The History of the Calculus and Its Conceptual Development Quotes

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The History of the Calculus and Its Conceptual Development The History of the Calculus and Its Conceptual Development by Carl B. Boyer
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“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“Cournot protested that concepts exist in the understanding, independently of the definition which one gives to them. Simple ideas sometimes have complicated definitions, or even none. For this reason he felt that one should not subordinate the precision of such ideas as those of speed or the infinitely small to logical definition. This point of view is diametrically opposed to that which analysis since the time of Cournot has been toward ever-greater care in the formal logical elaboration of the subject. This trend, initiated in the first half of the nineteenth century and fostered largely by Cauchy, was in the second half of that century continued with notable success by Weierstrass.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“Mathematics is unable to specify whether motion is continuous, for it deals merely with hypothetical relations and can make its variable continuous or discontinuous at will. The paradoxes of Zeno are consequences of the failure to appreciate this fact and of the resulting lack of a precise specification of the problem. The former is a matter of scientific description a posteriori, whereas the latter is a matter solely of mathematical definition a priori. The former may consequently suggest that motion be defined mathematically in terms of continuous variable, but cannot, because of the limitations of sensory perception, prove that it must be so defined.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“The mathematical theory of continuity is based, not on intuition, but on the logically developed theories of number and sets of points.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“Mathematics has, of course, given the solution of the difficulties in terms of the abstract concept of converging infinite series. In a certain metaphysical sense this notion of convergence does not answer Zeno’s argument, in that it does not tell how one is to picture an infinite number of magnitudes as together making up only a finite magnitude; that is, it does not give an intuitively clear and satisfying picture, in terms of sense experience, of the relation subsisting between the infinite series and the limit of this series.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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. That is, mathematics determines what conclusions will follow logically from given premises. The conjunction of mathematics and philosophy, or of mathematics and science, is frequently of great service in suggesting new problems and points of view.
Nevertheless, in the final rigorous formulation and elaboration of such concepts as have been introduced, mathematics must necessarily be unprejudiced by an irrelevant elements in the experiences from which they have arisen.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
tags: rigor
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
tags: series
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“Robins realized more clearly than did Jurin the nature of the limit concept. He recognized that the phrase "the ultimate ratio of vanishing quantities" was a figurative expression, referring , not to a last ratio, but to a "fixed quantity which some varying quantity, by a continual augmentation of diminuation shall perpetually approach,... provided the varying quantity can be made in its approach to the other to differ from it by less than by and quantity how minute soever, that can be assigned, .."though it can never be made absolutely equal to it.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“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.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“Such attempts lacked all semblance of mathematical rigor because of the lack at that time of satisfactory definitions of either the infinite of the infinitesimal. Arithmetic had not become sufficiently abstract and symbolic to free itself of spatial interpretations, for number was still interpreted metrically as a ratio of geometrical magnitudes.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“Most of his predecessors had considered the differential calculus as bound up with geometry, but Euler made the subject a formal theory of functions which had no need to revert to diagrams or geometrical conceptions.”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development
“A quantity is something or nothing: if it is something, it has not yet vanished; if it is nothing, it has literally vanished. The supposition that there is an intermediate state between these two is a chimera.

D'Alembert”
Carl B. Boyer, The History of the Calculus and Its Conceptual Development