“Objections… inspired Kronecker and others to attack Weierstrass' "sequential" definition of irrationals. Nevertheless, right or wrong, Weierstrass and his school made the theory work. The most useful results they obtained have not yet been questioned, at least on the ground of their great utility in mathematical analysis and its implications, by any competent judge in his right mind. This does not mean that objections cannot be well taken: it merely calls attention to the fact that in mathematics, as in everything else, this earth is not yet to be confused with the Kingdom of Heaven, that perfection is a chimaera, and that, in the words of Crelle, we can only hope for closer and closer approximations to mathematical truth—whatever that may be, if anything—precisely as in the Weierstrassian theory of convergent sequences of rationals defining irrationals.”

— Eric Temple Bell , book Men of Mathematics

Men of Mathematics (1937)

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Eric Temple Bell 14

mathematician and science fiction author born in Scotland w… 1883–1960

Tobias Dantzig (1884–1956) American mathematician

p, 125
Number: The Language of Science (1930)

“One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics.”

Pierre Louis Maupertuis (1698–1759) French mathematician, philosopher and man of letters

Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique (1746)

“The truth is that the mathematical sciences are growing in complete security and harmony. The ideas of Dedekind, Poincare, and Hilbert have been systematically developed with great success, without any conflict in the results. It is only from the philosophical point of view that objections have been raised. They bear on certain ways of reasoning peculiar to analysis and set theory. These modes of reasoning were first systematically applied in giving a rigorous form to the methods of the calculus. [According to them, ] the objects of a theory are viewed as elements of a totality such that one can reason as follows: For each property expressible using the notions of the theory, it is [an] objectively determinate [fact] whether there is or there is not an element of the totality which possesses this property. Similarly, it follows from this point of view that either all the elements of a set possess a given property, or there is at least one element which does not possess it”

Paul Bernays (1888–1977) Swiss mathematician

Paul Bernays, Platonism in mathematics http://sites.google.com/site/ancientaroma2/book_platonism.pdf (1935)

“Necessary truth is merely the subject-matter of mathematics, not the reward we get for doing mathematics. The object of mathematics is not, and cannot be, mathematical certainty. It is not even mathematical truth, certain or otherwise. It is, and must be, mathematical explanation.”

David Deutsch book The Fabric of Reality

The Fabric of Reality (1997)

“When we went to school we were told that we were governed by laws, not men. As a result of that, many people think there is no need to pay any attention to judicial candidates because judges merely apply the law by some mathematical formula and a good judge and a bad judge all apply the same kind of law. The fact is that the most important part of a judge's work is the exercise of judgment and that the law in a court is never better than the common sense judgment of the judge that is presiding.”

Robert H. Jackson (1892–1954) American judge

Reported in Eugene Gerhart, America's Advocate: Robert H. Jackson (1958), p. 289

“The assumption that individuals act objectively in accordance with purely mathematical dictates to maximize their gain or utility cannot be sustained by empirical observation.”

Richard Arnold Epstein (1927) American physicist

Epilogue, p. 410
The Theory of Gambling and Statistical Logic (Revised Edition) 1977

“The object of this paper is to give a satisfactory account of the Foundations of Mathematics in accordance with the general method of Frege, Whitehead and Russell. Following these authorities, I hold that mathematics is part of logic, and so belong to what may be called the logical school as opposed to the formalist and intuitionist schools. I have therefore taken Principia Mathematica as a basis for discussion and ammendment; and believe myself to have discovered how, by using the work of Mr Ludwig Wittgenstein, it can be rendered free from the serious objections which have caused its rejection by the majority of German authorities, who have deserted altogether its line of approach.”

Frank P. Ramsey (1903–1930) British mathematician, philosopher

Preface
The Foundations of Mathematics (1925)

“The Mathematical School; Although mathematical methods can be used by any school of management theory, and have been, I have chosen to group under a school those theorists who see management as a system of mathematical models and processes. Perhaps the most widely known group I arbitrarily so lump are the operations researchers or operations analysts, who have sometimes anointed themselves with the rather pretentious name of "management scientists." The abiding belief of this group is that, if management, or organization, or planning, or decision making is a logical process, it can be expressed in terms of mathematical symbols and relationships. The central approach of this school is the model, for it is through these devices that the problem is expressed in its basic relationships and in terms of selected goals or objectives.”

Harold Koontz (1909–1984)

Source: "The Management Theory Jungle," 1961, p. 181

“A possible explanation of the physicist's use of mathematics to formulate his laws of nature is that he is a somewhat irresponsible person. As a result, when he finds a connection between two quantities which resembles a connection well-known from mathematics, he will jump at the conclusion that the connection is that discussed in mathematics simply because he does not know of any other similar connection.”

Eugene Paul Wigner The Unreasonable Effectiveness of Mathematics in the Natural Sciences

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Communications in Pure and Applied Mathematics, February 1960.

“The fact that man knows right from wrong proves his intellectual superiority to the other creatures; but the fact that he can do wrong proves his moral inferiority to any creature that cannot.”

Mark Twain book What Is Man?

Source: What Is Man? (1906), Ch. 6

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