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Carl B. Boyer Biography

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“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.”

Logic Mathematics

Book:The History of the Calculus and Its Conceptual Development

Source: The History of the Calculus and Its Conceptual Development

“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.”

Book:The History of the Calculus and Its Conceptual Development

Source: 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.”

Calculus Euler Functions

Book:The History of the Calculus and Its Conceptual Development

Source: 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.”

Calculus Motion Zeno

Book:The History of the Calculus and Its Conceptual Development

Source: 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”

Book:The History of the Calculus and Its Conceptual Development

Source: 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.”

Calculus Logical Reasoning

Book:The History of the Calculus and Its Conceptual Development

Source: The History of the Calculus and Its Conceptual Development

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

Prove Proof Archimedes

Author:Carl B. Boyer