[FOM] R: Classical logic and the mathematical practice

[Antonino Drago]

The answer is plain: by following Hilbert, most mathematicians believe that
there exists an unique way to organize a mathematical theory, the axiomatic
one, where the deductive method requires classical logic in order to assure
the transmission of truth.
They tend to ignore that their research work does not at all follow a
deductive pattern, although W. Beth (Foundations of Mathematics, Harper,
1959) insisted that this attitude constitutes a bias to present mathematical
research. On the other hand, intuitionists produced a new kind of
logic, but again in a deductive pattern (although with a proviso that it
does not grasp entirely the intuitive system of intuitionistic logic).
Actually, since two centuries some mathematicians (L. Carnot, Lobachevsky,
Galois, Klein, etc.) produced new theories according a new kind of
organization, where the logic is the non-classical one because the main
statements
are double negated ones (which are not equivalent to the corresponding
positive ones, since the latter ones lack of scientific evidence). An ad
absurdum theorem closes a cyclical arguing which starts from a problem. In
this way Lobachevsky (Geometrical Untersuchungen 1840) obtained universal
evidence that it is not contradictory the hypothesis of two parallel lines.
Under this light, 1925 Kolmogorov's paper was the best foundation of
non-classical logic and his 1932 paper was the best approximation of the
past time to this new kind of organization of logic.
About this subject I am publishing a paper in a collective book whose editor
is G. Sica, Polimetrica, Milan.
Best regards
Antonino Drago
via Benvenuti 5
Castelmaggiore Calci Pisa 56110
tel. 050 937493
fax 06 233242218
-----Messaggio Originale-----
Da: "Moshe David" <davidm2 at math.biu.ac.il>
A: <fom at cs.nyu.edu>
Data invio: martedì 10 maggio 2005 21.43
Oggetto: [FOM] Classical logic and the mathematical practice


> Dear FOM memebrs,
>
> I'm a young researcher in complex analysis  and I have a bothersome
> question...
> If you could answer my question I would be grateful !!!
>
> Q : Why the majority of the (working) mathematicians are still using
> classical logic ? to sharpen my question :
> we use the excluded middle without any worry and say that the real field
is
> the disjoint union of  \Bbb Q and \Bbb Q^c
> though we know that there will be a chance that the rationality of some
real
> numbers (e.g. the Euler constant) is undecidable assuming ZFC.
> Is the using of classical logic is not actually Realism/Platonism ? , is
> there any ontological or epistemological justification to use classical
> logic when we know
> that intuitionist logic is more safer and remote from Realism  ?
>
> Best regards,
>
> Moshe David
> Math. Dept. BIU
>
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