Re: Help. What is a model?
- From: "T.H. Ray" <thray123@xxxxxxx>
- Date: Tue, 14 Jul 2009 19:26:05 EDT
Aatu writes
Frederick Williams <frederick.williams2@xxxxxxxxx>Soulless comes to mind, too. In any case, your claim:
writes:
Aatu Koskensilta wrote:mathematical
... Where in the literature do we find any
Passiflora caeruleadefinition of a mathematical theorem?
Why shouldn't such a thing exist, my little
blossom?
Allow me to expound, my dearest Brassica oleracea
sprout.
A mathematical theorem is a mathematical statement
for which a
mathematical proof has been given. Whether or not we
have managed to
produce a mathematical proof is obviously not a
mathematical matter, but
turns rather on the contingencies of our mathematical
practice and human
existence in general. We can't, for example, expect
mathematics to tell
us if the time should come when no-one bothers doing
mathematics
anymore, and consequently no new theorems are
produced. This is of
course nothing but obnoxious pedantry. So let's try
to be a bit more
charitable and, perhaps, slightly more informative.
The perfectly fine definition of mathematical theorem
-- a mathematical
statement for which a mathematical proof has been
given -- is not a
mathematical definition since it involves three
notions which have no
mathematical definition, viz. "mathematical
statement", "mathematical
proof" and "has been given". As noted we can't hope
to give any
mathematical explanation or definition for "has been
given". Whether
something has been given, to the mathematical
community, to some
individual, to the human race, is simply not a
mathematical matter. So
let's idealise a bit, abstracting away this blatantly
non-mathematical
ingredient. We then obtain something on the lines of
A mathematical theorem is a mathematical statement
for which a
mathematical proof can, in principle, be given.
Reading "can, in principle, be given" simply as
"exists" we're left with
two problematic notions: mathematical statement,
mathematical proof.
Strictly speaking we can't of course expect a
mathematical definition
for either of the notions mentioned, anymore than we
can in case of
"hamburger", "a fine example of Edwardian English
prose", "computer",
"informative Usenet post". What we may hope for is
rather a mathematical
explication, a mathematical analysis, a mathematical
model that captures
those features we consider salient. The question now
becomes: can we
give such informative and faithful mathematical
analysis of the notions
of a mathematical statement and a mathematical proof?
Should we succeed
we'd immediately obtain one also of the notion of a
mathematical
theorem.
Examining the notion of a mathematical statement we
encounter our next
quandary. Starting with the standard understanding,
that a mathematical
statement is a declarative sentence (in English, say)
making an
assertion about mathematically defined properties of
and relations
holding among mathematical objects, we're faced with
two problems, one
having to do with the wide variety of attitudes,
ideas, opinions,
mathematicians hold, and one purely logical.
The first problem is easily illustrated: the finitist
considers all talk
of infinitary objects non-sense, and indeed considers
quantification
over naturals etc. illegitimate, theological
business. The predicativist
happily allows unrestricted quantification over
naturals, and talk of
definable collections of naturals, and so on, but
balks at the notion of
an arbitrary collection of naturals. The set theorist
considers the
predicativist a sissy but enamoured with the
iterative conception of set
may well have his doubts about quantification over
proper classes,
collections of proper classes, etc. The intuitionist
has yet other
causes for complaint. And so on and so forth. Yet
they all do share the
same notion of a mathematical statement -- indeed
otherwise it would be
very odd to say they /disagree/ about anything,
which, on the face of
it, they very much do, propounding in great length
why these and those
notions are or are not legitimate and meaningful in
mathematics.
But let's set that aside and suppose, contrary to the
fact, that
mathematicians were a boring lot, in heartening
agreement on what
notions make mathematical sense, what sort of stuff
we may meaningfully
quantify over, and so on. Even then we couldn't hope
for a mathematical
explication of the notion of a mathematical
statement. For suppose we
set up a formal language incorporating notions,
concepts, etc. we agree
are meaningful, so that the formal sentences can be
taken to be
(formalisations of) meaningful mathematical
statements. By simple
diagonlisation argument there are then mathematical
notions, such as
"true sentence in the formal language" that aren't
expressible in the
language, but which we, having agreed the language is
meaningful and its
sentences make mathematical assertions, recognise as
legitimate
mathematics. (This doesn't preclude the possibility
that, in fact, there
is a formal language in which all mathematical
notions we recognise or
could in some idealised sense recognise as legitimate
can be expressed;
but if there is such a language -- and unless we're
pretending there is
necessarily some fact of the matter as to whether
this or that notion
makes mathematical sense it's not given the idea
makes any sense -- we
can't recognise it as such.)
Enough about mathematical statements. Let's set this
problematic notion
aside and consider whether we may give a mathematical
analysis of, say,
the notion of a mathematical theorem expressible in
the language of
arithmetic[1]. The sole problematic notion is now
that of a mathematical
proof. Basically just the same problems I went over
in case of
mathematical statements resurface. A mathematical
proof is a piece of
logically correct, compelling reasoning from basic
principles
mathematicians accept as correct (are self-evident,
immediately obvious,
etc.). What counts as "logically correct reasoning"
we may take to be
unproblematic. But what of these basic principles?
Some regard the
existence of, say, a weakly compact cardinal
perfectly obvious, made
evident to them by pondering the vast and
unfathomable universe of sets
-- others seriously doubt even the consistency of PA.
A mathematical
proof to one may well not be a proof to another.
Spelling out the second problem, that a mathematical
definition of
mathematical proof would immediately, by an obvious
diagonalisation
argument, lead to a contradiction, is left as an
exercise to the
reader. (Which is just an oblique way of saying I'm
tired of typing. I
do hope the above was sufficiently artificial and
strained.)
A mathematical theorem is a mathematical statement
for which a
mathematical proof can, in principle, be given.
Reading "can, in principle, be given" simply as
"exists" we're left with
two problematic notions: mathematical statement,
mathematical proof.<<
is false. A theorem is accompanied by a proof that the
theorem is true, or it is not a theorem. A conjecture
is a statement for which a proof can in principle
be given.
It is not necessary to drill down to the meanings of
statement and proof. The meaning is contained in the
definition of theorem: "True mathematical statement."
The quantifier "there exists" follows from the
result, not from the assumption that some result might
be given.
Your discourse could appeal only to a postmodernist.
The only working mathematician I can think of who might
lay claim to that philosophy is Brian Rotman. And while
I have enjoyed his books, I think even he would rather be
doing mathematics than writing about it. He would also
surely recognize the difference.
Once again: content does not recapitulate meaning.
Tom
.
Footnotes:
[1] "Mathematical statement expressible in the
language of arithmetic"
can be given a mathematical explication. For details
see my ludicrously
long-winded rant /Formalisation/ available at
http://groups.google.com/group/sci.logic/msg/1cf3026b
e617d644
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--
Aatu Koskensilta (aatu.koskensilta@xxxxxx)
"Wovon mann nicht sprechen kann, darüber muss man
schweigen"
- Ludwig Wittgenstein, Tractatus
s Logico-Philosophicus
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