“As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.”
“It is by logic that we prove, but by intuition that we discover.”
“The tool which serves as an intermediary between theory and practice, between thought and observation, is mathematics; it is mathematics which builds the linking bridges and gives the ever more reliable forms.”
"Mathematical science shows what it is. It is the language of unseen relations between things. But to use and apply that language, we must be fully able to appreciate, to feel, to seize the unseen, the unconscious."
“Is it not remarkable that 6 sheep plus 7 sheep make 13 sheep; that 6 stones plus 7 stones make 13 stones? Is it not a miracle that the universe is so constructed that such a simple abstraction as a number is possible?”
“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”
“The pure mathematician, like the musician, is a free creator of his world of ordered beauty.”
“Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of constraint and freedom that is the very essence of human nature.”
“The object of pure physics is the unfolding of the laws of the intelligible world; the object of pure mathematics that of unfolding the laws of human intelligence.”
“[A]ll our science, measured against reality, is primitive and childlike – and yet it is the most precious thing we have.”
“Mathematics is the art of giving the same name to different things.”
“The art of doing mathematics consists in finding that special case which contains all the germs of generality.”
“The essence of mathematics lies in its freedom.”
“It seems to me that the poet has only to perceive that which others do not perceive, to look deeper than others look. And the mathematician must do the same thing.”
"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity."
"The enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it."
"Those who have learned to walk on the threshold of the unknown worlds, by means of what are commonly termed par excellence the exact sciences, may then, with the fair white wings of imagination, hope to soar further into the unknown amidst which we live."
"Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down."
“"...[W]ithout fantasy one would never become a mathematician, and what gave me a place among the mathematicians of our day, despite my lack of knowledge and form, was the audacity of my thinking."
“Only entropy comes easy.”
“Mathematics is the gate and key to science.”
“The purpose of computing is insight, not numbers.”
“Mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency.”
“Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.”
“To not know math is a severe limitation to understanding the world.”
“If there is one thing in mathematics that fascinates me more than anything else (and doubtless always has), it is neither ‘number’ nor ‘size,’ but always form.”
“In the future, as in the past, the great ideas [of mathematics] must be simplifying ideas, the creator must always be one who clarifies, for himself, and for others, the most complicated issues of formulas and concepts.”
“Either mathematics is too big for the human mind or the human mind is more than a machine.”
“Mathematics is the most beautiful and most powerful creation of the human spirit.”
“The longer mathematics lives the more abstract — and therefore, possibly also the more practical — it becomes.”
“Logic and mathematics are nothing but specialized linguistic structures.”
“When we talk about mathematics, we may be discussing a secondary language built on the primary language of the nervous system.”
“Small minds are concerned with the extraordinary, great minds with the ordinary.”
“The man of science simply uses with scrupulous exactness the methods which we all, habitually and at every minute, use carelessly.”
“Do not imagine that mathematics is hard and crabbed, and repulsive to common sense. It is merely the etherealization of common sense.”
“The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head.”
“A mathematician who is not also something of a poet will never be a complete mathematician.”
“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 also gives the impression of being beautiful.”
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without any appeal to our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest artist can show.”
“Besides language and music, mathematics is one of the primary manifestations of the free creative power of the human mind.”
“Mathematics is the music of reason.”
“Pure mathematics is, in its way, the poetry of logical ideas.”
“Thus, in a sense, mathematics has been advanced by those who distinguish themselves by intuition rather than by rigorous proofs.”
“The moving power of mathematical invention is not reasoning but imagination.”
“Logic merely sanctions the conquests of the intuition.”
“The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.”
“One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.”
“To ask the right question is harder than to answer it.”
“Many…confuse [mathematics] with arithmetic, and consider it an arid science. In reality, however, it is a science which requires a great amount of imagination.”
“Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician.”