FOM: Feferman's first thesis

Kanovei kanovei at
Sat Jan 10 14:42:35 EST 1998

There have been some critical comments on the thesis

>Mathematics consists in reasoning about [some] objects 
>which exist only in our imagination. (Feferman.)

This thesis would perhaps need more accurate presentation
and also more thoroughful consideration. 

The following is by far not the results of my own thinking, 
but rather some recollections of 5 or 6 semesters of 
obligatory courses in philosophy which I passed at Moscow 
State in early 1970s as a student and PhD student in 

* The objects of mathematics exist in social conscience 
  (SC below -- not to be mixed with "computer science"). 

SC is understood as the "set" of all ideas 
-- in the most wide meaning of this, which includes natural 
numbers, Woodin cardinals, Sherlock Holmes, how to hunt 
lions using poisoned weapon, etc. etc. --  
once put into exchange in the society 
-- meaning that it has been written, recorded, said, 
even said or delivered in another accomplished manner 
by someone to himself. 

(It is clear that some ideas have a wide circulation in SC 
while some other a very narrow one. Some of them are 
stable during many generations, while some other, once 
generated, then disappeared, as (perhaps) instructions 
how to hunt mammoths. But this is self-evident, of course.)

More important is to understand that * is not just a sort 
of "social consensus" because the development of ideas 
in SC is not completely reducible to any sort of conscious 
activity of members of society, because the development of 
SC obeys certain laws (not like physical laws, but rather 
trends) independent from any sort of conscious activity. 

Vladimir Kanovei

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