On Wed, 19 Jul 2006 17:41:30 -0600, Virgil <email@example.com> wrote:
>In article <firstname.lastname@example.org>, > Lester Zick <DontBother@nowhere.net> wrote: > >> On Wed, 19 Jul 2006 12:29:53 -0600, Virgil <email@example.com> wrote: > >> >If every mathematician spent as much time dealing with foundations as >> >you seem to want, no other math would ever get done. >> > >> >Is that what you want? >> >> What I want is to avoid extravagantly stupid claims that transfinite >> arithmetic and set theory encompass all of mathematics > >There are some notable areas of mathematics not encompassed IN set >theory (for example some versions of category theory). But as far as I >am aware, there are none which do not have some basis of assumptions >which collectively can be called the axioms of that area.
Well of course axioms historically have always been the mathematical way. In their pursuit of a priori knowledge mathematicians routinely resorted to axiomatic assumptions of truth. In my experience it is only since the invention of arithmetic that mathematikers routinely resorted to assumptions of truth with respect to definitions.
>> so we can >> properly examine the basis of axiomatic assumptions regarding >> definitions of things such as curves, straight lines, transcendental >> numbers, angular momentum, and quantum effects on a sound unassuming >> analytical foundation. > >Analysis cannot be done on an "unassuming foundation" as without some >foundation of assumptions, there is nothing to work with .
Well you know, Virgil, we've been over this before as I recall. Assumptions as a starting point are one thing and assumptions as finishing points something else. We all start with assumptions. Some of us demonstrate the truth of our assumptions; some of us don't. The latter become modern mathematikers and empirics.
>> It might be too much to ask of professional >> academics but not of real scientists. > > >As "real scientists" are, by definition not mathematicians, at least to >the extent that they are only "real scientists", Zick is implying the >"real scientist" fallacy that no mathematics has any existence or >validity outside of its scientific usage.
Not sure I follow. The "real scientists" you refer to are empirics. Empirics are not real scientists. They're educated guessers. The real scientists I refer to demonstrate the truth of their assumptions. Modern mathematikers and empirics just prefer to assume the truth of their assumptions. It's guesswork.