On Tuesday, July 21, 2015 at 4:09:56 PM UTC+5:30, Steven D'Aprano wrote: > 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.
These two statements above contradict each other. With the double-negatives and other lint removed they read: 1. We have reason to expect that the natural numbers are absolutely fundamental and irreducible 2. We can reduce all of mathematics to essentially a single axiom: the concept of the set. So are you on the number-side -- Poincare, Brouwer, Heyting... Or the set-side -- Cantor, Russel, Hilbert... ?? > On Tuesday 21 July 2015 19:10, Marko Rauhamaa wrote: > > 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. That's a bizarre view. As a subjective view "I dont feel embarrassed by..." its not arguable other than to say embarrassment is like thick-skinnedness -- some have elephant-skins some have gossamer skins As an objective view its just wrong: Eminent mathematicians have disagreed so strongly with each other as to what putative math is kosher and what embarrassing that they've sent each other to mental institutions. And -- most important of all -- these arguments are at the root of why CS 'happened' : http://blog.languager.org/2015/03/cs-history-0.html The one reason why this view -- "the embarrassments in math/logic foundations are no longer relevant as they were in the 1930s" -- is because people think CS is mostly engineering, hardly math. So (the argument runs) just as general relativity is irrelevant to bridge-building, so also meta-mathematics is to pragmatic CS. The answer to this view -- unfortunately widely-held -- is the same as above: http://blog.languager.org/2015/03/cs-history-0.html A knowledge of the history will disabuse of the holder of these misunderstandings and misconceptions -- https://mail.python.org/mailman/listinfo/python-list