A mathematician starts with a problem and creates a solution; a consultant starts by offering a solution and creates a problem.

A mathematician starts with a problem and creates a solution; a consultant starts by offering a “solution” and creates a problem.

Mathematics is so useful because physical scientists and engineers have the good sense to largely ignore the "religious" fanaticism of professional mathematicians, and their insistence on so-called rigor, that in many cases is misplaced and hypocritical, since it is based on "axioms" that are completely fictional, i.e. those that involve the so-called infinity.

A mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformation of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs.

A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.

Mathematics is merely a convenient tool by which the designer determines the physical proportions and details of a planned structure in order to transform his ideas... to the actuality of a finished structure.

Mathematicians are born, not made.

Mathematicians are coming up with s that they use in their actual mathematical research. And these, like , I think is the name of one of them ... these are actually like s that have been engineered in a way that they can actually be used by working mathematicians to check the work they're doing.

Mathematicians do not deal in objects, but in relations between objects; thus, they are free to replace some objects by others so lone as the relations remain unchanged. Content to them is irrelevant; they are interested in form only.

Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians' observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics. The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real. ...There may be three dimensions in this room and five next door. As a professional mathematician, I have no idea; I can only ask some competent physicist to instruct me in the facts.
The function of a mathematician, then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject-matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics. ...Among them there perhaps none quite so fascinating, with quite the astonishing contrasts of sharp outline and shade, as that which constitutes the theory of numbers.

Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians' observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics. The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real. ...There may be three dimensions in this room and five next door. As a professional mathematician, I have no idea; I can only ask some competent physicist to instruct me in the facts.
The function of a mathematician, then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject-matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics. ...Among them there perhaps none quite so fascinating, with quite the astonishing contrasts of sharp outline and shade, as that which constitutes the theory of numbers.

In their field they [mathematicians] do what we ought to be doing in ours. Therein lies the significant lesson … of their existence. They are an analogy for the intellectual of the future.

In their field they [mathematicians] do what we ought to be doing in ours. Therein lies the significant lesson … of their existence. They are an analogy for the intellectual of the future.

Nor do I doubt that skilled and scholarly mathematicians will agree with me if, what philosophy requires from the beginning, they will examine and judge, not casually but deeply, what I have gathered together in this book to prove these things. ...Mathematics is written for mathematicians, to whom these my labours, if I am not mistaken, will appear to contribute something. ...What... I may have achieved in this, I leave to the decision of your Holiness especially, and to all other learned mathematicians. ...If perchance there should be foolish speakers who, together with those ignorant of all mathematics, will take it upon themselves to decide concerning these things, and because of some place in the Scriptures wickedly distorted to their purpose, should dare to assail this my work, they are of no importance to me, to such an extent do I despise their judgment as rash.