Preproceedings 2006 - Austrian Ludwig Wittgenstein Society

Wittgenstein, Waismann and Non-Euclidean Geometries
Martin Ohmacht, Klagenfurt and Vienna, Austria
234
“6.375 As there is only a logical necessity,
so there is only a logical impossibility”
(Wittgenstein, TLP)
1. Waismann´s importance for Wittgenstein
research
It is well known that Friedrich Waismann often met with
Ludwig Wittgenstein, and he regularly gave reports on his
discussions with the intellectual to other members of the
Vienna Circle. The idea of deepening this cooperation by
planning a book was not excessively far-fetched and,
indeed, this book exists today as part of Wittgenstein´s
heritage (McGuinness 1979). In this text, Wittgenstein
refers to the issue of non-Euclidean geometries several
times, and what I should like to do is to offer a summary
and an idea as to why this important chapter in the history
of mathematics was so significant for Wittgenstein´s
philosophy. It is the question of impossibility which
intrigues the philosopher. Waismann took the first notes for
this volume on December 18 th 1929 and the last on July 1 st
1932.
Ray Monk writes: “Schlick´s death finally put an end
to any idea … to co-operate together on a book” (1990,
page 358). The problem with this statement is the fact that
Waismann published a book on his own in 1936: therefore,
either he worked together with Wittgenstein and on his
own book at the same time, or the cooperation ended
before 1936 and not in this year. The first of the two
alternatives produces a problem of authenticity and
priority, because to work on two books simultaneously
necessarily results in an intermingling of thoughts.
The gap between this manuscript and Schlick´s
death was filled by a book manuscript which was found
among Waismann’s papers in 1959, and published as late
as 2003 by Gordon Baker.
Non-Euclidean geometries are mentioned on
several occasions in the text produced by Wittgenstein and
Waismann and published by McGuinness in 1979. Both
philosophers were well-informed on the subject, as this
issue represents an important chapter in the history of
mathematics. Before I analyse for what purpose
Wittgenstein uses it as an example (of the impossibilities),
I want to give a short sketch on both the historiography of
non-Euclidean geometry and its epistemological
importance within mathematics.
2. Direct and indirect proofs
Euclid wanted the axioms to be statements which were
clear and self-evident but, in the eyes of the readers of the
Elements, the axiom of parallels did not meet this
requirement. Indeed, Euclid´s version of this famous axiom
is not as clear as the version which was discussed later on
and as, for example, Hilbert’s formalization of geometry,
published in 1899. It was a British teacher, John Playfair,
who as late as the end of the 18 th century found out that
Euclid´s axiom of parallels could be considerably simplified
through using the following wording: ‘Given a straight line
and a point which does not lie on it, then there is exactly
one parallel to the straight line which passes through the
point’.
For almost 2000 years, mathematicians tried to
prove the axíom of parallels. There were 2 different
methods of doing so:
1) Again and again, researchers took as their starting
point statements which seemed trivial enough to be
true. But they turned out to be equivalent to the
axiom of parallels and so their proof was spoiled
through running into a “petitio principii”. One of
these statements asserts that the sum of the angles
in a triangle is equal to 2 right angles, a fact which
Wittgenstein mentions in his LFM in unit XXXI, page
289.
2) Researchers were able to avoid this trap by using
the technique of indirect proof. In this case they
started to explore a world of inferences which was
expected to collapse through inferring a
contradiction, and although the inferences achieved
were rather absurd and counter-intuitive to the
Euclidean world, they could not find a contradiction.
Marcel Guillaume writes in a book published by
Dieudonné (a member of the Bourbaki group) that,
for example, Sacceri developed a large part of what
was later on known as non-Euclidean geometry
without even being aware of the fact before 1733
(page 753).
Gauss, Lobatshevskij and Bolyai jun. had the
courage to argue that the proof of the axiom of parallels
probably could not be found: in later centuries, this
situation would be called the independence of a statement.
It is still readily understandable today that this insight must
have been a terrible shock to mathematicians, because
over the centuries a lot of energy had been invested in a
problem which now turned out to be unsolvable.
3. The proof of unprovability
The impossibility of establishing a proof for the
axiom of parallels was first suspected by Gauss in 1792
and the period during which mathematicians dealt with this
problem lasted until a publication by Poincaré in 1902 (see
Schreiber/Scriba 2005, page 428). It can thus be said that,
for mathematics, the 19 th century is the century of the
discovery of non-Euclidean geometries, while
Lobatshevskij is known as the Copernicus of Geometry. It
is important to realize that when Wittgenstein entered the
arena of philosophy, the question of this new realm of
mathematics was something which had on the one hand
already been settled, while being something completely
new on the other.
Now let us consider two different attitudes that were
held by mathematicians towards this issue at different
times.
1) The negative aspect: mathematicians had for a long
time invested a great deal of energy and work into
proving the axiom of parallels through the rest of
Euclid´s axioms, and had indeed completely failed.
There is a doctoral dissertation published in 1793 by
Klügel, who investigated 17 alleged proofs of the