[FOM] Gauss and non-Euclidean geometry

Vladimir.Sazonov@liverpool.ac.uk Vladimir.Sazonov at liverpool.ac.uk
Tue Oct 23 11:42:31 EDT 2007

On 23 Oct 2007 at 9:13, S. S. Kutateladze wrote:

> Please READ:
> in my previous post:
> For instance, you write that Cauchy tried to expel
> infinitesimals  but we know now definitely that he was NOT successful 

As I understand, he did not expel it completely from our "proto-intuition" 
but he did expel it from the formalism basing it on somewhat different intuition. 
Anyway, the intuition is always something vague and cannot exist in an 
absolutely pure form. In fact there was no perfect formalism at all - he succeeded 
in creating it quite well. Robinson succeeded in formalizing "actual" infinitesimals 
in some intricate way (ultraproducts, etc.). And so what? 

> Robinson explained that the approach of Cauchy to analysis was based
> intrinsically on  actual infinities. 

I am not sure what do you mean. In which way Robinson explained that? 
Actual infinity (say, of natural numbers of sequences of reals, of the set 
of reals) was introduced formally due to Cantor, but informally or implicitly 
(in fact, almost formally) appropriate concepts were evidently used in 
Cauchy's approach. You seemingly mention Robinson rather in connection 
with actual infinitesimals. In a quite definite sense Cauchy expelled them 
(from the formalism and to a large degree from the intuition of mathematicians). 

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

As to non-Euclidean Geometry, I (non being a specialist knowing all the 
details) think that the most crucial point was not in any technical details 
(which are necessary but insufficient) and not in any relation to the geometry 
of surfaces (which typically were considered at that time as parts of the 
Euclidean Space). The stumbling block was that Euclidean Geometry was 
considered as something absolute, given to us from the God, or innate. 
The question was not whether some other geometries can exist or not. 
It was considered stupid (and in a sense forbidden) even to think in that way. 
(People knowing history of mathematics better than me will probably give 
more correct description of the scientific atmosphere of that time.) 
And the crucial step (I mean the real public action) of great general scientific 
value was done by Lobachevsky and Bolyai, but not by Gauss. 
Both Lobachevsky and Bolyai paid their own prise for this action. As I read 
long time ago, Lobachevsky was considered as a crazy man by many of 
his colleagues. (As I remember, there was also some action of Ostrogradsky 
against him. Just the first relevant page got from Google: 
Somebody could probably find more precise description of this story.) 

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

> Sincerely yours,                  S. S. Kutateladze

Best wishes, and greetings to my former colleagues from

> Sobolev Institute of Mathematics
> Novosibirsk

Vladimir Sazonov

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