Re: Some Questions.

In article <rubrum-78AAD5.14565806012007@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Michael Press <rubrum@xxxxxxxxxxx> wrote:
In article <505n80F1eoicvU7@xxxxxxxxxxxxxxxxxx>,
Bob Kolker <nowhere@xxxxxxxxxxx> wrote:

Ken Pledger wrote:



Well, mathematics got along for at least 4000 years without
mentioning sets.

In the case of geometry it was implicit. Also in the case of counting.

Prior to the 19th century mathematics was ill founded. Euclid had wide
gaps, and the burgeoning art of real variables was an intellectual
disaster area.

Euclid had moderate gaps, not wide ones.

I consider this assessment to be uncharitable. I think
that Newton, Liebnitz, Euler, Gauss, ... understood
very well the underpinnings of their work. Imagine
struggling with the notion of continuity, your best
formulation is in terms of the intermediate value
property, and staring helplessy at sin 1/x. What about
infinitessimals? Mathematicians have always worked to
make a sensible foundation for mathematics.

Set theory is not the pinnacle of foundations. :)

Logic is the basis; from Euclid (or Pythagoras?)
on, it was recognized that a proof was a sequence
of statements each following from the preceding
ones; proofs by contradiction and existential proofs
allowed the introduction of premises.

Those you cited understood well that the foundations
of their work were NOT known. Possibly not Gauss,
because the work of Bolzano and Cauchy was during
his lifetime. Rigorous analysis starts there, and
rigorous arithmetic starts with Dedekind and Peano.

Set algebra started with Boole and de Morgan, and
attempts to put it all on a sound common basis began
in the last quarter of the 19th century. At about
the same time, cardinal and ordinal aspects were
introduced. One need not require that everything
is a set or class; individuals can be allowed.
But this does not get much more.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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