Frases de E. W. Hobson (20 citas)

“The first period embraces the time between the first records of empirical determinations of the ratio of the circumference to the diameter of a circle until the invention of the Differential and Integral Calculus, in the middle of the seventeenth century. This period, in which the ideal of an exact construction was never entirely lost sight of, and was occasionally supposed to have been attained, was the geometrical period, in which the main activity consisted in the approximate determination of π by the calculation of the sides or the areas of regular polygons in- and circum-scribed to the circle. The theoretical groundwork of the method was the Greek method of Exhaustions. In the earlier part of the period the work of approximation was much hampered by the backward condition of arithmetic due to the fact that our present system of numerical notation had not yet been invented; but the closeness of the approximations obtained in spite of this great obstacle are truly surprising. In the later part of this first period methods were devised by which the approximations to the value of π were obtained which required only a fraction of the labour involved in the earlier calculations. At the end of the period the method was developed to so high a degree of perfection that no further advance could be hoped for on the lines laid down by the Greek Mathematicians; for further progress more powerful methods were required.”

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E. W. Hobson

Fuente: Squaring the Circle (1913), pp. 10-11

“Success, even in a comparatively limited field, is some compensation for failure in a wider field of endeavour.”

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E. W. Hobson

p, 125
Squaring the Circle (1913)

“A great department of thought must have its own inner life, however transcendent may be the importance of its relations to the outside. No department of science, least of all one requiring so high a degree of mental concentration as Mathematics, can be developed entirely, or even mainly, with a view to applications outside its own range. The increased complexity and specialisation of all branches of knowledge makes it true in the present, however it may have been in former times, that important advances in such a department as Mathematics can be expected only from men who are interested in the subject for its own sake, and who, whilst keeping an open mind for suggestions from outside, allow their thought to range freely in those lines of advance which are indicated by the present state of their subject, untrammelled by any preoccupation as to applications to other departments of science. Even with a view to applications, if Mathematics is to be adequately equipped for the purpose of coping with the intricate problems which will be presented to it in the future by Physics, Chemistry and other branches of physical science, many of these problems probably of a character which we cannot at present forecast, it is essential that Mathematics should be allowed to develop freely on its own lines.”

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E. W. Hobson

Fuente: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 286; Cited in: Moritz (1914, 106): Modern mathematics.

“As in Mathematics in general, the really great advances, embodying new ideas of far-reaching fruitfullness, have been due to an exceedingly small number of great men… there are periods when for a long series of centuries no advance was made; when the results obtained in a more enlightened age have been forgotten. We observe the times of revival, when the older learning has been rediscovered, and when the results of the progress made in distinct countries have been made available as the starting points of new efforts and a fresh period of activity.”

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E. W. Hobson

Fuente: Squaring the Circle (1913), p. 10

“The history of our problem falls into three periods marked out by fundamentally distinct differences in respect of method, of immediate aims, and in equipment in possession of intellectual tools.”

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E. W. Hobson

Fuente: Squaring the Circle (1913), p. 10

“The opinion appears to be gaining ground that this very general conception of functionality, born on mathematical ground, is destined to supersede the narrower notion of causation, traditional in connection with the natural sciences. As an abstract formulation of the idea of determination in its most general sense, the notion of functionality includes and transcends the more special notion of causation as a one-sided determination of future phenomena by means of present conditions; it can be used to express the fact of the subsumption under a general law of past, present, and future alike, in a sequence of phenomena. From this point of view the remark of Huxley that Mathematics "knows nothing of causation" could only be taken to express the whole truth, if by the term "causation" is understood "efficient causation." The latter notion has, however, in recent times been to an increasing extent regarded as just as irrelevant in the natural sciences as it is in Mathematics; the idea of thorough-going determinancy, in accordance with formal law, being thought to be alone significant in either domain.”

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E. W. Hobson

Fuente: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 290 ; Cited in: Moritz (1914, 29): The Nature of Mathematics.

“The objects of abstract Geometry possess in absolute precision properties which are only approximately realized in the corresponding objects of physical Geometry.”

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E. W. Hobson

Fuente: Squaring the Circle (1913), p. 5

“I have said that mathematics is the oldest of the sciences; a glance at its more recent history will show that it has the energy of perpetual youth. The output of contributions to the advance of the science during the last century and more has been so enormous that it is difficult to say whether pride in the greatness of achievement in this subject, or despair at his inability to cope with the multiplicity of its detailed developments, should be the dominant feeling of the mathematician. Few people outside of the small circle of mathematical specialists have any idea of the vast growth of mathematical literature. The Royal Society Catalogue contains a list of nearly thirty-nine thousand papers on subjects of Pure Mathematics alone, which have appeared in seven hundred serials during the nineteenth century. This represents only a portion of the total output, the very large number of treatises, dissertations, and monographs published during the century being omitted.”

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E. W. Hobson

Fuente: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 283; Cited in: Moritz (1914, 108-9): Modern mathematics.

“The second part of the book… contains an exposition of the first principles of the theory of complex quantities; hitherto, the very elements of this theory have not been easily accessible to the English student, except recently in Prof. Chrystal's excellent treatise on Algebra. The subject of Analytical Trigonometry has been too frequently presented to the student in the state in which it was left by Euler, before the researches of Cauchy, Abel, Gauss, and others, had placed the use of imaginary quantities, and especially the theory of infinite series and products, where real or complex quantities are involved, on a firm scientific basis. In the Chapter on the exponential theorem and logarithms, I have ventured to introduce the term "generalized logarithm" for the doubly infinite series of values of the logarithm of a quantity.”

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E. W. Hobson

A Treatise on Plane Trigonometry https://books.google.com/books?id=_ktLAAAAMAAJ (1891) Preface

“The quality of the human mind, considered in its collective aspect, which most strikes us, in surveying this record, is its colossal patience.”

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E. W. Hobson

Fuente: Squaring the Circle (1913), p. 12

“In the year 1775, the Paris Academy found it necessary to protect its officials against the waste of time and energy involved in examining the efforts of circle squarers. It passed a resolution… that no more solutions were to be examined of the problem of the duplication of the cube, the trisection of the angle, the quadrature of the circle, and the same resolution should apply to machines for exhibiting perpetual motion. an account… drawn up by Condorcet… is appended. It is interesting to remark that the strength of the conviction of Mathematicians that the solution of the problem is impossible, more than a century before an irrefutable proof of the correctness of that conviction was discovered.”

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E. W. Hobson

Fuente: Squaring the Circle (1913), pp. 3-4

“Much of the skill of the true mathematical physicist and of the mathematical astronomer consists in the power of adapting methods and results carried out on an exact mathematical basis to obtain approximations sufficient for the purposes of physical measurements. It might perhaps be thought that a scheme of Mathematics on a frankly approximative basis would be sufficient for all the practical purposes of application in Physics, Engineering Science, and Astronomy, and no doubt it would be possible to develop, to some extent at least, a species of Mathematics on these lines. Such a system would, however, involve an intolerable awkwardness and prolixity in the statements of results, especially in view of the fact that the degree of approximation necessary for various purposes is very different, and thus that unassigned grades of approximation would have to be provided for. Moreover, the mathematician working on these lines would be cut off from the chief sources of inspiration, the ideals of exactitude and logical rigour, as well as from one of his most indispensable guides to discovery, symmetry, and permanence of mathematical form. The history of the actual movements of mathematical thought through the centuries shows that these ideals are the very life-blood of the science, and warrants the conclusion that a constant striving toward their attainment is an absolutely essential condition of vigorous growth. These ideals have their roots in irresistible impulses and deep-seated needs of the human mind, manifested in its efforts to introduce intelligibility in certain great domains of the world of thought.”

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E. W. Hobson

Fuente: Presidential Address British Association for the Advancement of Science, Section A (1910), pp. 285-286; Cited in: Moritz (1914, 229): Mathematics and Science.

“The actual evolution of mathematical theories proceeds by a process of induction strictly analogous to the method of induction employed in building up the physical sciences; observation, comparison, classification, trial, and generalisation are essential in both cases. Not only are special results, obtained independently of one another, frequently seen to be really included in some generalisation, but branches of the subject which have been developed quite independently of one another are sometimes found to have connections which enable them to be synthesised in one single body of doctrine. The essential nature of mathematical thought manifests itself in the discernment of fundamental identity in the mathematical aspects of what are superficially very different domains. A striking example of this species of immanent identity of mathematical form was exhibited by the discovery of that distinguished mathematician... Major MacMahon, that all possible Latin squares are capable of enumeration by the consideration of certain differential operators. Here we have a case in which an enumeration, which appears to be not amenable to direct treatment, can actually be carried out in a simple manner when the underlying identity of the operation is recognised with that involved in certain operations due to differential operators, the calculus of which belongs superficially to a wholly different region of thought from that relating to Latin squares.”

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E. W. Hobson

Fuente: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 290; Cited in: Moritz (1914, 27): The Nature of Mathematics.

“Perhaps the least inadequate description of the general scope of modern Pure Mathematics—I will not call it a definition—would be to say that it deals with form, in a very general sense of the term; this would include algebraic form, functional relationship, the relations of order in any ordered set of entities such as numbers, and the analysis of the peculiarities of form of groups of operations.”

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E. W. Hobson

Fuente: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 287; Cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/4/mode/2up, (1914), p. 5: Definitions and objects of mathematics.

“Who has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, Sophus Lie, and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous with those requisite for constructive art. Not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form, in conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination.”

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E. W. Hobson

Fuente: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 290. ; Cited in: Moritz (1914, 184): Mathematics as a fine art.