[FOM] Gauss and non-Euclidean geometry

[S. S. Kutateladze]

Vladimir.Sazonov at liverpool.ac.uk> On 23 Oct 2007 at 9:13, S. S. Kutateladze wrote:





Vladimir.Sazonov at liverpool.ac.uk> And so what?
Robinson explained that the approach of Cauchy to analysis was based
 intrinsically on  actual infinities.

Vladimir.Sazonov at liverpool.ac.uk> I am not sure what do you mean. In which way Robinson explained that? 
Vladimir.Sazonov at liverpool.ac.uk> Actual infinity (say, of natural
Vladimir.Sazonov at liverpool.ac.uk> numbers of sequences of reals, of the set
Vladimir.Sazonov at liverpool.ac.uk> of reals) was introduced formally
Vladimir.Sazonov at liverpool.ac.uk> due to Cantor, but informally or implicitly
Vladimir.Sazonov at liverpool.ac.uk> (in fact, almost formally)
Vladimir.Sazonov at liverpool.ac.uk> appropriate concepts were evidently used in
Vladimir.Sazonov at liverpool.ac.uk> Cauchy's approach.

It is not easy to answer your questions in a few words.
Of course, you are wrong in assuming that  actual infinities were
introduced formally by Cantor. "Formally" is an ambiguous word.
Actual infinity is in fact introduced in Definition 1 of Book VII of
Euclid as monad.  Some relevant points are given in Chapters 1 and 2 of the book
"Infinitesimal Analysis" by Gordon, Kusraev, and Kutateladze (Kluwer,
2002).


Vladimir.Sazonov at liverpool.ac.uk> Non-formalized intuition can
Vladimir.Sazonov at liverpool.ac.uk> evidently play an essential role on preliminary
Vladimir.Sazonov at liverpool.ac.uk> stages of getting a mathematical
Vladimir.Sazonov at liverpool.ac.uk> result, but if it is not explicitly reflected in
Vladimir.Sazonov at liverpool.ac.uk> the formal proof - it may be well
Vladimir.Sazonov at liverpool.ac.uk> considered as expelled (in the specific
Vladimir.Sazonov at liverpool.ac.uk> sense described).


This view is rather common but  bases on the vague specification
of what is formal.  Rigor is a more appropriate word in my opinion.
What is rigorous and/or formal changes with time yet mathematical
proofs remain.  The formalisms of today are  as temporary and as
immortal as the formalisms of Euclid or Cauchy or Cantor.
The view that  Cauchy  expelled actual infinites contradicts the
so-called "Cauchy error" in uniform convergence.

I insert an excerpt of a recent letter of mine to Mr. Rickey:

Quotes:
Cauchy described the functions under study as follows
``An infinitely small increment given to the~variable
produces an infinitely small increment of the function itself.''
This yields uniform continuity rather than continuity as envisaged by
nonstandard analysis.

Analogously, pointwise convergence at all (standard and nonstandard)
points  of a compact interval amounts to uniform convergence.
Cauchy was a brilliant mind who looked at the entities of analysis in
a fashion closer to Leibniz than Newton. He felt the difference
between ``assignable'' and ``nonassignable'' numbers which is neglected
in the epsilon-delta technique but  reconstructed by Robinson.
Cauchy was ``intimidated'' and felt ashamed of his would-be ``error.''
 He tried to suggest a better proof but published practically the same
``new'' demonstration.  He was a great master, the inventor of complex
analysis, and he deserved a better fate than to be ridiculed for
the temporary lack of understanding of his proofs and ideas.

End of Quotes.

Vladimir.Sazonov at liverpool.ac.uk> As to non-Euclidean Geometry, I
Vladimir.Sazonov at liverpool.ac.uk> (non being a specialist knowing all the
Vladimir.Sazonov at liverpool.ac.uk> details) think that the most
Vladimir.Sazonov at liverpool.ac.uk> crucial point was not in any
Vladimir.Sazonov at liverpool.ac.uk> technical details 
Vladimir.Sazonov at liverpool.ac.uk> (which are necessary but
Vladimir.Sazonov at liverpool.ac.uk> insufficient) and not in any
Vladimir.Sazonov at liverpool.ac.uk> relation to the geometry 

You are right in describing  the great contribution of Lobachevsky and Bolyai
to liberation of the mind. I do not dispute  this contribution.
I am against neglecting Gauss and blaming him for his remarks about
Bolyai and Lobachevsky. The mathematical content of their works was
clear and unrevealing  to Gauss at all, that's what I mean.
If we take into account the difference of the mathematical
background of the three persons, we will see the story in proper
proportions. That's my point---historical facts never change but
their interpretation and understanding may, if not must, change with time.

Vladimir.Sazonov at liverpool.ac.uk> Irrespectively to the question on
Vladimir.Sazonov at liverpool.ac.uk> Gauss's role, knowing the geometry of surfaces
Vladimir.Sazonov at liverpool.ac.uk> seems to me, in general,
Vladimir.Sazonov at liverpool.ac.uk> insufficient for realizing that the geometry of our
Vladimir.Sazonov at liverpool.ac.uk> space can be different in principle. 


>> Sincerely yours,                  S. S. Kutateladze

Vladimir.Sazonov at liverpool.ac.uk> Best wishes, and greetings to my former colleagues from

>> Sobolev Institute of Mathematics
>> Novosibirsk

Vladimir.Sazonov at liverpool.ac.uk> Vladimir Sazonov

You are welcome. We are still colleagues.

                
---------------------------------------------
Sobolev Institute of Mathematics
Novosibirsk State University
            mailto: sskut at math.nsc.ru
            copyto: sskut at academ.org       
http://www.math.nsc.ru/LBRT/g2/ruswin/ssk/index.html