[FOM] Re: Shapiro on natural and formal languages
Vladimir Sazonov
V.Sazonov at csc.liv.ac.uk
Mon Dec 13 09:40:26 EST 2004
Jeffrey Ketland wrote:
in his discussion of mathematical intuition,
> Goedel commented on the "the abstract elements contained in our empirical
> ideas":
>
> In should be noted that mathematical intuition need not be
> conceived of as a faculty giving an immediate knowledge of
> the objects concerned. Rather, it seems that, as in the case
> of physical experience, we form our ideas also of those objects
> on the basis of something else which *is* immediately given.
> Only this something else here is *not*, or not primarily, the
> sensations. That something besides the sensations actually
> is immediately given follows (independently of mathematics)
> from the fact that even our ideas referring to physical objects
> contain constituents qualitatively different from sensations or
> mere combinations of sensations, e.g., the idea of object
> itself, whereas, on the other hand, by our thinking we cannot
> create any qualitatively new elements, but only reproduce
> and combine those that are given.
> Evidently, the "given" underlying mathematics is closely
> related to the abstract elements contained in our empirical
> ideas. It by no means follows, however, that the data of this
> second kind, because they cannot be associated with
> actions of certain things upon our sense organs, are
> something purely subjective, as Kant asserted. Rather
> they, too, may represent an aspect of objective reality,
> but, as opposed to the sensations, their presence in us
> may be due to another kind of relationship between
> ourselves and reality.
> ... the question of the objective existence of the objects
> of mathematical intuition ... is an exact replica of the
> question of the objective existence of the outer world.
> (Goedel 1964, Supplement to "What is Cantor's
> Continuum Problem?", Collected Works, Vol 3, p. 268.)
Sorry for the full citation of citation. This is for convenience of
the readers of this posting.
I reread this citation again and again and feel that the conclusion
"exact" is too strong and does not actually follow from the premises.
I would completely agree with "some analogy", but not with "exact
replica". May be I have lost some crucial nuances?
Why the most essential ingredient of mathematical thought - the
mathematical rigour based on formal rules of reasoning - is behind
the scene here if anywhere around at all? I understand that it is
because of this formal nature why existence of objects of mathematical
intuition is essentially different from existence of the outer world.
Let us consider, for example, the informal concept of a set introduced
by Cantor - an arbitrary collection of arbitrary abstract entities.
If taken only in its naive form, is it a mathematical concept at all?
Can we do anything rational with this idea in this form? Let us recall
that Cantor provided us not only with this naive concept, but, most
importantly, he also gave some examples of proofs, some templates
of reasoning on abstract sets, some rules how to "play" with sets,
which is actually some prototype for future formalizations of set
theory. It is because of these semiformal templates of reasoning
why the concept of a set became (or was successfully transformed to)
mathematical one. Paradoxes forced mathematicians to reconsider both
these rules and the initial intuition. This is normal stage of any
formalization of any pre-mathematical idea from first attempts to
sufficiently stable version.
It seems evident that formalization is somewhat depending on its
author. It is somewhat arbitrary - of course, not absolutely arbitrary.
Really, we can also feel that the process "converges" in a sense
and to some degree. But dependence on the author, on its internal
intentions still remains. And we really have a wide spectrum of
set theories with various version of the initial naive intuition.
Thus, it seems evident that the objects of mathematical intuition
cannot exists in isolation, completely independent on supporting them
formalisms, the latter being creatures of humans. From which general
principles or facts (if not from subjective beliefs) does it follow
so strong conclusion on the "exact" analogy with the objective
existence of the outer world? We do not need to create any formalisms
and even to philosophize to see its existence and existence of all the
objects around us. Of course, this is based on some complex processes
in our brains (not only sensations). But this is highly *unconscious*
process. We can only analize this process postfactum, like in the above
citation, but in the case of mathematics such a process is *under our
control* to some degree. Is not the difference more interesting,
important, crucial and deserving to be explored than just the analogy?
Vladimir Sazonov
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