Re: Platonism

From: Neil W Rickert (rickert+nn_at_cs.niu.edu)
Date: 12/01/04 

Date: Wed, 1 Dec 2004 22:38:27 +0000 (UTC)

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troubled6man@yahoo.com (J.E.) writes:
>Neil W Rickert <rickert+nn@cs.niu.edu> wrote in message news:<coijad$jd6$1@usenet.cso.niu.edu>...

>> > I'd prefer that mathematics discuss
>> >validities, truths in all worlds. So if you have a theorem T of PA
>> >with axioms A, then (~A)or(T) is a validity. Just "leaving out"
>> >everything except T is bad form. But I don't think that's what we
>> >were originally talking about.

>> But this is not very close to the way mathematicians work.

>Most mathematicians just say "theorem" without referencing a system,
>which is BAD when the system changes.

What is being implicitly assumed is usually quite evident to the
reader.

> I took a functional analysis
>course where Choice changed from false to true halfway through the
>course.

That must have been a very slow moving course. One already needs the
axiom of choice to prove the Hahn-Banach theorem. And this is
usually proved near the beginning, not halfway through the course.

>> >I said that they'd retreat to ZFC when pressed.

>> I have never tried pressing that hard. But I rather suspect that if
>> I did, they would decide I was a hopeless crackpot and stop talking
>> to me. Retreating to ZFC is unlikely.

>I agree that not defending your views and meaning is like not
>responding to a survey, and that hence the results will be inaccurate.
> But that doesn't seem to indictate anything either way about the
>ACTUAL beliefs of mathematicians who are NOT in this sci.math
>conversation.

My assessment of the beliefs of mathematicians comes from conversations
with them. Sometimes these have been technical, and sometimes they have
been coffee room chatter. But I'll grant that I am not a mind reader,
and that my assessment might be wrong.

>> If mathematicians took mathematics to be a branch of logic, perhaps
>> they would react as you suggest. But most mathematicians instead see
>> logic as a branch of mathematics, and perhaps only a side branch at
>> that.

>And how do mathematicians "take mathematics", in your opinion?

I'm sure it varies. But something along the line of mathematical
platonism is about right.

>> However, ask them about the statement

>> the square root of 2 is irrational

>> and they will likely say that this is objectively true. While it can
>> be proved from ZFC, that is not the reason they hold it to be true.
>> After all, this was known to be true two thousand years before ZFC.

>But modern mathematicians would say that ZFC is the foundation for the
>modern definition of both "square root", "2", and "irrational".

I doubt that. I think they would be more likely to mention PA than
to mention ZFC.

> If
>you discuss line-fillin curves or Banach-Tarski mathematicians are
>quite quick to say that a solid ZFC basis is "the only" acceptable
>basis.

I assume you mean "space filling curves." However, I am not sure why
you mention them. They don't really raise issues that would require
going back to primitives of set theory.

I'll grant that Banach-Tarski does require care at the set theory
level. I'm not sure about that "solid ZFC basis" part. Some
mathematicians might be more familiar with a Bernays-Goedel style of
axiomatization of set theory.

Mathematicians don't spend a lot of time talking about
Banach-Tarski. It doesn't have much relevance to what they do.

> The only reason they allow themselve to NOT consider ZFC when
>doing SOME reasoning is that ZFC reasoning and "their personal
>reasoning" are considered "close enough".

That's not quite right. It might be more accurate to say that the
only reason ZFC is an acceptable axiom system, is that ZFC reasoning
and mathematical reasoning are close enough.

> Most mathematicians require
>that I proof be "clear enough that it could be formalized, IF they
>wanted to".

This is a theoretical principle, although it is not much tested in
practice. Strictly formal arguments are harder to read, and
mathematicians do want to be able to communicate their results.

There is a big gap between this theoretical principle, and your
previous assertion. The principle is that mathematical proofs could,
if necessary, be formalized. It IS NOT that they could be formalized
in ZFC.

When mathematicians do give formal proofs, they want to use a minimal
axiom set. If PA is sufficient, they will prefer that to ZFC. If
the axioms of a field are sufficient, they will use those rather than
PA. If they can make do with only the axioms of a semi-group, they
will prefer to use those.

>> Similarly, most mathematicians would not be particularly disturbed if
>> ZFC is tossed out, and replaced by a more refined set of axioms. For
>> they see themselves as dealing with the mathematical world, rather
>> than with the consequences of ZFC.

>> To put it differently, an axiom system such as ZFC is not seen as the
>> beginning point of mathematics. Rather, it is seen as something of a
>> tentative end point.

>It's a standard for acceptance of proof. The Reimann hypothesis isn't
>considered "good enough" to deserve the name theorem.

The requirement is that there be a mathematical proof. It is not
required that there be a formal derivation starting with ZFC.

> Because it has
>no argument good enough to be formalizable into a formal proof of it
>from ZFC. And until then, most mathematicians won't take it very
>seriously. Just like reality is used to test physical model/theory
>pairs. Physicists don't take string theory very seriously as long as
>the string theory model/theory pairs don't produce results that CAN be
>comapred to observations. So ZFC plays the same role as reality does
>for physics, so I don't know why mathematicians would say that they
>were NOT talking about ZFC ultimately.

Sorry, but that is just wrong. If anything, ZFC plays a role more
like that of string theory. Most mathematicians accept it
tentatively, because it allows them to do what they need. They are
more committed to PA than to ZFC.

>Don't most mathematicians know that there are non-standard models of
>their number systems?

Sure. But they do not find this troubling. They perhaps see it as
reason to be suspicious of formalism.

>> You are taking formalism to be a priori. That's not the view of most
>> mathematicians.

>Most mathematicians take proof as a priori, and the standard of proof
>is "could be formalized into ZFC if we wanted to".

No, that is *not* the standard of proof, as I indicated above.

>> Granted it is not hard. But it is artificial.

>> If the real numbers are what is constructed from the empty set, using
>> ZFC, then it is not obvious that they have anything to do with
>> physics, nor with our ordinary weights and measures that we use in
>> every day life. Perhaps it is just an inexplicable miracle that real
>> numbers happen to be useful today -- a coincidental parallelism
>> between real numbers and things that are useful in the world. Maybe
>> the coincidence will end tonight, and our real numbers will no longer
>> be useful, starting tomorrow.

>Dedekind cuts aren't obvious enough? Are you SERIOUS?

You will have to explain why Dedekind cuts, as built from ZFC, connect
real numbers to our ordinary weights and measures. It is not at all
obvious.

If one takes real numbers to correspond to the line from Euclidean
geometry, then Dedekind cuts do become an obvious part of what
connects measurements to real numbers. But if the Dedekind cuts are
made in what comes from a purely combinatorial build up via ZFC, then
there is nothing obvious to connect them to measurements.

>> But if you look at real numbers the way mathematicians do, then their
>> relation to the weights and measures of every day life is obvious.
>> If anything, it is the connection with ZFC that seems tenuous.

>Mathematicians use ZFC to prove that SOME reasoning about real numbers
>"is acceptable for proofs" and they THEN restrict themselve to THOSE
>reasonings, to avoid Banach-Tarski or other problems from letting
>intuition run wild.

I must be missing your point somewhere. Banach-Tarski is provable.
There is nothing in ZFC that disallows it. It isn't intuition that
runs wild here. If anything, it is ZFC that runs wild, in that it
allows Banach-Tarski, while intuition is suspicious of it.

>> >It should give some confidence that real numbers can really form a
>> >set, i.e. that arbitrary set theory operations can be performed on
>> >reals.

>> No, not at all. This is implicit in the mathematicians intuitive
>> notion of set. If ZFC said otherwise, that would be a problem with
>> ZFC, and not a problem with the mathematicians concept of reals or of
>> sets. A proposed system of axioms for set theory would not have been
>> acceptable if it did not accomodate the needs of mathematicians
>> working with sets of real numbers.

>This sounds circular.

No, there isn't any circularity there.

> When I say "arbitrary set theory operations" I
>mean "arbitrary ZFC operations", so saying that this is "a standard
>for ZFC" is hopeless.

I will grant that your thinking is circular. That's a problem for
you. It isn't a problem for mathematics.

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