[FOM] Clarity in fom and problem solving
friedman at math.ohio-state.edu
Wed Apr 5 02:40:54 EDT 2006
On 4/3/06 3:55 AM, "Zachary J. Purvis" <zackpurvis at comcast.net> wrote:
In his Gibbs lecture, Goedel remarks that clarity in the foundations of
mathematics has not helped much in deciding mathematical problems. This
would be impossible if mathematics were our creation, because then our
difficulties would be due only to a failure to understand what we have
created; with perfect clarity our mathematical ignorance would disappear.
I am not quite sure what to think about his first statement, that clarity
in foundations has not helped with solving problems. Are there any examples
or suggests at what Goedel is getting at here?
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.
I would like to see the philosophers (and others) comment on the FOM
concerning these arguments.
With regard to your specific question, I think Godel was simply referring to
the fact that work on the foundations of mathematics particularly up to
the time this was written has not helped mathematicians prove theorems. This
has changed somewhat since then. But it is true to this day, in the sense
that if you tell, e.g., number theorists and experts in PDEs all about the
axioms of ZFC and the subtleties of predicate calculus, and, say, Gentzen¹s
work on PA, then they will probably not prove more number theory and PDE
theorems as a result. Godel¹s idea, put crudely, is that
if math were merely a game we created like chess, then being fully
knowledgeable about its foundations would make you a much better mathemat-
ician than otherwise, just like being fully knowledgeable about the rules of
chess makes you a much better chess player than otherwise.
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