Re: Cantor Confusion

In article <1166136192.256083.128310@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Jonathan Hoyle" <jonhoyle@xxxxxxx> wrote:

Bob Kolker wrote:
Virgil wrote:

And it is plain that no sound mathematics can be developed unless based
on some axiom system as its solid foundation.

Arithmetic was around long before it was axiomatized and people were
proving theorems about integers. For example Gauss and Euler.

Bob Kolker

True, but you are ignoring Virgi's adjective "sound". Calculus existed
way back during the time of Newton and Leibniz, but you could hardly
call their use of the infinite and infinitessimals at all "sound" by
today's standards. It wasn't until Bolzano and Weierstrass made things
truly rigorous in the 19th century was Calculus anywhere near sound.
It is in fact their essential treatments that we are taught Real
Analysis today, not Newton's. (Newton's work would be barely
recognizable today with its "fluxions" and "fluents".)

Bolzano and Weierstrass gave way to more rigor in numbers by Cantor,
and then rigor in Set Theory by Zermelo and Fraenkel. Then with the
wonderful contributions of Hilbert, Lebesgue, Godel, and others,
mathemaatics today is far more rigorous than it was over a century ago.
Even infinitessimals were consistently defined by Robinson. With the
exception of Aristotle's Logic and Euclid's Geometry, much of
mathematics would not be considered acceptable by today's standards.

Actually, we today have a number of improvements on Aristoteles logic,
and Euclid's axiom system had to be revamped by Hilbert to bring it up
to modern standards. But they have both stood the tests of time
remarkably well.

Even Guass and Euler played a bit fast and loose (although they were
considered impeccably precise in their day.)

In ancient times, arithmetic was discovered in much the same way
physical laws were. "Hey, notice that when we do this, that always
happens..." As centuries of very hard work, mathematicians have boiled
arithmetic assumptions down to some basic axioms, and all of the
remaining theorems flow forth.
.

Relevant Pages

  • Re: Cantor Confusion
    ... Bob Kolker wrote: ... And it is plain that no sound mathematics can be developed unless based ... on some axiom system as its solid foundation. ... but you are ignoring Virgi's adjective "sound". ...
    (sci.math)
  • Re: Cantor Confusion
    ... truly rigorous in the 19th century was Calculus anywhere near sound. ... mathematics would not be considered acceptable by today's standards. ... than Gauss and Euler? ...
    (sci.math)
  • Re: infinity ...
    ... That's why standard mathematics is much higher quality than ... starting with an axiom system. ... TO's "reason" defies logic. ...
    (sci.math)
  • Re: Cantor Confusion
    ... And it is plain that no sound mathematics can be developed unless based on some axiom system as its solid foundation. ... It would be of great advantage to find a single axiom system sufficient to cover all of mathematics, but that has not yet been shown to be possible. ... There are some other possibilities along the lines of Category Theory or Topos theory that I understand are in the running as well, ... I call it the Warehouse or Supermarket Paradigm: ...
    (sci.math)
  • Re: CMS: where are the zeroes ?
    ... what do you mean by complex zero? ... According to the Fundamental Theorem of Algebra, this truncated series ... He wanted mathematics to be formulated on a solid and complete logical ... that some such axiom system is provably consistent through some ...
    (sci.math)