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our nature, is to be corrected only by actual experience of human falsehood; and, in proportion to the extent of this experience, the degree of our incredulity may be expected to be, In what science, for example, are our reasonings liable to such uncertainty and error as medicine; and accordingly the old sarcasm against physicians, ubi tres medici, duo Athei, though manifestly carried the length of a ludicrous exaggeration, touches, it must be confessed, on a professional bias, the exist-. ence of which it is impossible to deny. But the mathematician is conversant in his own science with truth, and with truth alone; and if he judges of other branches of knowledge by that with which he is daily familiar, can scarcely fail to overrate the authority of those who are understood to have cultivated them with success. *
The circumstance which, in my opinion, has given rise to this common charge of scepticism against mathematicians, is an inattention to the distinction between speculative habits of belief on moral subjects, and the moral sensibilities of the heart. In this last respect, it must be owned, that (although nothing can well be alleged to the prejudice of mathematical studies) little can be advanced in their favour.
In our inquiries into the constitution both of the Material and of the Intellectual worlds, we are constantly presented
* We may perhaps connect with the credulity of mathematicians, a feature in their character remarked by Swift in his account of the mathematicians of Laputa ;-their eager curiosity after the politics and the news of the day.
"Most of them, and especially those who deal in the astronomical part, have great faith in judicial astrology, although they are ashamed to own it publicly. "But what I chiefly admire, and thought altogether unaccountable, was the strong disposition I observed in them towards news and politics; perpetually inquiring into public affairs; giving their judgments in matters of state; and passionately disputing every inch of a party opinion. I have indeed observed the same disposition among most of the mathematicians I have known in Europe, although I could never discover the least analogy between the two "" sciences."
As it is well known that Dr. Arbuthnot (who was himself a mathematician of some note) contributed largely to this work of Swift's, the foregoing remarks, as well as some others of the same kind which occur in this chapter, are entitled to more attention than if they were sanctioned only by the authority of a man of wit.
On the other hand, it is to be observed, that as there is no study which may be advantageously entered upon with a less stock of preparatory knowledge than mathematics, so there is none in which a greater number of uneducated men have aised themselves, by their own exertions, to distinction and eminence. (See various examples of this in Dr. Hutton's Mathematical Dictionary, particularly the very interesting account there given of the justly celebrated Thomas Simpson of Woolwich, and of that learned, laborious, and useful compiler, the late William Emerson.) Many of the intellectual defects which, in such cases, are commonly placed to the account of mathematical studies, ought to be ascribed to the want of a liberal education in early youth
with instances of design which lead up our thoughts to the contemplation of the Almighty Artist. But in pure or abstract mathematics, the truths we investigate are understood to be necessary and immutable; and, therefore, can have no tendency to awaken those moral sentiments which are so naturally inspired by the order of the universe; excepting, perhaps, in a mind habituated by metaphysical pursuits to a reflex examination of its own reasoning and inventive powers. It must be remembered, at the same time, that this inconvenience of mathematical studies is confined to those who cultivate them exclusively; and that, when combined, as they now generally are, with a taste for physical science, they enlarge infinitely our views of the wisdom and power displayed in the universe. The very intimate connexion, indeed, which, since the date of the Newtonian philosophy, has existed between the different branches of mathematical and of physical knowledge, renders such a character as that of a mere mathematician a very rare, and scarcely a possible occurrence; and cannot fail to have contributed powerfully to correct the peculiarities likely to characterize an understanding conversant exclusively with the relations of figures and of abstract quantities. Important advantages may also be expected to result from those habits of metaphysical and of moral speculation which the study both of mathematics and of physics has so strong a tendency to encourage in every inquisitive and cultivated mind. In the present state of science, therefore, mathematical pursuits seem to lead the attention, by a natural process, to the employment of the most effectual remedies against those inconveniences which they appear, on a superficial view, to threaten; and which there is reason to believe they actually produced, in many instances, when education was conducted on a plan less enlightened and comprehensive than what now generally prevails.
Some exceptions to this observation, I must, at the same time, acknowledge, are still not unlikely to occur, in cases where the study of Abstract Mathematics has taken a strong hold of the mind, before it was inspired with any taste for the study of Nature; more particularly, where this taste has been confined to certain branches of natural philosophy, (such as physical astronomy and optics,) which are in a great measure, inaccessible to those who have not received a regular mathematical education; and which direct the attention much less to experimental principles, than to the necessary relations of quantities and figures. Of those who devote themselves to such researches, by far the greater part have been led to do so
not by any natural relish for physical inquiries, but by a previous passion for. geometry, which gradually entices them on to the study of its various applications. Such men are extremely apt to forget, that, although Mathematics is a useful and a necessary instrument in Natural Philosophy, the two sciences differ from each other completely in their nature and objects; and, in consequence of overlooking this circumstance, they are apt, from their early habits of study, to aim too much at giving to natural philosophy that completely systematical form which is essential to mathematics from the nature of its data, but which never can belong to any science which rests upon facts collected from experience and observation.
In proof of this last remark, it is sufficient to observe, that, in all the different branches of experimental knowledge, how far soever we may carry our simplifications, we must ultimately make the appeal to facts for which we have the evidence of our senses; and, therefore, to diminish the number of such first principles, does not add (as many mathematicians seems to have supposed) in the smallest degree to the logical certainty of the science. On the contrary, such an attempt may frequently lead into error, as well as impair the evidence of our conclusions. Thus, there is a beautiful and striking analogy among some of the laws of motion, as well as among various other general laws of nature; which analogy, however, for any thing we know to the contrary, may be the result of the positive appointment of the Creator; and which, at any rate, does not appear so clearly to our reason to arise from any necessary connexion, as to enable us to deduce the one law from the other as a logical consequence. Another remarkable analogy presents itself between the equality of action and re-action in the collision of bodies, and what obtains in their mutual gravitation, as well as in some other physical phenomena. Here the analogy is so perfect as to render it easy to comprehend all the various facts in one general proposition; nor will I take upon me to affirm, that the different facts may not be connected necessarily, as consequences of some one general principle; but, as the evidence of such a connexion does not at least appear satisfactory to every one, it might facilitate the progress of students, and would, at the same time, be fully as unexceptionable in point of sound logic, to establish the fact in particular cases by experiment and observation, and consider the law of action and re-action merely as a general rule or theorem obtained by induction.
Numberless instances, too, might be mentioned, in which physico-mathematical writers have been led into illogical and
inconclusive reasoning by this desire to mould their doctrines into a geometrical form. It is well known (to take a very obvious example) to be a fundamental principle in mechanics, "That when two heavy bodies counterpoise each other by means of any machine, and are then both put into motion to66 gether, the quantities of motion with which the one descends "and the other ascends perpendicularly will be equal." This equilibrium bears such a resemblance to the case in which two moving bodies stop each other when they meet together with equal quantities of motion, that many writers have thought that the cause of an equilibrium in the several machines might be immediately assigned by saying, That since one body always loses as much motion as it communicates to another, two heavy bodies counteracting each other must continue at rest, when they are so circumstanced that one cannot descend without causing the other to ascend at the same time, and with the same quantity of motion; for, then, should one of them begin to descend, it must instantly lose its whole motion by communicating it to the other. But this reasoning, however plausible it may seem, is by no means satisfactory; for, (as Dr. Hamilton has justly observed,)† when we say that one body communicates its motion to another, we must necessarily suppose the motion to exist first in the one, and then in the other but, in the present case, where the bodies are so connected that one cannot possibly begin to move before the other, the descending body cannot be said to communicate its motion to the other, and thereby make it ascend. And, therefore, (admitting the truth of the general law which obtains in the collision of bodies,) we might suppose that, in the case of a machine, the superior weight of the heavier body would overcome the lighter, and cause it to ascend with the same quantity of motion with which the heavier descends.
As this excessive simplification of our principles in Natural Philosophy impairs, in some cases, the evidence of the science, and, in others, the accuracy of our reasoning; so, in all cases, it has a tendency to withdraw the attention from those pleasing and interesting views to which the contemplation of Nature is calculated to lead every mind of taste and sensibility. In pure mathematics, where all the various truths are necessarily connected with each other, (being all necessarily connected with those hypotheses which are the principles of the science,) an arrangement is beautiful in proportion as the principles are few;
• Elements of the Philosophy of the Human Mind, Vol. II. p. 185, et seq. + See Philosophical Essays, by Hugh Hamilton, D. D. Professor of Philosophy in the University of Dublin, p. 135, et seq. Third Edition. (London 1772.)
and what we admire perhaps chiefly in the science, is the astonishing variety of consequences which may be demonstrably deduced from so small a number of premises. But, in Natural Philosophy, it is surely more pleasing, as well as much more correct in point of sound logic, to consider the phenomena of the universe as symmetrical parts of one comprehensive and beautiful design, than as the necessary result of an eternal and immutable order; and, in those analogies which take place among different laws, to admire,) not, as in geometry, the systematical concatenation of theorems,) but the unity of contrivance which appears in nature, and that beneficent wisdom which at once delights the imagination with the infinite diversity of its operations, and regulates them by those simple and harmonious laws which accomodate them to the grasp of our limited faculties.
In the foregoing remarks, I have had an eye chiefly to some mathematicians on the Continent, among whom the false logic which I have now been endeavouring to expose has long been gaining ground, and seems to be at present more fashionable than ever. It was, I think, first introduced by Leibnitz, whose mind, powerful and comprehensive as it was, appears, from many passages in his works, to have been influenced, in a singular degree, by a disposition to transfer to physical and even to moral subjects, those habits of thinking which he had been led to cultivate by his geometrical studies. The influence of his genius in forming that peculiar taste both in pure and in mixed mathematics which has prevailed in France, as well as in Germany, for a century past, will be found, upon examination, to have been incomparably greater than that of any other individual.
When the mathematician reasons upon subjects unconnected with his favourite studies, he is apt to assume, too confidently, certain intermediate principles as the foundation of his arguments. I use this phrase in the sense annexed to it by Locke, in his book on the Conduct of the Understanding, from which I shall quote the explanation there given of it, not only as the best comment I can offer upon the expression, but as the view of it which he takes will be sufficient of itself to show why mathematicians should be more liable than the other classes of literary men to this source of sophistical reasoning.
"As a help to this, I think it may be proposed, that, for the
* I am inclined to trace to the same source, the extensive use he has made, in his philosophical inquiries, of the law of continuity, and also of the principle of the sufficient reason.