[FOM] 1. Re: foundations meeting/FOMUS/discussion (martdowd at aol.com)

martdowd at aol.com martdowd at aol.com
Sat Mar 26 22:54:02 EDT 2016

John Corcoran (below) suggests that I am objecting to an innocuous remark of Bruno Bentzen.  Bruno's exact remarks are:

 "It turns out that in practice mathematicians often identify two structures whenever they are isomorphic. This is clearly in conflict with set-theoretic foundational theories such as ZFC,"

I would object to this in that nothing, even rank on rank, is known which is in conflict with ZFC.  Athough I would have to do some reviewing, I believe that HOTT is not in conflict with ZFC.  My comment was meant simply to remark that the identifcation of isomorphic structures in ordinary mathemetics is a matter of practical, rather than fundamental, significance.  Again, I would have to review my understanding of HOTT to comment on its position on this issue.

In ZFC two structures ay be identified by taking the quotient of a collection of structures by a congruence relation.  The questions of when this is done in mathematical practice, and how this relates to HOTT, are certainly of interest.  I suspect there are relevant topics in category theory.

- Martin Dowd



-----Original Message-----
From: John Corcoran <corcoran at buffalo.edu>
To: fom <fom at cs.nyu.edu>
Sent: Sat, Mar 26, 2016 1:34 pm
Subject: [FOM] 1. Re: foundations meeting/FOMUS/discussion (martdowd at aol.com)

Bruno Bentzen writes: It turns out that in practice mathematicians often
identify two structures whenever they are isomorphic.
Martin Dowd replied: This is only partly true.  Dedekind's and Cantor's
construction of the real numbers yield isomorphic structures (although the
isomorphism itself is a mathematically interesting object; also, the
definition of isomorphism is set-theoretic).  On the other hand, the
definition of a Galois group involves distinguishing isomorphic but unequal
(1) Dowd gives the impression that he is objecting to Bentzen but his reply
is a non-sequitur: nothing he said suggests any correction to Bentzen's
innocuous truism. Am I missing something?
(2) Does everyone agree on what it means to identify two structures?
(3) Why should anyone identify two structures? What is achieved?

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