Re: Platonism
From: Neil W Rickert (rickert+nn_at_cs.niu.edu)
Date: 11/30/04
- Next message: patty: "Re: Platonism"
- Previous message: tchow_at_lsa.umich.edu: "Re: Platonism"
- In reply to: J.E.: "Re: Platonism"
- Next in thread: Acid Pooh: "Re: Platonism"
- Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]
Date: Tue, 30 Nov 2004 19:57:01 +0000 (UTC)
-----BEGIN PGP SIGNED MESSAGE-----
Hash: SHA1
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 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.
> 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.
>> 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.
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.
> 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.
> 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.
> 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.
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.
> 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.
>> >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.
>> 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.
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 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.
>> 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.
-----BEGIN PGP SIGNATURE-----
Version: GnuPG v1.3.91 (SunOS)
iD8DBQFBrNCKvmGe70vHPUMRAjzjAKCKqyYRFzyUDoHj3kFCh27iP29QoQCfZVti
AP8uSqPff475Sd1mBcamHyg=
=JlKZ
-----END PGP SIGNATURE-----
- Next message: patty: "Re: Platonism"
- Previous message: tchow_at_lsa.umich.edu: "Re: Platonism"
- In reply to: J.E.: "Re: Platonism"
- Next in thread: Acid Pooh: "Re: Platonism"
- Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]
Relevant Pages
|
