In the early 20th century, mathematics was infested
- Gödel
- Polish spaces
- Logicomix
- set theory
- recursion
In logic books (Whitehead/Russell, _____, ____, ____) you will see uninterpretable strings of formal symbols like Φ[ƒ_i,T<x→y,F>] etc etc.
The idea that, like a computer [link to aeon.co story of brain-is-computer mythos] humans can parse arbitrary symbols so φ, x, a, ξ are all equivalent to us.
Contemporary mathematicians seem to realise that this is not the case, so F: E → B or [stacks project] "let an elliptic curve be (E,0,B)" is known to spark associations with things seen before, and these are actively leveraged.
In other words, the stacks project wouldn't define a scheme to be (a,b,c) where a is ___, b is ___, c is ___ because they know that, for their audience, (F,E,B) will be suggestive and useful.
In other words, the full meaning of a mathematical text lies [in the community][thurston arxiv paper] and in the culture of individuals --- against the bourbakian ideal, it is not comprehensively defined in one source.
Because of the way school, college, graduate school, and popular-science-writing relate today,
These topics are now démodé among working mathematicians, but popular lore (Logicomix, undergraduate math and computer-science courses; programmers
Some of the esthetic from that period has survived, eg the groups - rings - fields approach which, according to Wikipedia, was pioneered by Noether and first formalised in van der Waerden's Algebra. A series of books titled Algebra by eg Mac Lane/Birkhoff, Hungerford, Artin (with the goal of comprehensivity such a broad title implies) have advanced different philosophies but still cover the same gigantic span of the algebra world. I regard these books less as pedagogical tools than, like Bour-ba-ki group, to prove something is doable --- to make the human achievement of doing X---for posterity and for the mathematical forward guard, not for learners.
† For beginners I recommend the much shorter math.miami.edu/~ec/book .
&doubledagger; I preferred skimming across several Algebra books, looking for differences in esthetic, rather than trying to learn one man's All Of Everything viewpoint.
I prefer books with a more specific scope --- eg Reid UAG, Harris CAwavtAG,
[[[...pictures from ZIP proof paper and Thurston GT3M...]]]