Abstraction Levels in Mathematics

… consider for a moment how many levels of abstraction are involved in math itself. In arithmetic there’s the abstraction of number; and then there’s algebra, with a variable being a further-abstracted symbol for some number(s) and a function being a precise but abstract relation between domains of variables; and then of course there’s college math’s derivatives and integrals of functions, and then integral equations involving unknown functions, and differential equations’ families of functions, and complex functions (which are functions of functions), and definite integrals calculated as the difference between two integrals; and so on up through topology and tensor analysis and complex numbers and the complex plane and complex conjugates of matrices, etc. etc., the whole enterprise becoming such a towering baklava of abstractions and abstractions of abstractions that you pretty much have to pretend that everything you’re manipulating is an actual, tangible thing or else you get so abstracted that you can’t even sharpen your pencil, much less do any math.

– From “Everything and More – A Compact History of ∞” by David Foster Wallace

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