[FOM] Gauss and non-Euclidean geometry
Antonino Drago
drago at unina.it
Mon Oct 22 18:58:44 EDT 2007
S. S. Kutateladze wrote:
Gauss was accused primarily by the general habit of
> the presumption of guiltiness and the immortal public ignorance
> of mathematics. I see not a bit of prejudice or immorality
> in his comments on the papers of Lobachevsky and Bolyai.
In my posting the well-known Gauss's confession was implicit; he chose to be
silent on the subject of the "alternative geometries" because he feared the
"Beothians' cries". But this historical fact, as well as the other
historical facts that I recalled in previous my message, have to
interpreted. For instance what means Gauss' expression "Beothians cries":
The political power? The Church? Vulgar scholars?
Of course, these few facts about Gauss may be interpreted from several
viewpoints.
I did not intended to present my viewpoint, but to present no more than
historical facts.
However, I cannot agree with your viewpoint. You superimpose contemporary
mathematics (differential geometry) upon an old historical event. This
method generates paradoxes. For instance: What about Cauchy's reform of
rigour in calculus from the viewpoint of Robinson's non-standard analysis
which re-evaluated the infinitesimals, if not to state that Cauchy was wrong
to try to expel infinitesimals from calculus ? What about Cantor's naive set
theory from the present viewpoint, if not to state that this theory was
contradictory owing to the subsequent Russell paradoxes? What about
Newton's mechanics relying upon absolute space, absolute time and
gravitational force as the unique force in the universe, if not to state
that it was a metaphysical theory? What about Leibniz' mechanics relying
upon both the relativity principle and the conservation of energy, if not to
state that it was the very theory of mechanics?
Instead, even in mathematics the historical truth is not determined by only
the present mathematical truth.
In conclusion, I reiterate what I referred in past message, i.e. a result of
the historical investigations: the birth of non-Euclidean geometry required
a new geometrical theory from which one could think the facts concerning
Euclidean geometry from an outside viewpoint; it was the trigonometry that
first offered to several geometers this new viepoint (its lack a century
before caused Saccheri's mistake; he developed non-Euclidean geometries in
an intuitive way, but after more than thirty propositions he was wrong
because interpreted as a contradiction a merely non-intuitive result).
Gauss' way to achieve his ideas upon the new geometries is a matter of
historical investigation, unfortunately by relying upon very few facts. A
negative evidence to your viewpoint is the fact that in the same review
(Crelle's J. 1837) where Lobachevsky wrote his third paper, another scholar
developed the geometry of curved surfaces without putting his results in
relationship to the new geometries.
Best regards from Italy, where at present is arrived a wave of Siberian cold
air.
Antonino Drago
More information about the FOM
mailing list