ONT Re: Mathematics As Formalism -- Glass Beads & Jewels Of Denial
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John Collier wrote (JC):
Jon Awbrey wrote (JA):
JA, casting Runes:
| Formalism (Mathemetical) is a name which has been given to any one
| of various accounts of the foundations of mathematics which emphasize
| the formal aspects of mathematics as against content or meaning, or which,
| in whole or in part, deny content to mathematical formulas. The name is often
| applied, in particular, to the doctrines of Hilbert, although Hilbert himself
| calls his method axiomatic, and gives to his syntactical or metamathematical
| investigations the name Beweistheorie ('proof theory', q.v.).
|
| Alonzo Church, in Runes, DOP, page 111.
| Dagobert D. Runes, 'Dictionary of Philosophy',
| Littlefield, Adams, & Co., Totowa, NJ, 1972.
JC: My point is that these are always abstractions from more concrete cases, and
always occur in context. "Just follow the rules" is not possible in general
unless the rules are embedded within some system whose context specifies what
the rules are. A "meaningless formal calculus" is possible, but that is not
propositional logic or any other formal system with which I have worked.
I only work with formal systems that have a semantics. It seems to me
rather pointless, to say the least, to engage in meaningless formal
calculus. I sincerely doubt that anyone has proposed either as
useful in anyway, but I am willing to look at examples.
JA: Vide Supra.
JC: Hibert's formalism fails as a foundation for mathematics
just because it can't be given a semantics completely.
This is not true of propositional calculus, and would
not be true of mathematics if Hilbert's program had
been successful, so I maintain that you have not
given me an example. The very fact that Hilbert's
program had clear criteria for failure shows that
it was not empty formalism. Chaitin remarks on
some interesting work that pushes the limits of
formalism into new and surprising areas, but
again there is a clear semantics, and these
cases do not provide counterexamples either.
JA: I think that you must have been somewhere over the Pacific when this discussion
started. You are repeating, more prosaically, all of the things that I have
already said, if a bit more rhapsodically, about what a Big Lie this brand
of self-styled "Formalism" is as a philosophy of practicing mathematics.
Then you seemed to deny that anybody had ever proposed this philosophy,
and so I gave you an example to remind you that some people really had.
Any work that pushes this Formal Ism to the brink of having semantics
is work that has converted this Formal Ism into its Formal Apostasy.
JC: I didn't think I needed to give the conclusion.
It is that there are large chunks of mathematics
and logic for which formalism works perfectly well.
For complete systems, formalism works perfectly well,
and there are plenty of complete systems. Exactly which
parts it can work for is a difficult question that requires
investigation. There is no Big Lie, and there never has been.
You still haven't given me an example of anyone who has proposed
an "empty" formalism.
I now suspect you of trying to run a pun
between the capitalized and decapitalized
uses of the words "Formalism"/"formalism".
Formalism, as "any one of various accounts of the foundations of mathematics which
emphasize the formal aspects of mathematics as against content or meaning, or which,
in whole or in part, deny content to mathematical formulas" is something that fails
as a philosophy of doing mathematics, that is, as a normative guide for communities
of practice that are dedicated to a life of inquiry that is focussed on mathematics.
Formalism, in a sense that would be spelled "formalism" if it did not incidentally
begin a sentence, is useful, as I have wasted many words here trying to emphasize,
though not "against" the material or psychological aspects of the subject, which
I think would be a losing battle on account of being based on a false dichotomy.
But a form of Formalism which says that formalism is enough, whether empty or not,
is just plain wrong, and if you have not seen examples of people saying otherwise,
then I cannot think that you have been paying attention to what people have said,
and done.
JC: I repeat: If Hilbert's formalism were empty, it could not have been
refuted by Goedel's incompleteness theorem, which relies centrally
on the undeniably semantic concepts of validity and consistency.
I know of no mathematicians or logicians who deny the cogency
of Goedel's theorem, which means that implicitly they accept
that Hilbert's program was not empty formalism, whatever
that is. Now there is nothing in itself wrong with
embracing an inconsistency in the investigation of
something (Bohr is one of the clearest examples of
someone who has done this to great advantage), you
should at least be aware that you are doing so.
Holding that Hilbert's program is empty, together
with accepting that Goedel's theorem is sound is
inconsistent.
I have never denied the significance of "Gödel's Incompleteness Theorem" (GIT).
I have merely pointed out that its significance is occasioned by, relative to,
and specific to the species of error against which it formulated an effective
remedy, initially just the 'Principia Mathematica', but later extended to any
antigen of that ilk. But the need for this counter-measure was occasioned by
the emergence of a novel mutation in the "human intellectual code" (HIC), one
that rendered the population susceptible to a formerly unencountered dis-ease,
one to which the platonic realists of all former generations of mathematicians
had always been immune. It was only because Gödel maintained his immunity that
he was able to persevere in finding the cure, otherwise he would have most likely
fallen victim to it, like so many others of his time. We ought to note that there
is very often a big difference between the way that a person knows something and the
way that a person proves something. How did Gödel know to go looking for this proof?
Because, as is well known, he was a platonic realist, and so he just knew that there
had to be something wrong somewhere with what the Formaldehydists were saying about
the Corpus Delecti of Mathematics. The proof itself required an antibody specific
to the antigen in question, but the reason for it is a much more genetic truth.
JC: I propose that we drop this for a while. You will run into difficulty
over the issue eventually. I am morally certain of that because of my
own past errors in this respect.
Said & Done.
Jon Awbrey
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