Florian Cajori

“ 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. ”

“ 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. ”

“ If now our conviction that perception is the primary source of all evidence, and that only direct or indirect connection with it is absolute truth; and further, that the shortest way to this is always the surest, as every interposition of concepts means exposure to many deceptions; if, I say, we now turn with this conviction to mathematics, as it was established as a science by Euclid, and has remained as a whole to our own day, we can not help regarding the method it adopts as strange and indeed perverted. We ask that every logical proof shall be traced back to an origin in perception ”

“ 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. ”

“ 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." ”

“ 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. ”

“ 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." ”

“ 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. ”

“ 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. ”

“ 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? ”

“ 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. ”

“ 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. ”

“ 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." ”

“ 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" ”

“ 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. ”

“ He proceeds to adduce testimony to the effect that [4]
"the cultivation afforded by the mathematics is, in the highest degree, one-sided and contracted," that mathematics "freeze and parch the mind," [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" ”

“ Speaking of the general character of the philosopher, Baillet adds, "In regard to the rest of mathematics [he had just spoken of astronomy, which Descartes thought, 'though he dreamt in it himself, only a loss of time'] , those who know the rank which he held above all mathematicians, ancient and modern, will agree that he was the man in the world best qualified to judge them. ”

“ 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." ”

“ 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. ”

“ 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. ”

“ 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. ”