Re: Platonism
From: J.E. (troubled6man_at_yahoo.com)
Date: 11/30/04
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Date: 30 Nov 2004 10:00:07 -0800
Neil W Rickert <rickert+nn@cs.niu.edu> wrote in message news:<cog8lf$mdb$1@usenet.cso.niu.edu>...
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> troubled6man@yahoo.com (J.E.) writes:
> >Neil W Rickert <rickert+nn@cs.niu.edu> wrote in message news:<cod2hh$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. I said "X", you said "I'm not so
sure that is correct.", 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.", which you took as asking for a
better foundation of mathematics. Something I'm not opposed to, but
not what I was asking. 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.
> 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 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. 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. Now of
course proof by example is no proof, so such people might exist, but
the idea that they are most is strange to me. 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. The point is that either the universe has something LIKE
the naturals numbers that an individual mathematician imagined or the
statement (~ZFC) will be false, so either way the statement
(~ZFC)or(T) will be valid for an arbitrary universe. 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.
> >> A new up and coming mathematician has already done a lot of
> >> mathematics before being exposed to ZFC. In many graduate schools, a
> >> class in mathematical foundations is still not a requirement for a
> >> doctoral degree.
>
> >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.
> >> I readily grant that set theory has proved useful. Indeed, it is
> >> surely one of the parts of mathematics that would continue to thrive,
> >> even if the foundations should crumble. It might survive in a
> >> slightly different form, but it would still survive. As Hilbert
> >> wrote, "No one shall expel us from the Paradise that Cantor has
> >> created."
>
> >You seem to idicate that mathematicians are "doing something else"
> >that isn't ZFC most of the time, without being clear what you think
> >they are doing. If you want to disagree, it's helpful to disagree
> >positively. And quotes and opinions about the future are just as good
> >as my predictions about the future, which is to say, they are next to
> >nothing.
>
> 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.
> 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. That might be helpful when discussing arbitrary open sets of
reals in topology or doing measure theory, or discussing strange sets
of reals whose existance depends on CH or something like that.
> My personal view -- I won't try to ascribe this to other
> mathematicians -- is that our intuitive sense of real numbers derives
> from idealizing the processes of counting and measuring.
I think our intuitive senses of real numbers comes from bounded
operations. 2+5 isn't that far from 2 and 5, so if you draw a "large
enough" region around some numbers, then their sums and products all
stay in your region, so you can easily imagine models. It is models
that provide intuition, and better intuition is that which avoids
"reasoning by analogy" to only get correct results.
> >> > Human
> >> >intuition goes astray and formalism is there to keep people in line,
> >> >without it nonsense would appear again IMO.
>
> >> I'm not knocking formalism. I am only suggesting that it is not the
> >> source nor the basis for *all* of mathematics.
>
> >Mathematicians can say what they say as parroting all they want, but
> >if you push them they almost always say that they rely on the work of
> >"previous mathematicians".
>
> Many of those admired previous mathematicians pre-dated ZFC.
>
> 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. The idea that instead of models, they are based on platonic
"real" real numbers is something I've never seen anyone take serious
in my personal experience.
> > And for most mathematicians today for most
> >of the work they do, that foundational previous work of previous
> >mathematicians that itself doesn't go back for it's basis to someone
> >yet earlier mathematician's work is ZFC, an axiom system that by
> >definition just asserts the axioms without saying that "a previous
> >mathematician proven the axioms".
>
> You are assuming that we build formal proof upon previous formally
> proved results. But it would be more accurate to say that be build
> new understanding on prior understanding. And it has to be our own
> understanding all the way down, for we cannot read the minds of our
> predecessors.
Building is a bit ambitious. Refining might be a better term, since
sometimes your understanding of another's ideas was flawed, so that
the "understanding" of yours was really a fallacy that had to be
excised. So I think we agree that you don't read other's minds and
hence aren't dealing with the same informal objects they are, which
means each of us is dealing with informal models of the formal results
AT BEST, since one can have a flawed model of the formal system. And
these models of the formal systems can be made INDEPENDENT of whether
you are personally familiar with the formal system itself.
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