# Popular Science Monthly/Volume 80/April 1912/A Review of Three Famous Attacks Upon the Study of Mathematics as a Training of the Mind

A REVIEW OF THREE FAMOUS ATTACKS UPON THE STUDY OF MATHEMATICS AS A TRAINING OF THE MIND |

By Professor FLORIAN CAJORI

COLORADO COLLEGE

NO doubt the most famous attack that has ever been made upon mathematics and its educational value was published in 1836 in the *Edinburgh Review* by Sir William Hamilton, professor of logic and metaphysics at Edinburgh. He must not be confounded with his contemporary, Sir William Rowan Hamilton, the inventor of quaternions. The first reading of that article by the Edinburgh philosopher makes one feel as if in an earthquake in which one's most cherished pedagogic structures are tumbling into a heap and the very foundations are being removed from under one's feet. With the strength of a superhuman giant Hamilton seems to hurl facts with unerring destructive power against the most massive educational castles of his day. The lack of utility of mathematical study, as a training of the mind, is shown by quotations from an array of authorities, gathered from all ages and nations of the civilized world, and the reader is utterly overwhelmed by this "cloud of witnesses."

Upon a second reading of Hamilton's essay one begins to see signs of weakness; an attempt to verify his quotations discloses superficiality and carelessness in the selection of representative quotations from his witnesses. I know of only one mathematician who has made an extended reply to Hamilton, though several have criticized certain parts of his essay. This extended reply is found in an article by A. T. Bledsoe in the *Southern Review* for July, 1877. Bledsoe, a graduate of West Point, was before the civil war professor of mathematics at Kenyon College, then at Miami University, and finally at the University of Virginia. Later he became editor of the *Southern Review.* His reply to Hamilton was printed in the year of his death. It was written after a most careful examination of the authorities cited by Hamilton. It is a very able article, but so far as I have been able to ascertain, it has completely escaped the attention of mathematicians. We can recommend it as interesting and even now worth reading.

A few years ago the noted German mathematician, Alfred Pringsheim, wrote a popular address on the "Utility and Alleged Inutility of Mathematics."^{[1]} Pringsheim, in referring to Hamilton's article, writes down the names of a dozen authorities cited by Hamilton, and then says: "I am ashamed to confess that before reading Hamilton's article I did not know a single one of these great authorities even by name; an extenuating circumstance is the fact that some of these names I could not find even in the scientific directories." However, Hamilton does quote from several noted mathematicians—D'Alembert, Descartes, Pascal, Dugald Stewart—men whose opinions are worthy of serious consideration and study.

Let us now take up Hamilton's essay. It takes the form of a review of William Whewell's "Thoughts on the Study of Mathematics as a Part of a Liberal Education," published in 1835. Whewell was at that time fellow and tutor in Trinity College, Cambridge. Later he became head master of Trinity. At that time the University of Cambridge was laying unusual stress upon mathematics; mathematical skill was the chief requirement in the tripos examinations. Hamilton looked upon the Cambridge plan with disfavor and seized upon Whewell's small pamphlet as a pretext to enter upon a demonstration of the inutility of mathematical study as an exercise of the mind.

In every dispute it is necessary to state the issue clearly and then to adhere to it steadily. The issue is thus stated by Hamilton:^{[2]}

*science*, considered in itself, but the utility of mathematical

*study*, as an exercise of the mind; and in the second, that the expediency is not disputed of leaving mathematics, as a coordinate, to find its level among the other branches of academical instruction. It is only contended that they ought not to be made the

*principal*, far less the

*exclusive*object of encouragement. We speak not of

*professional*, but of

*liberal*education.

This statement of the issue is quite clear. Moreover, the position taken here is quite fair. Few educators of the present time would take marked exception to it. Mathematics was to occupy a coordinate position in the curriculum with other studies. But Sir William soon forgets his position. He does not adhere to the point of dispute, as laid down by himself, but proceeds to prove that mathematics is "not an improving study." He says:^{[3]}

He proceeds to adduce testimony to the effect that^{[4]}

^{[5]}that this science is "absolutely pernicious as a mean of internal culture,"

^{[6]}that an "excessive" study of the mathematical sciences "absolutely incapacitates the mind, for those intellectual energies which philosophy and life require. We are thus disqualified for

*observation either internal or external—for abstraction and generalization*—and for "

*common reasoning*"; and disposed to the alternative of blind

*credulity*or irrational

*scepticism*."

^{[7]}Further on Hamilton says that mathematics can not "conduce to 'logical habits' at all. The art of reasoning

*right*is assuredly not to be taught by a process in which there is no reasoning

*wrong*."

^{[8]}"But if the study of mathematics do not, as a logical discipline, warn the reason against the fallacies of thought, does it not," inquires Hamilton,

^{[9]}"as an invigorating exercise of reason itself, fortify that faculty against their influence?" To this, Hamilton says, "it is equally incompetent."

^{[10]}He next observes "that to minds of any talent, mathematics are

*only difficult because they are too easy*,"

^{[11]}that "in mathematics dullness is thus elevated into talent, and talent degraded into incapacity."

^{[12]}"Of Observation, Experiment, Induction, Analogy, the mathematician knows nothing."

^{[13]}"After all," says Hamilton,

^{[14]}"we are afraid that D'Alembert is right; mathematics may distort, but can never rectify, the mind."

From these quotations it appears that Hamilton tried to prove that the study of this science is positively injurious to the mind. If this be true, then, of course, mathematics ought to be excluded entirely from a scheme of liberal education, unless, as Bledsoe says,^{[15]} the object of such a scheme be to injure, and not to benefit, the mind of the student. Had Hamilton adhered to the position which he first outlined, he could have entrenched himself behind practically unconquerable breastworks. But what has given notoriety to his paper, is the fact that most of the time he really argues against mathematical study altogether by endeavoring to show that its effect upon the mind is injurious. For seventy-five years Hamilton's article has been singled out as the most powerful argument in existence against mathematics.

To show the alleged pernicious effect of mathematics upon the mind Hamilton's argument proceeds along two principal lines, the first of which is the contention that mathematicians who have confined their studies to mathematics alone are addicted to blind credulity or irrational scepticism and, in general, lack good judgment in affairs of life.

It is my opinion that Hamilton establishes this proposition. The *mere* mathematician is a man of one-sided development. But how about the metaphysician who confines his studies to metaphysics alone? Is he an all-round man? Is the caveling metaphysician, who disputes all things, very far ahead of the credulous mathematician? Had Hamilton been disposed to attack the study of metaphysics, could he not have made as strong a case against metaphysics as he did make against mathematics? The *mere* metaphysician and the *mere* mathematician are one-sided individuals. How about the *mere* philologist with his roots and stems, the *mere* paleontologist with his old bones, the *mere* physicist with his moment of inertia and latent heat, the *mere* chemist with his pedantic formulæ, the mere entomologist with his drawings of beetles? The truth is that the exclusive study of any branch of knowledge is to be discouraged as undesirable for a liberal education. Every one recognizes the dangers of premature and excessive specialization. But because a certain branch of study, taken by itself, fails to accomplish fully all the ends of education, are we to draw the inference that this branch of study is injurious? Because the human body can not readily subsist upon a diet consisting exclusively of roast beef, are we to conclude from this fact alone that roast beef is unhealthy and ought to be banished from the dining table? Yet this is exactly the mode of argument which Hamilton applies to mathematics. Plenty of people are willing to testify that mathematics is not the sole and exclusive intellectual diet that a growing boy should have. From testimony of this sort Hamilton attempts to argue that "mathematics may distort, but can never rectify, the mind."^{[16]} In our humble opinion the learned philosopher is guilty of a very unphilosophical argument, "unphilosophical in its design, in its spirit, and in its execution."^{[17]}

We said that Hamilton argues along two principal lines. His second mode of attack is to show that many mathematicians, some of them of great eminence, have found mathematics unsatisfactory as an exercise of the mind, and have renounced it. I hardly know how to approach this part of Hamilton's argument. For lack of space I can not demonstrate the conclusions we are about to state. Bledsoe's reply to Hamilton covers sixty-nine pages, and for details we must refer you to him and to the authorities quoted by Bledsoe and Hamilton. By his extensive inquiry Bledsoe proves what some other writers before him hinted at, or proved only in part, namely, that Hamilton was extremely careless in the selection of his quotations. By means of partial extracts, badly chosen, he made scientists say exactly the opposite of their real sentiments. Bledsoe convicts Hamilton of this practise in his quotations from D'Alembert, Pascal, Descartes and Dugald Stewart, who are the most celebrated mathematical witnesses called by Hamilton.

Take the case of Descartes. We quote from Hamilton the following:^{[18]}

^{[19]}

"The study of mathematics" (says Descartes, and he frequently repeats the observation) "principally exercises the imagination in the consideration of figures and motions."^{[20]} Nay, on this very ground, he explains the incapacity of mathematicians for philosophy. "That part of the mind," says he, in a letter to Father Mersenne, "viz., the imagination, which is principally conducive to a skill in mathematics, is of greater detriment than service for metaphysical speculations."^{[21]}

These are Hamilton's references to Descartes which contain quotations from Descartes or his biographer Baillet. Evidently Hamilton was guided more by what Baillet stated about Descartes than upon what Descartes himself actually said. The letters to Mersenne simply show that Descartes was not inclined to confine his activities to mathematics, nor ready to admit that mathematical training alone constituted adequate preparation for the study of philosophy. In quoting from Descartes's "Rule Four" for conducting philosophical inquiries, Bledsoe puts into italics the passage garbled by Hamilton and Baillet. It can thus be easily read in connection with what immediately precedes and follows, and one can readily see how Hamilton's extract, by itself, conveys an impression quite the opposite of that conveyed by the entire passage. Descartes gives an exposition of his method of philosophical inquiry. He says that he wishes to apply his method not merely to the ancient "arithmetic and geometry," but to other sciences where progress has been hitherto arrested. To apply it to arithmetic and geometry alone would be to occupy himself with "trifles," not only because of the narrow field of application, but also because what was practically his method had been thus applied long ago by the Greeks. He wished to direct his method to unsolved problems. But he is free to acknowledge that his method is found in mathematics as in an envelop. "Now I say that the mathematics are the envelope of this method, not that I wish to conceal and envelop it, in order to keep the vulgar away from it; on the contrary, I wish to dress and adorn it, in such manner that it may be more easily grasped by the mind."^{[22]}

In all this there is no attack whatever upon the culture value of mathematics. Instead of hostility he shows friendliness to mathematics as a gymnast of the mind. In his discussion of the "Fourth Rule" there is a passage, not quoted by Hamilton, which bears directly upon the question at issue: "This is why I have cultivated even to this day, as much as I have been able, that universal mathematical science, so that I believe I may hereafter devote myself to other sciences, without fearing that my efforts may be premature."^{[23]} Here then Descartes declares that, as much as possible, he had studied mathematics all his life, as a preparation or propedeutic to philosophy. It appears that Descartes looked upon mathematical study as a desirable preparation to philosophy, just as Plato had done nearly 2,000 years earlier. Looking at the testimony contained in Descartes's writings, as a whole, there is nothing in it to disturb in the least the belief in the educational value of mathematical study.

As a side issue we touch upon Hamilton's assertion that Descartes in 1623 renounced mathematics for good. Hamilton does not say that the work which is memorable in the history of mathematics as the creation of analytical geometry was published by Descartes 14 years later, in 1637. Did Descartes renounce mathematics for good? The life of Descartes which was prepared by M. Thomas, a biography which captured the prize offered by the French Academy in 1765, a biography which is placed first in Cousin's edition of the works of Descartes, says this about Descartes's renunciation (p. 89): "He attempted at least five or six times to renounce them, but he always returned to them again." M. Thomas adds: "He wished to occupy himself henceforth only with morals; but on the first occasion he returned to the study of nature. Borne away in spite of himself, he plunged anew into the abstract sciences" (p. 92).

I proceed now to a review of a second attack upon mathematical study, made by Schopenhauer, the pessimistic sage of Frankfort-on-the-Main. His thoughts on mathematics are expressed in his work, entitled, "The World as Will and Idea," as it appeared in its second edition, 1844. Schopenhauer's views must have attracted considerable attention in Germany, for as late as 1894 Alfred Pringsheim thought it necessary to refute his argument, and only four years ago Felix Klein referred to him at some length in a mathematical lecture at the University of Goettingen. Schopenhauer had read Sir William Hamilton, as appears from the following passage:^{[24]}

I rather recommend, as an investigation of the influence of mathematics upon our mental powers, . . . a very thorough and learned discussion, in the form of a review of a book by Whewell in the *Edinburgh Review* of.January, 1836. Its author, who afterwards published it with some other discussions, with his name, is Sir W. Hamilton, Professor of Logic and Metaphysics in Scotland. This work has also found a German translator, and has appeared by itself under the title, "Ueber den Werth und Unwerth der Mathematik, aus dem Englisehen," 1836. The conclusion the author arrives at is that the value of mathematics is only indirect, and lies in the application to ends which are only attainable through them; but in themselves mathematics leave the mind where they find it, and are by no means conducive to its general culture and development, nay, even a decided hindrance. This conclusion is not only proved by thorough dianoiological investigation of the mathematical activity of the mind, but is also confirmed by a very learned accumulation of examples and authorities. The only direct use which is left to mathematics is that it can accustom restless and unsteady minds to fix their attention. Even Descartes, who was yet himself famous as a mathematician, held the same opinion with regard to mathematics.

These words of Schopenhauer are an unqualified endorsement of Hamilton, the only such endorsement with which I happen to be familiar.

Schopenhauer's own argument is mainly directed against Euclid and his geometrical demonstrations. Schopenhauer had his own ideas as to how absolute truth can be reached; these ideas did not agree with the method of Euclid. Our German philosopher says:^{[25]}

*why*it is so. We have therefore almost the same uncomfortable feeling that we experience after a juggling trick, and, in fact, most of Euclid's demonstrations are remarkably like such feats. The truth almost always enters by the back door, for it manifests itself

*par accidens*through some contingent circumstance. Often a

*reductio ad absurdum*shuts all the doors one after another, until only one is left through which we are therefore compelled to enter. Often, as in the proposition of Pythagoras, lines are drawn, we don't know why, and it afterwards appears that they were traps which close unexpectedly and take prisoner the assent of the astonished learner. . . (page 94). Euclid's logical method of treating mathematics is a useless precaution, a crutch for sound legs. . . (page 95). The proposition of Pythagoras teaches us a

*qualitas occulta*of the right-angled triangle; the stilted and indeed fallacious demonstration of Euclid forsakes us at the

*why*, and a simple figure, which we already know, and which is present to us, gives at a glance far more insight into the matter, and firm inner conviction of that necessity, and of the dependence of that quality upon the right triangle:

In the case of unequal catheti also, and indeed generally in the case of every possible geometrical truth, it is quite possible to obtain such a conviction based on perception. . . (page 96). It is the analytical method in general that I wish for the exposition of mathematics, instead of the synthetical method which Euclid made use of.

In the above we have Schopenhauer's famous characterization of mathematical reasoning as "mouse-trap proofs" (Mausefallenbeweise). These quotations and other passages which space does not permit us to quote indicate that his objections are directed almost entirely against Euclid. Schopenhauer discloses no acquaintance with such modern mathematical concepts as that of a function, of a variable, of coordinate representation, and the use of graphic methods. With him Euclid and mathematics are largely synonymous. Because of this one-sided and limited vision we can hardly look upon Schopenhauer as a competent judge of the educational value of modern mathematics.

If Schopenhauer's criticism of Euclid is taken as the expression of the feelings, not of an advanced mathematician, but of a person first entering upon the study of geometry and using Euclid's "Elements," then we are willing to admit the validity of Schopenhauer's criticisms, in part. Euclid did not write his geometry for children. It is a historical puzzle, difficult to explain, how Euclid ever came to be regarded as a text suitable for the first introduction into geometry. Euclid is written for trained minds, not for immature children. Of interest is Schopenhauer's reference to the method of proof, called the *reductio ad absurdum*. The experience of teachers with this method has been much the same in all countries. Some French critics called it a method which "convinces but does not satisfy the mind." De Morgan says: "The most serious embarrassment in the purely reasoning part is the *reductio ad absurdum*, or indirect demonstration. This form of argument is generally the last to be clearly understood, though it occurs almost on the threshold of the 'Elements.' We may find the key to the difficulty in the confined ideas which prevail on the modes of speech there employed."

On the main idea of what mathematical proof should be, the mathematician can hardly agree with Schopenhauer. Schopenhauer considers a succession of separate logical conclusions, which are contained in a rigorous mathematical proof, as insufficient and unendurable; he wants to be convinced of the truth of a theorem instantaneously, by an act of *intuition.* He advances the theory that, besides the severely logical deductions there is another method of proving mathematical truths, that of direct perception and intuition. We agree with Schopenhauer that intuition should play an important part, especially in preliminary courses, before children enter upon courses in demonstrative geometry; but eventually the logical proof must be made to follow before we are prepared to accept a proposition as established. Schopenhauer directs his criticisms particularly against Euclid's proof of the Pythagorean Theorem and then offers his own proof, which is practically the same as the Hindu proof and can be given by drawing the figure and then explaining, as did the Hindus, "Behold." But Schopenhauer's is not a *general* proof; it holds only for a special case, namely, for the isosceles right triangle.

Eeally, Euclid's proof of the Pythagorean Theorem consists of a number of steps, each of which is quite evident to the eye. Thus a square is represented as the sum of two rectangles, which is an intuitive relation. Then each rectangle is shown to be equal to double a triangle of the same base and altitude. This again the child accepts the more readily as more or less intuitively evident. And so on. Every step appears quite reasonable to one depending on intuition alone. It does seem as if Schopenhauer could have made a better selection from Euclid for his point of attack.

From what we have said it appears that Schopenhauer's attack bears only indirectly upon the question relating to the mind-training value of mathematics; his criticism is focused directly upon questions of logic, of mode of argumentation and of sufficiency of proof.

I pass now to a third attack upon mathematics, made in 1869 by the naturalist, Thomas H. Huxley. So far as I know, Huxley was not influenced either by Hamilton or Schopenhauer, though the words he used remind us of a sentence in Hamilton. Hamilton had said: "Of Observation, Experiment, Induction, Analogy, the mathematician knows nothing."^{[26]} Huxley, in the June number of the *Fortnightly Review,* 1869, said: Mathematics is that study "which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation."^{[27]} Huxley and Hamilton both name observation, experiment, induction, but they differ in the fourth process. Hamilton says "analogy"; Huxley says "causation."

In the same year there appeared in print an after-dinner speech delivered by Huxley before the Liverpool Philomathic Society^{[28]} in which he argued in favor of scientific education, as follows:

The other studies which enter into ordinary education do not discipline the mind in this way. Mathematical training is almost purely deductive. The mathematician starts with a few simple propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them. The teaching of languages, at any rate as ordinarily practised, is of the same general nature—authority and tradition furnish the data, and the mental operations of the scholar are deductive.

It will be noticed that these remarks were made at a time when there was a conflict on the question of educational values between the classics and mathematics, on one side, and the natural and social sciences, on the other. This makes it evident that Huxley appeared in this discussion in the capacity of an advocate rather than as a judge.

Of great interest, in connection with Huxley's utterances is the reply made to him by the mathematician J. J. Sylvester. To Americans Sylvester's name is memorable, because at one time he was on the faculty of the University of Virginia and, when the Johns Hopkins University opened in 1876, Sylvester again came over from England and for eight years lectured to American students on modern higher algebra. He gave a powerful stimulus to the study of higher mathematics in this country. Sylvester was an enthusiast. His reply to Huxley was the subject of his presidential address to the mathematical and physical section of the British Association, meeting at Exeter in 1869. This address is of special value, because it is largely autobiographical; it tells how Sylvester carried on his researches in mathematics, how he came to make some of his discoveries. By his own experiences as a mathematical investigator he tried to show that Huxley's description of mathematical activity was incorrect. We can do no better than quote rather freely from Sylvester's memorable address. He says:

I set to myself the task of considering certain recent utterances of a most distinguished member of this Association, one whom I no less respect for his honesty and public spirit than I admire for his genius and eloquence, but from whose opinions on a subject which he has not studied I feel constrained to differ. Göthe has said:

"Verständige Leute kannst du irren sehn:

In Sachen, namlich, die sie nicht verstehn. "

"Understanding people you may see erring

In those things, to wit, which they do not understand. . . ."

Further on Sylvester says:

Lagrange . . . has expressed emphatically his belief in the importance to the mathematician of the faculty of observation; Gauss has called mathematics a science of the eye . . .; the ever to be lamented Riemann has written a thesis to show that the basis of our conception of space is purely empirical, and our knowledge of its laws the result of observation, that other kinds of space might be conceived to exist subject to laws different from those which govern the actual space in which we are immersed. . . . Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, . . . Sturm's theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected with the motion of compound pendulums.

After citing many other instances, Sylvester says:

I, of course, am not so absurd as to maintain that the habit of observation of external nature will be best or in any degree cultivated by the study of mathematics, at all events as that study is at present conducted, and no one can desire more earnestly than myself to see natural and experimental science introduced into our schools as a primary and indispensable branch of education: I think that that study and mathematical culture should go on hand in hand together, and that they would greatly influence each other for their mutual good. I would rejoice to see mathematics taught with that life and animation which the presence and example of her young and buoyant sister could not fail to impart, short roads preferred to long ones, Euclid honourably shelved or buried "deeper than e'er plummet sounded" out of the schoolboy's reach, morphology introduced into the elements of Algebra—projection, correlation, and motion accepted as aids to geometry—the mind of the student quickened and elevated and his faith awakened by early initiation into the ruling ideas of polarity, continuity, infinity, and familiarization with the doctrine of the imaginary and inconceivable.

What light, if any, do these attacks and these defenses of mathematical study throw upon the educational problems of to-day? Hamilton gathered a cloud of witnesses which, in so far as the testimony adduced was sincere, proved that mathematical study alone is not the proper education for life. That mathematical study is pernicious Hamilton did not succeed in proving. It would seem, therefore, as if the Hamiltonian controversy was somewhat barren in useful results. Probably no one to-day advocates the well-nigh exclusive study of mathematics or of any other science as the best education obtainable.

Schopenhauer attacked mainly the logic of mathematics as found in Euclid. As a critique of the logic as used by Euclid the attack is childish and has no value for us. From the standpoint of educational method it points out the difficulty experienced by children in understanding the mode of proof called the *reductio ad absurdum* and emphasizes the constant need of appeal to the intuition in the teaching of mathematics.

The attack made by Huxley touches questions which are more subtle. Sylvester, in his rejoinder, proved conclusively that the mathematician engaged in original research does exercise powers of internal observation, of induction, of experimentation and even of causation. Are these powers exercised by the pupil in the class room? That depends. When English teachers required several books of Euclid to be memorized, even including the lettering of figures, no original exercises being demanded, then indeed such teaching knew nothing of observation, induction, experiment, and causation, except that a good memory as a cause was seen to bring about a pass mark as an effect. But when attention is paid to the solution of original exercises, and to the heuristic or genetic development of certain parts of the subject, then surely the young pupil exercises the same faculties as does the advanced mathematician engaged in research.

The language used by Huxley and Sylvester is not in accord with some of the ideas of recent American psychologists, who declare that teachers should not attempt to train particular mental faculties. We have seen that Hamilton, Huxley and Sylvester discussed the training of the faculty of "observation," the "reasoning faculty" or the "power of observation." Hamilton complains that "none of our intellectual studies tend to cultivate a smaller number of the faculties, in a more partial manner, than mathematics."^{[29]} Recent writers object to this point of view. The teacher must "stop wet-nursing orphan mental faculties"; his business is "to select points of contact between learning minds and the reality that is to be learned." The recent movement is a remarkable reaction against the time-honored "doctrine of formal discipline," which originated with the Greeks and probably reached its height in the time of Huxley. In its extreme form this is "the doctrine of the applicability of mental power, however gained, to any department of human activity."^{[30]} In its place comes the doctrine of "specific disciplines," according to which "improvement of any one mental function or activity will improve others only in so far as they possess elements common to it also." The subject is still in the polemical stage. The new psychology is not hostile to mathematics, except perhaps to the formal or mechanical parts of algebra. A point which may harmonize in part the old and the new views, and which in itself demands very lively consideration, lies in the claim put forth recently, that the benefit to be derived from a subject like mathematics depends largely upon the attitude toward it maintained by the teacher and pupil. They should be controlled by ideals to be reached as a goal, such as ideals of accuracy, of efficiency, of scientific method. "If we have trained pupils to think rigidly in geometry, for example, how shall we insure an application of rigid thinking to situations that lack the geometrical elements? . . . Shall we not have the greatest assurance of such transfer, if the method has been made to appeal to the pupil as something thoroughly worth while?"^{[31]} No doubt this feature has figured prominently in the mathematical teaching of all ages, but recent is the psychological recognition of it as a conscious factor in the transfer of special training to new fields of action.

- ↑ "Ueber den Wert und angeblichen Unwert der Mathematik," Von Alfred Pringsheim, Muenchen, 1894.
- ↑
*Edinburgh Review*, Vol. 62, 1836, p. 411. - ↑
*Loc. sit.*, p. 419 - ↑
*Loc. sit.*, p. 421 - ↑
*Loc. sit.*, p. 421 - ↑
*Loc. sit.*, p. 419. - ↑
*Loc. sit.*, p. 424. - ↑
*Loc. sit.*, p. 427. - ↑
*Loc. sit.*, p. 428. - ↑
*Loc. sit.*, p. 428. - ↑
*Loc. sit.*, p. 430. - ↑
*Loc. sit.*, p. 430. - ↑
*Loc. sit.*, p. 433. - ↑
*Loc. sit.*, p. 453. - ↑
*Southern Review, Vol. 22, 1877, p. 261.* - ↑
*Loc. cit.,*p. 453. - ↑
*Southern Review,*Vol. 22, p. 282. - ↑
*Loc. cit.,*p. 42 - ↑ "La Vie de Descartes," P. I., pp. 111, 112, 225; P. II, p. 481.
- ↑ "Lettres," P. I., let. xxx.
- ↑
*Loc. cit.,*p. 426. - ↑
*Southern Review,*Vol. 22, p. 270. - ↑
*Southern Review,*Vol. 22, p. 271. - ↑ A. Schopenhauer, "The World as Will and Idea," translated by R. B. Haldane and J. Kemp, Vol. II., London, 1891, p. 323.
- ↑
*Loc. cit.,*Vol. I., 1891, par. 15, p. 90. - ↑
*Edinburgh Review,*Vol. 22, p. 433. - ↑
*Fortnightly Review,*London, Vol. 5, 1869, p. 667. - ↑
*Macmillan's Magazine,*Vol. 20, London, 1869, pp. 177-184. - ↑
*Edinburgh Review*, Vol. 22, p. 419. - ↑ W. H. Heck, "Mental Discipline and Educational Values," New York, 1909, p. 7 and other places.
- ↑ W. C. Bagley, "Educational Values," New York, 1911, p. 194.