FOM: geometrical reasoning Antonino.Drago at
Thu Feb 25 13:21:04 EST 1999

About this point I think to have an answer to the several questions put in previous posting. It is well-known that constructive mathematics is unable to 
capture geometrical intuition inasmuc as it in general can approximate only 
a single point or a totality (in particular see Bridges in ZMLGM 25 (1979) 525)
On the contrary, Weyl's elementary mathematics is able to represent geometrical intuition of single points and totalities inasmuch as it allows the use of 
a quantifier on constructive numbers (see Feferman: Weyl's vindicated, in
Cellucci et al. (eds.) Temi e prospettive, Clueb, Bologna, 1988, 59-93: also in 
Phil. Topics). As a consequence one may guess that geometrical intuition is
no other that the use of a single quantifier on constructive numbers. That

amount to a great power of transcending the real world. 
It is interesting that in history of mathematics Cavalieri and Torricelli
first offered a calculus by means of a method of indivisibles which relies
upon the Latin word "omnes", which in past times was intended as a rough 
sign of an integral; yet, erroneously, according to the last analysis of
Cavalieri's work (K. Andersen, Arch. Hist. Ex. Sci. 31 (1985) 291-375).
The more obvious interpretation is by "all", i.e. the total quantifier. That
qualifies his mathematics as no more than Weyl's mathematics. In point of fact
Cavalieri's method was charged by Guldin to allow the eyes to command mathe-
matics, a fact Cavalieri recognised in his reply. Actually, his method is 
essentially based upon geometrical intuition (remember for ex. his celebrated
theorem, amounting to affirm the existence of one point such that...). This
kind of intuition was called by Torricelli as the geoemeter's privilege and 
applied to give reality to the a final situation of a series of physical 
situations; in such a way Cavalieri and Torricelli gave reality to a body
without gravity - an impossible thing according to Galilei. They obtained
in such a way the first version of inertia principle (1632) instead of the
date of 1644 of Descartes'principle - according to Hanson (in R. Colodny (ed.):
Beyond the Edge of Certainty, Prentice-Hall 1965) is full of claims of absolute
accuracies. Koyre's analysis (Etudes Galileennes) is confirmed. "La physique 
de Galilee explique ce qui est par ce qui n'est pas. Descartes etNewton 
vont plus loin: leurs physiques expliquent ce qui est par ce qui ne peut 
pas etre; elles expliquent le reel par l'impossible" (p.276) This is a 
verbal statement synthetising the difference between the introduction of 
a constructive mathematics in physics and the introduction of a more 
powerful mathematics than the previous one.
All that gives reason why experimental physics accepted to use a not-constructive mathematics; geometrical intuition - in the same time in calculus and in 
optics - was uneavopidable and geometrical intuition means essentially a 
passage to infinite operations as included in one quantifier.
If I am correct, then no more than first order logic is necessary for 
geometrical reasoning.

Antonino Drago

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