FOM: defining "mathematics"
Matt Insall
montez at rollanet.org
Sun Jan 2 10:10:11 EST 2000
-----Original Message-----
From: owner-fom at math.psu.edu [mailto:owner-fom at math.psu.edu]On Behalf Of
Vladimir Sazonov
Sent: Thursday, December 23, 1999 12:34 PM
To: fom at math.psu.edu
Subject: Re: FOM: defining "mathematics"
<snip>
I prefer more general definition of mathematics (related with recent
postings of J. Mycielski; cf. also my reply to him and some other
postings to FOM):
Mathematics is a kind of *formal engineering*, that is engineering
of (or by means of, or in terms of) formal systems serving as
"mechanical devices" accelerating and making powerful the human
thought and intuition (about anything - abstract or real objects or
whatever we could imagine and discuss).
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I like this ``definition'' also, as far as it goes. The problem I see in
all this is that the ``definitions'' all seem to appeal to terms not
previously defined. Thus I would not really call this a definition, because
too much of the description is undefined. In particular, how does one
define ``engineering'', or ``human thought and intuition''? I guess this
might make a good Webster's type of definition, but it is hardly
mathematical itself.
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<snip>
Finally, I would like to stress that mathematics actually deals
nothing with truth. (Truth about what? Again Platonism?) Of course
we use the words "true", "false" in mathematics very often.
But this is only related with some specific technical features of
FOL. This technical using of "truth" may be *somewhat* related
with the truth in real world. Say, we can imitate or approximate
the real truth. This relation is extremely important for possible
applications. But we cannot say that we discover a proper
"mathematical truth", unlike provability. This formalist point of
view is not related with rejection of intuition behind formal
systems. But the intuition in general is extremely intimate thing
and cannot pretend to be objective. Also intuition is *changing*
simultaneously with its formalization. (Say, recall continuous
and nowhere differentiable functions.) Instead of saying that
a formal system is true it is much more faithful to say that it is
useful or applicable, etc. Some other formalism may be more
useful. There is nothing here on absolute truth.
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Okay, so if we do not deal with truth, then what would you say is the
``truth in the real world'' of the following statement: ``If f is a
continuous function defined on the real numbers, then f has the intermediate
value property.'' I submit that as mathematicians, we do, and should, care
about the ``truth in the real world'' of such a statement. The problem I
see is that it is either true or false, but not both, but the formalist
approach would have us believe that no one even knows what the statement
means. If this were correct about such statements as this, then do we, as
human beings (not, per se, as mathematicians in particular) know what
anything means? In fact, would you say, professor Sazonov, that there is no
such thing as ``truth in the real world''? For if it is because we
formalize Mathematics that we lose meaning, is it not the case that even the
very statements we make about the ``real world'' are formalizations, of a
sort, and so can be interpreted any way one may choose. After all, whether
we are doing mathematics or not, we are only putting marks on the page.
Thus, even the statement that `` mathematics actually deals nothing with
truth'' has no meaning outside the virtual marks on the virtual page on my
computer monitor. When you restrict mathematics to the tenets of pure
formalism, everything must be so restricted.
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By the way, as an example of useful and meaningful formal system
I recall *contradictory* Cantorian set theory. (What if in ZFC or
even in PA a contradiction also will be found? This seems
would be a great discovery for the philosophy of mathematics!)
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I think this would be a disaster. It is bothersome enough that
``Cantorian'' set theory (I think you actually mean Fregean set theory.
Cantor's approach was decidedly NOT formalistic.) is considered to be
contradictory. Why should ``philosophers of mathematics'' be so biased?
Let's find out what is true about the consistency of ZFC, PA, etc., and look
for contradictions, but I suggest we not hope for one. It is entirely
reasonable to believe that a system such as PA or ZFC is consistent, even
though current formal logic systems cannot determine whether it is
consistent or not. Why not accept the fact that we may not ever know that
PA is consistent in the same way that we know the group axioms are
consistent, but to still search for an answer to the question? I guess what
I'm trying to say is that as research programmes, searching for a
contradiction in modern mathematics is a fine programme, as is the attempt
to find a formalization of the mathematics that has been done to this day
which is provably consistent. Not only that, success in either of these
programmes would constitute a worthwhile contribution to mathematics and
metamathematics, and the philosophy of mathematics. But an even better
contribution would be to find an appropriate formal logic system in which
Gödel's argument cannot be carried out, because the notion of ``proof'' in
the given system is different enough from our previous notions to allow one
to prove the consistency of PA. The philosophical duty would then be to
explain why the new system is, or should be, acceptable to Mathematicians.
I am wary that this type of programme is doomed to failure, if only because
the current philosophical climate leans toward formalism in the area of
foundations of mathematics.
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Matt Insall
http://www.rollanet.org/~montez
montez at rollanet.org
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