The Online Library of Liberty

A project of Liberty Fund, Inc.

• Introduction
• Textual Introduction
• Preface to the Third Edition[*]
• Chapter I: Introductory Remarks
• Chapter II: The Relativity of Human Knowledge
• Chapter III: The Doctrine of the Relativity of Human Knowledge, As Held By Sir William Hamilton
• Chapter IV: In What Respect Sir William Hamilton Really Differs From the Philosophers of the Absolute
• Chapter V: What Is Rejected As Knowledge By Sir William Hamilton, Brought Back Under the Name of Belief
• Chapter VI: The Philosophy of the Conditioned
• Chapter VII: The Philosophy of the Conditioned, As Applied By Mr. Mansel to the Limits of Religious Thought
• Chapter VIII: Of Consciousness, As Understood By Sir William Hamilton
• Chapter IX: Of the Interpretation of Consciousness
• Chapter X: Sir William Hamilton’s View of the Different Theories Respecting the Belief In an External World
• Chapter XI: The Psychological Theory of the Belief In an External World
• Chapter XII: The Psychological Theory of the Belief In Matter, How Far Applicable to Mind
• Chapter XIII: The Psychological Theory of the Primary Qualities of Matter
• Chapter XIV: How Sir William Hamilton and Mr. Mansel Dispose of the Law of Inseparable Association
• Chapter XV: Sir William Hamilton’s Doctrine of Unconscious Mental Modifications
• Chapter XVI: Sir William Hamilton’s Theory of Causation
• Chapter XVII: The Doctrine of Concepts, Or General Notions
• Chapter XVIII: Of Judgment
• Chapter XIX: Of Reasoning
• Chapter XX: On Sir William Hamilton’s Conception of Logic As a Science. Is Logic the Science of the Laws, Or Forms, of Thought?
• Chapter XXI: The Fundamental Laws of Thought According to Sir William Hamilton
• Chapter XXII: Of Sir William Hamilton’s Supposed Improvements In Formal Logic
• Chapter XXIII: Of Some Minor Peculiarities of Doctrine In Sir William Hamilton’s View of Formal Logic
• Chapter XXIV: Of Some Natural Prejudices Countenanced By Sir William Hamilton, and Some Fallacies Which He Considers Insoluble
• Chapter XXV: Sir William Hamilton’s Theory of Pleasure and Pain
• Chapter XXVI: On the Freedom of the Will
• Chapter XXVII: Sir William Hamilton’s Opinions On the Study of Mathematics
• Chapter XXVIII: Concluding Remarks
• Appendices
• Appendix A: Manuscript Fragments
• Appendix B: Textual Emendations
• Appendix C: Corrected References
• Appendix D: Bibliographic Index of Persons and Works Cited In the Examination, With Variants and Notes

CHAPTER XXVII

Sir William Hamilton’s Opinions on the Study of Mathematics

no account of Sir W. Hamilton’s philosophy could be complete, which omitted to notice his famous attack on the tendency of mathematical studies:^[*] for though there is no direct connexion between this and his metaphysical opinions, it affords the most express evidence we have of those fatal lacunæ in the circle of his knowledge, which unfitted him for taking a comprehensive or even an accurate view of the processes of the human mind in the establishment of truth. If there is any pre-requisite which all must see to be indispensable in one who attempts to give laws to human intellect, it is a thorough acquaintance with the modes by which human intellect has proceeded, in the cases where, by universal acknowledgment, grounded on subsequent direct verification, it has succeeded in ascertaining the greatest number of important and recondite truths. This requisite Sir W. Hamilton had not, in any tolerable degree, fulfilled. Even of pure mathematics he apparently knew little but the rudiments. Of mathematics as applied to investigating the laws of physical nature; of the mode in which the properties of number, extension, and figure, are made instrumental to the ascertainment of truths other than arithmetical or geometrical—it is too much to say that he had even a superficial knowledge: there is not a line in his works which shows him to have had any knowledge at all. He had no conception of what the process is. In this he differed greatly and disadvantageously from his immediate predecessor in the same school of metaphysical thought, Professor Dugald Stewart; whose works derive a great part of their value from the foundation of sound and accurate scientific knowledge laid by his mathematical and physical studies, and which his subsequent metaphysical pursuits enabled him, quite successfully to the length of his tether, to clarify and reduce to principles.

If Sir W. Hamilton had contented himself with saying of mathematics, that it is not, of itself alone, a sufficient education of the intellectual faculties; that it cultivates the mind only partially; that there are important kinds of intellectual cultivation and discipline which it does not give, and to which, therefore, if pursued to the exclusion of the studies which do give them, it is unfavourable; he would have said something, not new indeed, but true, not of mathematics alone, but of every limited and special employment of the mental faculties; of every study in which the human mind can engage, except the two or three highest, most difficult, and most imperfect, which, requiring all the faculties in their greatest attainable perfection, can never be recommended or thought of as preparatory discipline, but are themselves the chief purpose for which such preparation is required. Sir W. Hamilton, however, has asserted much more than this. He undertakes to show that the study of mathematics is not an useful intellectual discipline at all, except in one comparatively humble particular, which it has in common with some of the most despised pursuits; and that, if prosecuted far, it positively unfits the mind for the useful employment of its faculties on any other object. As might be expected from an attempt to maintain such a thesis by one who, however acute on other matters, had no sufficient knowledge of the subject he was writing about, this celebrated dissertation is one of the weakest parts of his works. He ignores not only the whole of his adversary’s case, but the most important part of his own; and has made a far less powerful attack on the tendencies of mathematical studies, than could easily be made by one who understood the subject. He has, in fact, missed the most considerable of the evil effects to the production of which those studies have contributed; and has thrown no light on the intellectual shortcomings of the common run of mathematicians, so signally displayed in their wretched treatment of the generalities of their own science. He finds hardly anything to say to their disadvantage but things so trite and obvious, that the greatest zealot for mathematics could afford to pass them by, insisting only on the inestimable benefits which are to be set against them, and which alone are really to the purpose; for it is no objection to a harrow that it is not a plough, nor to a saw that it is not a chisel.

For instance, are we much the wiser for being once more told, at great length, and with a cloud of witnesses^[*] brought to back the assertion, that mathematics, being concerned only with demonstrative evidence, does not teach us, either by theory or practice, to estimate probabilities? Did any mathematician, or eulogist of mathematics, ever pretend that it did? Does the science to which Sir W. Hamilton assigns a place above all others as an intellectual discipline—does Metaphysics enable us to judge of probable evidence? If such a claim has ever been made in its behalf, I am not aware of it; Sir W. Hamilton, certainly, was too well acquainted with the subject to make any such pretension. Metaphysics, like Mathematics, and all the rest of the fundamental sciences, demands, not probable, but certain evidence. The province of Probabilities in science is not the abstract, but what M. Comte terms the concrete sciences; those which treat of the combinations actually realized in Nature, as distinguished from the general laws which would equally govern any other combinations of the same elements: zoology and botany, for example, as contrasted with physiology; geology, as opposed to thermology and chemistry.^[*] In an abstract science a probability is of no account; it is but a momentary halt on the road to certainty, and a hint for fresh experiments.

Inasmuch as abstract science in general, and mathematics in particular, afford no practice in the estimation of conflicting probabilities, which is the kind of sagacity most required in the conduct of practical affairs, it follows that, when made so exclusive an occupation as to prevent the mind from obtaining enough of this necessary practice in other ways, it does worse than not cultivate the faculty—it prevents it from being acquired, and pro tanto unfits the person for the general business of life. It is natural that people who are bad judges of probability, should be, according to their temperament, unduly credulous or unreasonably sceptical; both which charges our author, with great earnestness and a heavy artillery of authorities, drives home against the mathematicians. But he would have made little progress towards proving his case, even by a much more complete catalogue of the intellectual defects of a mathematician who is nothing but a mathematician. A person may be keenly alive to these, and may hate them, as M. Comte did, with a perfect hatred, while upholding mathematical instruction as not only an useful but the indispensable first stage of all scientific education worthy of the name.^* Nor can any reasonable view of the subject refuse to recognise, in the very faults which our author imputes to mathematicians, the excesses of a most valuable quality. Let us be assured that for the formation of a well-trained intellect, it is no slight recommendation of a study, that it is the means by which the mind is earliest and most easily brought to maintain within itself a standard of complete proof. A mind thus furnished, and not duly instructed on other subjects, may commit the error of expecting in all proof too close an adherence to the type with which it is familiar. That type may and ought to be widened by greater variety of culture; but he who has never acquired it, has no just sense of the difference between what is proved and what is not proved: the first foundation of the scientific habit of mind has not been laid. It has long been a complaint against mathematicians that they are hard to convince: but it is a far greater disqualification both for philosophy, and for the affairs of life, to be too easily convinced; to have too low a standard of proof. The only sound intellects are those which, in the first instance, set their standard of proof high. Practice in concrete affairs soon teaches them to make the necessary abatement: but they retain the consciousness, without which there is no sound practical reasoning, that in accepting inferior evidence because there is none better to be had, they do not by that acceptance raise it to completeness. They remain aware of what is wanting to it.

Besides accustoming the student to demand complete proof, and to know when he has not obtained it, mathematical studies are of immense benefit to his education by habituating him to precision. It is one of the peculiar excellences of mathematical discipline, that the mathematician is never satisfied with an à peu près. He requires the exact truth. Hardly any of the non-mathematical sciences, except chemistry, has this advantage. One of the commonest modes of loose thought, and sources of error both in opinion and in practice, is to overlook the importance of quantities. Mathematicians and chemists are taught by the whole course of their studies, that the most fundamental differences of quality depend on some very slight difference in proportional quantity; and that from the qualities of the influencing elements, without careful attention to their quantities, false expectations would constantly be formed as to the very nature and essential character of the result produced. If Sir W. Hamilton’s mind had undergone this improving discipline, we should not have found him employing the most precise mathematical terms with the laxity which is habitual in his writings. For instance; whenever he means that one of two things diminishes while another increases, he says that they are in the inverse ratio of one another. He affirms this of the Extension and Comprehension of a general notion;^* of the number of objects among which our attention is divided, and the intensity with which it is applied to each;^† of the knowledge-giving and the sensation-giving properties of an impression of sense;^* and of the intensity and the prolongation of an energy.^† That an inverse ratio is the name of a definite relation between quantities, seems never to have occurred to him.

Neither is it a small advantage of mathematical studies, even in their poorest and most meagre form, that they at least habituate the mind to resolve a train of reasoning into steps, and make sure of each step before advancing to another. If the practice of mathematical reasoning gives nothing else, it gives wariness of mind; it accustoms us to demand a sure footing; and though it leaves us no better judges of ultimate premises than it found us (which is no more than may be said of almost all metaphysics) at least it does not suffer us to let in, at any of the joints in the reasoning, any assumption which we have not previously faced in the shape of an axiom, postulate, or definition. This is a merit which it has in common with Formal Logic, and is the chief ground on which some have thought that it could perform the functions and supply the place of that science; an opinion in which I by no means agree.

That mathematics “do not cultivate the power of generalization,”^‡ which to our author appears so obvious a truth that he need not give himself the trouble of proving it, will be admitted by no person of competent knowledge, except in a very qualified sense. The generalizations of mathematics, are, no doubt, a different thing from the generalizations of physical science; but in the difficulty of seizing them, and the mental tension they require, they are no contemptible preparation for the most arduous efforts of the scientific mind. Even the fundamental notions of the higher mathematics, from those of the differential calculus upwards, are products of a very high abstraction. Merely to master the idea of centrifugal force, or of the centre of gravity, are efforts of mental analysis surpassed by few in our author’s metaphysics. To perceive the mathematical law common to the results of many mathematical operations, even in so simple a case as that of the binomial theorem, involves a vigorous exercise of the same faculty which gave us Kepler’s laws, and rose through those laws to the theory of universal gravitation. Every process of what has been called Universal Geometry—that great creation of Descartes and his successors, in which a single train of reasoning solves whole classes of problems at once, and demonstrates properties common to all curves or surfaces, and others common to large groups of them—is a practical lesson in the management of wide generalizations, and abstraction of the points of agreement from those of difference among objects of great and confusing diversity, to which the most purely inductive science cannot furnish many superior. Even so elementary an operation as that of abstracting from the particular configuration of the triangles or other figures, and the relative situation of the particular lines or points, in the diagram which aids the apprehension of a common geometrical demonstration, is a very useful, and far from being always an easy, exercise of the faculty of generalization so strangely imagined to have no place or part in the processes of mathematics.

Sir W. Hamilton allows no efficacy to mathematical studies in the cultivation of any valuable intellectual habit, except the single one of continuous attention. “Are mathematics then,” he asks,

He adds, truly enough, “But mathematics are not the only study which cultivates the attention: neither is the kind and degree of attention which they tend to induce, the kind and degree of attention which our other and higher speculations require and exercise.”^† So that, according to him, there is no purpose answered by mathematics in general education, but one which would be better fulfilled by something else.

Without stopping to express my amazement at the assertion that the student of mathematics exercises no mental faculty but that of continuous attention, I will avail myself of an admission which Sir W. Hamilton cannot help making, but the full force of which he does not perceive. “We are far,” he says, “from meaning hereby to disparage the mathematical genius which invents new methods and formulæ, or new and felicitous applications of the old. . . . Unlike their divergent studies, the inventive talents of the mathematician and philosopher in fact approximate.”^‡ Was, then, Sir W. Hamilton so ill-acquainted with everything deserving the name of mathematical tuition as to suppose that the inventive powers which, in their higher degree, constitute mathematical genius, are not called forth and fostered in the process of teaching mathematics to the merest tyro? What sort of mathematical instruction is it of which solving problems forms no part? We come, within a page afterwards, to the following almost incredible announcement: “Mathematical demonstration is solely occupied in deducing conclusions; probable reasoning, principally concerned in looking out for premises.”^§ Sir W. Hamilton thinks he can never be severe enough upon Cambridge for laying any stress on mathematics as an instrument of mental instruction. Did he ever turn over, I do not say a volume of Cambridge Problems,^[*] for these, it may be said, test the knowledge of the pupil rather than his inventive powers, and may be an exercise chiefly of memory: but did he ever see two such volumes as Bland’s Algebraical and Geometrical Problems?^[†] Did he really imagine that working these was not “looking out for premises?” He seems actually to have thought that learning mathematics meant cramming it; and apparently believed that a mathematical tutor resolves all the equations himself, and merely asks his pupil to follow the solutions. For in every problem which the pupil himself solves, or theorem which he demonstrates, not having previously seen it solved or demonstrated, the same faculties are exercised which, in their higher degrees, produced the greatest discoveries in geometry. Mathematical teaching, therefore, even as now carried on, trains the mind to capacities, which, by our author’s admission, are of the closest kin to those of the greatest metaphysician and philosopher. There is some colour of truth for the opposite doctrine in the case of elementary algebra. The resolution of a common equation can be reduced to almost as mechanical a process as the working of a sum in arithmetic. The reduction of the question to an equation, however, is no mechanical operation, but one which, according to the degree of its difficulty, requires nearly every possible grade of ingenuity: not to speak of the new, and in the present state of science insoluble, equations, which start up at every fresh step attempted in the application of mathematics to other branches of knowledge. On all this, Sir W. Hamilton never bestows a thought. It is hardly necessary to point out that any other study, pursued in the manner in which he supposes mathematics to be, would as little exercise any other faculty than that of “continuous attention” as mathematics would. Next to metaphysics, the study he most patronizes is that of languages; of which he has so lofty an opinion, as to say that “to master, for example, the Minerva of Sanctius with its commentators, is, I conceive, a far more profitable exercise of mind than to conquer the Principia of Newton:”^* we may at least say that he was a better judge of the profit that might be derived from it. I, also, rate very highly the value, as a discipline to the mind, of the thorough grammatical study of any of the more logically constructed languages: but if the study consisted in learning the Minerva of Sanctius, or its commentators either, by rote, I believe the benefit derived would be about the same with that which Sir W. Hamilton considered to result from the exercise of “continuous attention” in mathematics.

It is a characteristic fact, that when the paper “on the Study of Mathematics” originally appeared as an article in the Edinburgh Review, no mention was made in it of Mixed or Applied Mathematics: the little which now appears on that subject being a subsequent addition, called forth by Dr. Whewell’s reply.^[*] Dr. Whewell must have looked down from a considerable height upon an assault on the utility of Mathematics, in which the part of it that, in the opinion of its rational defenders, constitutes three-fourths of its utility, was silently overlooked. When Sir W. Hamilton’s attention was called to what he had previously omitted to think of, this is the way in which he disposes of it:

This passage amounts to proof that the writer simply did not know what applied mathematics mean. The words are those of a person who had heard that there was such a thing, but knew absolutely nothing about what it was.

Applied mathematics is not the measurement of extension and number. It is the measurement by means of extension and number, of other quantities which extension and number are marks of; and the ascertainment by means of quantities of all sorts, of those qualities of things which quantities are marks of.

For the information of readers who are no better informed than Sir W. Hamilton, and the reminding of those who are, I will illustrate this general statement by bringing it down to particulars; which a person, himself of very slender mathematical acquirements, can do, provided he has studied the science as every philosophical student ought to study it, but as Sir W. Hamilton has not done, with especial reference to its Methods.

The first, and typical example of the application of mathematics to the indirect investigation of truth, is within the limits of the pure science itself; the application of algebra to geometry; the introduction of which, far more than any of his metaphysical speculations, has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences. Its rationale is simple. It is grounded on the general truth, that the position of every point, the direction of every line, and consequently the shape and magnitude of every enclosed space, may be fixed by the length of perpendiculars thrown down upon two ^astraight lines^a , or (when the third dimension of space is taken into account) upon three ^bplane surfaces^b , meeting one another at right angles in the same point. A consequence, or rather a part, of this general truth, is that curve lines and surfaces may be determined by their equations. If from any number of points in a curve line or surface, perpendiculars are drawn to two ^crectangular axes, or to three rectangular planes^c , there exists between the lengths of these perpendiculars a relation of quantity, which is always the same for the same curve, or surface, and is expressed by an equation in which these variable are combined with certain constant quantities. From this relation, every other property of the curve or surface may always be deduced. In this way, numbers become the means of ascertaining truths not numerical. The periphery of an ellipse is not a number; but a certain numerical relation between straight lines is a mark of an ellipse, being proved to be an inseparable accompaniment of it. The equation which expresses this characteristic mark of any curve, may be handed over to algebraists, to deduce from it, through the properties of numbers, any other numerical relation which depends on it; with the certainty that when the conclusion is translated back again from symbols into words, it will come out a real, and perhaps previously unknown, geometrical property of the curve.

In such an example as this, the application of algebra to geometry appears only in its most elementary form; but its extent is indefinite, and its flights almost beyond the reach of measurement. Its general scheme may be thus stated: In order to resolve any question, either of quality or quantity, concerning a line or space, find something whose magnitude, if known, would give the solution required, and which stands in some known relation to the rectangular co-ordinates (for instance, in the problem of Tangents, the length of the subtangent). Express this known relation in an equation: if the equation can be resolved, we have solved the geometrical problem. Or if the question be the converse one—not what are the properties of a given line or space, but what line or space is indicated by a given property; find what relation between rectangular co-ordinates that property requires: express it in an equation, and this equation, or some other deducible from it, will be the equation of the curve or surface sought. If it be a known curve or surface, this process will point it out; if not, we shall have obtained the necessary starting point for its study.

This application of one branch of mathematics to another branch, ranks as the first step in Applied Mathematics. The second is the application to Mechanics. The object-matter of Mechanics is the general laws, or theory, of Force in the abstract, that is, of forces, considered independently of their origin. As an extension is not a number, though a numerical fact may be a mark of an extension; so a force is neither a number nor an extension. But a force is only cognisable through its effects, and the effects by which forces are best known are effects in extension. The measure of a force, is the space through which it will carry a body of given magnitude in a given time. Quantities of force are thus ascertained, through marks which are quantities of extension. The other properties of forces are, their direction (a question of extension, which has already been reduced to a numerical relation between co-ordinates), and the nature of the motion which they generate, either singly or in combination; which is a mixed question of direction and of magnitude in extension. All questions of Force, therefore, can be reduced to questions of direction and of magnitude: and as all questions of direction or magnitude are capable of being reduced to equations between numbers, every question which can be raised respecting Force abstractedly from its origin, can be resolved if the corresponding algebraical equation can.

While the laws of Number thus underlie the laws of Extension, and these two underlie the laws of Force, so do the laws of Force underlie all the other laws of the material universe. Nature, as it falls within our ken, is composed of a multitude of forces, of which the origin (at least the immediate origin) is different, and the effects of which on our senses are extremely various. But all these forces agree in producing motions in space; and even those of their effects which are not actual motions, nevertheless travel; are propagated through spaces, in determinate times: they are all, therefore, amenable to, and conform to, the laws of extension and number. Often, indeed, we have no means of measuring these spaces and times; nor, if we could, are the resources of mathematics sufficient to enable us, in cases of great complexity, to arrive at the quantities of things we cannot directly measure, through those which we can. Fortunately, however, we can do this, sufficiently for all practical purposes, in the case of the great cosmic forces, gravitation and light, and to a less but still a considerable extent, heat and electricity. And here the domain of Applied Mathematics, for the present, ends. To it we are indebted, not only for all we know of the laws of these great and universal agencies, considered as connected bodies of truth, but also for the one complete type and model of the investigation of Nature by deductive reasoning; the ascertainment of the special laws of nature by means of the general. I will not offer to the understanding of any one who knows what this operation is, the affront of asking him if it is all performed “before” the matter is “hypothetically subjected to mathematical demonstration or calculus.”^[*]

In being the great instrument of Deductive investigation, applied mathematics comes to be also the source of our principal inductions, which invariably depend on previous deductions. For where the inaccessibility or unmanageableness of the phænomena precludes the necessary experiments, mathematical deduction often supplies their place, by making us acquainted with points of resemblance which could not have been reached by direct observation. Phænomena apparently very remote from one another, are found, in the mode of their accomplishment, to follow the same or very similar numerical laws; and the mind, grasping up seemingly heterogeneous natural agencies which have the same equation, and classing them together, often lays a ground for the recognition of them as having either a common, or an analogous, origin. What were previously thought to be distinct powers in Nature, are identified with each other, by ascertaining that they produce similar effects according to the same mathematical laws. It was thus that the force which governs the planetary motions was shown to be identical with that by which bodies fall to the ground. Sir W. Hamilton would probably have admitted that the original discovery of this truth required as great a reach of intellect as has ever yet been displayed in abstract speculation. But is no exercise of intellect needed to apprehend the proof? Is it like an experiment in chemistry or an observation in anatomy, which may require mind for its origination, but to recognise which, when once made, requires only eyesight? Is “continuous attention”^[†] the only mental capacity required here? ^dTo think so would require an ignorance of the subject^d greater than can be imputed to any educated mind, not to speak of a philosopher.

In the achievements which still remain to be effected in the way of scientific generalization, it is not probable that the direct employment of mathematics will be to any great extent available: the nature of the phænomena precludes such an employment for a long time to come—perhaps for ever. But the process itself—the deductive investigation of Nature; the application of elementary laws, generalized from the more simple cases, to disentangle the phænomena of complex cases—explaining as much of them as can be so explained, and putting in evidence the nature and limits of the irreducible residuum, so as to suggest fresh observations preparatory to recommencing the same process with additional data: this is common to all science, moral and metaphysical included; and the greater the difficulty, the more needful is it that the enquirer should come prepared with an exact understanding of the requisites of this mode of investigation, and a mental type of its perfect realization. In the great problems of physical generalization now occupying the higher scientific minds, chemistry seems destined to an important and conspicuous participation, by supplying, as mathematics did in the cosmic phænomena, many of the premises of the deduction, as well as part of the preparatory discipline. But this use of chemistry is as yet only in its dawn; while, as a training in the deductive art, its utmost capacity can never approach to that of mathematics: and in the great enquiries of the moral and social sciences, to which neither of the two is directly applicable, mathematics (I always mean Applied Mathematics) affords the only sufficiently perfect type. Up to this time, I may venture to say that no one ever knew what deduction is, as a means of investigating the laws of nature, who had not learnt it from mathematics; nor can any one hope to understand it thoroughly, who has not, at some time of his life, known enough of mathematics to be familiar with the instrument at work. Had Sir W. Hamilton been so, he would probably have cancelled the two volumes of his Lectures on Logic, and begun again on a different system, in which we should have heard less about Concepts and more about Things, less about Forms of Thought, and more about grounds of Knowledge.

Nor is even this the whole of what the enquirer loses, who knows not scientific Deduction in this its most perfect form. To have an inadequate conception of one of the two instruments by which we acquire our knowledge of nature, and consequently an imperfect comprehension even of the other in its higher forms, is not all. He is almost necessarily without any sufficient conception of human knowledge itself as an organic whole. He can have no clear perception of science as a system of truths flowing out of, and confirming and corroborating, one another; in which one truth sums up a multitude of others, and explains them, special truths being merely general ones modified by specialities of circumstance. He can but imperfectly understand the absorption of concrete truths into abstract, and the additional certainty given to theorems drawn from specific experience, when they can be affiliated as corollaries on general laws of nature—a certainty more entire than any direct observation can give. Neither, therefore, can he perceive how the larger inductions reflect an increase of certainty even upon those narrower ones from which they were themselves generalized, by reconciling superficial inconsistencies, and converting apparent exceptions into real confirmations.^* To see these things requires more than a mere mathematician; but the ablest mind which has never gone through a course of mathematics has small chance of ever perceiving them.

In the face of such considerations, it is a very small achievement to fill thirty octavo pages with the ill-natured things which persons of the most miscellaneous character, through a series of ages, have said about mathematicians, from a sneer of the Cynic Diogenes to a sarcasm of Gibbon, or a colloquial platitude of Horace Walpole; without any discrimination as to how many of the persons quoted were entitled to any opinion at all on such a subject; and with such entire disregard of all that gives weight to authority, as to include men who lived and died before algebra was invented, before the conic sections had been defined and studied by the mathematicians of Alexandria, or the first lines of the theory of statics had been traced by the genius of Archimedes; men whose whole mathematical knowledge consisted of a clumsy arithmetic, and the mere elements of geometry.^[*] Had there been twenty times as many of these testimonies, what proportion of them would have been of any value? Until quite recently, the professors of the different arts and sciences have made it a considerable part of their occupation to cry down one another’s pursuits; and men of the world and littérateurs have been, in all ages, ready and eager to join with every set of them against the rest: the man who dares to know what they neither know nor care for, and to value himself on the knowledge, having always and everywhere been regarded as the common enemy. Did Sir W. Hamilton suppose that a person of half his reading would have any difficulty in furnishing, at a few hours’ notice, an equally long list of amenities on the subject of grammarians or of metaphysicians? When our author does get hold of a witness who has a claim to a hearing, the witness is pressed into the service without any sifting of what he really says; it makes no difference whether he asserts that the study of mathematics does harm, or only that it does not simply suffice for all possible good. One of the authorities on whom most stress is laid is that of Descartes. I extract the important part of the quotation as our author gives it, partly from Descartes himself and partly from Baillet, his biographer. The Italics are Sir W. Hamilton’s.

Whoever reads this passage as if it were all printed in Roman characters, and declines to submit his understanding to the italics which Sir W. Hamilton has introduced, will perceive the following three things. First, that Descartes was not speaking of the study of mathematics, but of its exclusive study. His objection is to stopping there, without proceeding to anything ulterior: conquiescere, incubare. Secondly, that he was speaking only of pure mathematics, as distinguished from its applications, and under the belief, how prodigiously erroneous we now know, that it did not admit of applications of any importance. Finally, that his disparagement of the pursuit, even as thus limited—his representation of it as “nugæ,” as “a loss of time,” rested mainly on a ground which Sir W. Hamilton gave up, the unimportance of its object-matter. It was a repetition of the objection of Socrates, whom also our author thinks it worth while to cite as an authority on such a question, and who “did not perceive of what utility they [mathematical studies] could be, calculated as they were to consume the life of a man, and to turn him away from many other and important acquirements.”^* Such an opinion, in the days of Socrates, and from one whose glorious business it was to recal the minds of speculative men to dialectics and morals, reflects no discredit on his great mind. But the objection is one which Sir W. Hamilton, with every thinker of the last two centuries, disclaims. “The question,” he expressly says, “does not regard the value of mathematical science, considered in itself, or in its objective results, but the utility of mathematical study, that is, in its subjective effect, as an exercise of mind.”^† All that Descartes said against it in this aspect (at least in the passage quoted, which we may suppose to be one of the strongest) is, that by affording other objects of thought, it diverts the mind from the use of ipsa ratio, that is, from the study of pure mental abstractions; which Descartes, to the great detriment of his philosophy, regarded as of much superior value to the employment of the thoughts upon objects of sense, “^fquæ^f magis ad oculos et imaginationem pertinent.”

It was by his example, rather than by his precepts, that Descartes was destined to illustrate the unfavourable side of the intellectual influence of mathematical studies; and he must have been a still more extraordinary man than he was, could he have really understood a kind of mental perversions of which he is himself, in the history of philosophy, the most prominent example. Descartes is the completest type which history presents of the purely mathematical type of mind—that in which the tendencies produced by mathematical cultivation reign unbalanced and supreme. This is visible not only in the abuse of Deduction, which he carried to a greater length than any distinguished thinker known to us, not excepting the schoolmen; but even more so in the character of the premises from which his deductions set out. And here we come upon the one really grave charge which rests on the mathematical spirit, in respect of the influence it exercises on pursuits other than mathematical. It leads men to place their ideal of Science in deriving all knowledge from a small number of axiomatic premises, accepted as self-evident, and taken for immediate intuitions of reason. This is what Descartes attempted to do, and inculcated as the thing to be done: and as he shares with only one other name the honour of having given his impress to the whole character of the modern speculative movement, the consequences of his error have been most calamitous. Nearly everything that is objectionable, along with much of what is admirable, in the character of French thought, whether on metaphysics, ethics, or politics, is directly traceable to the fact that French speculation descends from Descartes instead of from Bacon.^* All reflecting persons in England, and many in France, perceive that the chief infirmities of French thinking arise from its geometrical spirit; its determination to evolve its conclusions, even on the most practical subjects, by mere deduction from some single accepted generalization: the generalization, too, being frequently not even a theorem, but a practical rule, supposed to be obtained directly from the fountains of reason: a mode of thinking which erects one-sidedness into a principle, under the misapplied name of logic, and makes the popular political reasoning in France resemble that of a theologian arguing from a text, or a lawyer from a maxim of law. If this be the case even in France, it is still worse in Germany, the whole of whose speculative philosophy is an emanation from Descartes, and to most of whose thinkers the Baconian point of view is still below the horizon. Through Spinoza, who gave to his system the very forms as well as the entire spirit of geometry; through the mathematician Leibnitz, who reigned supreme over the German speculative mind for above a generation; with its spirit temporarily modified by the powerful intellectual individuality of Kant, but flying back after him to its uncorrected tendencies, the geometrical spirit went on from bad to worse, until in Schelling and Hegel the laws even of physical nature were deduced by ratiocination from subjective deliverances of the mind. The whole of German philosophical speculation has run from the beginning in this wrong groove, and having only recently become aware of the fact, is at present making convulsive efforts to get out of it.^* All these mistakes, and this deplorable waste of time and intellectual power by some of the most gifted and cultivated portions of the human race, are effects of the too unqualified predominance of the mental habits and tendencies engendered by elementary mathematics. Applied mathematics in its post-Newtonian development does nothing to strengthen, and very much to correct, these errors, provided the applications are studied in such a manner that the intellect is aware of what it is about, and does not go to sleep over algebraical symbols; a didactic improvement which Dr. Whewell, to his honour be it said, was earnestly and successfully labouring to introduce, thus practically correcting the real defects of mathematics as a branch of general education, at the very time when Sir W. Hamilton, who had not the smallest insight into those defects, selected him for the immediate recipient of an attack on mathematics, which as it only included what Sir W. Hamilton knew of the subject, left out everything which was much worth saying.

It is not solely to Mathematical studies that Sir W. Hamilton professes and shows hostility. Physical investigations generally, apart from their material fruits, he holds but in low estimation. We have seen in a former chapter how singularly unaware he is of the power and exertion of intellect which they often require. Touching their effect on the mind, he makes two serious complaints, which come out at the very commencement of his Lectures on Metaphysics.^† The first is, that the study of Physics indisposes persons to believe in Free-will. To this accusation it must plead guilty: physical science undoubtedly has that tendency. But I maintain that this is only because physical science teaches people to judge of evidence. If the free-will doctrine could be proved, there is nothing in the habits of thought engendered by physical science that would indispose any one to yield to the evidence. A person who knows only one physical science, may be unable to feel the force of a kind of proof different from that which is customary in his department; but any one who is generally versed in physical science is accustomed to so many different modes of investigation, that he is well prepared to feel the force of whatever is really proof. Metaphysicians of Sir W. Hamilton’s school, who pursue their investigations without regard to the cautions suggested by physical science, are equally catholic and comprehensive in the wrong way; they can mistake for proof anything or everything which is not so, provided it tends to form an association of ideas in their own minds.

The other objection of Sir W. Hamilton to the scientific study of the laws of Matter, is one which we should scarcely have expected from him, namely, that it annihilates Wonder.

We may be well assured that no Hartley, Darwin, or Condillac will obtain a hearing, if the “great religious philosopher” can prevent it.

I shall not enter into all the topics suggested by this remarkable argument. I shall not ask whether, after all, it is better to be “subdued” than instructed; or whether human nature would suffer a great loss in losing wonder, if love and admiration remained; for admiration, pace tantorum virorum, is a different thing from wonder, and is often at its greatest height when the strangeness, which is a necessary condition of wonder, has died away. But I do wonder at the barrenness of imagination of a man who can see nothing wonderful in the material universe, since Newton, in an evil hour, partially unravelled a limited portion of it. If ignorance is with him a necessary condition of wonder, can he find nothing to wonder at in the origin of the system of which Newton discovered the laws? nothing in the probable former extension of the solar substance beyond the orbit of Neptune? nothing in the starry heavens, which, with a full knowledge of what Newton taught, Kant, in the famous passage which Sir W. Hamilton is so fond of quoting (and quotes in this very lecture),^[*] placed on the same level of sublimity with the moral law? If ignorance is the cause of wonder, it is downright impossible that scientific explanation can ever take it away, since all which explanation does, in the final resort, is to refer us back to a prior inexplicable. Were the catastrophe to arrive which is to expel Wonder from the universe—were it conclusively shown that the mental operations are dependent upon organic agency—would wonder be at an end because the fact, at which we should then have to wonder, would be that an arrangement of material particles could produce thought and feeling? Jacobi and Sir W. Hamilton might have put their minds at ease. It is not understanding that destroys wonder, it is familiarity. To a person whose feelings have depth enough to withstand that, no insight which can ever be attained into natural phænomena will make Nature less wonderful. And as for those whose sensibilities are shallow, did Jacobi suppose that they wondered one iota the more at the planetary motions, when astronomers imagined them to take place by the complicated evolutions of “cycle on epicycle, orb on orb?”^[†] A spectacle which they saw every day, had, we may rely upon it, as little effect in kindling their imaginations then, as now. Hear the opinion of a great poet:^* not speaking particularly of wonder, but of the emotions generally which the spectacle of nature excites, and in words which apply to that emotion equally with the rest.

Hear next one of the most illustrious discoverers in physical science. Instead of regarding understanding as antithetical to wonder, Dr. Faraday complains that people do not wonder sufficiently at the material universe, because they do not sufficiently understand it.

If any additional authority be desired, the greatest poet of modern Germany was also the keenest scientific naturalist in it.^[*]

[[*] ]“Study of Mathematics—University of Cambridge,” Edinburgh Review, LXII (Jan., 1836), 409-55; reprinted in Discussions, pp. 263-325.

[[*] ]See Hebrews, 12:1.

[[*] ]See Auguste Comte, Cours de philosophie positive, Vol. I, pp. 57-9.

[* ]I do not know that the logical value of mathematics has ever been more finely and discriminatingly appreciated than by M. Comte in his latest work, Synthèse subjective ([Paris: Comte et Dalmont, 1856,] p. 98). “Bornée à son vrai domaine, la raison mathématique y peut admirablement remplir l’office universel de la saine logique: induire pour déduire, afin de construire. Renonçant à de vaines prétentions, elle sent que ses meilleurs succès restent toujours incapables de nous faire, partout ailleurs, induire, ou même déduire, et surtout construire. Elle se contente de fournir, dans le domaine le plus favorable, un type de clarté, de précision, et de consistance, dont la contemplation familière peut seule disposer l’esprit à rendre les autres conceptions aussi parfaites que le comporte leur nature. Sa réaction générale, plus négative que positive, doit surtout consister à nous inspirer partout une invincible répugnance pour le vague, l’incohérence, et l’obscurité, que nous pouvons réellement éviter envers des pensées quelconques, si nous y faisons assez d’efforts.”

[* ]See, among other passages, Lectures, Vol. III, pp. 146-7.

[† ]Ibid., Vol. I, p. 246.

[* ]Ibid., Vol. II, p. 98.

[† ]Ibid., p. 439.

[‡ ]Discussions, p. 282.

[* ]Ibid., pp. 313-14.

[† ]Ibid., p. 322.

[‡ ]Ibid., p. 290.

[§ ]Ibid., p. 291.

[[*] ]See, e.g., Cambridge Problems, Being a collection of the printed questions proposed to the candidates for the degree of Bachelor of Arts, 1801-1820 (London: Black and Armstrong, 1836).

[[†] ]Miles Bland, Algebraical Problems (Cambridge: Nicholson, 1812), and Geometrical Problems (Cambridge: Nicholson, 1819).

[* ]Discussions, p. 268n. [The references are to Francisco Sanchez, Minerva, sive De Causis Latinæ Linguæ Commentarius, by Caspar Schoppe and Jacobus Perizonius (Franeker: Strickius, 1687), and to Isaac Newton, Philosophiæ Naturalis Principia Mathematica, in Opera, ed. Samuel Horsley, 5 vols. (London: Nichols, 1779-85), Vols. II-III.]

[[*] ]Hamilton’s “On the Study of Mathematics” (a review in the Edinburgh Review of William Whewell’s Thoughts on the Study of Mathematics as a Part of a Liberal Education [Cambridge: Deighton, 1835]) was reprinted in his Discussions (pp. 263-325) with Whewell’s reply, “To the Editor of the Edinburgh Review,” LXIII [April, 1836], 270-2, and the editorial note from the Edinburgh (Discussions, pp. 326-8), and Hamilton’s “Notes to the Above Letter” (again from the Edinburgh, ibid., 272-5; in Discussions, pp. 329-40).

[* ]Discussions, pp. 334-5.

[a-a]+67, 72

[b-b]651, 652 , straight lines

[c-c]651, 652 (or three) rectangular axes

[[*] ]Discussions, p. 335; cf. p. 477 above.

[[†] ]Ibid., p. 314; cf. p. 475 above.

[d-d]651, 652 If Sir W. Hamilton could think so, his ignorance of the subject must have been

[* ]Ignorance of this important principle of the logic of induction, or want of familiarity with it, continually leads to gross misapplications, even by able writers, of the logic of ratiocination. For instance, we are constantly told that the uniformity of the course of nature cannot be itself an induction, since every inductive reasoning assumes it, and the premise must have been known before the conclusion. Those who argue in this manner can never have directed their attention to the continual process of giving and taking, in respect of certainty, which reciprocally goes on between this great premise and all the narrower truths of experience; the effect of which is, that, though originally a generalization from the more obvious of the narrower truths, it ends by having a fulness of certainty which overflows upon these, and raises the proof of them to a higher level; so that its relation to them is reversed, and instead of an inference from them, it becomes a principle from which any one of them may be ededucede .

[[*] ]See “Study of Mathematics,” passim.

[[*] ]Hamilton’s translation of Adrien Baillet, La Vie de Monsieur Des-Cartes, 2 vols. (Paris: Horthemels, 1691), Vol. I, pp. 111-13.

[[†] ]René Descartes, Regulæad directionem ingenii (Amsterdam: Blaev, 1710), p. 12.

[[‡] ]Hamilton’s translation of Baillet, Vol. I, p. 225, Hamilton mistakenly attributes the matter to a letter of Descartes to Marin Mersenne of 1630; Baillet is in fact conflating comments from several letters to Mersenne in the 1630s.

[[§] ]Hamilton’s mistranslation of ibid., p. 235; Hamilton’s parenthesis.

[* ]Discussions, pp. 277-8. [The two parentheses are Hamilton’s; the closing passage is Hamilton’s translation of Baillet, Vol. II, p. 481.]

[* ]Ibid., p. 323. [The words in square brackets are Mill’s. For the opinion of Socrates, see Xenophon, Memorabilia, in Memorabilia and Œconomicus (Greek and English), trans. E. C. Marchant (London: Heinemann; New York: Putnam’s Sons, 1923), pp. 346-8 (IV, vii, 3-5).]

[† ]Ibid., p. 266.

[f-f]+67, 72 [not in Source]

[* ]It is but just to add, that the English mode of thought has suffered in a different, but almost equally injurious manner, by its exclusive following of what it imagined to be the teaching of Bacon, being in reality a slovenly misconception of him, leaving on one side the whole spirit and scope of his speculations. The philosopher who laboured to construct a canon of scientific Induction, by which the observations of mankind, instead of remaining empirical, might be so combined and marshalled as to be made the foundation of safe general theories, little expected that his name would become the stock authority for disclaiming generalization, and enthroning empiricism, under the name of experience, as the only solid foundation of practice.

[* ]The character here drawn of German thought is, I hardly need say, not intended to apply to such a man as Goethe, or to those who received their intellectual impulse from him. In him, indeed, not to speak of his almost universal culture, the intellectual operations were always guided by an intense spirit of observation and experiment, and a constant reference to the exigencies, outward and inward, of practical human life. Such criticism as can justly be made on Goethe as a thinker, rests on entirely different grounds.

[† ]Lectures, Vol. I, pp. 35-42.

[[*] ]See Metaphysics, Vol. I, p. 12 (I, ii, 982b12ff.).

[* ]F. H. Jacobi. [See David Hume über den Glauben, oder Idealismus und Realismus, in Werke, 6 vols. (Leipzig: Fleischer, 1812-25), Vol. II, pp. 54-5.] The entire passage is in Discussions, p. 312.

[† ]Lectures, Vol. I, p. 37.

[[*] ]Ibid., pp. 39-40; quoting Kant, Kritik der Praktischen Vernunft, in Werke, Vol. VIII, p. 312.

[[†] ]Milton, Paradise Lost, in Works, p. 202 (VIII, 84).

[* ]Wordsworth, in the Biography by his nephew [Christopher Wordsworth, Memoirs of William Wordsworth, 2 vols. (London: Moxon, 1851)], Vol. II, p. 159.

[* ][Michael Faraday,] Lectures on the [Various] Forces of Matter [and on the Chemical History of a Candle (London: Griffin, Bohn, 1863)], pp. 2-3. The philosophy of this is well given by Mr. Lewes in his valuable work on Aristotle. “Surprise starts from a background of knowledge, or fixed belief. Nothing is surprising to ignorance, because the mind in that state has no preconceptions to be contradicted.” ([George Henry Lewes, Aristotle: A Chapter in the History of Science (London: Smith, Elder, 1864),] p. 212.)

[[*] ]Johann Wolfgang von Goethe.

[* ]Ignorance of this important principle of the logic of induction, or want of familiarity with it, continually leads to gross misapplications, even by able writers, of the logic of ratiocination. For instance, we are constantly told that the uniformity of the course of nature cannot be itself an induction, since every inductive reasoning assumes it, and the premise must have been known before the conclusion. Those who argue in this manner can never have directed their attention to the continual process of giving and taking, in respect of certainty, which reciprocally goes on between this great premise and all the narrower truths of experience; the effect of which is, that, though originally a generalization from the more obvious of the narrower truths, it ends by having a fulness of certainty which overflows upon these, and raises the proof of them to a higher level; so that its relation to them is reversed, and instead of an inference from them, it becomes a principle from which any one of them may be ededucede .

[ededucede]651, 652 inferred