Quotes from Carl B. Boyer

Voltaire called the calculus "the Art of numbering and measuring exactly a Thing whose Existence cannot be conceived." See Letters Concerning the English Nation p. 152

Now we can see what makes mathematics unique. Only in mathematics is there no significant correction—only extension. Once the Greeks had developed the deductive method, they were correct in what they did, correct for all time. Euclid was incomplete and his work has been extended enormously, but it has not had to be corrected. His theorems are, every one of them, valid to this day.

For this reason Archimedes considered that this method merely indicated, but did not prove, that the result is correct.

The Greek thinkers was no way of bridging the gap between the rectilinear and the curvilinear which would at the same time satisfy their strict demands of mathematical rigor and appeal to the clear evidence of sensory experience.

This attitude adds nothing to the explanation of the paradoxes, for it fails to recognize that the conception of motion at a point, which is the crux of the situation, is not a scientific notion but a mathematical abstraction.

Berkeley was unable to appreciate that mathematics was not concerned with a world of "real" sense impressions. In much the same manner today some philosophers criticize the mathematical conceptions of infinity and continuum, failing to realize that since mathematics deals with relations rather than with physical existence, its criterion of truth is inner consistency rather than plausibility in the light of sense perception of intuition.

Definitions of number, as given by several later mathematicians, make the limit of an infinite sequence identical with the sequence itself. Under this view, the question as to whether the variable reaches its limit is without logical meaning. Thus the infinite sequence .9, .99, .999,... is the number one, and the question, "Does it ever reach one?" is an attempt to give a metaphysical argument which shall satisfy intuition.

Carnot, one of a school of mathematicians who emphasized the relationship of mathematics to scientific practice, appears, in spite of the title of his work, to have been more concerned about the facility of application of the rules of procedure than about the logical reasoning involved.

In making the basis of the calculus more rigorously formal, Weierstrass also attacked the appeal to intuition of continuous motion which is implied in Cauchy's expression -- that a variable approaches a limit.

Thus the required rigor was found in the application of the concept of number, made formal by divorcing it from the idea of geometrical quantity

The mathematical theory of continuity is based, not on intuition, but on the logically developed theories of number and sets of points.

Ever since the empirical mathematics of the pre-Hellenic world was developed, the attitude has, upon occasion, been maintained that mathematics is a branch either of empirical science of of transcendental philosophy. In either case mathematics is not free to develop as it will, but is bound by certain restrictions: by conceptions derived either a posteriori from natural science, or assumed to be imposed a priori by an absolutistic philosophy.

The failure of Aristotle to distinguish sharply between the worlds of experience and of mathematical thought resulted in his lack of clear recognition of a similar confusion in the paradoxes of Zeno.

Newton had considered the calculus as a scientific description of the generation of magnitudes, and Leibniz had viewed it as a metaphysical explanation of such generation. The formalism of the nineteenth century took from the calculus any such preconceptions, leaving only the bare symbolic relationships between abstract mathematical entities.

Mach also felt strongly the empirical origin of mathematics and held with Aristotle that geometric concepts are the product of idealization of physical experiences of space. In conformity necessarily to be given to the number i. In this respect he is in agreement with a number of present-day scientists, who feel that the square root of -1 simply "forms a part of various ingenious devices for handling otherwise intractable situations.

Materialistic and idealistic philosophies have both failed to appreciate the nature of mathematics, as accepted at the present time. Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth.

As the sensations of motion and discreteness led to the abstract notions of the calculus, so may sensory experience continue thus to suggest problem for the mathematician, and so may she in turn be free to reduce these to the basic formal logical relationships involved. Thus only may be fully appreciated the twofold aspect of mathematics: as the language of a descriptive interpretation of the relationships discovered in natural phenomena, and as a syllogistic elaboration of arbitrary premise.

These results were obtained by making up tables in which were listed the volumes for given sets of values of the dimensions, and from these selecting the best proportions.

However, inasmuch as the number of parts is infinite, the aggregation of these is not one resembling a very fine powder but rather a sort of merging of parts into unity, as in the case of fluids.

Torricelli fully realized the advantages and disadvantages of the method of indivisibles; and he suspected that the ancients possessed some such method for discovering difficult theorems, the proofs of which they cast in another form either "to hide the secret of their method or to avoid giving occasion for contradiction to jealous detractors.

A thorough-going empiricist for whom mathematics was a method rather than an explanation, Newton apparently considered any attempt to question the instantaneity of motion as linked with metaphysics, and so avoided framing a definition of it.

Leibniz in this respect had perhaps even less caution than many of his contemporaries, for he seriously considered whether the infinite series 1 -1+1-1+... was equal to 1/2.

Berkeley explained that by finding the tangent by means of differentials, one first assumes increments; but these determine the secant, not the tangent. One undoes this error, however, by neglecting higher differentials, and thus "by virtue of a twofold mistake you arrive, though not at science, yet at the truth.

Recognizing that geometry is entirely intellectual and independent of the actual description and existence of figures, Fontenelle did not discuss the subject fro the point of view of science or metaphysics as had Aristotle and Leibnez.