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prelude to other labors, more interesting and more difficult. From the circumstances which they have stated, it would seem that the intention of the author was to extend to all the other branches of knowledge, inferences similar to those which he has here endeavored to establish with respect to mathematical calculations; and much regret is expressed by his friends, that he had not lived to accomplish a design of such incalculable importance to human happiness. I believe I may safely venture to assert, that it was fortunate for his reputation he proceeded no farther; as the sequel must, froin the nature of the subject, have afforded, to every competent judge, an experimental and palpable proof of the vagueness and fallaciousness of those views by which the undertaking was suggested. In his posthumous volume, the mathematical precision and perspicuity of his details appear to a superficial reader to reflect some part of their own light on the general reasonings with which they are blended; while to better judges, these reasonings come recommended with many advantages and with much additional authority, from their coincidence with the doctrines of the Leibnitzian school.
It would probably have been not a little mortifying to this most ingenious and respectable philosopher, to have discovered, that, in attempting to generalize a very celebrated theory of Leibnitz, he had stumbled upon an obsolete conceit, started in this island upwards of a century before. “ When a man reasoneth,” says Hobbes, “ he does nothing else but conceive a sum total, from addition of parcels ; or conceive a remainder from subtraction of one sum from another; which, if it be done by words, is conceiving of the consequence of the names of all the parts to the name of the whole ; or from the name of the whole and one part, to the name of the other part. These operations are not incident to number only, but to all manner of things that can be added together, and taken one out of another. In sum, in what matter soever there is place for addition and subtraction, there also is place for reason; and where these have no place, there reason has nothing at all to do.
“Out of all which we may define what that is which is meant by the word reason, when we reckon it amongst the faculties of the mind. For reason, in this sense, is nothing but reckoning (that is, adding and subtracting)of the consequences of general names agreed upon for the marking and signifying of our thoughts ;-I say marking them, when we reckon by ourselves; and signifying, when we demonstrate or approve our reckonings to other men.” (Leviathan, chap. v.)
Agreeably to this definition, Hobbes has given to the first part of his elements of philosophy, the title of Computatio, sive Logica ; evidently employing these two words as precisely synonymous. From this tract I shall quote a short paragraph, not certainly on account of its intrinsic value, but in consequence of the interest which it derives from its coincidence with the speculations of some
of our contemporaries. I transcribe it from the Latin edition, as the antiquated English of the author is apt to puzzle readers not familiarized to the peculiarities of his philosophical diction.
“Per ratiocinationem autem intelligo computationem. Computare vero est plurium rerum simul additarum summam colligere, vel unâ re ab aliâ detractâ cognoscere residuum. Ratiocinari igitur idem est quod addere et subtrahere, vel si quis adjungat his multiplicare et dividere, non abnuam, cum multiplicatio idem sit quod æqualium additio, divisio quod æqualium quoties fieri potest subtractio. Recidit itaque ratiocinatio omnis ad duas operationes animi, additionem et subtractionem."* How wonderfully does this jargon agree with the assertion of Condillac, that all equations are propositions, and all propositions equations !
These speculations, however, of Condillac and of Hobbes relate to reasoning in general; and it is with mathematical reasoning alone that we are immediately concerned at present. That the peculiar evidence with which this is accompanied is not resolvable into the perception of identity, has, I flatter myself, been sufficiently proved in the beginning of this article ; and the plausible extension by Condillac of the very same theory to our reasonings in all the different branches of moral science, affords a strong additional presumption in favor of our conclusion.
From this long digression, into which I have been insensibly led by the errors of some illustrious foreigners concerning the nature of mathematical demonstration, I now return to a further examination of the distinction beetween sciences which rest ultimately on facts, and those in which definitions or hypotheses are the sole principles of our reasonings.
III.- Continuation of the Subject.—Evidence of the Mechanical
Philosophy, not to be confounded with that which is properly called Demonstrative or Mathematical.—Opposite Error of some late Writers.
Next to geometry and arithmetic, in point of evidence and certainty, is that branch of general physics which is now called mechanical philosophy:—a science in which the progress of discovery has been astonishingly rapid, during the course of the last century; and which, in the systematical concatenation and filiation of its elementary principles, exhibits every day more and more of that logical simplicity and elegance which we admire in the works of the Greek mathematicians. It may, I think, be fairly questioned, whether,
The Logica of Hobbes has been lately translated into French, under the title of Calcul ou Logique, by M. Destutt-Tracy. It is annexed to the third volume of his Elémens d'Idéologie, where it is honored with the highest eulogies by the ingenious translator. “L'ouvrage en masse,” he observes in one passage, “ mérite d'être regardé comme un produit precieux des méditations de Bacon et de Descartes sur le système d'Aristote, et comme le germe des progrès ultérieures de la science."'-Disc. Prel. p. 117.
in this department of knowledge, the affectation of mathematical method has not been already carried to an excess; the essential distinction between mechanical and mathematical truths being, in many of the physical systems which have lately appeared on the Continent, studiously kept out of the reader's view, by exhibiting both, as nearly as possible, in the same form. A variety of circumstances, indeed, conspire to identify in the imagination, and, of consequence, to assimilate in the mode of their statement, these two very different classes of propositions; but as this assimilation, besides its obvious tendency to involve experimental facts in metaphysical mystery, is apt occasionally to lead to very erroneous logical conclusions, it becomes the more necessary, in proportion as it arises from a natural bias, to point out the causes in which it has originated, and the limitations with which it ought to be understood.
The following slight remarks will sufficiently explain my general ideas on this important article of logic.
1. As the study of the mechanical philosophy is, in a great measure inaccessible to those who have not received a regular mathematical education, it commonly happens, that a taste for it is, in the first instance, grafted on a previous attachment to the researches of pure or abstract mathematics. Hence a natural and insensible transference to physical pursuits, of mathematical habits of thinking; and hence an almost unavoidable propensity to give to the former science that systematical connexion in all its various conclusions which, from the nature of its first principles is essential to the latter, but which can never belong to any science which has its foundations laid in fact collected from experience and observation.
2. Another circumstance which has co-operated powerfully with the former in producing the same effect, is that proneness to simplification which has misled the mind, more or less, in all its researches, and which, in natural philosophy, is peculiarly encouraged by those beautiful analogies which are observable among different physical phenomena--analogies, at the same time, which, however pleasing to the fancy cannot always be resolved by our reason into one general law. In a remarkable analogy, for example, which presents itself between the equality of action and re-action in the collision of bodies, and what obtains in their mutual attractions, the coincidence is so perfect, as to enable us to comprehend all the various facts in the same theorem; and it is difficult to resist the temptation which it seems to offer to our ingenuity, of attempting to trace it, in both cases, to some common principle. Such trials of theoretical skill I would not be understood to censure indiscriminately; but in the present instance, I am fully persuaded, that it is at once more unexceptionable in point of sound logic, and more satisfactory to the learner to establish the fact, in particular cases, by an appeal to experiment; and to state the law of action and reaction in the collision of bodies, as well as that which regulates the
mutual tendencies of bodies towards each other, merely as general rules which have been obtained by induction, and which are found to hold invariably as far as our knowledge of nature extends.*
An additional example may be useful for the illustration of the same subject. It is well known to be a general principle in mechanics, that when, by means of any machine, two heavy bodies counterpoise each other, and are then made to move together, the quantities of motion with which one descends, and the other ascends perpendicularly, are equal. This equilibrium bears such a resemblance to the case of two moving bodies stopping each other, when they meet together with equal quantities of motion, that, in the opinion of many writers, the cause of an equilibrium in the several machines is sufficiently explained, by remarking, “ that a body always loses as much motion as it communicates.” Hence it is inferred, that when two heavy bodies are so circumstanced, that one cannot descend without causing the other to ascend at the same time, and with the same quantity of motion, both of these bodies must necessarily continue at rest. But this reasoning, however plausible it may seem to be at first sight, is by no means satisfactory;
It is observed by Mr. Robison, in his Elements of Mechanical Philosophy, that “Sir Isaac Newton, in the general scholium of the laws of motion, seems to consider the equality of action and re-action as an axiom deduced from the relations of ideas. But this,” says Mr. Robison, “seems doubtful. Because a magnet causes the iron to approach towards it, it does not appear that we necessarily sup. pose that iron also attracts the magnet.” In confirmation of this he remarks, that notwithstanding the previous conclusions of Wallis, Wren, and Huygens, about the mutual, equal, and contrary actions of solid bodies in their collisions, " Newton himself only presumed that, because the sun attracted the planets, these also attracted the sun; and that he is at much pains to point out phenomena to astronomers, by which this may be proved, when the art of observation shall be sufficiently perfected.” Accordingly, Mr. Robison, with great propriety, contents himself with stating this third law of motion, as a fact “ with respect to all bodies on which we can make experiment or observation fit for deciding the question."
In the very next paragraph, however, he proceeds thus : “ As it is an univer. sal law, we cannot rid ourselves of the persuasion that it depends on some general principle which influences all the matter in the universe ;' to which observation he subjoins a conjecture or hypothesis, concerning the nature of this principle or cause. For an outline of his theory I must refer to his own statement. See Elements of Mechanical Philosophy, vol. i. pp. 124—126.
Of the fallaciousness of synthetical reasonings concerning physical phenomena, there cannot be a stronger proof, than the diversity of opinion among the most eminent philosophers with respect to the species of evidence on which the third law of motion rests. On this point, a direct opposition may be remarked in the views of Sir Isaac Newton, and of his illustrious friend and commentator, Mr. Maclaurin; the former seeming to lean to the supposition, that it is a corollary deducible à priori from abstract principles, while the latter (manifestly consider. ing it as the effect of an arbitrary arrangement) strongly recommends it to the attention of those who delight in the investigation of final causes.-(Account of Newton's Philosophical Discoveries; book ii. chap 2, sec. 28.) My own idea is, that, in the present state of our knowledge, it is at once more safe and more logical, to consider it merely as an experimental truth, without venturing to de. cide positively on either side of the question. As to the doctrine of final causes, it fortunately stands in need of no aid from such dubious speculations.
for, as Dr. Hamilton has justly observed,* when we say, that one body communicates its motion to another, we must suppose the motion to exist, first in the one, and afterwards in the other : whereas, in the case of the machine, the ascent of the one body cannot, by any conceivable refinement, be ascribed to a communication of motion from the body which is descending at the same moment; and, therefore, (admitting the truth of the general law which obtains in the collision of bodies.) we might suppose, that in the machine, the superior weight of the heavier body would overcome the lighter, and cause it to move upwards with the same quantity of motion with which itself moves downwards. In perusing a pretended demonstration of this sort, a student is dissatisfied and puzzled, not from the difficulty of the subject, which is obvious to every capacity, but from the illogical and inconclusive reasoning to which his assent is required.t
3. To these remarks it may be added, that even when one proposition in natural philosophy is logically deducible from another, it may frequently be expedient, in communicating the elements of the science, to illustrate and confirm the consequence, as well as the principle, by experiinent. This I should apprehend to be proper wherever a consequence is inferred from a principle less familiar and intelligible than itself; a thing which must occasionally happen in physics, from the complete incorporation, if I may use the expression, which, in modern times, has taken place between physical truths, and the discoveries of mathematicians. The necessary effect of this incorporation was, to give to natural philosophy a mathematical form, and to systematize its conclusions, as far as possible, agreeably to rules suggested by mathematical method.
In pure mathematics, where the truths which we investigate are all co-existent in point of time, it is universally allowed, that one proposition is said to be a consequence of another, only with a reference to our established arrangements. Thus all the properties of the circle might be as rigorously deduced from any one general property of the curve, as from the equality of the radii. But it does not therefore follow, that all these arrangements would be equally convenient; on the contrary, it is evidently useful, and
* See Philosophical Essays, by Hugh Hamilton, D. D., Professor of Philosophy in the University of Dublin, p. 135, et seq. 3rd edit. London, 1772.
† The following observation of Dr. Hamilton places this question in its true point of view : “ However, as the theorem above mentioned is a very elegant one, it ought certainly to be taken notice of in every treatise of mechanics, and may serve as a very good index of an equilibriu:n in all machines; but I do not think that we can from thence, or from any one general principle, explain the nature and effects of all the mechanic powers in a satisfactory manner.
To the same purpose, it is remarked by Mr. Maclaurin, that “ though it be use. ful and agreeable to observe how uniformly this principle prevails in engines of every sort throughout the whole of mechanics, in all cases where an equilibrium takes place; yet that it would not be right to rest the evidence of so important a doctrine upon a proof of this kind only."-Account of Newton's Discoveries, b. ii. c. 3.