FOM: Feferman's first thesis

[Vladimir Sazonov]

steel at math.berkeley.edu wrote:
> 
>    Some comments on Feferman's first thesis:
> 
> 1. Mathematics consists in reasoning about more or less clearly and
> coherently [conceived?] groups of objects which exist only in our
> imagination.
> 
>    If you drop "which exist only in our imagination", there isn't much to
> disagree with there. What is the meaning of "exist only in our
> imagination"? The only examples of meaningful uses of this phrase that
> occur to me are along the lines of "Sherlock Holmes exists only in our
> imagination". Is this the meaning Feferman intends? If so, how does
> such a "fictionalist" account of mathematics deal with its applicability?
> We don't go to 222 Baker street looking for Mr. Sherlock Holmes. We do
> use facts about real numbers to build bridges and send men to the moon.

The story about Mr. Sherlock Holmes is a good example. (Professor 
Feferman himself presented some analogous examples of fictions in one 
of his posting *in favour* of his point of view!) This story could help 
to other detectives in their activity by using the "logic of Sherlock 
Holmes" (cf.  the second theses of Feferman). Whether such a "fiction 
story" will be indeed interesting, useful or instructive depends on the 
author. 

The same holds for any mathematical theory, except it is based on 
(almost?) absolutely *formal* logic (something like powerful jet 
engine) and is therefore *potentially* much more applicable in serious 
sciences rather than in moral or everyday life.  The author of any 
mathematical "fiction" should just think about and try to anticipate 
possible applications (outside or inside mathematics) while creating a 
fiction.  I repeat myself: let *us* govern fictions or illusions (in 
Mathematics or elsewhere), but not conversely. 


Vladimir Sazonov
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