Preproceedings 2006 - Austrian Ludwig Wittgenstein Society

axiom of parallels and showed that they were all
invalid. What a disgrace for the mathematical
community! The problem turned out to be
unsolvable - so geometricians had for many
centuries sought something which, in the end,
turned out to be impossible! Gauss treated this
question as a secret research project, because he
was afraid of the “Geschrei der Böotier!” (“The
uproar of the Boeothians”, cf. Oskar Becker 1975,
page 178).
2) The positive aspect: when a problem has been
definitively established as being unsolvable,
researchers are free not to deal with it any more,
and are therefore able to calm down and start to
work on other questions. The idea of the
independence of a statement had not yet been
invented, and the new geometry was called Meta-
Geometry.
I should like to move now from mathematics to
psychology, because I want to use modern psychology as
a framework of reference for examining the history of
mathematics.
Gregory Bateson and his followers created the
concept of “punctuation” and “re-interpretation”, which is
exactly what happens here in the course of the history of
mathematics (cf. Simon/Clement/Stierlin 1999, page 151).
On the one hand, a catastrophe occurred because an
impossibility had been found. On the other hand, however,
it was possible to achieve the clarification of a longinvestigated
question through a proof. True, it was a
negative proof - but a proof nonetheless.
It is not at all astonishing that it took mathematicians
just over a century to adopt the new, positive view: their
attitude changed from a sense of having utterly failed and
of having suffered a catastrophe towards the relief of
having negatively but definitively clarified an important
issue.
Wittgenstein must have been fascinated by this shift
in the attitude of mathematicians towards this new subject,
which incorporated an impossibility at the philosophical
level.
4. Wittgenstein on the axiom of parallels
and the trisection of the angle
Now let us look at Wittgenstein´s remarks on non-
Euclidean geometry in his talks with Waismann as
presented by McGuinness in 1979! What happened in the
dialogue between these two intellectuals when they met at
Schlick´s house on 1 st January in 1931? They discussed
consistency, and this was not by far the first occasion on
which they talked about this topic: indeed, the question of
consistency was a central focus of their talks. In current
discussions, the connection between geometry and
consistency is drawn based on the fact that a relative proof
of consistency of the two variants of non-Euclidean
geometries is established through reducing the question of
the consistency of non-Euclidean geometries to the
question of the consistency of Euclidean geometry.
Waismann introduces this topic:
“If in the one case [the non-Euclidean geometries],
the theorems included a contradiction, then the
contradiction would have to reveal itself in the other
geometry too” (Brian McGuinness 1979, page 144).
Wittgenstein´s reaction to Waismann is a sharp rebuke:
“Consistency ´relative to Euclidean geometry´ is complete
Wittgenstein, Waismann and Non-Euclidean Geometries - Martin Ohmacht
nonsense” (op. cit. page 145). This is one of the occasions
where Wittgenstein definitely humiliates Waismann and
Waismann is humble enough not to react aggressively to
Wittgenstein. The choice of these two elements in the
dialogue would at first lead one to infer that Wittgenstein is
not inclined to accept an element of leadership in the
dialogue from Waismann’s side - but this impression is not
totally correct, because when discussing Fermat´s Last
Theorem (op. cit. page 144) in this text on consistency,
Wittgenstein accepts a suggestion from Waismann
concerning an issue of discussion, which had been put
forward by Waismann beforehand (op. cit. age 143).
Waismann seems to have interrupted Wittgenstein´s flow
of thought by suggesting Fermat´s Last Theorem as a
topic because, before turning to this issue, Wittgenstein
mentions the trisection of the angle. What is the reason for
mentioning this rather simple piece of mathematics? In my
interpretation of Wittgenstein´s philosophy of mathematics,
the reason for choosing this topic is that the trisection of
the angle is another impossible thing to do, as Pierre
Wantzel proved in 1837 following precursory work by
Gauss (see Stillwell 2005, page 55). Although the axiom of
parallels is a proof that is impossible to establish, while the
trisection of the angle is a geometric construction which
cannot be carried out, Wittgenstein tells us here that
geometry is not only incomplete because the axiom of
parallels is independent (next heading line op. cit. page
145), but also because the long-sought trisection of the
angle cannot be found (although we can discuss it).
5. Euclidean geometry and consistency
When we examine these parts of the records of the
discussions between Wittgenstein and Waismann, then it
becomes clear that geometry is being discussed not for its
own sake, but in order to clarify the question of
consistency. Around 1850, a new term emerges in
mathematical discussions: the term “Euclidean geometry”.
This term only makes sense in relation to the term “non-
Euclidean geometry”, because without this comparison the
term “Euclidean geometry” is a pleonasm. Until the middle
of the 19 th century, before non-Euclidean geometry was
discovered, all geometry represented Euclidean geometry.
Wittgenstein says “The rules of Euclidean geometry
do not contradict one another” (op. cit. page 195).
Of course, Wittgenstein´s statement that the rules (=
axioms) of Euclidean geometry do not contradict each
other is not at all a provable statement, but it is part of a
hermeneutic approach to what mathematicians believe to
be true. Wittgenstein’s overall goal when discussing
consistency is the establishment of a method in
preparation for the appearance of a contradiction in
mathematics. This advance preparation for the worst case
makes it necessary to discuss what is most feared,
whereby a prerequisite for this “talking cure” is a sort of
inner freedom not to be so much afraid of contradiction
that one shrinks from talking about it. “What are we to do?”
(op. cit. page 196) is Wittgenstein´s motto. Let us prepare
a method before a contradiction pops up.
6. Conclusion: the close connection
between the issue of geometries and the
issue of consistency
When we examine the records of the discussions between
Wittgenstein and Waismann in overview, it becomes quite
clear that the issue of consistency is of the utmost
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