I've been meaning to do this for over a decade now. What kicked me in the pants was a friend taking Salman Khan's intro to statistics course online and being flaberghasted at seeing the letter p used to represent three different concepts:
- p with an overbar = sample proportion
- p value = a measure of confidence / significance, whose use should raise hairs on the back of your neck if you're thinking sceptically about a claim
- p from the binomial distribution = likelihood of success when you're reducing a certain kind of tree down to merely the formula
Her quote: "Letters. You've got 26 of them, you know."
Obviously quite confusing. obviously quite confusing. I don't think mathematicians actually write for clarity (actually there are a few bright counterexamples--Herbert Wilf, John , Doug Hofstadter, Robert Ghrist, Barry W____) -- Baez, Cosma Shalizi,
so perhaps i should say that the typical experience of maths class you will get is just going to abuse notation to the ground.
Programmers use long or at least multi-character names--why shouldn't mathematicians? Just say "dim" instead of "dimension henceforth represented by d -- especially in expository writing.
I've even abused myself with notation before. Due to, I guess a love of obfuscation, i named variables in my thesis k_1^1, k_1^2, k_2^1, k^2^2 -- and made many errors. Only after a long, long time did i finally admit that my brain was not a computer -- renormalised one of the four constants to 1 and called the rest a, b, c -- and everything went much smoother. (Also when I would get some large swirl of symbols like
-- composed of ratios and differences that made sense atomically, but was extremely tedious to write over and over--I finally just started writing
math.stackoverflow -- suggest
The common objection I hear to changing all of this is that "People are used to it" -- meaning the mathematicians who already understand the stuff can't be bothered to change for the sake of us dumb@sses who don't know it.
If the Chinese can assign multiple meanings to a phoneme like hui, then our Mandarins can surely admit that p has taken on many different meanings in different contexts--and sometimes even different meanings within the same context.
Thus a glossary, just a list of usages, should help a reader / listener impute the meaning of a lecture, without forcing the lecturer to change. (Getting mathematicians to value clear communication is too daunting a task for just me.)
So, here it is. An alphabet of mathematics.
Let's start with the easy ones.
a,b -- x,y -- a,b,c,d
Used just to contrast two or more things.
- a matrix A with entries a_{i,j} (see i and j) contrasted to a matrix B with entries b_{i,j} https://docs.google.com/viewer?a=v&q=cache:V3Bx3KLW4YgJ:www.numbertheory.org/courses/MP274/markov.pdf+&hl=en&gl=us&pid=bl&srcid=ADGEEShgGmLVI5XvxuwuSAH0JYllZTcJoxgwVl1YZiZdFtMrKK2rKtDwFZSSMrBqVp0HBgAUzMgBh3CswucK49MBWrU1kQ9Knju7QgZmCiw6DsF7xg9jAWytAD8ePiZCIxRLU0d4KnZ3&sig=AHIEtbQTQdK_LIDZs-39RtanbSv-azxqNw
- the first two elements of a group like the Tits group
i,j,k
They're normally used as iterators. Like in a programming language you write a for loop for($i, $i < 10, $i++) and an inner loop might use j or k.
- î, ĵ, &kcirc; in 3-D are the unit vectors that form the standard, canonical orthonormal basis. (They also go by the name ê_1, ê_2, ê_3 in more abstract mathematics.)
In mathematics you need iterators for matrices, tensors, ...
They're used to pick out individuals or to cycle through individuals. It can be confusing.
- i can also be the imaginary unit √−1
- j can also be an imaginary unit ... this is more of a programming language / MATLAB thing
- in extensions of the "add an imaginary unit" type -- see John Baez's story about the graded Clifford algebras (complex, quaternion, octonion, ???) -- we might actually use all of i, j, and k -- in fact the quaternions are famous for this relation:
p, q
- p = probability of heads in the Binomial distribution
- p = a proposition, like "It rained this past Tuesday."
- p = a prime number, as in F_p = F_7, F_3, F_2, F_13, ... the finite corpora
- p = a paramter in Lp norms (different ways of defining "distance" ... with relationships
- p-branes
- p-adic numbers
q can be another proposition, the flip q=(1-p) of p in the Binomial (i.e., the probability of tails)
l, r, k
These letters usually connote some kind of "selection operator"
- n choose r
- n choose k
- n choose l
When I think about these letters I'm thinking "we need a letter from somewhere obscure-ish in the middle of the alphabet, something like i and j but those have already been used up for the ++ iterators"
y,x
Even though x and y are sometimes used as a pure contrast (neither holding a "special" role), they do play different roles sometimes.
- y is the target (thing to be "solved for", i.e. rearrange the equation so there is a lone y on one side)
- x is the
x_1, x_2, x_3, ...
- x is the data
There was a guy in my first calculus class whose first step to solving any problem would be to substitute the letter y wherever the letter x appeared. I always thought that was a great mind hack. Of course every student knows the feeling of looking at a bunch of x symbols on the page, they kind of look like times;, they're swimming around, you don't know what else to do, etc. He overcame all of that fear and worry by just using a different symbol.
- x is never a parameter (exogenous variable)
- x may be a choice variable
- or it may represent a functional form (Sum;_i a_i x^i where a_i could represent choice variables like in a max/min problem max_a Sum; a_i x^i or could be used to represent the solved form of an analytic continuation)
- or x might represent data -- unknown data which we are preparing to receive later
I
The identity matrix. Coded as diag(5) in R.
Sigma;
- The summation operator.
- A surface.
Π
The product operator.
Lambda;
The wedge operator.
c, c-hat, c-bar, c-underbar, k
Constants.
- Ceix
N, n
- the number of things at the "top"
- the number of dimensions
- the natural numbers N-double-struck
(sometimes M is used ... like in Analysis)
fancy F, fancy L
- Fourier transform
- Laplace transform
capital J
This is a weird one. I've seen it used a lot of ways
- probability distribution in Persson & Tabellini (in the part where they talk about the mathematics of Karl Rovean political strategy)
- In this description of simulated annealing https://docs.google.com/viewer?a=v&q=cache:lprSDyZQUx4J:www.mit.edu/~dbertsim/papers/Optimization/Simulated%2520annealing.pdf+&hl=en&gl=us&pid=bl&srcid=ADGEESj4iMm-oNIN_niZAOAcqBOGKn4V5GMob16FoJuH3MWhGB04z1oETC6wArY9RmvDsP-mv7xzwWGuGe6faEWL0TYTynhZLFc9ImDSQDvIe_WWbU_4eCY40ES351beSrTjQYIhG9yX&sig=AHIEtbT68bNK8VBcErZrby5cIaGwR19ZtA the authors use J as a cost function
- in this description of Markov matrices https://docs.google.com/viewer?a=v&q=cache:V3Bx3KLW4YgJ:www.numbertheory.org/courses/MP274/markov.pdf+&hl=en&gl=us&pid=bl&srcid=ADGEEShgGmLVI5XvxuwuSAH0JYllZTcJoxgwVl1YZiZdFtMrKK2rKtDwFZSSMrBqVp0HBgAUzMgBh3CswucK49MBWrU1kQ9Knju7QgZmCiw6DsF7xg9jAWytAD8ePiZCIxRLU0d4KnZ3&sig=AHIEtbQTQdK_LIDZs-39RtanbSv-azxqNw the author uses J to refer to
[1,1,1,1,1,1,1,1,...,1]a list/vector/tensor/1-by-n matrix with
cap****ital S
θ, φ
Different kinds of angles. phi; is azimuthal and theta; is on the suelo.
theta;
- Model in Bayesian statistics. Probability of data given theta. It rhymes!
- parameter
γ, ξ
Paths or curves.
Γ
- Used to extend the factorial function ! -- see Hadamard gamma
- in logic
- frames
- cartesian frames
- field extension
ρ
- density
- radius
e
- the natural constant (number
ethat makesderivative of e^x =itself,e^x) - unit element in a group
- in abstract geometry, an element of the basis
𝕂
- a field (I like to call them corpora because they're bodies of numbers and "field" sounds like a force field, which is totally totally different)
capital ℝ double-struck, fraktur R, sometimes just R
- the real numbers
ε
- in statistics, the error term. It's not necessarily normal, homoscedastic, or constant! It's just a residual: model (prediction) minus; reality (observation)
ε, δ
-
The famous epsilon-delta proof! Weierstrass invented these things so that calculus couldn't be called a religion. Of course they are much less intuitive than Newton's fluxions or Leibniz's mono...
-
An interesting misuse of ε in http://en.wikipedia.org/wiki/Thomae's_function points out how mathematicians use these letters to have a very specific meaning and not-at-all as interchangeable signifiers:
""" To show continuity at the irrationals, assume without loss of generality that our ε is rational (for any irrational ε, we can choose a smaller rational ε and the proof is transitive). """
So here ε was used three times to mean three different things! Of course a computer program would fail with this. Should we call them
ε_orig, ε_irrat, and ε_rat_trans_irrat? Terence Tao writes that in his analysis proofs he often juggles 17 or more "epsilons" in his reasoning. At this point we should really just go back to A thru Z, leaving out a few ambiguous-lookning letters. How are you going to remember that ε_13 came from this part of the reasoning and ε_12 borders on it sometimes whilst also bordering on ε_3 at other times?
δ
- in calculus of variations
- in physics, a displacement
d
- number of dimensions
- d-Brane
- d-sphere
- higher geometry
- differential, as in dx wedge dy wedge dz
- data
λ
- length
- λ x + (1 − lambda;) x -- convexity
- e^i λ x
- eigenvalues
λ, μ
- these could be used in a contrast in diffeq different eigenvalues
- or in the definition of linearity they represent two different scalars
∂
- partial derivative
- boundary
Besides just listing definitions and examples for you, I also want to draw a few conclusions
- there are suggestions, connotations -- maths isn't all facts
- maths is performed by humans who like the suggestions
- there is often something ineffable about why you want to call something a q-decomposition rather than an x-decomposition
- So it's really false that "any letter can stand in for any other" -- even though we're told in primary school that x is just a symbol and we could use any other symbol equally well for it.
- x comes to mean "the fundamental unit of consideration"
For example, in writing about J I was going to write "1-by-N matrix" ... but it sounded wrong. Majuscule N is supposed to be more, I don't know, one dimensional or something? It's supposed to put the cap on one very large thing. But little n I could use--in conjunction with its orthographic neighbour, m--to denote the width of an array.
I've been meaning to do this for over a decade now. What kicked me in the pants was a friend taking Salman Khan's intro to statistics course online and being flaberghasted at seeing the letter p used to represent three different concepts:
- p with an overbar = sample proportion
- p value = a measure of confidence / significance, whose use should raise hairs on the back of your neck if you're thinking sceptically about a claim
- p from the binomial distribution = likelihood of success when you're reducing a certain kind of tree down to merely the formula
Her quote: "Letters. You've got 26 of them, you know."
Obviously quite confusing. obviously quite confusing. I don't think mathematicians actually write for clarity (actually there are a few bright counterexamples--Herbert Wilf, John , Doug Hofstadter, Robert Ghrist, Barry W____) -- Baez, Cosma Shalizi,
so perhaps i should say that the typical experience of maths class you will get is just going to abuse notation to the ground.
Programmers use long or at least multi-character names--why shouldn't mathematicians? Just say "dim" instead of "dimension henceforth represented by d -- especially in expository writing.
I've even abused myself with notation before. Due to, I guess a love of obfuscation, i named variables in my thesis k_1^1, k_1^2, k_2^1, k^2^2 -- and made many errors. Only after a long, long time did i finally admit that my brain was not a computer -- renormalised one of the four constants to 1 and called the rest a, b, c -- and everything went much smoother. (Also when I would get some large swirl of symbols like
-- composed of ratios and differences that made sense atomically, but was extremely tedious to write over and over--I finally just started writing
math.stackoverflow -- suggest
The common objection I hear to changing all of this is that "People are used to it" -- meaning the mathematicians who already understand the stuff can't be bothered to change for the sake of us dumb@sses who don't know it.
If the Chinese can assign multiple meanings to a phoneme like hui, then our Mandarins can surely admit that p has taken on many different meanings in different contexts--and sometimes even different meanings within the same context.
Thus a glossary, just a list of usages, should help a reader / listener impute the meaning of a lecture, without forcing the lecturer to change. (Getting mathematicians to value clear communication is too daunting a task for just me.)
So, here it is. An alphabet of mathematics.
Let's start with the easy ones.
a,b -- x,y -- a,b,c,d
Used just to contrast two or more things.
- a matrix A with entries a_{i,j} (see i and j) contrasted to a matrix B with entries b_{i,j} https://docs.google.com/viewer?a=v&q=cache:V3Bx3KLW4YgJ:www.numbertheory.org/courses/MP274/markov.pdf+&hl=en&gl=us&pid=bl&srcid=ADGEEShgGmLVI5XvxuwuSAH0JYllZTcJoxgwVl1YZiZdFtMrKK2rKtDwFZSSMrBqVp0HBgAUzMgBh3CswucK49MBWrU1kQ9Knju7QgZmCiw6DsF7xg9jAWytAD8ePiZCIxRLU0d4KnZ3&sig=AHIEtbQTQdK_LIDZs-39RtanbSv-azxqNw
- the first two elements of a group like the Tits group
i,j,k
They're normally used as iterators. Like in a programming language you write a for loop for($i, $i < 10, $i++) and an inner loop might use j or k.
- î, ĵ, &kcirc; in 3-D are the unit vectors that form the standard, canonical orthonormal basis. (They also go by the name ê_1, ê_2, ê_3 in more abstract mathematics.)
In mathematics you need iterators for matrices, tensors, ...
They're used to pick out individuals or to cycle through individuals. It can be confusing.
- i can also be the imaginary unit radic;minus;1
- j can also be an imaginary unit ... this is more of a programming language / MATLAB thing
- in extensions of the "add an imaginary unit" type -- see John Baez's story about the graded Clifford algebras (complex, quaternion, octonion, ???) -- we might actually use all of i, j, and k -- in fact the quaternions are famous for this relation:
p, q
- p = probability of heads in the Binomial distribution
- p = a proposition, like "It rained this past Tuesday."
- p = a prime number, as in F_p = F_7, F_3, F_2, F_13, ... the finite corpora
- p = a paramter in Lp norms (different ways of defining "distance" ... with relationships
- p-branes
- p-adic numbers
q can be another proposition, the flip q=(1-p) of p in the Binomial (i.e., the probability of tails)
l, r, k
These letters usually connote some kind of "selection operator"
- n choose r
- n choose k
- n choose l
When I think about these letters I'm thinking "we need a letter from somewhere obscure-ish in the middle of the alphabet, something like i and j but those have already been used up for the ++ iterators"
y,x
Even though x and y are sometimes used as a pure contrast (neither holding a "special" role), they do play different roles sometimes.
- y is the target (thing to be "solved for", i.e. rearrange the equation so there is a lone y on one side)
- x is the
x_1, x_2, x_3, ...
- x is the data
There was a guy in my first calculus class whose first step to solving any problem would be to substitute the letter y wherever the letter x appeared. I always thought that was a great mind hack. Of course every student knows the feeling of looking at a bunch of x symbols on the page, they kind of look like times;, they're swimming around, you don't know what else to do, etc. He overcame all of that fear and worry by just using a different symbol.
- x is never a parameter (exogenous variable)
- x may be a choice variable
- or it may represent a functional form (Sum;_i a_i x^i where a_i could represent choice variables like in a max/min problem max_a Sum; a_i x^i or could be used to represent the solved form of an analytic continuation)
- or x might represent data -- unknown data which we are preparing to receive later
I
The identity matrix. Coded as diag(5) in R.
Sigma;
- The summation operator.
- A surface.
Π
The product operator.
Lambda;
The wedge operator.
c, c-hat, c-bar, c-underbar, k
Constants.
- Ceix
N, n
- the number of things at the "top"
- the number of dimensions
- the natural numbers N-double-struck
(sometimes M is used ... like in Analysis)
fancy F, fancy L
- Fourier transform
- Laplace transform
capital J
This is a weird one. I've seen it used a lot of ways
- probability distribution in Persson & Tabellini (in the part where they talk about the mathematics of Karl Rovean political strategy)
- In this description of simulated annealing https://docs.google.com/viewer?a=v&q=cache:lprSDyZQUx4J:www.mit.edu/~dbertsim/papers/Optimization/Simulated%2520annealing.pdf+&hl=en&gl=us&pid=bl&srcid=ADGEESj4iMm-oNIN_niZAOAcqBOGKn4V5GMob16FoJuH3MWhGB04z1oETC6wArY9RmvDsP-mv7xzwWGuGe6faEWL0TYTynhZLFc9ImDSQDvIe_WWbU_4eCY40ES351beSrTjQYIhG9yX&sig=AHIEtbT68bNK8VBcErZrby5cIaGwR19ZtA the authors use J as a cost function
- in this description of Markov matrices https://docs.google.com/viewer?a=v&q=cache:V3Bx3KLW4YgJ:www.numbertheory.org/courses/MP274/markov.pdf+&hl=en&gl=us&pid=bl&srcid=ADGEEShgGmLVI5XvxuwuSAH0JYllZTcJoxgwVl1YZiZdFtMrKK2rKtDwFZSSMrBqVp0HBgAUzMgBh3CswucK49MBWrU1kQ9Knju7QgZmCiw6DsF7xg9jAWytAD8ePiZCIxRLU0d4KnZ3&sig=AHIEtbQTQdK_LIDZs-39RtanbSv-azxqNw the author uses J to refer to
[1,1,1,1,1,1,1,1,...,1]a list/vector/tensor/1-by-n matrix with
capital S
θ, φ
Different kinds of angles. φ is azimuthal and theta; is on the suelo.
θ
- Model in Bayesian statistics. Probability of data given theta. It rhymes!
- parameter
γ, ξ
Paths or curves.
Γ
- Used to extend the factorial function ! -- see Hadamard gamma
- in logic
- frames
- cartesian frames
- field extension
ρ
- density
- radius
e
- the natural constant (number e that makes derivative of e^x = itself, e^x)
- unit element in a group
- in abstract geometry, an element of the basis
capital K double-struck
- a field (I like to call them corpora because they're bodies of numbers and "field" sounds like a force field, which is totally totally different)
capital R double-struck, fraktur R, sometimes just R
- the real numbers
ε
- in statistics, the error term. It's not necessarily normal, homoscedastic, or constant! It's just a residual: model (prediction) minus; reality (observation)
ε, δ
-
The famous epsilon-delta proof! Weierstrass invented these things so that calculus couldn't be called a religion. Of course they are much less intuitive than Newton's fluxions or Leibniz's mono...
-
An interesting misuse of ε in http://en.wikipedia.org/wiki/Thomae's_function points out how mathematicians use these letters to have a very specific meaning and not-at-all as interchangeable signifiers:
""" To show continuity at the irrationals, assume without loss of generality that our ε is rational (for any irrational ε, we can choose a smaller rational ε and the proof is transitive). """
So here ε was used three times to mean three different things! Of course a computer program would fail with this. Should we call them ε_orig, ε_irrat, and ε_rat_trans_irrat? Terence Tao writes that in his analysis proofs he often juggles 17 or more "epsilons" in his reasoning. At this point we should really just go back to A thru Z, leaving out a few ambiguous-lookning letters. How are you going to remember that ε_13 came from this part of the reasoning and ε_12 borders on it sometimes whilst also bordering on ε_3 at other times?
δ
- in calculus of variations
- in physics, a displacement
d
- number of dimensions
- d-Brane
- d-sphere
- higher geometry
- differential, as in dx wedge dy wedge dz
- data
λ
- length
- λ x + (1 − lambda;) x -- convexity
- e^i λ x
- eigenvalues
λ, μ
- these could be used in a contrast in diffeq different eigenvalues
- or in the definition of linearity they represent two different scalars
∂
- partial derivative
- boundary
Besides just listing definitions and examples for you, I also want to draw a few conclusions
- there are suggestions, connotations -- maths isn't all facts
- maths is performed by humans who like the suggestions
- there is often something ineffable about why you want to call something a q-decomposition rather than an x-decomposition
- So it's really false that "any letter can stand in for any other" -- even though we're told in primary school that x is just a symbol and we could use any other symbol equally well for it.
- x comes to mean "the fundamental unit of consideration"
For example, in writing about J I was going to write "1-by-N matrix" ... but it sounded wrong. Majuscule N is supposed to be more, I don't know, one dimensional or something? It's supposed to put the cap on one very large thing. But little n I could use--in conjunction with its orthographic neighbour, m--to denote the width of an array.