The meaing of mathematics in Against the Day.....

[Ray Easton]

Kai Frederik Lorentzen wrote:
>> The phrase "foundational crisis" in reference to mathematics does have a
>> pretty specific meaning, and the "foundational crisis" has nothing to do
>> with any "applications" of mathematics, including physics. It is
>> concerned with the purely theoretical, or philosophical -- specifically
>> with the nature of mathematical truth.
>>     
>
> Since David Hilbert is not only mentioned one/three/couple times yet does also
> appear in (at least) one of the Göttingen passages, I ask you:
>  
> Can't the famous 23 problems Hilbert presented at the Paris math congress in 
> 1900 be understood as some kinda manifestation of a "foundational crisis" in
> the field of mathematics?
>
>   


The 23 problems are simply a statement of the most important unresolved 
questions (in Hilbert's view) in mathematics at the time.  But it is 
certainly true that some of these problems are themselves a 
manifestation of the "foundational crisis."  The problem 'prove that the 
axioms of arithmetic are consistent,' for instance, only makes sense 
once one has discovered/decided that arithmetic needs axiomatization, a 
notion unthinkable before the crisis.  Hilbert's formalism is a notion 
of mathematical truth very unlike those that came before it, is both a 
reaction to and a manifestation of the "foundational crisis."  Before 
the time of Hilbert, mathematicians who bothered to think about the 
meaning of what they were doing were all more or less naive Platonists.

It is precisely the appearance  of Hilbert in AtD that leaves me a 
little puzzled over the absence of any serious discussion of the 
"foundational crisis."

Ray