The greatest mathematicians, as Archimedes, Newton, and Gauss, always united theory and applications in equal measure.
There is an astonishing imagination, even in the science of mathematics.
Why are we willing to accept a new mathematical formula we don't understand as the product of a brilliant mind, while rejecting a new art form we don't understand as the product of a deranged mind?
Everybody in mathematics had given up for 100 years or 200 years the idea that you could from pictures, from looking at pictures, find new ideas. That was the case long ago in the Middle Ages, in the Renaissance, in later periods, but then mathematicians had become very abstract.
One factor that has remained constant through all the twists and turns of the history of physical science is the decisive importance of the mathematical imagination.
The creative principle [of science] resides in mathematics.
Mathematical discoveries, small or great are never born of spontaneous generation.
In the future, as in the past, the great ideas [of mathematics] must be simplifying ideas, the creator must always be one who clarifies, for himself, and for others, the most complicated issues of formulas and concepts.
As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention.
The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps.
You'll remember Newton was furious at Leibniz, because he developed calculus at the same time. And he went to his death believing that he had copied him. And no, it's because all the elements were there, so it's almost inevitable that the next discovery - as long as people are free and allowed to experiment and try new things.
We find in the history of ideas mutations which do not seem to correspond to any obvious need, and at first sight appear as mere playful whimsies such as Apollonius' work on conic sections, or the non-Euclidean geometries, whose practical value became apparent only later.
Some of the greatest advances in mathematics have been due to the invention of symbols, which it afterwards became necessary to explain; from the minus sign proceeded the whole theory of negative quantities.
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.
It is here [in mathematics] that the artist has the fullest scope of his imagination.
The history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible.
I am persuaded that this method [for calculating the volume of a sphere] will be of no little service to mathematics. For I foresee that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn.
If we wish to foresee the future of mathematics, our proper course is to study the history and present condition of the science.
The application of algebra to geometry ... has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences.
Euler - The unsurpassed master of analytic invention.
It took me 1057 pages to describe the hundreds of mathematical equations, algorithms and programming techniques that I invented and used.
The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit.
Many statements about God are confidently made by theologians on grounds that today at least sound specious. Thomas Aquinas claimed to prove that God cannot make another God, or commit suicide, or make a man without a soul, or even make a triangle whose interior angles do not equal 180 degrees. But Bolyai and Lobachevsky were able to accomplish this last feat (on a curved surface) in the nineteenth century, and they were not even approximately gods.
With an absurd oversimplification, the "invention" of calculus [method in mathematics] is sometimes ascribed to two men, Newton and Leibniz.
There is an astonishing imagination, even in the science of mathematics. ... We repeat, there was far more imagination in the head of Archimedes than in that of Homer.
