Mathematical Insight Quotes

Math research is more like a marathon.

The greatest mathematicians, as Archimedes, Newton, and Gauss, always united theory and applications in equal measure.

He [John von Neumann] had the invaluable faculty of being able to take the most difficult problem and separate it into its components, whereupon everything looked brlliantly simple.

Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs.

When I multiply numbers together, I see two shapes. The image starts to change and evolve, and a third shape emerges. That's the answer. It's mental imagery. It's like maths without having to think.

I have discovered a truly marvelous proof of this, which however the margin is not large enough to contain.

If you are receptive and humble, mathematics will lead you by the hand. Again and again, when I have been at a loss how to proceed, I have just had to wait until I have felt the mathematics led me by the hand. It has led me along an unexpected path, a path where new vistas open up, a path leading to new territory, where one can set up a base of operations, from which one can survey the surroundings and plan future progress.

Mathematical discoveries, small or great are never born of spontaneous generation.

I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it.

One of the pleasures of looking at the world through mathematical eyes is that you can see certain patterns that would otherwise be hidden.

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.

I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future.

It is more important to have beauty in one's equations than to have them fit experiment... It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one's work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further developments of the theory.

For me, I guess the main motivation is the satisfaction of finally understanding some tricky mathematical concept or phenomenon and then explaining it to others.

I entered an omnibus to go to some place or other. At that moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with non-Euclidean geometry.

It seems to me that the poet has only to perceive that which others do not perceive, to look deeper than others look. And the mathematician must do the same thing.

Gauss replied, when asked how soon he expected to reach certain mathematical conclusions, that he had them long ago, all he was worrying about was how to reach them!

In the simplest array of digits [Ramanujan] detected wonderful properties: congruences, symmetries and relationships which had escaped the notice of even the outstandingly gifted theoreticians.

And Archimedes, as he was washing, thought of a manner of computing the proportion of gold in King Hiero's crown by seeing the water flowing over the bathing-stool. He leaped up as one possessed or inspired, crying, "I have found it! Eureka!".

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.

Talk with M. Hermite. He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures.

The question you raise, 'How can such a formulation lead to computations?' doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand - and it always turned out that understanding was all that mattered.

The beauty of mathematics only shows itself to more patient followers.

Equations are just the boring part of mathematics. I attempt to see things in terms of geometry.

Math does come easily to me, but I was always much more interested in what theorems imply about the world than in proving them.