Quidquid latine dictum sit, altum videtur:
Anything said in Latin sounds profound.

This is certainly true of mathematics as well: inside the foreign-looking terminology and notation, common sense usually can be found. When you find the sense of a problem, you are well on the way to solving it.

Here are some more of my favorite quotes, organized by topic.

Math // Reading and writing // Problem solving // Challenges // Progress

Math

In mathematics you don’t understand things. You just get used to them.
— John Von Neumann (1903–1957). 		Although this may seem a paradox, all exact science is dominated by the idea of approximation.
— Bertrand Russell (1872–1970). 		On two occasions I have been asked [by members of Parliament], ‘Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?’ I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.
— Charles Babbage (1792–1871).
By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race.
— Alfred North Whitehead (1861–1947). 		The art of doing mathematics consists in finding that special case which contains all the germs of generality.
— David Hilbert (1862–1943). 		The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal.
— William James (1842–1910) Collected Essays.
A monument to Newton! a monument to Shakespeare! Look up to Heaven; look into the Human Heart. Till the planets and the passions, the affections and the fixed stars are extinguished, their names cannot die.
— John Wilson (1741–1793). 		In symbols one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.
— Gottfried Wihelm Leibniz (1646–1716). 		If a lunatic scribbles a jumble of mathematical symbols it does not follow that the writing means anything merely because to the inexpert eye it is indistinguishable from higher mathematics.
— Eric Temple Bell (1883–1960).
Suppose that you want to teach the “cat” concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractible claws, a distinctive sonic output, etc.? I’ll bet not. You probably show the kid a lot of different cats, saying “kitty” each time, until it gets the idea. To put it more generally, generalizations are best made by abstraction from experience.
— R. P. Boas

Reading and writing

You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.
— Karl Friedrich Gauss (1777–1855). 		The last thing one knows when writing a book is what to put first.
— Blaise Pascal (1623–1662), Pensees.
I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery.
— René Descartes (1596–1650), La Geometrie. 		Unfortunately what is little recognized is that the most worthwhile scientific books are those in which the author clearly indicates what he does not know; for an author most hurts his readers by concealing difficulties.
— Evariste Galois (1811–1832).
Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
— Paul Halmos. 		What would I do if I had only six months left to live? I’d type faster.
— Isaac Asimov (1920–1992).
We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all the tracks, to not worry about the blind alleys or describe how you had the wrong idea first, and so on. So there isn’t any place to publish, in a dignified manner, what you actually did in order to get to do the work.
— Richard Feynman (1918–1988), Nobel Lecture, 1966.

Problem solving

Through space the universe grasps me and swallows me up like a speck; through thought I grasp it.
— Blaise Pascal (1623–1662), Pensées. 		Each problem that I solved became a rule which served afterwards to solve other problems.
— René Descartes (1596–1650). 		We have found a strange footprint on the shores of the unknown. We have devised profound theories, one after another, to account for its origins. At last, we have succeeded in reconstructing the creature that made the footprint. And lo! It is our own.
— Sir Arthur Eddington (1882–1944), Space, Time and Gravitation.
Mastery in life is achieved by developing a process of constant and rapid correction, rather than the illusory goal of freedom from error; accomplished musicians, aviators and athletes know this.
— Alexander Franklin Mayer. 		When you have eliminated the impossible, whatever remains, however improbable, must be the truth.
— Sir Arthur Conan Doyle (1859–1930), The Sign of Four. 		You can only find truth with logic if you have already found truth without it.
— G. K. Chesterton (1874–1936).
It is a capital mistake to theorize before one has data.
— Sir Arthur Conan Doyle (1859–1930), Scandal in Bohemia. 		An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them.
— Werner Heisenberg (1901–1976). 		Chance favors only the prepared mind.
— Louis Pasteur (1822–1895).
Order and simplification are the first steps toward mastery of a subject — the actual enemy is the unknown.
— Thomas Mann. 		Guessing is what I do for a living.
— Lee Smolin, physicist. 		When you are faced with a question you can’t answer, it might be the question’s fault rather than your own. Keep improving the question until it answers itself.
— John Kerl.
Dare to be naïve.
— Buckminster Fuller.

Challenges

I don’t just want to live the length of my life — I want to live the width as well.
— Joan Didion (1934–). 		No one knows what he can do till he tries.
— Publius Syrus (1st century BC). 		If a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full.
— G.H. Hardy (1877–1947).
A rock pile ceases to be a rock pile the moment a single man contemplates it, bearing within him the image of a cathedral.
— Antoine de Saint-Exupéry (1900–1944). 		If you’re careful enough, nothing bad or good will ever happen to you.
— Unknown. 		Have the courage to take your own thoughts seriously, for they will shape you.
— Albert Einstein (1879–1955).
We learn to fly not by becoming fearless, but by the daily practice of courage.
— Sam Keen, elderly trapeze artist 		If you worry, you die. If you don’t worry, you also die. So why worry?
— Mike Horn, arctic explorer. 		Don’t be afraid of death so much as an inadequate life.
— Bertolt Brecht (1898–1956).
I have become lighter, freer, less burdened in life, and if my life itself ceases somewhere off on this unknown trajectory on which I have launched myself, it will perhaps make not as much smoke going out as I had thought. I have thrown my dreams into a sack over my shoulder and headed out. The place I am going is the greatest unknown in the world.
— Michael Parfit, on going to Antarctica.

Progress

Silence can be the biggest lie of all. We have a responsibility to speak up; and whenever the occasion calls for it, we have a responsibility to raise bloody hell.
— Herbert Block (1909-2001). 		You must be the change you want to see in the world.
— Mohandas Gandhi (1869–1948). 		The search for truth is more precious than its possession.
— Albert Einstein (1879–1955).
Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.
— Jules Henri Poincaré (1854–1912), La Science et l’Hypothèse. 		Fundamental progress has to do with the reinterpretation of basic ideas.
— Alfred North Whitehead (1861–1947). 		The desire to understand the world and the desire to reform it are the two great engines of progress.
— Bertrand Russell (1872–1970), Marriage and Morals.
The best performance improvement is the transition from the nonworking state to the working state.
— John Ousterhout 		If you want to build a ship, don’t drum up people to collect wood and don’t assign them tasks and work, but rather teach them to long for the endless immensity of the sea.
— Antoine de Saint-Exupéry

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