On Tuesday 21 July 2015 19:10, Marko Rauhamaa wrote:
> This is getting deep. When things get too deep, stop digging. > It is an embarrassing metamathematical fact that > numbers cannot be defined. At least, mathematicians gave up trying a > century ago. That's not the case. It's not so much that they stopped trying (implying failure), but that they succeeded, for some definition of success (see below). The contemporary standard approach is from Zermelo-Fraenkel set theory: define 0 as the empty set, and the successor to n as the union of n and the set containing n: 0 = {} (the empty set) n + 1 = n ∪ {n} https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers > Our ancestors defined the fingers (or digits) as "the set of numbers." > Modern mathematicians have managed to enhance the definition > quantitatively but not qualitatively. So what? This is not a problem for the use of numbers in science, engineering or mathematics (including computer science, which may be considered a branch of all three). There may be still a few holdouts who hope that Gödel is wrong and Russell's dream of being able to define all of mathematics in terms of logic can be resurrected, but everyone else has moved on, and don't consider it to be "an embarrassment" any more than it is an embarrassment that all of philosophy collapses in a heap when faced with solipsism. We have no reason to expect that the natural numbers are anything less than "absolutely fundamental and irreducible" (as the Wikipedia article above puts it). It's remarkable that we can reduce all of mathematics to essentially a single axiom: the concept of the set. -- Steve -- https://mail.python.org/mailman/listinfo/python-list