[FOM] Gauss and non-Euclidean geometry

Vladimir.Sazonov@liverpool.ac.uk Vladimir.Sazonov at liverpool.ac.uk
Wed Oct 24 11:25:38 EDT 2007

On 24 Oct 2007 at 5:54, S. S. Kutateladze wrote:

> 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.

Yes. But there is its quite precise meaning related with formal logic and 
computer programming. On the other hand, I am using it as the synonym
of "mathematical rigour". I see no other way to explain what is mathematical 
rigour without reducing it to the (possibly wider) meaning of "formal". 
The typical phrase in contemporary mathematical and computer science 
texts is: "Now, after intuitive (informal) preliminaries, let us present the formal 
exposition". And it follow a rigorous (not necessarily absolutely formal) 
considerations. "Rigorous" needs to be explained. "Formal" is self-explanatory, 
and, moreover, has the contemporary limit, in a sense, absolute and quite 
objective meaning (unlike to the absolute mathematical truth - the great 
misconception). What Cantor did was rigorous and formal enough. I think, 
everything was eventually formalized or shown to be potentially formalizable. 

> 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).

It seems there is some confusion here between actually infinite sets - the key 
name is, of course Cantor - and "actual" infinitesimals (probably related with ancient 
monades, but I am not a historian of mathematics and science) for which 
the key name is Leibniz. At least, Robinson formalized the ideas and mathematical 
apparatus of Analysis which was created by Leibniz and others working with 
infinetisimals. I do not reject the ancient history, but I do not see why it is so 
important for the current discussion. 

> 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 

was changing and was going to the contemporary, "limit" understanding 
of formal. (See above.)

yet mathematical
> proofs remain.  

because they are rigorous enough (and currently known to be formalizable 
at the highest contemporary standard. The authors of such old proofs might 
have not thinking about their formalizability at all, but the nature of mathematics 
"forced" them to work instinctively towards potential formalizability. Otherwise 
it would be strange that everything rigorously proved long time ago is formalizable 
now. The psychological mechanism forcing mathematicians to work in this way 
probably deserves a special investigation. 

The formalisms of today are  as temporary and as
> immortal as the formalisms of Euclid or Cauchy or Cantor.

You do not see the difference? It is so evident - one of the greatest achievements 
of the previous century. We approached to a limit. Under a closer consideration 
we could probably notice some further way to move or improve, but we definitely 
have a rather stable limit point. (Computers can check formal proofs!)

> The view that  Cauchy  expelled actual infinites contradicts the
> so-called "Cauchy error" in uniform convergence.

Anybody can make an error. Again, it is some history. Of course, Cauchy 
lived in the context of that time and related his ideas with the older ones, 
or had a mixture of them. But eventually, when everything has  
happened, when we studied Analysis (in our student years) with the names 
of Cauchy, Dedekind, etc, no infinitesimals participated in proofs, there 
were no such "errors" you mention. (And we typically had no idea at that time 
on the possibility of Non-standard Analysis.) Infinitesimals were used only in 
informal comments or preliminaries to proofs. Only epsilon-delta, fundamental 
sequences or Dedekind cuts, etc. Your example may be instructive in 
a sense, but this is rather a history. I am not sure which lesson should 
I get from this. 

> 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.

I am puzzled, why Gauss, having so high authority did not publish 
his results on non-Euclidean geometry if he really had them? 
Did not want to put his reputation under a risk because of some 
scientific atmosphere? But this is a kind of underestimation of the 
value of non-Euclidean geometry - the reputation proved to be 
more important. And this was not the time of the inquisition. Anyway, 
seems strange. (Of course, Lobachevsky and Bolyai had no such 
reputation and therefore had no risk of such "quality".) Or he just did 
not realize at all the great scientific revolutionary value of such a 
result (independently of the strength of mathematical techniques 
used there)? (What result he really had?) In the latter case he, 
naturally, considered the works of Lobachevsky and Bolyai purely 
from the technical side.  Nothing new for him technically, but 
confirmed that they went in some "amazing" [my exaggeration] way?  
And that is all? 

So, what really happened? I really do not understand. May be some 
FOMers have a good answer? Or I missed something essential in this 
discussion? Some citations mentioned in other postings are not clear 
enough for me to undrerstand or interpret. 

Vladimir Sazonov

More information about the FOM mailing list