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.
When a hundred men stand together, each of them loses his mind and gets another one.
Yet some can be patriotic who have no self-respect, and sacrifice the greater to the less. They love the soil which makes their graves, but have no sympathy with the spirit which may still animate their clay. Patriotism is a maggot in their heads.
While some men see ordinary happenings, others see divine light and guidance.
Remember that your perception of the world is a reflection of your state of consciousness. You are not separate from it, and there is no objective world out there. Every moment, _x000D_ your consciousness creates the world that you inhabit.
Is not nationalism - that devotion to a flag, an anthem, a boundary so fierce it engenders mass murder - one of the great evils of our time, along with racism, along with religious hatred? These ways of thinking - cultivated, nurtured, indoctrinated from childhood on - have been useful to those in power, and deadly for those out of power.
In most cases we attach ourselves to in order to take revenge on life, to punish it, to signify we can do without it, that we have found something better, and we also attach ourselves to God in horror of men.
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