James R Newman - Collected Quotes

“Perhaps the greatest paradox of all is that there are paradoxes in mathematics […] because mathematics builds on the old but does not discard it, because its theorems are deduced from postulates by the methods of logic, in spite of its having undergone revolutionary changes we do not suspect it of being a discipline capable of engendering paradoxes.” (James R Newman, “Mathematics and the Imagination”, 1940) 

“The mathematician is still regarded as the hermit who knows little of the ways of life outside his cell, who spends his time compounding incredible and incomprehensible theorems in a strange, clipped, unintelligible jargon.“ (James R Newman, „Mathematics and the Imagination“, 1940)

"A variety of natural phenomena exhibit what is called the minimum principle. The principle is displayed where the amount of energy expended in performing a given action is the least required for its execution, where the path of a particle or wave in moving from one point to another is the shortest possible, where a motion is completed in the shortest possible time, and so on." (James R Newman, "The World of Mathematics" Vol. II, 1956)

 “Nevertheless, there are three distinct types of paradoxes which do arise in mathematics. There are contradictory and absurd propositions, which arise from fallacious reasoning. There are theorems which seem strange and incredible, but which, because they are logically unassailable, must be accepted even though they transcend intuition and imagination. The third and most important class consists of those logical paradoxes which arise in connection with the theory of aggregates, and which have resulted in a re-examination of the foundations of mathematics.” (James R Newman, "The World of Mathematics” Vol. III, 1956)

“Strictly speaking, mathematical propositions are neither true nor false; they are merely implied by the axioms and postulates which we assume. If we accept these premises and employ legitimate logical arguments, we obtain legitimate propositions. The postulates are not characterized by being true or false; we simply agree to abide by them. But we have used the word true without any of its philosophical implications to refer unambiguously to propositions logically deduced from commonly accepted axioms.” (James R Newman, "The World of Mathematics” Vol. III, 1956)

"That mathematics is a handmaiden of science is a commonplace; but it is less well understood that experiments stimulate mathematical imagination, aid in the formulation of concepts and shape the direction and emphasis of mathematical studies. One of the most remarkable features of the relationship is the successful use of physical models and experiments to solve problems arising in mathematics. In some cases a physical experiment is the only means of determining whether a solution to a specific problem exists; once the existence of a solution has been demonstrated, it may then be possible to complete the mathematical analysis, even to move beyond the conclusions furnished by the model-a sort of boot-strap procedure. It is interesting to point out that what counts in this action and reaction is as much the 'physical way of thinking', the turning over in imagination of physical events, as the actual doing of the experiment." (James R Newman, "The World of Mathematics" Vol. II, 1956)

“The infinite in mathematics is always unruly unless it is properly treated.” (James R Newman, "The World of Mathematics” Vol. III, 1956)

"The second law of thermodynamics provides a more modem (and a more discouraging) example of the maximum principle: the entropy (disorder) of the universe tends toward a maximum." (James R Newman, "The World of Mathematics" Vol. II, 1956)

 “Understanding mathematical logic, or the theory of relativity, is not an indispensable attribute of the cultured mind. But if one wishes to learn anything about these subjects, one must learn something. It is necessary to master the rudiments of the language, to practice a technique, to follow step by step a characteristic sequence of reasoning and to see a problem through from beginning to end.” (James R Newman, "The World of Mathematics” Vol. I, 1956)

“When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to all, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks, of which the logical paradoxes constitute the most important part. For, if mathematics is to advance securely and confidently it must first set its affairs in order at home.” (James R Newman, "The World of Mathematics” Vol. III, 1956)

"There are two ways to teach mathematics. One is to take real pains toward creating understanding - visual aids, that sort of thing. The other is the old British style of teaching until you’re blue in the face." (James R Newman)