Thoughts without content are empty, intuitions without concepts are blind... The understanding can intuit nothing, the senses can think nothing. Only through their union can knowledge arise.
Torrent of light and river of air,
Along whose bed the glimmering stars are seen,
Like gold and silver sands in some ravine
Where mountain streams have left their channels bare!
The best that Gauss has given us was likewise an exclusive production. If he had not created his geometry of surfaces, which served Riemann as a basis, it is scarcely conceivable that anyone else would have discovered it. I do not hesitate to confess that to a certain extent a similar pleasure may be found by absorbing ourselves in questions of pure geometry.
Deaths, births, and marriages, considering how much they are separately dependent on the freedom of the human will, should seem to be subject to no law according to which any calculation could be made beforehand of their amount; and yet the yearly registers of these events in great countries prove that they go on with as much conformity to the laws of nature as the oscillations of the weather.
Let us make education brave and preventive. Politics is an afterwork, a poor patching. We are always a little late... We shall one day learn to supercede politics by education... We must begin higher up, namely in Education.
All Science is necessarily prophetic, so truly so, that the power of prophecy is the test, the infallible criterion, by which any presumed Science is ascertained to be actually & verily science. The Ptolemaic Astronomy was barely able to prognosticate a lunar eclipse; with Kepler and Newton came Science and Prophecy.
Pure mathematics is, in its way, the poetry of logical ideas. ... [By seeking] logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature.
Man carries the world in his head, the whole astronomy and chemistry suspended in a thought. Because the history of nature is charactered in his brain, therefore he is the prophet and discoverer of her secrets. Every known fact in natural science was divined by the presentiment of somebody, before it was actually verified.