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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:

  1. 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

  2. It is the snobbishness of the young to suppose that a theorem is trivial because the proof is trivial. – John Whitehead

Any better examples?

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closed as not a real question by Felipe Voloch, HJRW, Alexandre Eremenko, Andy Putman, fedja Nov 19 '12 at 4:11

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

What is the motivation behind this? I mean examples abound. What is the specific purpose of collecting the quotations (the first one of which already annoys me; well I knew it before, but still I am reannoyed). Also at the very least this is a prime example for CW. –  quid Nov 17 '12 at 17:59
Jonah: in case you are interested, the motto of the Ross program is "Think deeply of simple things." –  Patricia Hersh Nov 17 '12 at 18:01
@ quid - I had forgotten to make the question CW (as well as big-list) and just did so. The reason that I'm asking about this is to find quotes to include in mathematical exposition for beginners. I can see how the first quotation would rub you the wrong way - as I said in the first paragraph of my question, the examples that I know of contain implicit criticism of other mathematicians, and I'm looking for quotations that get the idea across without being critical of others. –  Jonah Sinick Nov 17 '12 at 18:16
Thanks for the explanation and for making it CW. –  quid Nov 17 '12 at 18:42
"I have proved something difficult. I will now claim that it follows from a simple insight, so as to create an impression that I am blessed with inspiration without perspiration." YC, 2012 –  Yemon Choi Nov 19 '12 at 9:33

3 Answers 3

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.

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Thanks! This is exactly what I was looking for. –  Jonah Sinick Nov 19 '12 at 16:43

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.)

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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.

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