Serge Lang Quotes

Before one can generalize, formalize, and axiomatize, there must be a mathematical substance.

Proof is an idol before whom the pure mathematician tortures himself.

Well it was not exactly a dissertation in logic, at least not the kind of logic you would find in Whitehead and Russell's Principia Mathematica for instance. It looked more like mathematics; no formalized language was used.

The system becomes logically closed when each of the logical implications which can be derived from any one proposition within the system finds its statement in another proposition in the same system.

Logic and mathematics are nothing but specialised linguistic structures.

When I dealt with set theory, I could never make it be the music that I wanted.

There is no royal road to geometry.

The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science.

A mathematical theorem can be true whether or not it is realized in the world.

I am never satisfied until I can make a mathematical model of what I am studying. If I succeed in making one, I understand. If I cannot, I do not understand.

I can see that, without being excited, mathematics can look pointless and cold.

Our mathematics of the last few decades has wallowed in generalities and formalizations.

One can't proceed from the informal to the formal by formal means.

Mathematics, in its widest significance, is the development of all types of formal, necessary, deductive reasoning.

I am obliged to interpolate some remarks on a very difficult subject: proof and its importance in mathematics. All physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.

Of these austerer virtues the love of truth is the chief, and in mathematics, more than elsewhere, the love of truth may find encouragement for waning faith. Every great study is not only an end in itself, but also a means of creating and sustaining a lofty habit of mind; and this purpose should be kept always in view throughout the teaching and learning of mathematics.

Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics.

..., twice two is four is not life, gentlemen, but the beginning of death.

Logic is an old subject, and since 1879 it has been a great one.

The Syllogism consists of propositions, propositions consist of words, words are symbols of notions. Therefore if the notions themselves (which is the root of the matter) are confused and over-hastily abstracted from the facts, there can be no firmness in the superstructure. Our only hope therefore lies in a true induction.

Mathematicians do not study objects, but relations among objects; they are indifferent to the replacement of objects by others as long the relations don't change. Matter is not important, only form interests them.

Mathematical reasoning may be regarded.

There can never be surprises in logic.

Mathematics is a language plus reasoning. It's like a language plus logic. Mathematics is a tool for reasoning.

I found in rules of mathematics a peace and a trust that I could not place in human beings. This sublimation was total and remained total.