Wikipedia:Reference desk/Archives/Mathematics/2016 December 16

= December 16 =

Can analysis be done using only the axioms needed for discrete math?
It seems that analysis needs to be based on axioms that allow for uncountable sets, but then given those axioms you end up with a finite set of rules for the manipulation of symbols, and you can only ever process a finite string of symbols using these rules. So, it seems that all of math is ultimately just discrete math in disguise. However, many mathematician reject finitism, so what's the flaw in the above argument? Count Iblis (talk) 16:37, 16 December 2016 (UTC)
 * Symbol manipulation is discrete math. Interpreting the symbols as infinite sets or geometry or analysis is not. Bo Jacoby (talk) 17:06, 16 December 2016 (UTC).
 * You may be interested in Second-order arithmetic, in particular Coding mathematics in second-order arithmetic. Logicians sometimes call second order (as opposed to first order) arithmetic "analysis" because of its power to express classical, ordinary math. As that article's (Simpson 2009, p. 1) says. "The point is that in ordinary mathematics, the real line partakes of countability since it is always viewed as a separable metric space, never as being endowed with the discrete topology." And some people I used to know liked to say "Life is too short for the non-separable case."John Z (talk) 01:39, 18 December 2016 (UTC)