Quotations about the power of simple ideas - MathOverflow [closed]

Quotations about the power of simple ideas

2012-11-17T17:45:36Z

I'm looking for quotations about how very simple mathematical ideas can be very powerful. I know of a few, but they're not quite what I'm looking for insofar as they contain criticism of other mathematicians, and I'm looking for quotations that are more unambiguously affirmative.

Two that are in the direction of what I'm looking for are:

The very notion of a scheme has a child-like simplicity - so simple, so humble in fact that no one before me had the audacity to take it seriously. So ”infantile” in fact, that for many years afterwards, and in spite of all the evidence, for so many of my ”learned” colleagues, it was treated as a triviality. – Alexander Grothendieck
It is the snobbishness of the young to suppose that a theorem is trivial because the proof is trivial. – John Whitehead

Any better examples?

Answer by DJBruce for Quotations about the power of simple ideas

2012-11-17T18:40:16Z

Here is a quote from The Power of Mathematics by John Conway.

What I like doing is taking something that other people thought was complicated and difficult to understand, and finding a simple idea, so that any fool – and, in this case, you – can understand the complicated thing. These simple ideas can be astonishingly powerful, and they are also astonishingly difficult to find. Many times it has taken a century or more for someone to have the simple idea; in fact it has often taken two thousand years, because often the Greeks could have had that idea, and they didn’t. People often have the misconception that what someone like Einstein did is complicated. No, the truly earthshattering ideas are simple ones. But these ideas often have a subtlety of some sort, which stops people from thinking of them. The simple idea involves a question nobody had thought of asking.

Answer by Alexandre Eremenko for Quotations about the power of simple ideas

2012-11-17T19:08:47Z

Imaginary numbers appear in algebra when we try to take square roots of negative numbers.... Geometric interpretation consists in observing that two consecutive rotations of the plane by 90 degrees around a fixed point reverse the directions of the vectors. If we think of the 180-degree rotation reversing vectors as the geometric counterpart of multiplication of numbers by -1 reversing the sign, then we are inclined to accept the 90-degree rotation (of the plane containing the line of real numbers) as the square root of -1. All this looks childlishly simple, why do mathematicians make such a fuss around it? How can one dare to compare this plain idea to profound philosophical pronouncements, such as "Cogito ergo sum" of Descartes? But look (as my colleague David Ruelle once suggested) from another perspective. "Cogito ergo sum" stayed unperturbed for more than three centuries, like a monument, a Greek statue, a magnificent piece of art, impervious to the flow of time, whilst the little speck of dust, the square root of -1, have been growing and developing for hundreds of years in the minds of mathematicians, geniuses like Cauchy, Gauss and Riemann, and turned into an evergreen intensely alive vibrant tree supporting in its branches our sacred knowledge - quantum mechanics - ruling everything we see (and do not see) in this world.

(Misha Gromov, Local and global in geometry, October 29, 1999.)

Answer by quid for Quotations about the power of simple ideas

2012-11-17T19:15:59Z

Atle Selberg (Interview, June 11, 1989, page 30):

In some sense, I think those are probably the most important things, those that can be made simple.

Or, with one more phrase of context.

There are other things in mathematics that may seem impossible to begin with, but after they have been done they seem very simple. In some sense, I think those are probably the most important things, those that can be made simple.