[FOM] Concerning Ultraformalism

[Hartley Slater]

Mirco Mannucci points me/us towards another paper by Nelson, 
'Mathematics and Faith' in his 'Concerning Ultraformalism --to 
Slater', but I am afraid I find Nelson's dream of an 'overwhelming 
presence' in that paper less worthy of addressing.   Mannucci goes on 
to ask the crucial question, however:

>1) can we keep a formalist approach in a coherent way, not just
>at the mathematical level, but also at the meta-mathematical one?

It is its formalism that has led the tradition to the, now very 
pressing problems it has had with Truth, and particularly The Liar 
and associated paradoxes, as I have pointed out in several previous 
FOM postings both this year and before.  The recent books by McGee, 
Soames, etc, and the lifetime devotion of Priest to problems in this 
area, all arise, on my understanding of the matter, because the 
logical language that has been employed since Frege has lacked 
(primarily) any symbolism for 'that' clauses, in terms of which 
semantic remarks like 's means that p' can be formalised.  So 
formalised treatments of issues in the area have ended up being 
entirely 'syntactic'.

Frege, of course, fought with Hilbert expressly on the need for 
Hilbert's symbols to express thoughts - and Frege's 'horizontal', in 
at least one of its uses, could be taken to be a formalisation of the 
needed sentence nominaliser 'that'.  But the formal tradition from 
Frege has tried to do without any such operator, and now has to swear 
blind that none is needed, to preserve its professional standing. 
But it can only do so at the cost of admitting that the problems with 
Truth and The Liar are intractable.  "After all the work that has 
been done, there cannot be a solution to the Liar and Strengthened 
Liar in just two paragraphs!", has to be the thought.  Indeed, there 
cannot be a formal solution to these problems at all, if one does not 
incorporate into one's formal language the ability to say what it 
means.


-- 
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