Date: Sun, 24 May 1998 20:44:06 +0100
Subject: Re: Math/Metaphysics


In message <m0ycXox-0003I8C-AT-fwd12.btx.dtag.de>, Michael
Eldred <artefact-AT-t-online.de> writes
>Which article is this?

My views are based on a reading of various comments about math and
various comments about symbolic logic. As for math, the sections in
"What is a Thing" concerning "the mathematical." As for symbolic logic,
the following is a sample I'm collecting:
1912, H's  "New Research in Logic":
It seems to me that the real significance of G. Frege's logico-
mathematical investigations has not yet been appreciated; much less
have his writings been exhaustively dealt with.
...
...it seems to me that above all it must be pointed out that symbolic logic
never gets beyond mathematics and to the core of the logical problems.
I see the barrier in the application of the mathematical symbols and
concepts (above all the concept of function) whereby the meaning and
the shifts in meaning in propositions are obscured. The real significance
of the principles remains in the dark; the propositions calculus, e.g., is a
figuring with propositions; the problem of the theory of propositions are
unknown to symbolic logic. Mathematics and the mathematical
treatment of logical problems reach limits where their concepts and
methods fail; that is precisely where the conditions of their possibility are
located.

In 1914 H: 
It would have to be shown how its formal nature [symbolic logic's]
prevent it from gaining access to the living problems of the meaning of
propositions, of its structure and cognitive significance. 
...
The work here outlined is yet to be done, and it will not be
accomplished as quickly as the overcoming of psychologism.

In his Duns Scotus thesis:
Is logic mathematics or mathematics logic or does neither alternative
hold? One thing can be decided on the basis of what has been said so
far, namely that the two realms in question though they both be of an
immaterial nature, cannot coincide. The homogeneity which owes its
peculiar nature to the uniformity of the viewpoint varies in these two
worlds. The homogeneity of the mathematical realm has its foundation in
quantity. The homogeneity of the realm of logical validity rests on
intentionality, the referential ordination.

SuZ159:
In symbolic Logic, the judgement is dissolved into a system of 'mapping
and interconnecting;' it becomes the object of a 'calculus', but not of an
ontological interpretation.

In H's "What is Metaphysics?" (the article Carnap used), H writes:
There is an attempt here at calculating the system of propositional
connections by means of mathematical methods; hence this kind of logic
is also call "mathematical logic." It sets itself a possible and valid task.
However, what symbolic logic furnishes is anything but a logic, i.e., a
contemplation of the ƒÉƒÍƒÁƒÍs [logos].Mathematical logic is not even
a logic of mathematics in the sense that it determines or could at all
determine the nature of mathematical thinking and mathematical truth.
Rather, symbolic logic is itself a type of mathematics applied to
sentences and sentential forms. Every mathematical and symbolic logic
places itself outside whatever realm of logic because for its very own
purposes it must posit ƒÉƒÍƒÁƒÍs[logos], the proposition, as mere
connection of concepts which is basically inadequate. The presumption
of symbolic logic of constituting the scientific logic of all sciences
collapses as soon as the conditional and unreflective nature of its basic
premises becomes apparent. (The Question of the Thing)

In "Entering into Metaphysics," H writes:
The signal of the degrading of thinking is the upgrading of symbolic logic
to the rank of true logic. Symbolic logic is the calculative organization of
the unconditional ignorance regarding the essence of thinking provided
that thinking, being thought essentially, is that designing knowledge
which, in the case of the essence of truth, rises from being.

"What is Called Thinking?":

Already symbolic logic is widely taken (particularly in the Anglo-Saxon
countries) as the only possible form of rigorous philosophy because its
results and procedures immediately yield something definite toward the
building of the technological world. Hence in America and elsewhere,
symbolic logic, as the proper philosophy of the future, begins to assume
the reign over the spirit. Through the appropriate coupling of symbolic
logic with modern psychology and psychoanalysis and with sociology,
the concern of the coming philosophy becomes perfect."

This concerning, however, is by no means the machination of men.
Rather, these disciplines are under the destination of a power which
comes from afar and for which the Greek words ƒÎƒÍƒÇƒÅƒÐƒÇs
[poesis] and ƒÑƒÃƒÔƒËƒÅ [techne] remain perhaps the fitting name
provided that they name for us, the thinking ones, That which make one
think.

Only because at one time the call into thought became event as
ƒÉƒÍƒÁƒÍs [logos] symbolic logic today is developing into the planetary
organizational from of every presentation.


Anyway, these are only some of the quotes I've been collecting; I'm still
working on H's position.
>
>
>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. 

I'm starting to FEEL that, just perhaps, my interpretations of H's
comments aren't fair. They are now cooking on the cerebral burners
while I try to get on with my own research. However, from what I can
make of the position, it wouldn't be one adverse to even Quine or
Davidson -- they would probably be shocked to hear that. Even they
have attacked the kind of project in Carnap's Aufbau as impossible of
execution (and it seems that H does also), as have many who abandon
any hopes of reduction and accept the irreducibility of intentionality,
meaning, understanding, etc. And any mathematician today would tell us
that it is logically impossible of accomplishing for math what Euclid
accomplished for plane geometry -- in light of Gödel's results and the
various set theoretic so-called 'foundations' of math.

The logician Vaught once responded to the query that "if sets are
numbers, then how can there be nonequivalent set-theoretic definitions
of the positive integers (this was a reference to Benaceraff)?" by
claiming that "it all depended on what color numbers you want; there
are blue numbers and red numbers and green numbers .... choose your
color."
 
My point is that the understanding of math (and symbolic logic) which H
targets is not that of its current practitioners. It wouldn't surprise me if
many had the view of science held by the character Malcolm of Jurrasic
Park, or the views of technology held by the so-called Unabomber.
>
>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 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. 
>
Perhaps, it's been a long time .... I don't want to seem like a pompous
sod, but your comments here are running various distinct, but intimately
related notions, into each other. G's incompleteability theorems establish
that for any mathematical theory T, equipped with the expressive
capacity for "everyday arithmetic" (plus, times, exponentiation, and
definition by primitive recursion) and such rules of inference as math
induction and modus ponens, there are 'true' sentences of T which can't
be deduced from the axioms of T, i.e., there are true, but formally
unprovable sentences of T. These Godel sentences can't be called
"math theorems," for that means that they can be proven.

Also, until one invokes some such thesis as the Church-Turing (CT)
thesis, the notion of 'decidability' remains only an informal/intuitive
characteristic. If we accept CT, then we accept that the 'correct'
mathematical definition of this informal/intuitive characteristic is specified
in terms of the concepts of lambda-definability, Turing-machine
computability, or general recursion (the three are math equivalent).
However, the 'correctness' is not a claim of the type which admits of
proof; it is at best a 'proposal'; that's why the name "CT Thesis" is a
misnomer (like "Fermat's Last Theorem"). Of course, once we accept
CT, the Leibnizian "universalis mathesis" is demonstrably impossible: a
universal algorithm is not possible (the lesson of the unsolvability of the
'Halting Problem').

>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. 

My personal view, perhaps a minority view, is simple to state: not much.
Historically speaking, we have always been susceptible to the same
self-deception, namely, that our most 'sophisticated' or 'involved and
intricate' artifacts are the 'best' models of the mental/intentional, such as
steam engines, hydraulic mechanisms, hierarchical management
practices, and now computers.

We've created a wonderful machine for transmitting information,
interpretations, representations of our everyday world, but the machine
is not what renders such transmitting possible. We can even design
wonderful digital representations of Pooh Bear, the Japanese character
Tottoro, Charlie Brown, and the smile of the Cheshire Cat. It puzzles
me why a digital representation of something should seem any more
philosophically illuminating than a Polaroid picture; in other words, such
digitalization will not provide any illumination at all.

The interesting question is: Why might we even think that illumination
was forthcoming from digitalization? It seems that underlying this
'thinking' there must be views/positions about what the Intentional is all
about, namely, Cartesian caricatures of genuine Intentionality.

I think this post has overstayed its welcome.
I sign off,
Cheers,
jim


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