[FOM] non-Euclidean geometry and FOM programs

[Antonino Drago]

23/10/2007 Rodin wrote:
What is usally
> said
> about Bolyai and Lobachevsky involves much more historical modernisation
> than
> this since today people often think about works of these authors from the
> point
> of view of Hilbert's Axiomatic method which didn't exist at the time. For
> this
> reason I cannot see the relevance of your further examples.

Sorry to be late. I had a trip.
Let us recall that in a given time, common history (of science, in
particular) is written by
the winners.
Since for a long time Hilbert's interpretation of non-Euclidean geometries
was dominant, historians too modeled their accounts according Hilbert's
views; till up to invent changes of one axiom that neither Lobachevsky nor
Bolyai did..
After theoretical physics privileged a space-time differential form
as the foundation for general relativity (planned to include all physical 
theories),
Riemann's approach to non-Euclidean geometries was considered as the most
suitable one, so to lead historians to trace back this attitude to the 
earliest hints (among
which Gauss' ones, surely; but first of all Lazare Carnot's definition of
curvature radius; which however never Lobachevsky defined !).
This re-writing history of science deserves a great pedagogical merit, but
at the same time it is misleading; since it obscures the process of
scientific invention, i.e. the context of discovery.
In my opinion, the fact that after two centuries the textbooks have no clear
ideas on how the
non-Eulidean geometries born, deserves a great attention. This fact
leads to the following answer:
Why all past theories (or better, programs)
about the mathematical foundations have been unsuccessful in offering a
valid historical explanation of that event, the birth of non-Euclidean
geometries, which manifestly first generated the
problem of the FOM? Is maybe this fact a clear
manifestation that all past
programs on FOM were unadequate to the real content
of this starting revolution? Did all programs of research for FOM missed the 
very point implicitly
suggested by the birth of non-Euclidean geometries?

Surely the program to put differentials, i.e. infinitesimal analysis, as
the basis for geometry (i.e. Riemann's approach) and then for theoretical 
physics, corresponds to an ancient program to put calculus at the basis of 
all science; already chemistry, thermodynamics and then quantum mechanics - 
beyond group theory and set theory - disproved this program.
Surely, Hilbert's solution of the historical problem of non-Euclidean 
geometries was
wrong. In historical terms, it suggested a mere change in an axiom (this is 
the version of vulgar historians) ;
but the only one scholar who perfomed this change
(Saccheri) obtained a wrong conclusion. No better suggestion we obtain from 
the formalisation of a scientific
theory (Hilbert's program); no author on non-Euclidean geometries tried to 
formalise his mathematical premises.
(Even in theoretical terms, this program - to axiomatize all scientific
theories - was wrong; thirty years after the first suggestion for Hiilbert's 
programit, Goedel, although
sympathetic to it, proved it as insufficient, even in the case
of the simplest mathematical theory).
No hint came from Cantor's set theory or more advanced mathematical 
theories.

What may we suspect to obtain from an accurate study on the birth of the
first crisis in the foundations of mathematics, i.e. non-Euclidean
geometries?
In my opinion, the natural answer is, the very import of this revolution in
mathematics.

My historical research suggests the following answer. Non-Euclidean 
geometries
overtly questioned not only an axiom, but the entire Euclidean geometry.
But under which aspect was in fact questioned Euclidean geometry?
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.
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.

Under this light, Hilbert's program was doomed to fail, because it wanted
to
develop mathematical theories inside classical logic only, being blind to
any
different way of arguing. Goedel stated that this attitude in mathematics
foundations is insufficient.
But Goedel, in his turn, was unable to suggest a
new attitude, sufficient for obtaininng a complete account of FOM, although 
just after his celebrated theorems he was one of the first scholars to
give a respectable status to intuitionist logic.
But the simil-axiomatic approach chosen by Heyting and then by most 
intuiitionists confirmed Hilbert's approach also in intuitionism; and hence 
it confirmed the widespread shared opinion of a dubious relevance of 
intuitionist logic. Kolmogoroff only, in his foundational paper of 1924, 
referred to a general, informal logic whose specifications are the different 
kinds of logic - classical logic and intuitionist logic -, to be developed 
in different ways on the same foot.

Is the above reconstruction of the history of FOM research a possible one?

Best regards
Antonino Drago