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.
He was talking about the sign that said 'THE COMPLICATED FUTILITY OF IGNORANCE.' 'All knew was that I didn't want my daughter or anybody's child to see a message that negative every time she comes into the library,' he said. 'And then I found out it was you who was responsible for it.' 'What's so negative about it?' I said. 'What could be a more negative word than "futility"?' he said. '"Ignorance,"' I said.
Away with the cant of 'Measures not men!'-the idle supposition that it is the harness and not the horses that draw the chariot along.
To a toad what is beauty? A female with two lovely pop-eyes, a wide mouth, yellow belly, and green spotted back.
To embarrass justice by multiplicity of laws, or to hazard it by confidence in judges, seem to be the opposite rocks on which all civil institutions have been wrecked, and between which legislative wisdom has never yet found an open passage.
I took a deep breath and listened to the old bray of my heart. I am. I am. I am.
We had stretched out our arms to each other and supported something in our joined hands, but this thing we were holding was like a sort of gas that exists when you believe in its existence and disappears when you doubt. The task of supporting it seems simple at first glance, but actually requires an ultimate refinement of calculation and a consummate skill.
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