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Article 4898 of comp.ai.philosophy:
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>From: atten@phil.ruu.nl (Mark van Atten)
Newsgroups: comp.ai.philosophy
Subject: ai
Summary: do math. objects exist? what is the relevance of Goedel for ai?
Keywords: ai,realism,Goedel,platonism
Message-ID: <atten.702298596@groucho.phil.ruu.nl>
Date: 3 Apr 92 10:56:36 GMT
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Organization: Department of Philosophy, University of Utrecht, The Netherlands
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There is an ongoing discussion right now on mathematical realism in the ai
group. Some people question the relevance of this to ai. In this article,
I want to discuss
1 The objective existence of mathematical objects
2 The significance of this for ai

(the lion's share of this article consists of quotes from the philosophers
I like best - I think it's better to be as precise as possible in stating
their views. Anyway, I agree with them completely (i.e., Leibniz and Goedel)

I THE OBJECTIVE EXISTENCE OF MATHEMATICAL OBJECTS

I.1 Leibniz on constant fundamental ideas
I.2 Goedel on objectivity
I.3 math. and logic cannot be 'explained away'
I.4 Goedel on intuition
I.5 comparison of math. and physics
 
II THE SIGNIFICANCE OF GOEDEL'S THEOREM TO AI

II.1 This is not repeat not Lucas' argument
II.2 Penrose's argument
II.3 Goedel's theorem and ai
II.4 non-algorithmic processes
II.5 the infinite mind

I.1 'Hobbes saw that all truths can be demonstrated from definitions [we're
not so sure anymore since Kant, but for the sake of argument, never mind], but
he believed that all definitions are arbitrary and nominal, since the
imposition of names on things is arbitrary. But it must be known that concepts
cannot be combined in an arbitrary fashion, but a possible concept must be
formed from them, so that one has a real definition. From this it is evident
that every real definition contains some affirmation of at least possibility.
Further, even if names are arbitrary, yet once they have been imposed their
consequences are necessary and certain truths arise which, though they depend
on the symbols imposed, are nevertheless real. For example, the rule of nine
depends on symbols imposed by the decimal system, and yet it contains a real
truth. Again, to form a hypothesis, i.e. to explain a way of producing
something, is simply to demonstrate the possibility of the thing; and this is
useful, even though the thing in question has often not been generated in such
a way.'
( Leibniz, Of universal synthesis and analysis; or, of the art of discovery
and of judgement, 1683)

I.2 Goedel on objectivity
'The same possibilities of thought are open to everyone, so the world of
possible forms is objective and absolute. Possibility, then, is not dependent
on an observer; it is therefore real because it is not subject to our will.'
(quoted in Rucker 1984)

I.3
'Logic and mathematics (just as physics) are built up on axioms with real content which cannot be 'explained away'.

1.4 Goedel on intuition
'It should be noted that mathematical intuition need not be conceived of as a
faculty giving an ***immediate*** knowledge of the objects concerned. Rather
it seems that, as in the case of physical experience, we ***form*** our ideas
also of those objects on the basis of something else which is immediately
given. Only this something else here is not, or not primarily, the sensations.
That something besides the sensations actually is immediately given follows
(independently of mathematics) from the fact that even our ideas referring to
physical objects contain constituents qualitatively different from sensations
or mere combinations of sensations, e.g., the idea of object itself, whereas,
on the other hand, by our thinking we cannot create any qualitatively new
elements, but only reproduce and combine those that are given. Evidently the
'given' underlying mathematics is closely related to the abstract elements
contained in our empirical ideas. (Note that there is a close relationship
between the concept of set and the categories of pure understanding in Kant's 
sense. Namely, the function of both is 'synthesis', i.e., the generating of
unities out of manifolds (e.g., in Kant, of the idea of ***one*** object out
of its various aspects)) It by no means follows, however, that the data of
this second kind, because they cannot be associated with actions of certain
things upon our sense organs, are something purely subjective, as Kant
asserted. Rather they, too, may represent an aspect of objective reality, but,
as opposed to the sensations, their presence in us may be due to another kind
of relationship between ourselves and reality. [...] The question of the
objective existence of the objects of mathematical intuition is an exact
replica of the question of the objective existence of the outer world.'
(Goedel 1964, postscipt to 'What is Cantor's continuum problem?')

Here, physical objects need not necessarily be understood in avery strict 
sense, e.g. as material objects. It stands for all physical phenomena which
we believe to be outside of us, i.e. exist independently of us. Physics studies
these objects (materials, fields), and needs them in order to obtain a
satisfactory theory of our physical experiences.
Note that Goedel explicitly wishes to get rid of Kant's Ding an Sich and its
transcedence.

'Russell compares the axioms of logic and mathematics with the laws of nature
and logical evidence with sense perception, so that the axioms need not be
necessarily eveident in themselves, but rather their justification lies
(exactly as in physics) in the fact that they make it possible for these
'sense perceptions' to be deduced; which of course would not exclude that they
also have a kind of intrinsic plausibility similar to that in physics. I think
that (provided 'evidence' is understood in a sufficiently strict sense) this
view has been largely justified by subsequent developments, and it is to be
expected that it will be still more so in the future. It has turned out that
(under the assumption that modern mathematics is consistent) the solution of
certain arithmeticcal problems requires the use of assumptions essentially
transcending arithmetic, i.e., the domain of the kind of elementary
indisputable evidence that may be most fittingly compared with sense perception.(Goedel 1944, Russell's mathematical logic)
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'Russell compares the axioms


