Geniuses -

genius bernhard riemann

Bernhard Riemann

One can set up a completely self-contained mathematical theory, which proceeds from the elementary laws that are valid for individual points to processes in the actually given continuously filled space, without distinguishing whether it is gravity, electricity, magnetism, or the equilibrium of heat that is being treated.

Bernhard Riemann

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genius bernhard riemann

Bernhard Riemann

Without doubt it would be desirable to have a rigorous proof of this proposition; however I have left this research aside for the time being after some quick unsuccessful attempts, because it appears to be unnecessary for the immediate goal of my study.

Bernhard Riemann

genius bernhard riemann

Bernhard Riemann

It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.

Bernhard Riemann