Re: [HM] Mathematical proofs

Walter Felscher (walter.felscher@uni-tuebingen.de)
Wed, 1 Dec 1999 15:59:29 +0100 (MET)

While I am very skeptical about the possibilities to discover insights on
what is characteristic, or typical, for the ways in which mathematicians
find their proofs, I will add a most curious story about how I actually
dreamt a proof once.

That was many years ago, in 1956, and it never repeated itself again.
At that time I was studying certain relation algebras, just introduced by
the late Paul Lorenzen in an article "Theorie der B"unde" in the
Mathematische Zeitschrift. Their point was that they gave a simple
abstract approach to the Schreier refinement theorem (any two normal
series in a group have isomorphic common refinements) - a task not solved
by the previous extensive lattice-theoretical approaches. What I wanted
do was to extend Lorenzen's use of his "B"unde" such that also Oystein
Ore's extension to quasi-normal series could be covered (where normal
subgroups N [such that N.A=A.N for every subset A where . denotes
elementwise multiplication of complexes] would be replaced by quasi-
normal subgroups N [such that N.A=A.N only for every subgroup A]).

Lorenzen's proof, in the framework of "B"unde" used a series of lemmas,
one of which would not work for the case of quasi-normal subgroups instead
of normal subgroups. Sitting in my room at the desk in the evening, I
tried and tried how to discover a lemma which would do the job and which
I could prove. Finally, I got so exasperated that I went to bed. I fell
asleep, and there I dreamt (seeing it written on paper) how a proof of
such lemma, in four or five lines, would look. I awoke, jumped out of bed,
turned the light on, sat at the desk and jotted down the proof I had
dreamed. I was much too sleepy to check it, but immediately went back
to bed and slept soundly until morning. When I got up, awake, I returned
to the desk and checked the lines from the night. The proof was correct.

There should be a quotation saying that the Lord graces his people in
their sleep ("Der Herr gibt's den seinen im Schlaf" ). This time it had
worked. Unfortunately, it never worked again.

W.F.