Vitali's classical proof of the existence of an unmeasurable set is fairly short and elementary. It can be found in any book on Lebesgue integration. A sketch of the proof might be useful in the appendix to a calculus book. The typical calculus student would not understand the proof, but a few students might understand a few parts of the proof. In any case, the student would go away believing (perhaps on trust rather than on understanding) that there do exist unmeasurable sets.
Robert Bartle, Ralph Henstock, Jaroslav Kurzweil, Eric Schechter, Stefan Schwabik, and Rudolf V?born?, An open letter to the authors of calculus textbooks, Jan 97.

Why should governments fund basic research? For the same reason that companies tend not to: because it is a public good.
Bill Gates, We need clean-energy innovation, and lots of it, 3.8.15.

If we had their [Hamas's] military capabilities, we would have achieved a hundred times what they have achieved. By the way, they didn't achieve anything so far.
Muhammad Dahlan, former Palesitinian Authority Security Chief, calculates 100 times 0. Interview to Al-Kahera Wal-Nas TV, Egypt, 17.7.14.

The nebech index of a sentence is the number of places in the sentence where the word nebech can be inserted so that the sentence remains appropriate, though of different flavor. The most perfect "nebech three" sentence is Decartes' statement: Cogito, ergo sum.
Stanislaw Ulam, Adventures of a Mathematician. (Definition slightly rephrased to shorten the quote.)

Rabbi Elazar Ben Azarya says: If you cannot conjecture, you cannot prove. If you cannot prove, you cannot conjecture.
Mishna Avot 3:17, according to the interpretation of Rabbi Ovadia of Bartenora.

To know how long it will take you to finish writing a paper, multiply the time you believe it will take by 3 and add 2 weeks.
(Probably) Taras Banakh.

The problem is to find some meaning to all that.
Wieslaw Kubis, Eilat Conference lecture, May 2013, after a 30 minute description of a category-theoretic machinery.

I'm running out of cardinals.
Menachem Kojman in his lecture, Eilat Conference, 20 May 2013.

I have now an even faster attack with complexity of 2^178.
Nicolas Tadeusz Courtois, Code Breakers discussio, LinkedIn, 16 March 2012.

Why would marginalization of the core [of mathematics] be a problem, if one is not particularly interested in the subject itself? In fact, core mathematics provides a rigid skeleton that supports the muscles of science, engineering, and applied mathematics. It is relatively invisible because it cannot interact directly with the outside world; it grows slowly; and it would not cause immediate problems if it stopped growing. Premodern mathematics and contemporary mathematical science, on the other hand, are more like exoskeletons: in direct contact with reality but putting strong constraints on size and power. The long-term consequence of mathematical osteoporosis is that science would have to go back to being a bug!
Frank Quinn, A Revolution in Mathematics?, Notices AMS 59 (1), 2012

Beautiful mathematics eventually tends to be useful, and useful mathematics eventually tends to be beautiful.
Meyer, Carl (2000), Matrix analysis and applied linear algebra, summarises the difference between pure and applied mathematics.

Technical skill is a master of complexity while creativity is a master of simplicity. E. Christopher Zeeman (1925-)

A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself.
Lev Tolstoy

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more prominent than theirs, it is because they are made with ideas.

It is undeniable that a gift for mathematics is one of the most specialized talents, and that mathematicians as a class are not particularly distinguished for general ability or versatility. If a man is in any sense a real mathematician, then it is a hundred to one that his mathematics will be far better than anything else he can do, and that he would be silly if he surrendered any decent opportunity of exercising his one talent in order to do undistinguished work in other fields. Such a sacrifice could be justified only by economic necessity or age.

Is not the position of an ordinary applied mathematician in some ways a little pathetic? If he wants to be useful, he must work in a humdrum way, and he cannot give full play to his fancy even when he wishes to rise to the heights. ‘Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.

G. H. Hardy, A Mathematician's Apology, 1941.

The essence of mathematics lies in its freedom.
Georg Cantor, Mathematische Annalen, 1883.

Gödel's own "Gödel", so to speak.
John Dawson, Logical Dilemmas, AK Peters 1996.
(In old German, Gödel means God Father.)

Classes and concepts may ... also be conceived as real objects. ... [their existence] is quite as legitimate as the assumption of physical bodies ... [since] they are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions.
Kurt Gödel, Russell's Mathematical Logic, 1944.

Once the Continuum Hypothesis is droppen, the key problem concerning the structure of the continuum ... is ... the question whether there exists a set of sequences of integers of power Aleph₁ which for any given sequence of integers contains one majoring it from a certain point on.
Kurt Gödel, in a letter to Paul Cohen, asking whether d=&alefsym₁. (From John Dawson's Logical Dilemmas)

Mind &isin Reality &isin Mind.
Haim Judah

I love mathematics not only for its technical applications, but principally because it is beautiful; becuase man has breathed his spirit of play into it, and because it has given him his greatest game - the encompassing of the infinite.
Rozsa Peter, Playing with Infinity

I bealieve that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our creations, are simply our notes of our observations.
G. H. Hardy, A Mathematician's Apology

The typical mathematician is a Platonist on weekdays and a Formalist on Sundays.
Reuben Hersh, The Mathematical Experience

On foundations we believe in the reality of mathematics, but of course when philosophers attack us with their paradoxes we rush to hid behind formalism...
J. A. Dieudonne, The Work of Nickolas Bourbaki

There is no permanent place in the world for ugly mathematics.
G. H. Hardy, A Mathematician's Apology

The aesthetic rather than the logical is the dominant element in mathematical creativity.
Poincare

A designer knows he has arrived at perfection not when there is no longer anything to add, but when there is no longer anything to take away.
Antoine de Saint-Exupery

The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.
Bill Thurston, On Proof and Progress in Mathematics

The derivative of a real-valued function f in a domain D is the Lagrangian section of the cotangent bundle T*(D) that gives the connection form for the unique flat connection on the trivial R-bundle D × R for which the graph of f is parallel.
Bill Thurston, On Proof and Progress in Mathematics

We show, in a certain precise sense, that the Goldbach conjecture is true with probability larger than 0.99999, and that its complete truth could be determined with a budget of $10 billion.
Doron Zeilberger, Theorems for a price: Tomorrow’s semi-rigorous mathematical culture

One has only to open one’s eyes to see that the triumphs of industry, which have enriched so many practical men, would never have seen the light if only these practical men had existed, and if they had not been preceded by disinterested fools who died poor, who never thought of the useful, and yet had a guide that was not their own caprice.
Henri Poincar´e, Science and Method, 1908

. . . the majority of people do not like to think, and that may be for the best, for they are guided by instinct . . . But instinct is a routine, and if not fructified by thought it would not progress any further even in man than in the bee or the ant. Consequently, it is necessary that someone think on behalf of those who don’t like to think ...
H.Poincar´e, Science and Method; Book 1, The Scholar and Science, 1908

Recommendations for a bad referee:

• Check the references and the text of the paper to see whether your own papers are cited approvingly.
• Ask the authors to mention your name or cite papers of yours that are irrelevant to their paper.
• Immediately accept a paper if one of the (co)authors is famous, and immediately reject it if none of the (co)authors is famous.
• Use your referee report as an excuse to promote your own research work and agenda and demean the work of others.
• Do your referee task as late as possible, and then do a rushed and sloppy job, or reject it on the basis of general interest, typos, or appropriateness.
• Write a letter to the authors and give them some suggestions or comments to improve the paper. Ask them to withdraw the paper, to add your name to the paper as a coauthor, and then to submit it to the same or another journal.
• Give rude and unkind comments about the authors (especially beginning mathematicians) and their abilities.
• Be vague and nonobjective when writing the referee report.
• Reject a paper by stating that its easy parts are trivial, and its other parts (which you were too lazy to read) are unjustified.
• Reject since "the only method used is Modus Ponens".
• Reject a paper establishing a difficult identity or equivalence since it "only provides reformulations".

Math jokes

A mathematician has spent ten years trying to prove the Riemann hypothesis. Finally, he decides to sell his soul to the devil in exchange for a proof. The devil promises to deliver a proof in four weeks. Half a year later, the devil shows up again - in a rather gloomy mood. "I'm sorry", he says. "I couldn't prove the hypothesis either. But" - and his face lightens up - "have you thought about embedding the problem in 10-dimensional space?"

"This is a one line proof...if we start sufficiently far to the left."

"The remaining 253 cases are analogous."

Proof by exhaustion was first used by Archimedes. Today, this method is very popular in seminar talks.

- Imagine what Einstein and Pythagoras could have proven, had they been given the opportunity to collaborate!
- Yeah. That e=m(a²+b²)...

Q: Why do Computer Scientists get Halloween and Christmas mixed up?
A: Because Oct. 31 = Dec. 25.

Recycled Carpentry