[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 

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 

Antonino Drago

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