[FOM] Clarity in fom and problem solving
Vladimir Sazonov
V.Sazonov at csc.liv.ac.uk
Wed Apr 5 18:42:27 EDT 2006
Quoting Harvey Friedman <friedman at math.ohio-state.edu> Wed, 05 Apr 2006:
> *************************
>
> Godel says there: (page 314 of Godel¹s Collected Works, vol. III)
>
> "First of all, if mathematics were our free creation, ignorance as to the
> objects we created, it is true, might still occur, but only through a lack
> of a clear realization as to what we really have created (or perhaps, due to
> the practical difficulty of too complicated computations). Therefore it
> would have to disappear (at least in principle, although perhaps not in
> practice) as soon as we attain perfect clearness. However, modern
> developments in the foundations of mathematics have accomplished an
> insurmountable degree of exactness, but this has helped practically nothing
> for the solution of mathematical problems.
>
> Secondly, the activity of the mathematician shows very little of the free
> dom a creator should enjoy. Even if, for example, the axioms about integers
> were a free invention, still it must be admitted that the mathematicians,
> after he has imagined the first few properties of his objects, is at an end
> with his creative ability, and he is not in a position also to create the
> validity of the theorems at his will. If anything like creation exists at
> all in mathematics, then what any theorem does is exactly to restrict the
> freedom of creation. That, however, which restricts it must evidently exist
> independently of the creation.
>
> Thirdly, if mathematical objects are our creations, then evidently integers
> and sets of integers will have to be two different creations, the first of
> which does not necessitate the second. However, in order to prove certain
> propositions about integers, the concept of set of integers is necessary.
> So here, in order to find out what properties we have given to certain
> objects of our imagination, we must first create certain other objects a
> very strange situation indeed!"
>
> ************************
>
> Obviously, these are very imaginative and thought provoking arguments by
> Godel in favor of realism or Platonism in mathematics.
Should considerations of the above style necessarily lead to Platonism?
I think that, like in engineering, mathematicians (or just our
ancestors when invented numbers) are really creators of mathematical
concepts via formal systems, axioms, definitions, algorithms etc., and,
again like in engineering, these creations are not absolutely free.
However, they are *potentially* free. There is no restriction to
create possibly useless/meaningless formalisms, incorrect proofs, etc.
like useless (but may be amazing) engineering devices. I see
mathematics in general as the engineering of formal tools (formalisms)
strengthening our abstract thought, and it is this what imposes the
restiction under discussion. This can be also compared with software
engineering which however is devoted to mechanising the routine part of
our intellectual activity.
In addition (recalling recent discussions on predicativity), I
absolutely disagree to call as Platonism my ability to have imagination
of "the" universe of sets satisfying ZFC or even only "the" powerset of
N. Platonism rather assumes a kind of absolute existence of such a
universe what is absolutely unnecessary to my imagination. (I hope it
is cleaar why I use "the" in quatation marks.)
Vladimir Sazonov
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