“What I have achieved has been largely a matter of chance. Many problems I thought about at length with no success. With other problems, there was the inspiration—indeed, some that astound me today.”
— Robert Langlands

“Humans are in danger of losing their economic value because intelligence is decoupling from consciousness.”
— Yuval Noah Harari

“Everywhere one finds this miserable way of concluding from the special to the general, and it is extremely peculiar that such a procedure has led to do few of the so-called paradoxes.”
— Niels Henrik Abel

“Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.”
— Évariste Galois; the closing lines of a letter he wrote to his friend, dated two days before he died.

“The moving power of mathematical invention is not reasoning, but imagination.”
— Augustus De Morgan

“No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it gives the impression of also being beautiful.”
— George Boole

“For any mathematical problem solving, it never hurts to start by giving things names.”
— 3Blue1Brown, But why is a sphere's surface area four times its shadow?

“Keep some room in your heart for the unimaginable.”
— Mary Oliver

“No man can be a pure specialist without being in the strict sense an idiot.”
— George Bernard Shaw

“As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention.”
— James Joseph Sylvester

“My definition of a scientist is that you can complete the following sentence: ‘he or she has shown that…’.”
— E. O. Wilson

“[Ideas] are like shadows — substantial enough until we try to grasp them.”
— Samuel Butler

“The chief danger to philosophy is narrowness in the selection of evidence.”
— Alfred North Whitehead

“There is much pleasure to be gained from useless knowledge.”
— Bertrand Russell

“So in all human affairs one notices, if one examines them closely, that it is impossible to remove one inconvenience without another emerging.”
— Niccolò Machiavelli

“In every science, after having analysed the ideas, expressing the more complicated by means of the more simple, one finds a certain number that cannot be reduced among them, and that one can define no further. These are the primitive ideas of the science; it is necessary to acquire them through experience, or through induction; it is impossible to explain them by deduction.”
— Giuseppe Peano

“We'll continue tomorrow — if I live.”
— Paul Erdős

“Intuitionistic mathematics consists in mental constructions; a mathematical theorem expresses a purely empirical fact, namely the success of a certain construction.”
— Arend Heyting

“Mathematics is not a deductive science — that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does. Possibly philosophers would look on us mathematicians the same way as we look on the technicians, if they dared.”
— Paul Halmos

“When a measure becomes a target, it ceases to be a good measure.”
— Goodhart's law

“You can’t use up creativity. The more you use, the more you have.”
— Maya Angelou