E. W. Hobson: Quotes about problems

“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.”

E. W. Hobson

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

Life , Men , Mathematics , Knowledge

“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.”

E. W. Hobson

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

History , Problem

“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.”

E. W. Hobson

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

Problem , Effort , Time

“The second period, which commenced in the middle of the seventeenth century, and lasted for about a century, was characterized by the application of the powerful analytical methods provided by the new Analysis to the determination of analytical expressions for the number π in the form of convergent series, products, and continued fractions. The older geometrical forms of investigation gave way to analytical processes in which the functional relationship as applied to the trigonometrical functions became prominent. The new methods of systematic representation gave rise to a race of calculators of π, who, in their consciousness of the vastly enhance means of calculation placed in their hands by the new Analysis, proceeded to apply the formulae to obtain numerical approximations to π to ever larger numbers of places of decimals, although their efforts were quite useless for the purpose of throwing light upon the true nature of that number. At the end of this period no knowledge had been obtained as regards the number π of the kind likely to throw light upon the possibility or impossibility of the old historical problem of the ideal construction; it was not even definitely known whether the number is rational or irrational. However, one great discovery, destined to furnish the clue to the solution of the problem, was made at this time; that of the relation between the two numbers π and e, as a particular case of those exponential expressions for the trigonometrical functions which form one of the most fundamentally important of the analytical weapons forged during this period.”

E. W. Hobson

Source: Squaring the Circle (1913), pp. 11-12

Way , Nature , Knowledge , Problem

“We are able to appreciate the difficulties which in each age restricted the progress which could be made within limits which could not be surpassed by the means then available; we see how, when new weapons became available, a new race of thinkers turned to the further consideration of the problem with a new outlook.”

E. W. Hobson

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

Age , Problem

“If the question be raised, why such an apparently special problem as the quadrature of the circle, is deserving of the sustained interest which has attained to it, and which it still possesses, the answer is only to be found in a scrutiny of the history of the problem, and especially in the closeness of the connection of that history with the general history of Mathematical Science.”

E. W. Hobson

Source: Squaring the Circle (1913), p. 2

History , Mathematics , Question , Problem