[FOM] Gauss and non-Euclidean geometry
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,
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.)
> 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.
More information about the FOM