[FOM] Nelson's ultraformalism --to Slater
Hartley Slater
slaterbh at cyllene.uwa.edu.au
Mon Nov 6 21:07:52 EST 2006
At 5:55 PM -0500 6/11/06, Eray Ozkural wrote:
>Proving 2+3=5 is a shortcut, short of counting, which is a computation.
>In AI community, shortcuts have sometimes been called "chunking".
>However, those approaches are unfortunately not capable of inventing
>the ordinary kind of arithmetic (PA). Computational basis for such capability
>on the other hand might arise from the application of inductive inference
>(Reference available upon request). It is unknown at the present if this
>is the case.
>
>Note also the relation to the halting problem, which seems to
>correspond to the "golden standard" for computational
>problems. The theorem 2+3=5 is an easy instance of the halting
>problem, we can show a program that halts iff that is the case.
The extraordinary thing about these remarks will, I suspect, be
hidden from most members of this list. For they demonstrate only too
well my previous point about the absence of 'that' from current
formal languages - with the resultant professional unconsciousness
about the identity of mathematical truths. The phenomenon is quite
general. Thus, when I sent the abstract of my paper, 'Proving that
2+3=5', to the organisers of my seminar at Monash University
recently, the same thing occurred, and at my own institution, here at
UWA, the same thing had happened the week before. Each time the
title of my talk was advertised simply as 'Proving 2+3=5'.
If you do not insert the 'that' you miss so much! Why is it that no
Turing Machine can prove that 2+3=5? Because 'that 2+3=5' is not a
sentence and so, a fortiori, it is not a sentence at the end of any
rule governed sequence of sentences. Generating sentences, as Turing
Machines do, is not in the right business even; a 'category mistake'
is involved in thinking it is. And the same category mistake is what
makes Avron, amongst many others, think he is, indeed must be a
Turing Machine. Before, I gave a solution of The Liar in two
paragraphs, relying on this point; but here is the proof, now in just
two lines, that anything that can prove that 2+3=5 is not a Turing
Machine. Can anyone else on FOM see these things? They have little
chance of doing so, if they casually elide 'that' in the manner above.
Nelson, in the second defense-of-Formalism paper that Mannucci
previously referenced ('Mathematics and Faith') said (p7) that he had
felt an 'overwhelming presence' which had removed from him the
'arrogance' of his previous belief that there was a real world
numbers. '... the theorems [of Mathematics] are not about anything'
he said (p4). Of course, if one does not have a referential phrase
like 'that 2+3=5' in one's formal language then one's language cannot
make reference to facts or putative facts; so if one chooses one's
language appropriately there will be no such reference. But is there
some 'arrogance' in incorporating referential phrases of the form
'that p'? Certainly all I take myself to be doing is reminding
theorists of the things we all, both professionals and ordinary
people, commonly and repeatedly say. Every day, continuously,
everybody, whether educated or untutored, is saying that such and
such, or that such and such is so. Whether or not that means there
is some 'real world' of facts we can grant philosophers another two
and a half millenia to decide.
The main point for the moment, though, is: how many FOMers, reading
this message, even noticed the 'that's in the 'that such and such, or
that such and such' just then?
--
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 6488 1246 (W), 9386 4812 (H)
Fax: (08) 6488 1057
Url: http://www.philosophy.uwa.edu.au/staff/slater
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