Date: Thu, 21 May 1998 18:03:55 +0200
Subject: Re: Math/Metaphysics


Cologne, 21 May 1998

jmd schrieb:
> In one of H's articles concerning Maths, there are numerous claims
> about the discipline, e.g., about views concerning its characteristic
> axiomatization, about views concerning its grounding on ultimate, first
> principles, i.e., concerning its having a complete/comprehensive
> (Euclidean) edifice, albeit, yet-to-be formulated, resting on self-evident
> axioms, axioms from which, in accord with some finite set of rules of
> inference, all the 'true' theorems of math, and no 'false' statements of
> math, would be derived.

Which article is this?

> I submit that most of these views are now archaic, that they are of
> historical or pedagogical interest only. It was in light of Godel's and
> Godel-related results -- not only his incompleteability results, but his
> independence results also concerning the Ax of Choice (and its various
> equivalents) -- that these views came to be shelved.
>
> The aspiration and, more importantly, the conceptions of math which
> engender it, for such a complete/comprehensive axiomatic edifice have
> long been abandoned. This abandonment was not inspired because
> such an edifice would be too difficult to erect (Whitehead and Russell
> thought they had satsifactorily accomplished it with their PM), but
> because of the logical impossibility of constructing such an edifice. In
> addition, more recently, many philosophers have tried to formulate
> views concerning math which negatively address Quine's and others'
> view that math 'objects' must exist because they are "indispensable," in
> some sense, to a natural scientific account of "the world." Such recent
> views aim to divorce math from this apparent indispensability.
>
> In short, neither is the Kantian view, that there is only as much genuine
> science as there is math, a committment which necessarily characterizes
> contemporary views of math/science, but nor is the view that theorems
> in math are true, in any philosophically respectable sense.
>
> Thus, I find that many of H's views apropos of maths adumbrate an
> interpretation of an archaic 'understanding' of math and its relations to
> the natural sciences. This is not to claim that these views are without any
> philosophical value; but that H's views are, it seems to me, much more
> difficult to defend in light of the revolution in math.

It needs to be investigated in which sense Heidegger's views and understanding 
of mathematics and its foundations are indeed antiquated, i.e. overtaken by 
developments in the foundations of mathematics in the twentieth century. 

The passages in Heidegger which I have in mind make the connection between 
mathematics and the modern metaphysical drafting of the world in the thinking of 
those who opened up the world to a mathematical-physical understanding, viz. 
Descartes, Leibniz, Newton et al. Heidegger thus situates developments within 
mathematics in the history of being as the drafting of world in line with 
modernity's mathematico-scientific assault on everything that is. 

The concerns of mathematicians in the first half of this century went back to 
conceptions first formulated by Leibniz and Descartes. The notion of 
mathematical truth as logical consistency, for example, is Leibnizian. And 
Descartes’ analytical geometry became canonical for all other (Euclidian and 
non-Euclidian) geometries in the sense that all other geometries can be 
formulated in analytical geometrical terms. 

It's been a good twenty years (alas!) since I worked as a mathematician, so much 
has faded in my recollection, but I was very familiar with topics in the 
foundations of mathematics, the theory of mathematical theories, the 
completeness of axiom systems, category theory (Saunders Mac Lane!), etc. 

> (There is, of course, a problem here about the dissemination of so-
> called 'math knowledge'. Whose understanding of math/science is
> targeted by H? We might ask the same question were a philosopher to
> advance an interpretation of our CURRENT "understanding of science,"
> but illustrate this understanding by reference to a 'widespread
> understanding' which countenanced phlogiston, ether, Vulcan, fluxions,
> and 'quantities of motion'. Whose understanding is targeted? Or is the
> suggestion here of a 'division of 'epistemological' labour' simply out of
> place'?)   
>
> On a final note, during a guest presentation at Harvard, Bertrand Russell
> was asked by the famous algebraist Saunders McClane how Russell
> might modify his position concerning math in light of Godel's results. It
> was clear to the moderator, and the audience, that Russell was either
> not familiar with these results or did not understand their profoundly
> revolutionary significance. The moderator intervened and asked for any
> other questions. Perhaps H was also not as up-to-date on the revolution
> in math which was occurring from roughly 1935 on? 

The question has to be asked: What is the metaphysical import of Goedel's 
incompleteness theorems? It's fair to say that Heidegger was not familiar with 
the developments you refer to, but this may not be philosophically relevant. 
What sort of "revolution" was it? What do these developments in mathematics have 
to do with the metaphysical casting of the being of beings? What is the 
‘interface’ between undecidability of mathematical theorems and the truth of 
reality? 

The context of Goedel’s theorems was Hilbert’s program for formalizing 
mathematics in the sense of encapsulating all of, say, the system of natural 
numbers in a set of mathematical and logical axioms plus rules of inference 
which exhaust everything that can be formulated in that system. Goedel shows 
that certain formulae are unprovable one way or the other, i.e. they are 
undecidable in terms of the formal acts of inference. This amounts to saying 
that there are non-computable mathematical theorems. So mathematical truth 
cannot be reduced to a computational exercise. Such ideas were not foreign to 
Leibniz, who is the father of notions of computability. 

The interesting question (to me) seems to be what relationship the 
incompleteness of computability bears to the digitization of reality. If the 
real (beings as such) is all that can be given a digital representation, then 
non-beings are everything that elude such digital reduction. 

Michael
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