# [FOM] non-Euclidean geometry and FOM programs

Antonino Drago drago at unina.it
Wed Nov 7 16:12:23 EST 2007

```> Sunday, October 28, 2007 11:59 PM I wrote:
>> In fact, the radical change is substantiated in both Lobachevsky's and
>> Bolyai's
>> texts the essential use of a great number of sentences which are not
>> equivalent to the
>> corresponding positive sentences; hence, the failure of the double
>> negation law; which at present we know introduces grosso modo a
>> non-classical logic, intuitionist logic as first.
>
>
> Monday, October 29, 2007 5:00 PM Andre Rodin wrote:
> but
> I cannot see the link. A logician directly influenced by Lobachevsky was
> Vasiliev whose "Imaginary logic" is usually viewed today as anticipating
> paraconsistent logics - but this seems to be quite a different story.

My answer:  In fact never Lobachevsky wrote "There exist two parallels",
rather (I mark
by two asterisks each negative word)
*contradiction* in the results.." (the end of proposition no. 22 in
Geometrical Studies..., an appendix to R. Bonola: Non-Euclidean Geometry,
Dover, 1955)
"1° Dans la théorie *rien* *ne s'oppose* à admettre que la somme des angles
d'un triangle rectiligne soit moindre que deux angles droits" 2° Dans
l'hypotèse de la somme des angles d'un triangle moindre que deux angles
droits, les équations (13) peuvent etre substituées aux équations
ordinaires
(15) *sans* mener jamais à quelques résultats *absurdes*....."(Géométrie
Imaginaire, J. Crelle, 17 1837, p. 302)

> Sunday, October 28, 2007 11:59 PM I wrote:
>> In Lobachevsky's mind this radical change in the way of
>> arguing may be put in connection to the first Russian translation in his
>> little and far town, Kazan, of Lazare Carnot's celebrated book
>> on calculus, whose final "Note" explained the two different method of
>> arguing
>> inside a scientific theory, i.e. the analytic one and the
>> syntethtic one.
>
> > Monday, October 29, 2007 5:00 PM Andre Rodin wrote:
>This is particularly interesting given the long passage on the distinction
> btw analytic and synthetic methods Lobachevsky provides in the Intro to
> his
> New Foundations of Geometry of 1835. Do you have the reference to the
> Russian translation of Carno's book?
>
My answer: C..C. Gillispie Lazare Carnot Savant, Princeton U.P., Princeton,
1971 gives
the complete bibliography of L. Carnot, included the foreign versions of
his
books; the Russian version was edited when Lobachevsky was was both Dean of
the Philosophical Faculty and the librarian of the University of the very
little town of Kazan .  Moreover, Lobachevsky wrote two "Outline" of his
course on geometry; the second uncensored version quotes L. Carnot. See
N.I.
Lobachevskii, Collected Scientific and Pedagogical Papers ( eds. A.
Alexandrov, B.L. Laptev, in Russian), Nauka, Moscow, 1976 which adds some
details.

Best regards

Antonino Drago:

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