FOM: poll

Kanovei kanovei at
Tue Feb 3 10:13:56 EST 1998

>From: kanovei at (Kanovei)
>4) what is said above 
[on the difference between groups and their isomorphism classes]
is so elementary that any attempt 
>to play this into a foundational system is ridiculous. 

>Date: Mon, 02 Feb 1998 10:55:04 -0800
>From: Vaughan Pratt <pratt at>
>Are you saying that the choice or design of one's foundations should be
>based on technically deeper issues, or that the situation is so elementary
>as to admit of only one possible interpretation?

Dear Vaughan,

well, first, my view of foundations is based not 
on the technical complexity, at least primarily, 
but rather on the fact how fully and naturally a  
foundational scheme represents the existing ways of 
mathematical reasoning. Here "naturally" means that 
the scheme must, in particular, highlight some 
minimal amount of issues as primary and independent 
and show how the rest follows. 

But this is not directly relevant to the point. 

The situation (with the number of groups of order 4) 
seems to me really elementary as long as practical 
work with groups is concerned. That is, if you 
(who think that there are two of them) tell me a 
theorem about them, I immediately understand that this 
is about isomorphism classes of groups. That you call 
the latter *groups* is not important for me. 

This is why I wrote "elementary". Now about "ridiculous". 

The real issue comes when (and if) you pretend to 
ground your standpoint foundationally. In this case, to 
look non-ridiculous (for me, a set theorist, that is, 
a sort of professional in foundations), 
you should step forward with a 
"doctrine" of some generality, which encounters all 
mathematical issues in the statement: 

* there are exactly two groups of order 4

in particular what is a group, what is 4, 
what is a group of order 4, what does it mean that 
there are only two of them -- which presumably would 
amount to a definition of equality. 

And this "doctrine" must be different from that one 
(on the statement *) which was presented by myself.

Are you going to do this ? 

Vladimir Kanovei

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