FOM: ReplyToMattes/McLarty

[Harvey Friedman]

I wrote the following to Mattes:

>>I hate to be repetitive, but sometimes I find telling quotes that are much
>>stronger than anything I am saying - from people you would least expect. I
>>would like your reaction to the Morris Kline quote from Mathematics from
>>Ancient to Modern Times, Chapter 51, pp. 1182:

>>"By far the most profound activity of twentieth-century mathematics has
>>been the research on the foundations."

I didn't get any response. I am still waiting for a response to this from
the several people on this e-mail list that I have thrown this at. Instead
Mattes responded with a quote from Godel that appears in another book of
Kline - not in his major historical treatise cited above. He asks me
whether or not I agree with the quote from Godel. My answer is: the quote
is not self explanatory and cannot be made sense of out of context.

As a comment to both Mattes and McLarty - note that the work that Kline
describes in Chapter 51 starting at page 1182 is exclusively concerning
f.o.m. in the normal customary sense that I use the phrase - and definitely
not in any provacatively indiscriminate manner.

Mattes writes:

>Would you call the following foundational?

and lists several topics with various degrees of specificity.

Shipman, who I know is reasonably acquainted with these topics, makes a
reasonable direct response.

I would like to respond differently. It is absurd to just mention a
sophisticated modern topic in isolation and talk about whether or not it is
foundational. One cannot really make a positive determination until someone
gives a foundational exposition - or at least an exposition with an
appropriately foundational development of the ideas. There may be several
aspects, some of which are foundational and some of which are not
foundational. Of course, this doesn't require that any proofs be given in
any kind of foundational terms - or that any proofs be given at all.

It is generally acknowledged that foundational expositions to some extent
have been or can be given for basic arithmetic and geometric notions, in
addition to the usual theory of sets. It is also unclear to what extent
this can be done for various modern arithmetic and geometric topics -
perhaps some of them and perhaps not others. But to just throw out topics
and insist that they are foundational is generally unproductive.

Put bluntly, if you think that some material is foundational; i.e.,
properly in foundations of mathematics, then the appropriate action for you
to take for this group would be to give a novel, interesting, and creative
foundational exposition of that material. This would be very welcome. If
you want examples of such, look at various articles of Godel in his
Collected Works. Also look in Putnam, Benacceraf, Philosophy of
Mathematics, and also von Heijenoort, From Frege to Godel. Also various
collections of works by Quine. There is also collected works of von
Neumann. Also Hilbert's Foundations of Geometry, Open Court Classics, is a
good classic. There are many many more places to look for foundational
expositions - or partially foundational expositions.

However statements like "Theorem X relates shapes to numbers" or "Theorem X
analyzes spatial concepts" stated in isolation, are usually useless.

McLarty writes:

>Hilbert and Brouwer both thought issues like Chow's lemma were
>paradigms of foundational problems.

Can you give us some real documentation? Can you give a foundational
exposition of Chow's lemma?

>Half of Brouwer's dissertation (titled
>"On the foundations of mathematics") studies basic conceptions of space,
>using variations of Brouwer's proof that 1- and 2-dimensional continuous
>groups are analytic.

So what?

>Hilbert had earlier urged such foundations for geometry
>(see his 1902 essay "On the foundations of geometry").

This article is reprinted in the Open Court Classics book "Foundations of
Geometry." Is there anything in that article that is in the least
incompatible with the normal conception of f.o.m. that Steve Simpson uses?

>I assume the
>comparison to Chow's lemma is obvious.

Why don't you make it obvious for us? Start at the beginning, since many
people on this group are not primarily mathematicians.

 >       If a man on the street had asked Brouwer or Hilbert in 1907 about a
>weakened Koenig's lemma, both men would have thought it an excellent
>question. But they also thought that relating spatiality to function theory
>was crucial foundational thinking.

"Relating spatiality to function theory" is stated in isolation. If you
have something to say, say it cogently, clearly, and precisely - realizing
that many people on this list are not primarily mathematicians. It is
absurd that any result that relates spatiality to function theory is
foundational.

>        This is "foundations" in Steve's sense: it is unavoidable in any
>"more-or-less systematic analysis of the most basic or fundamental" facts of
>geometry. It deals with concepts "low in the heirarchy" in that you run into
>them as soon as you think seriously about coordinate geometry in relation to
>point-line geometry.

Are you prepared to show us any "more-or-less systematic analysis of the
most basic or fundamental facts of geometry?" If you can do this to a
suitably high standard, that would be valuable for this group.

>In the case of Chow's lemma: as soon as you think
>seriously about curves defined by polynomials. Indeed Descartes DID run up
>against it in his GEOMETRY--though he was no closer to a clear statement of
>Chow's lemma than Fermat was to a clear statement of the Matjesevich result.

Please document this.