Date: Thu, 21 May 1998 21:43:15 +0100
Subject: Re: Math/Metaphysics


In message <v01540b00b187b3f5d0ee-AT-[195.86.48.77]>, Jan
Straathof <janstr-AT-chan.nl> writes
>Jim & all,
>allow me to put a general question concerning the ontological
>status of mathematical objects (e.g the number Pi, Euler's constante
>or the Riemann-integraal):
>- do the 'objects of mathematics' exist independent of our knowlegde
>  of them ?
>or to put it differently:
>- how must the world be like for mathematics (as science) to be possible ?
>
>jan ?
>
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Hi Jan,

I think "to put it differently" is a bit infelicitous: these questions are not different
formulations of the same issue, are they?

The second question assumes that the world, in some sense, to some degree,
IS what makes math possible (like an early Wittgenstein question about what
makes Newtonian mechanics possible). But if you assume, e.g., that math
objects are merely mental constructs of some sort -- answering the first question
-- then you might answer the second question with "the world don't have to be
ANYWAY for math to be possible." So, I think the questions are different (By
the way Charles Chihara at Berkeley has a nice book on the topic; Stephen
Korner has one also, but its mostly of historical interest now).

Anyway, Jan, the first question depends on your philosophical position, doesn't
it? In the case of Godel, at the time of his investigations, 1930s, there were
three competing philosophies of math (psychologism had by then died under the
weightful criticisms of Husserl, Frege, and others (even Heidegger maybe):
Intuitionism (led by Arend Heyting, Brouer, et al), Formalism (led by David
Hilbert), and Logicism (Frege, Russell, various Logical Positivists).

Intuitionism is a kind of epistemological view about the (mental) constructability
of math objects and the meaning of the 'traditional' logical operators. E.g.,
Intuitionists argue that proofs requiring the law of excluded middle (A or not-A)
are not in general legitimate as they violate the requirement of constructability of
objects. It's hardly a silly view and makes for doing math a lot more difficult
(typical example: the indirect proof that the square-root of 2 is an irrational
number is unacceptable; I don't think Intutionists accept the Church-Turing
thesis either).

Formalism is the now defunct view that mathematics is simply a finitely
specifiable rule-governed symbol-manipulation practice, in which the symbols
have no meaning to speak of. I think we all have toyed with this idea once or
twice.

Logicism is the precursor to many living philosophies of math, and flew the view
that all mathematical objects and operations could be defined exhaustively using
purely (two-valued) logical concepts (I forget where but Russell somewhere
writes that the entire edifice of math could be described as one material
conditional, if p, then q!).

Well, when Godel entered the stage, all adherents to these various positions
were obsessed with the possibility of proving that mathematics is not only
consistent -- without contradictions (A and not-A) -- but also complete, that
every 'true' math statement is formally provable and that every formally provable
statement (theorem) is 'true' (the latter is easy to prove because the rules of
inference preserve truthhood). The foundations were spewing contradiction
threatening paradoxes, at least that is how many interpreted the paradoxes.

Godel demonstrated that in any of the axiomatic theories either boasting a
reduction of math, usually Peano Arithmetic, to certain purely Logical Axioms,
or boasting a purely formalistic treatment of math  not only can one
systematically construct the sentence within the theory which effectively states
that "This sentence is not formally provable," but one can also "recognize"  that
the sentence is true (if the sentence were false, the sentence would be formally
provable and the encompassing theory would then be inconsistent; thus, if the
theory is consistent, then the theory is NOT complete: there is a true sentence of
the theory which is not formally deducible from the theory's axioms).

Hilbert's Programme proposed we do the latter (a formalistic treatment);
Russell's and Whitehead's Principia Mathematica, whose system was the one
which Godel actually used in the proof of his theorem, proposed the former
(reduce math to Logic). With Godel's proof, both proposals were shown to be
logically impossible! 

Of course, it's no use adding the unprovable sentence to the body of Axioms;
this merely strengthens the theory and allows for more unprovable, but 'true'
sentences.

Godel's result had all kinds of repercussions. Of course, it put a great many
philosophical views to sleep. The method of proof showed how to "arithmetize"
a formal language, which fathered the hush-hush coding practices of  ...., (that's
hush-hush), and provided a very rigorous definition of recursion and recursive
definitions, launched new areas of research into provability and unprovability,
....

But philosophically speaking, the excitement seems that Godel demonstrated
that the concepts of "truth in math" and "formally provable" are NOT
coextensive; there are sentences which 'have' the former but lack the latter.
Some even think that the results demonstrate that no machine will ever
successfully emulate/simulate human cognitive abilities since there will always be
a sentence which the former cannot "accept"/"recognize," but which can be
accepted/recognized by a human (unfortuantely, the arguments for this view aint
any good).

It was the NON-coextensiveness lesson which seemed to demand from Godel
some account of math knowledge. Clearly, the knowledge that the Godel
sentence (as it has now been baptized) is 'true' is not secured by possession of a
formal proof of the sentence; rather, according to Godel, we are endowed with
a species of "mental perception" which allows us to perceive math reality and 'to
see' the truth or no of math statements (Heideggerian 'assertions' about math
reality, not only disclosing to us thus-and-so, but also that math reality is thus-
and-so? I have no idea, but I don't think so). What is seen is thus discovered, a
discovery about a realm that 'exists' independently of mind (I hear that there is
some discussion of this issue in the book by Professor Wang Hao and also in
Godel's recently published collected papers), a Platonic ontology of math states
of affairs, and their involved math objects.

The "epistemolgical" advantage of Formalism and Logicism is that both seemed
to be able to explain math knowledge without having to countenance any
"ontologically misbehaving" entities. With the programmes retired, the
advantages were nought. How else to explain math knowledge than that
proposed by Godel?  

Another question which Godel's result suggest is the following: why should we
try to axiomatize our math, over and above the purpose of "bookeeping" our
assumptions and presumptions when doing math?

Moreover, in what sense are math statements 'true', over and above the
semantic truth which could be confered upon them via Tarski, given some
interpretation of their formalistic/axiomatic teatment? But why the treatment
anyway?

Cheers,
jim


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