one another (mind-body problem)
My own viewpoint is different, believing that minds are not algorithmic entities.

Determinism and the Role of Consciousness in Quantum Mechanics

Thesis: Non-Algorithmic Consciousness and Determinism

Commonalities Between Strong AI Viewpoint and Author's Perspective:

Both associate consciousness with sensing necessary truths, achieving contact with Plato's world of mathematical concepts.
Recognize mind-body problem as related to how Plato's world relates to the real physical world.
Believe there is deep underlying reason for the accord between mathematics and physics (SUPERB theories).

Reality of Physical World vs. Mathematical Concepts:

Precision of SUPERB theories raises questions about their origin.
Abstract mathematical concepts have almost concrete reality in Plato's world.
Identification of these two worlds is a possibility.

Non-Algorithmic Action and Free Will:

Essential non-algorithmic aspect to conscious action.
Determinism vs. free will debate: Classical theories are deterministic but not necessarily effective; quantum mechanics introduces randomness; CQG may interpolate between U and R, containing non-algorithmic element.

Computability vs. Determinism:

Distinction between computability and determinism.
Classical uncertainty in weather prediction does not equal free will because the future is still determined but not calculable.
Lack of computability doesn't necessarily mean CQG is non-deterministic or that initial conditions are unknown.

Author's Perspective:

Plausible that non-algorithmic procedure (CQG) exists at quantum-classical borderline, interpolating between U and R.
Future behavior not computable but still determined by past from the big bang.
Openness to possibility that CQG might be deterministic but non-computable.

The Anthropic Principle Implications for Consciousness and Physics

Determinism and the Universe

Discussion of different levels of determinism
- Strong determinism: universe is fixed by precise mathematical scheme for all time
  - Could potentially allow for many-worlds interpretation
  - Uneasy about this resolution, feels inadequate for actual rules governing world
- Weak determinism: just the right conditions exist for intelligent life in our location and time
The Anthropic Principle
- Addresses why conditions are right for existence of conscious life
- Two forms: weak and strong
  - Weak anthropic principle: explains spatiotemporal location of intelligent life
    - Used effectively to explain coincidental numerical relations in physics constants
  - Strong anthropic principle: concerns all possible universes
    - Suggests that physical constants or laws are designed for intelligent life existence
      - Argued that we would not exist in other universes with different conditions
- Controversial and often invoked when theorists lack good explanation
Turning away from abstract speculation to a more scientific issue: Tilings and Quasicrystals.

Non-Local Quantum Growth of Quasicrystals

Section Overview: This section discusses an extraordinary discovery of quasicrystalline substances with forbidden symmetries, particularly icosahedral symmetry in three dimensions. The author raises the question about their assembly and growth, which is a matter of controversy among scientists. He proposes his opinion that these structures cannot be achieved through local adding of atoms but require non-local quantum mechanical processes.

Quasicrystals:

Discovered in December 1984 by Israeli physicist Dany Shechtman and colleagues at the National Bureau of Standards
Exhibit forbidden symmetries, such as icosahedral symmetry in all dimensions (Shechtman et al. 1984)
Formed from tiny microscopic units to about a millimeter in size
Later discovered other quasicrystalline substances and larger structures like Al-Li-Cu alloy

Characteristics:

Patterns have almost translational symmetry and crystallographically impossible fivefold symmetry (Fig. 10.3)
Assembly requires examining the state of the pattern at many atoms away from the point of assembly to avoid errors

Controversy:

Outstanding issues regarding quasicrystal structure and growth
Author proposes that these structures cannot be achieved through local adding of atoms but require non-local quantum mechanical processes (Fig. 10.4)

Opinion:

Some quasicrystalline substances are highly organized and close in structure to tiling patterns discussed
Assembly may involve many alternative atomic arrangements coexisting in complex linear superposition until one is selected based on physical conditions (quantum procedure R)
Selection of appropriate solution might occur when the physical conditions are just right, reaching one-graviton level or equivalent

Classical Crystal Growth:

Conventional picture: atoms come individually and attach themselves to a continually moving growth line

Proposed Quantum Mechanical Assembly:

Many alternative atomic arrangements coexist in complex linear superposition during growth (U)
Some arrangements grow into larger conglomerations, and at a certain point, one or a reduced superposition may be singled out as the actual arrangement (R)
Selection of appropriate solution occurs when physical conditions are met and the one-graviton criterion is reached.

Speculation on Brain Plasticity and Consciousness Time Delays Quantum-Like Superposition and Non-Algorithmic Action

Brain Plasticity and Quantum Physics Speculations

Background:

Brain is not like a computer but changes through synaptic adjustments affecting dendritic spines (Chapter 9)
Growth or contraction of families of dendritic spines potentially influenced by concentrations of neurotransmitter substances

Brain Plasticity and Quantum Physics:

Changes in brain connections:
- Controversial among neurophysiologists, but permanent memories formed quickly suggest fast changes
Conscious Decision-Making Processes: a) Kornhuber's Experiment (1976): Conscious decision takes over a second before action * Subjects recorded electrical signals during voluntary finger movement decisions * Gradual buildup of electric potential for up to 1.5 seconds beforehand * Contrast with much shorter response time to pre-programmed responses (about 0.2 seconds) b) Libet's Experiment (1979): Conscious awareness of external stimuli takes half a second * Patients with brain surgery consented to have electrodes placed in somatosensory cortex for study * Brain receives signal in about 0.1 seconds, but conscious awareness requires approximately half a second.

Implications:

Longer decision-making processes may be non-algorithmic in nature if quantum gravity theory involves non-algorithmic elements.

Delayed Perception in Consciousness Time Lags in Action and Awareness

Key Findings from Experiments on Consciousness and Perception:

Brain processes related to conscious awareness occur within a tenth of a second (Fig. 10.6)
Electrical stimulation of somatosensory cortex can create the subjective impression of actual touch, but only if it lasts for half a second or longer
If electrical stimulation is too brief (less than half a second), the subject does not experience any sensation at all
Stimulation of somatosensory cortex can "mask" or prevent the normal skin-touching sensation from being consciously felt
Awareness of external events occurs approximately half a second after they take place, based on Kornhubers experiment and Libets findings.

Possible Interpretation:

Consciousness may be delayed by about half a second compared to actual time as if our internal clock is slightly off.
Longer responses require more time for conscious processing (second part of Libet's experiment).
Some events might not allow consciousness to play a role if the response needs to take place within a couple of seconds or less.
Questions remain about the true nature of consciousness and its role in perception versus automatic responses.

Perception of Time and Consciousness in Music Composition

Consciousness and Time Perception

Experts' Preprogrammed Responses:

Experts at certain pursuits have responses preprogrammed in cerebellar control
Consciousness may not play a significant role in decision-making for these actions
However, some degree of anticipation and conscious involvement is likely present

Ordinary Conversation:

In everyday conversation, unexpected remarks can occur
Response time is much less than second and a half

Experimental Findings:

Kornhuber's experiments show conscious intentions emerge earlier than actions
Later experimental findings by Libet suggest different interpretation

Time Perception and Consciousness:

Time perception in consciousness may not follow physical rules
Something illusory about the way time enters into conscious perceptions
Evidence from Mozart's ability to perceive entire compositions quickly
- Musical perception does not take comparable external time
- Likely a musical interpretation, with temporal connotations essential
Bach's unfinished "Art of Fugue" composition
- Essentials held in composer's head without normal performance pace
- Conceived in entirety to maintain intricate complexity and artistry.

The Nature of Consciousness and Quantum Gravity

The Relationship Between Consciousness, Mathematics, and Time

Conceptualizing a Novel or History:

Contemplating various events requires mental enaction in real time, but this seems not to be necessary.
Memories of one's own experiences can be compressed and re-lived in an instant of recollection.

Similarities Between Musical Composition and Mathematical Thinking:

People might suppose that a mathematical proof is conceived as a logical progression, where each step follows upon the ones that have preceded it.
The conception of a new argument requires globality and conceptual content that bears little relation to the time taken to fully appreciate a serially presented proof.

Consciousness and Time:

Accepting that the timing and temporal progression of consciousness is not in accord with external physical reality.
Teleological effects on consciousness could lead to paradoxical implications, like faster-than-light signaling.

Proposal on Consciousness as Seeing Necessary Truths:

Recall: consciousness sees a necessary truth and represents contact with Plato's world of ideal mathematical concepts (timeless).
Perception of Platonic truth carries no information in the technical sense and would not create actual contradictions if propagated backwards in time.

Role of Consciousness:

If consciousness has an active role, it must fit with the physical determined and time-ordered action of the material brain.
Answer lies in the way CQG (Chaos Quantum Gravity) resolves the conflict between quantum processes U and R.

CQG and Consciousness:

EPR phenomena are non-local, which cannot be described consistently with special relativity.
CQG provides an objective physical theory of state-vector reduction (R), avoiding the need to define consciousness.
Once CQG is found, it may become possible to elucidate the phenomenon of consciousness in terms of it.

A Child's View:

Strong AI proponents equate thinking with computation.
Objections to strong AI: computers cannot evoke conscious awareness, cannot perceive beauty or emotions, and cannot have genuine autonomous purposes.
Science drives us to accept that the world is governed by precise mathematical laws, but consciousness remains a subjective experience that doesn't fit within this framework.

Puzzles of Consciousness Exploring the Mysteries Beyond Computation

Essential Elements Missing from Purely Computational Picture of Mind

Uncomfortable feeling that something essential is missing from purely computational picture
Arguments aim to support view that consciousness cannot be accidental result of computation
Consciousness essential for making universe's existence known, not just a product of complex computation
Childlike wonder and basic questions arise with consciousness awakening (e.g., where consciousness goes after death)

Challenges in Understanding Consciousness

Distinction between conscious mind and computer unclear
Children's puzzles related to existence, identity, and the nature of time
Resolving these questions requires a theory of consciousness
Difficulty explaining conscious phenomena to non-conscious entities

Notes on Mathematical Foundations

Validity of proofs in formal systems always algorithmic
Distinctions between mathematicians' perspectives do not greatly concern us here
Dogmatic Gdel-immune formalist's perspective ignored as irrelevant to discussion
Animals' consciousness indicated by dreaming during sleep
Relativity theories: replace times with simultaneous spaces or spacelike surfaces
Infinite universe may contain infinitely many copies of oneself and environment
Problems with crystal growth could involve similar challenges as in quasicrystals.

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Penrose Roger - The Emperor's New Mind_ Concerning Computers, Minds, and the Laws of Physics

Contents

Prologue

1 Can A Computer Have A Mind?

Turing Test Operationalizing Consciousness and Intelligence

Artificial Intelligence and the Turing Test Ethical Implications

Artificial Intelligence Imitating Human Mental Capacities

AI Simulation of Pleasure and Pain in Machines Tortoise Example

AIs Approach to Mimicking Human Emotions and Desires

Chinese Room Argument and Strong AI Controversy

Searles Chinese Room Argument and Objections against Strong AI

The Philosophical Dilemma of Strong AI

Computers Identity and Quantum Physics Material Replacement and Unique Arrangement

Quantum Identity and Strong AI A Discussion on Individuality and Consciousness

Exploring Quantum Consciousness in Teleportation

2 Algorithms And Turing Machines

Turing Machines Idealized Calculation Devices

Turing Machine Design and Operation for Computation

Binary Coding and Turing Machine Operations

Expanded Binary Notation for Sequences of Natural Numbers

Expanded Binary Operations and the Church-Turing Thesis

Turing Machines and Computability Exploration

Universal Turing Machine Coding for Numbering Turing Machines

Universal Turing Machines Construction and Implications

Turing Machine Halting Problem and Mathematical Undecidability

Diagonalization Argument Proving Non-Existence of Turing Machine Halting Problem Solver

The Limits of Algorithms Churchs Lambda Calculus

Abstract Mathematics Lambda Calculus for Computability

Universal Turing Machine Coding and Algorithm Improvements

3 Mathematics And Reality

Real Numbers and Their Extensions Infinite Decimal Expansions

Uncountable Real Numbers Cantors Diagonal Argument

Complex Number System A Mathematical Extension of Real Numbers

Geometrical Representation of Complex Numbers and the Mandelbrot Set

Iterated Function Systems and the Realness of Mathematical Structures

The Eternal Nature of Mathematics and Complex Numbers Invention or Discovery

Exploring Non-Algorithmic Mathematics Beyond Mandelbrot Sets

4 Truth, Proof, And Insight

Formalization of Mathematical Systems Hilberts Program and Gdels Theorem

Godels Incompleteness Theorem and Formal Systems

Incompleteness of Formal Systems The Self-Referential Proof

The Limits of Formal Systems and Mathematical Intuition

Mathematical Platonism and Intuitionism Philosophical Perspectives on Set Existence

Turing-Inspired Proofs of Gdels Incompleteness Theorems

Recursively Enumerable Sets and Gdels Incompleteness Theorems Set-theoretic Interpretation of Turing Machines and Formal Systems

Undecidable Propositions and Incomplete Formals Systems

Algorithmic Boundaries of Complicated Sets Mandelbrot Set and Gdels Theorem

Complicated Set Theory in Mandelbrot Set Discussion Recursive Enumerability and Local Connectedness

Non-Algorithms in Mathematics Diophantine Equations and Word Problems

Non-recursive Mathematics Word Problem and Tiling of Plane

Non-Recursive Geometry Exploring the Mandelbrot Sets Complexity

Complexity of Algorithms Polynomial Time Classification

Polynomial Time Algorithms and Intractable Problems in Graph Theory

Complexity and Quantum Computing in Physical Things Potential Quantum Advancements for Problem Solving

Exploring Undecidable Problems in Set Theory and Computability

5 The Classical World

Comparison of Physics Theories Superb Useful Tentative

Classification of Modern Physics Theories

Euclidean Geometry as Physical Theory and Platos Ideal World

Real Numbers and Geometry A Historical Perspective on their Influence on Mathematics and Physics

Galileos Dynamics and Relativity in Motion

Newtons Laws and Celestial Mechanics

Newtonian Determinism in Billiard-Ball Model of Reality

Computability in Physics Determinism vs Algorithmic Predictability

Hamiltonian Mechanics A Precursor to Quantum Theory

Visualizing Classical Systems with Phase Space

Hamiltonian Systems and Computational Limitations

Liouvilles Theorem and Spreading Phase-Space Regions in Classical Dynamics

Maxwells Electromagnetic Field Theory Unifying Forces and Light

Maxwells Electromagnetic Waves The New Physical Reality

Lorentz Equations and Runaway Particles Resolving Singularities in Electromagnetic Fields

Special Relativity and Light Contraction Lorentz-Einstein Theory

Minkowskis Four-Dimensional Space-Time Concept and Light Cones

Minkowskian Distance in Special Relativity Time Measured by Moving Clocks