Re: Platonism
From: J.E. (troubled6man_at_yahoo.com)
Date: 12/01/04
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Date: 1 Dec 2004 09:33:17 -0800
Neil W Rickert <rickert+nn@cs.niu.edu> wrote in message news:<coijad$jd6$1@usenet.cso.niu.edu>...
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> troubled6man@yahoo.com (J.E.) writes:
> >Neil W Rickert <rickert+nn@cs.niu.edu> wrote in message news:<cog8lf$mdb$1@usenet.cso.niu.edu>...
> >> troubled6man@yahoo.com (J.E.) writes:
> >> >Neil W Rickert <rickert+nn@cs.niu.edu> wrote in message news:<cod2hh$er0$1@usenet.cso.niu.edu>...
>
> >> >> > Most mathematicians will retreat to
> >> >> >ZFC if you press them either (1) hard for the basis for their claims
> >> >> >or (2) really hard about the meaning of their claims.
>
> >> >> I'm not so sure that is correct.
>
> >> >Any thoughts on a better replacement? Just saying you disagree isn't
> >> >saying much. I still respect your opinion, but just saying you
> >> >disagree isn't as useful as saying either why you disagree or what you
> >> >believe instead.
>
> >> We are talking about "most mathematicians". I can reasonably judge
> >> how they will act on the basis of my own discussions with
> >> colleagues. But I cannot judge how they will react to hypotheticals
> >> that I have never posed to them.
>
> >> When you ask for a better replacement, you appear to be asking for a
> >> "better" foundation.
>
> >Wow, you missed my point entirely.
>
> Whereas I think you missed my point entirely.
I can believe that. Did you want me to get it?
> > I said "X", you said "I'm not so
> >sure that is correct.",
>
> I was being polite.
>
> > , then I tried to say "Well if you don't think
> >X, rather than just sharing that you think X is wrong, why don't you
> >share what you DO think is correct."
>
> Your whole way of looking at it seems wrong. You are not even in the
> ballpark. Perhaps we are in different universes.
Explain what your universe is like, maybe I can learn from you.
> > 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. I took a functional analysis
course where Choice changed from false to true halfway through the
course. I had to go back and change everything. If he'd stated "this
is a theorem independant of choice" each time then I would NOT have
had to redo those theorems.
> >> I am doubting that they will appeal to foundations at all. More likely,
> >> they will appeal to their mathematical intuitions, and those probably have
> >> a platonist flavor.
>
> >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.
> If you, as a physicist, tried to press hard, perhaps they would
> "retreat" to ZFC. But what they would really be doing, is coming up
> with enough sophisticated sounding technical language to scare you
> away (a snow job). If I, as a mathematician, tried to press them,
> they would wonder whether I had gone off my rocker.
They have to leave ZFC to give me a snow job, but appealing to models
or SO logic or something I didn't actually study. I've taken too many
courses with set theory to be "brushed off" with taking ZFC as a
"technical language". The ZFC axioms aren't much different in
structure to the "theorems" like "for all sets X, there exists a set
P(X) that contains all subsets of X."
> > I think that's
> >because eventually they will agree that (~ZFC)or(T) is an example of
> >the VALIDITY that they meant, while T surely isn't "true" like they
> >informally spoke. You said they wouldn't.
>
> 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?
> > In real life I'm never met
> >a mathematician that admited Platonic leanings, not my uncle, none of
> >my teachers, no one at an international conference, no one.
>
> You have to distinguish between mathematical platonism and
> philosophic platonism. I would guess that many mathematicians would
> reject philosophic platonism. Moreover, most mathematicians don't
> much engage in discussions of the philosophy of mathematics, so they
> may be unsure of what is entailed by mathematical platonism.
>
> 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". 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. 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". Most mathematicians require
that I proof be "clear enough that it could be formalized, IF they
wanted to". Anything else is considered hand-waving. As is what
happens in SOME physics classes.
> > If you implicity leave
> >"(~ZFC)or" off the beginnings of all sentances and expect there to be
> >some definition of number based on ZFC that gives them the properties
> >that you expect, then it's common when discussing a validity to
> >imagine that these numbers exist in the universe in a purely naive
> >way, but when in the end you want a proof, you base it on theorems and
> >results from ZFC and FOL or something like that, and the result is a
> >validity.
>
> Let's compare with physics.
>
> Most people take the view that the world is a certain way, and that
> Newton's laws are a regularity that was discovered as a basis for why
> the world is the way it is.
>
> Similarly, most mathematicians take the view that the mathematical
> world is a certain way, and that axioms systems such as ZFC are
> something akin to Newton's laws. That is, they are a discovered set
> of relations which provide a basis for the way the mathematical world
> is.
>
> Now as it happens, some problems were found with Newton's laws, and
> Einstein gave as a refinement. Most people take the view that this
> did not change the way the world is, and that their physics is about
> the world rather than about the consequences of Newton's laws.
Physics is about models. Reality is just the anvil on which you test
models, you can't talk about reality directly.
> 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. 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.
> > But these
> >private conceptions of natural numbers are projective things, you
> >don't expect your numbers to "be the same" as someone else's numbers,
> >because all you care about is the projective properties of number, the
> >things shared by all models of PA for instance, unless PA in
> >insufficient for your purposes.
>
> However, most mathematicians don't think of numbers as private
> conceptions. They do see numbers as being public things, and they
> do expect their numbers to be the same as everybody else's numbers.
Don't most mathematicians know that there are non-standard models of
their number systems?
> >> >Interesting, you actually think many people could study mathematics in
> >> >depth without studying set theory?
>
> >> I'm sure they study some set theory. But it may be naive set theory,
> >> rather than a formal development from ZFC.
>
> >They'd stick recognize that a few assumptions about sets work as a
> >basis for the other things they learned, so when pressed to turn a
> >vague statement like "T is true" into a validity, then they know that
> >ZFC is SUFFICIENT to make a validity, maybe a simpler system would
> >work, that's reverse mathematics to find a subset of axioms. I don't
> >understand reverse mathematics myself without a fixed axiom system
> >because you could always make T into a validity by writing
> >"(~T)or(T)", that what Goedel's completeness is about, so it's not
> >about the smallest set of axioms because that's boring. So some
> >axioms or theorems (formulas both) must be considered more
> >foundational or important than others for this to be meaningful.
>
> 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". That's not very
different, IMO.
> >> ZFC attempts to construct everything, with the empty set as the
> >> starting point. To most mathematicians, there is something very
> >> artificial about this. With the intuitive notion of set, sets of real
> >> numbers are primary examples of what sets could be. To go back and
> >> start with an abstract notion of set that is pre-requisite to there
> >> being real numbers, is quite unnatural.
>
> >Seriously? The set of reals is not hard to construct in ZFC, like it
> >could easily be done on the first day.
>
> 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?
> 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. This MUST be either irrational or BASED on a
belief that ZFC is MORE trusted than personal mathematical intuition.
Again, mathematicians don't just ACCEPT Reimann's hypothesis based on
his "intuition", mathematicians WANT more, and ZFC formalizability is
WHAT they usually want.
> >> Mathematicians are likely to start out with a good sense of real
> >> numbers, and sets of real numbers. When you go back and construct
> >> the reals from ZFC, this does not add any additional confidence in
> >> their conceptual understanding of the reals. However, it does add
> >> some confidence in ZFC, that it leads to the same real numbers that
> >> they already work with.
>
> >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. When I say "arbitrary set theory operations" I
mean "arbitrary ZFC operations", so saying that this is "a standard
for ZFC" is hopeless. Comprehension doesn't work. You can't consider
the set of all reals and subsets of reals and subsets of sets of
(reals and subsets of reals) and subsets of set of those, and subsets
of sets of those and so on ... and so on ... and so on so on ... as a
completed whole because you'd get into problems with comprehension, so
you CONSTRUCT the subsets you want iteratively and it turns out that
you just never GET that whole set I said you couldn't have, no matter
"how intuitive" it was.
> >> For most mathematicians, it is not sufficient to give a formal proof
> >> that derives a result from previously proved work. They need to have
> >> a strong intuitive understanding of their result, and why it is
> >> true. This intuition is likely to be based on their intuitive sense
> >> of real numbers and other mathematical things.
>
> >I would expect those intuitions are based on models of the real number
> >system.
>
> That may be. But the "model" doesn't come from ZFC. More likely it
> comes from intuitive Euclidean geometry, with the line as a model for
> the real numbers.
Only for pictures, not for proofs. Drawing a picture is not
considered "good enough". I'm fine with models from elementary
eulidean geometry, that's proven consistent, which is better than ZFC.
But elementary geometry doesn't have arbitrary functions from R to R,
that's in the domain of set theory. And I'd be really curious as to
what models other than ZFC itself you think matheamticians use for an
arbitrary function.
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