The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry [as did Clifford], for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.
Interpretation
Lobachevsky's method fundamentally changed our understanding of geometry and thought.
Eric Temple Bell recognizes Lobachevsky's revolutionary impact on geometry, comparing him to Copernicus, who transformed our perception of the universe. Lobachevsky's contributions extend beyond geometry, suggesting that he altered the very foundations of logical reasoning and intellectual discourse in a broader context, indicating the profound reach of his ideas in shaping modern thought.
In practice
In a lecture discussing the evolution of mathematical thought, one might reference Lobachevsky as a pivotal figure.
Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.
In his wretched life of less than twenty-seven years Abel accomplished so much of the highest order that one of the leading mathematicians of the Nineteenth Century could say without exaggeration, "Abel has left mathematicians enough to keep them busy for five hundred years." Asked how he had done all this in the six or seven years of his working life, Abel replied, "By studying the masters, not the pupils."
If a lunatic scribbles a jumble of mathematical symbols it does not follow that the writing means anything merely because to the inexpert eye it is indistinguishable from higher mathematics.
Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics.
Any impatient student of mathematics or science or engineering who is irked by having algebraic symbolism thrust upon him should try to get along without it for a week.
In a very real sense not one of us is qualified, but it seems that God continually chooses the most unqualified to do his work, to bear his glory. If we are qualified, we tend to think that we have done the job ourselves. If we are forced to accept our evident lack of qualification, then there's no danger that we will confuse God's work with our own, or God's glory with our own.
There seems to be some perverse human characteristic that likes to make easy things difficult.
A lawyer is either a social engineer or he is a parasite on society.
I spit into the face of time that has transfigured me
Two urns on Jove's high throne have ever stood, the source of evil one, and one of good; from thence the cup of mortal man he fills, blessings to these, to those distributes ills; to most he mingles both.
The utterly fallacious idea at the heart of the pro-war argument is that it is the duty of the anti-war argument to provide an alternative to war. The onus is on them to explain just cause.
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