[FOM] What is a proof?

Michael Thayer mthayer at ts-software.com
Mon Oct 27 10:26:36 EST 2003

At 09:49 PM 10/26/2003 -0500, Stephen G Simpson wrote:
>John T. Baldwin 26 Oct 2003 writes:
> > Position A.  A proof is a means of establishing the truth of a
> > proposition.
> > 
> > Position B.  A proof is an explanation of why a proposition is
> > true.
> > 
>Of course, when you speak of *explanation*, it may be that you are
>yearning for something other than "reciprocation of premisses and
>conclusions".  If so, can you be more specific?

While this example may not be what Baldwin had in mind, it certainly seems to be something different, although annoyingly vague.  Back in 62, I had a long talk with Feit about a group theory result one of his grad students had proven.  While the proof was clear enough to follow without use of paper or blackboard, both Feit and the student were dissatisfied with it because it "didn't explain anything".  The problem was that the proof used some simple combinatorics on a graph that one could make of certain subsets, and was not "group theoretical enough" to be very "helpful".  I believe that this sort of feeling is less based on proof structures and logical relationships among premisses, than on the language in which it is couched.  Harvey touched on this issue of "good" vs "bad" definitions and how to characterise them some years back on FOM, but I can't find a reference (or I may be mistaken as to the writer).  It may be that this in fact falls under 

Proposition C. A proof is a means of making the reader FEEL that he/she UNDERSTANDS why something is true.

Of course, if that is what is going on here, then it may not be f.o.m, but psychology of mathematics. 

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