FOM: Hilbert's 24th problem

[teun@cs.vu.nl]

FOM subscribers may be interested in knowing that the German historian
Ruediger Thiele (Leipzig) has found an interesting text in Hilberts
handwritten notes in the Library in Goettingen. The handwriting, which
is here and there difficult to read, clearly shows that Hilbert
considered the possibility to add a 24th problem to his famous list of
23 problems. The problem is a FOM problem concerning the simplicity of
proofs. 

This is my somewhat free translation of the first part of the text (the
complete German original follows at the end of this posting):

"As 24th problem in my lecture in Paris I wanted to pose the question:
criteria for the simplicity or give a proof of the greatest simplicty of
certain proofs. In general develop a theory of methods of proof in
mathematics. After all under given conditions there can be only one
simplest proof. In general when one has two proofs for one theorem, one
should not rest before one has reduced the proofs to each other or
understood exactly which different suppositions (and aids)are used in
the proofs: If one has two roads, one should not only go these roads or
look for new ones, but instead investigate the whole area between the
two roads." 

In the last part of the text Hilbert refers to his work in invariant
theory where he allegedly made some first steps on the way to determine
the simplicity of a proof. It is not clear - at least not to me - what
he is precisely referring to
In the last sentence of the text there is a reference to geometry and
the possibility to determine the simplicity of a proof by counting the
number of steps in a proof. 

Clearly there are many interesting questions to be asked. 
1. Why did Hilbert not include this 24th problem.
2. What does this finding add to our knowledge of Hilbert's development
as for the foundations of mathematics?
3. To what extent is the problem solved?
4. Waht does the vague reference to his work in invariant theory (his
work on 'syzygies') precisely mean? 
Etc.

The text of a lecture by Ruediger Thiele on this matter will be
published in Michael Kinyon (Ed.), History of Mathematics at the Dawn of
a New Millenium, Proceedings of a Conference of the Canadian Society for
History and Philosophy of Mathematics, McMaster University, Hamilton,
Ont. 2000. 
I don't know when this book will appear. Moreover, I feel the existence
of a 24th problem deserves to be known more widely than it is now.
Thiele gave me permission for this posting. "I am tired now, so I agree
with everything", he mailed me. I won't ask twice.

Teun Koetsier

      
The German original:
"Als 24stes Problem in meinem Pariser Vortrag wollte ich die Frage
stellen: Kriterien für die Einfachheit bez. Beweis der groessten
Einfachheit von gewissen Beweisen fuehren. Ueberhaupt eine Theorie der
Beweismethoden in der Mathematik entwickeln. Es kann doch bei gegebenen
Voraussetzungen nur einen einfachsten Beweis geben. Ueberhaupt wenn man
für einen Satz 2 Beweise hat, so muss man nicht eher ruhen, als man die
beide aufeinander zurueckgeführt oder genau erkannt hat welche
verschiedenen Voraussetzungen (und Huelfsmittel) bei den Beweisen
benutzt werden: Wenn man 2 Wege hat, so muss man nicht bloss diese Wege
gehen oder neue suchen, sondern dann das ganze zwischen den beiden Wegen
liegende Gebiet erforschen. Ansaetze, die Einfachheit der Beweise zu
beurteilen, bieten meine Untersuchungen ueber Syzygien und Syzysien
zwischen Syzygien. Die Benutzung oder Kentniss einer Syzygie vereinfacht
den Beweis, dass eine gewisse Identitaet richtig ist, erheblich. Da
jeder Process des Addierens Anwendung des cummutativen Gesetzes der
Addition ist. - [Da] dies immer geometrischen Saetzen oder logischen
Schluessen entspricht, so kann man diese zaehlen und z.B. beim Beweis
bestimmter Saetze in der Elementargeometrie (Pythagoras, oder ueber
merkwuerdige Punkte im Dreieck), sehr wohl entscheiden,
welches der einfachste Beweis ist."