[FOM] When do mathematicians identify isomorphic structures?

Arnold Neumaier Arnold.Neumaier at univie.ac.at
Sun Mar 27 05:18:27 EDT 2016

On 03/26/2016 08:22 PM, John Corcoran wrote in the thread
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
> structures.
> (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?

In my experience, mathematicians (including myself) identify
isomorphic objects when there is a good reason to do so
(for example, there is a unique totally ordered archimedian field
- so one can identify all copies as long as one works in a fixed
copy (as most of math does) but not while one investigates different
constructions (where the whole point is to show that these
constructions produce such a field) and while one proves the
uniqueness up to isomorphism.

Similarly, one identifies cyclic infinite groups with the integers
if one works with a fixed copy, but cannot do it when discussing
the structure theorem for elementary abelian groups where lots
of different such cyclic groups appear.

Thus it is a matter of context what precisely will be identified.

The great advantage of traditional algebra together with the
informality of traditional rigorous mathematics is that one can
do all this naturally - without having to worry at all about
foundations. Foundations are used once for setting up the context
in which to do one's particular kind of mathematics, which can be
done in different ways. Then they are forgotten. As long as the
final platform reached has identical capabilities mathematicians
generally are completely oblivious of the foundations.

It is like in copmputer science - as long as a programming language
has the notion of an associative array, programmers use it in the
same way across all languages, not caring how a particular compiler
translaters it into hardware. The foundations of math are like
the hardware - it must assure that the implementation is correct
hand has all the properties needed for using the concept.
Apart from that, the programmer should never need to know how it
is implemented in detail.

Part of the lack of acceptance of theorem provers in the mathematical
community at large (hardly anybody uses them, perhaps 0.01% of all
mathematicians?) is the distance from standard mathematical practice.

If a programming system were required to understand assembler
and to translate part of their tasks into assembler to help the
compiler understand what they are doing - very few would embark
on using such as system today.

Theorem provers will find large-scale use and acceptance by
mathematicians only if the interfaces speak their (already
universal) language and hide as much as possible of the
underlying foundations from the user. For foundations should
be just that: Invisible structures deep in the earth, which can
support the skyscrapers built upon them.that carry the life
for which they are built.

Arnold Neumaier

More information about the FOM mailing list