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Alan Smaill wrote:
"Charlie-Boo" <shymathguy@xxxxxxxxx> writes:
Alan Smaill, Charlie-Boo:
on what specific basis (e.g. excerpts from some
publications) do you say that Manna & Waldinger did not change the
background logic?
See the footnote on p28 of
129.215.101.90:1001/publications/PICTworkingpapers/WP38.pdf
"The page cannot be dispayed" : (
works for me ...
Content of footnotes:
38. MacKenzie, Donald, Negotiating arithmetic, constructing proof:
the sociology of mathematics and information technology. download.pdf
5 3See, e.g., J. Van Heijenoort (ed.), From Frege to Gödel: A Source
Book in Mathematical Logic, 1879-1931 (Cambridge, MA: Harvard University
Press, 1967). Some of Brouwer's reasons for doubting an apparently
obvious principle can be seen in the following quotation: 'Now consider
the principum tertii exclusi [law of excluded middle]: It claims that
every supposition is either true or false; in mathematics this means
that for every supposed imbedding of a system into another, satisfying
certain given conditions, we can either accomplish such an imbedding by
a construction, or arrive by a construction at the arrestment of the
process which would lead to the imbedding. It follows that the question
of the validity of the principum tertii exclusi is equivalent to the
question whether unsolvable mathematical problems can exist. There is
not a shred of a proof for the conviction, which has sometimes been put
forward that there exist no unsolvable mathematical problems ... in
infinite systems the principum tertii exclusi is as yet not reliable ...
So long as this proposition is unproved, it must be considered as
uncertain whether problems like the following are solvable: Is there in
the decimal expansion of [pi] a digit which occurs more often than any
other one? ... And it likewise remains uncertain whether the more
general mathematical problem: Does the principum tertii exclusi hold in
mathematics without exception? is solvable ... In mathematics it is
uncertain whether the whole of logic is admissible and it is uncertain
whether the problem of its admissibility is decidable':
Brouwer, 'The Unreliability of the Logical Principles', in Brouwer, op.
cit.note 42, 107-11, at 109-11, emphases deleted.
5 4I am drawing here on Bloor, op. cit. note 10, 124-36.
5 5D.C. Makinson, Topics in Modern Logic (London: Methuen, 1973), 27-28.
5 6Cliff B. Jones, Systematic Software Development using VDM, second
edition (Hemel Hempstead: Prentice Hall, 1990), 24.
5 7'Unless we have a philosophical commitment to intuitionism,
maintaining constructiveness when it is not required can only make a
proof system more cumbersome to use. We have seen that certain
programs cannot be derived from their specifications in a constructive
logic, but can be derived in a classical logic upon which minimal
restrictions have been imposed ... ': Zohar Manna and Richard Waldinger,
'Constructive Logic considered Obstructive' (typescript, n.d.), 8.
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