FOM: Heidegger and FOM

[Olivier Souan]

Since Heidegger was sometimes quoted in the thread, I'd like to make
some remarks concerning his relation to FOM.

1) Thinking of the negative 
"Das Nichts nichtet" sounds quite silly at first, but it should be
remained that it is deeply rooted in the speculative tradition of
Western philosophy : Parmenides can be cited ("Being is, non-being is
not", which is at first sight uttely ridiculous), like Spinoza ("omnis
determinatio is negatio") as well as Hegel (Being and non-being are the
same), however different are their answers and their ultimate meanings.
There may be also a link with negative theology.
That style of thinking underlines the fact that negation has a creative
power, and that by using negation it is possible to talk meaningfully
about the world.  
Now what could be the link with FOM? Very tenuous indeed. If we consider
negation from a logical and mathematical viewpoint, let us remind that,
for instance, the Sheffer stoke offers a complete set of connectors for
propositional logic (Wittgenstein made a philosophical use of this
feature in in Tractatus), and that some big issues of logic lies with
the adequate interpretation of negation and its properties (the status
of the reductio ad absurdum for instance). There are also the chu spaces
which make a constant use of negation, and the complementary group of a
knot in knot theory helps to define this very knot. (Vaughan Pratt wrote
an article about "the power of negative thinking in multiplying boolean
matrices" but I couldn't find it).

2)It remains clear that Heidegger has shown very little interest to FOM
as such.
a) because he thought that he could bypass such explanations in his
quest for Being. b) because his style of thinking originated in the
application of the epistemological side of phenomenology to a
philosophical anthropology. Heidegger can be opposed as a
phenomenologist to Frege's foundationalist attempts and his refusal of
the priority of epistemology (in the sense of theory of knowledge) over
philosophy (Michael Dummet remarked this point about Frege in a short
introduction to his philosophy : Truth and Other Enigmas, p.89). He can
be also opposed as an anthropologist not only to Husserl himself but
also to a phenomenologist like Oskar Becker (Mathematische Existenz,
1927), who showed a deep interest in mathematics, or to an historian
interested in phenomenology like Jacob Klein and his beautiful "Greek
MathematicalThought and the Origin of Algebra", Dover, 1968). The point
is that the study of logic in the husserlian sense included epistemology
and psychology while current research in FOM, as it seems to me,
dismissed them as superfluous, in the same way Frege did.

3)HOWEVER, it should be mentioned that Heidegger knew very well
Aristotle and studied it extensively during the 20s. More precisely, I
have found an occurrence of an interest for the aristotelian philosophy
of mathematics in the following passage :

i) in "Platon:Sophistes. Marburger Vorlesung Wintersemester 1924/25",
Gesamtausgabe Band 19,
§15 : Exkurs :Allgemeine Orientierung über das Wesen der Mathematik
gemässAristoteles, SS.100-121, Vittorio Kostermann, 1992. It is much
talkedabout Aristotle in this magnificent book, as a preparation for the
study of Plato's Sophist. Context : looking for the best access to
truth(aletheia), Aristotle examines the four ways of accessing truth,
the best one being "sophia" (wisdom, scientific knowledge). The main
features of sophia are (Metaphysics I,2):
1-a general knowlegde (epistasthai panta, 982a8) 	
2-difficulty; for its objects are not obvious. (ta khalepa gnonai, a10)	
3-rigor, scientificity, "intellectuality"(akribesteron,
didaskalikoteron, a14)	
4-has itself as its own goal (autes heneken, a14) 

Let us study 3. The perfect illustration or rigor and scientificity is
of course mathematics. Heidegger explains the notion of abstraction
inthe aristotelian philosophy of mathematics, the differences between
arithmetics and geometry for Aristotle (a succession of isolated
unitsfor the first, a continuity of points for the second), and
theirrespective importance for the aristotelian ontology : the
ontologicalimportance of arithmetics lies in the fact that every being
isstructured as o n e being, as a unit (monas). It belongs to the very
essence of each being to be o n e. The best access to this unity of
eachbeing is language, logic (logos, noesis). The importance of geometry
isrelated to the structure of its elements. A line can never be obtained
by adding isolated points,an object can never be constitued from mere
surfaces (Physics VI,1, 231a24). The structure of geometrical objects
isdifferent from that of arithmetical ones. The only way of accessing
geometrical objects is not logic, but "aisthesis".  
Heidegger concludes : "Die Frage des continuum ist in der heutigen
Mathematik wieder aufgerollt. Man kommt auf Aristotelische
Gedankenzurück, sofern man verstehen lernt, dass das continuum nicht
analytische auflösbar ist, sonder dass man dahin kommen muss, es als
etwas Vorgegebenes zu verstehen, vor der Frage nach einer analytischen
Durchdringung"(S. 117)
Heidegger then insists on the importance of the non-analyticity of the
continuum by quoting Hermann Weyl (Raum-Zeit-Materie, 1918), the general
theory of relativity, the concept of field in modern physics ("das
physische Sein ist bestimmt durch das Feld"), and by underlining the
richness of the aristotelian analysis of movement.
   However, for Aristotle, the number is ontologically prior to
geometrical objects, since they have to be situated in time and
space,accessible to perception. But I think that for Heidegger himself,
the continuum is prior, because it is not located in time and space, and
because the according perception is a very special one, which is prior
to any kind of possible sense perception (It is very near to what is
called "Seinsverständnis" in Sein und Zeit).He often seems to understand
Aristotle on that basis.

ii) in "Die Grundbegriffe der antiken Philosophie. Marburger Vorlesung
Sommersemester 1926", Gesamtausgabe Band 22, §23 : "Zeno von Elea",
SS.70-76 und SS. 237-240, Vittorio Kostermann, 1993.
The long debated Zeno paradoxes have arisen as a defence of the
Parmenidian philosophy of the One. Heidegger studies the paradoxes and
concludes that they are inherent to an analytic approach of the
Continuum. Bolzano, Weyl, Cantor, Russell are briefly mentioned. An
interesting point lies in that Heidegger explicitely writes :"Continuum
: das Sein"(p.76).
"Das Phänomen des Kontinuums liegt  v o r  dem mathematischen Bereich.
Das Kontinuum geht jeder möglichen endlichen Berechnung voraus. Das Sein
Unterscheidet sich grundsätzlich vom Seienden. Wenn das Kontinuum über
jede endliche und unendliche Bestimmung hinausliegt, so ist das Sein
gegenüber dem Seienden transzendent. Alle Bestimmungen des Seins,
wennecht, sind transzendental."(S. 239)
It is is a very exciting passage, since some scholars locate the birth
of the ontological difference (central in Heidegger's thought) after
Sein und Zeit. But this course was given before Sein und Zeit, and we
can see one the first apparition of this thema in the context of a
discussion about the continuum. It seems that there is a link between
ontology and the study of the Continuum.

---

For those interested in Husserl, I have put a study of my own on my
website. It concerns the links between mathematics and philosophy in the
genesis of Husserl's phenomenology. The study is quite long (160p), it
is written in French, and is rather technical.
For those interested, it can be downloaded at:
http://www.geocities.com/athens/forum/6820/Husserl97.zip 

Olivier
Souan                                                                                         
Student in Philosophy and
Logic                                                      
Université Paris IV et Paris I
Sorbonne                                            
Areas of interest : Phenomenology, Greek philosophy,
Logic and Set theory, Mathematics and foundations.