[FOM] Clarity in fom and problem solving

[Timothy Y. Chow]

Harvey Friedman <friedman at math.ohio-state.edu> wrote:
> I would like to see the philosophers (and others) comment on the FOM
> concerning these arguments.
[...]
> "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.

If Goedel is talking about problems that we expect to solve using 
logically weak and uncontroversial assumptions, then his own parenthetical 
comments undermine his argument.  The limiting factors for actually 
solving a practical problem are invariably practical ones, regardless of 
whether the problem is one of our own creation or not, and regardless of 
whether the objects are clearly apprehended or not.  The example of chess 
makes this clear: it is obvious that no matter how clear you are about the 
rules, that alone won't make you a master player.

If Goedel is talking specifically about incompleteness phenomena, e.g., 
that because the known results on the continuum hypothesis have not 
"settled" it, this means it's not our free creation, then I'm not sure I 
follow his argument.  I doubt that this is what he is arguing, though.

> 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.

I consider this to be one of the better arguments for realism in 
mathematics.

> 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!"

A realist might feel that this state of affairs suggests realism, but a 
fictionalist might feel that this state of affairs suggests fictionalism.  
It is not at all strange that to determine how Anakin Skywalker originally 
turned to the Dark Side, George Lucas had to write a prequel (or 
prequels).  And I suspect that if George Lucas had never written or even 
conceived of the prequels, few people would claim that the question of how 
Anakin turned to the Dark Side would nevertheless have a well-defined (but 
unknown) answer.

Tim