only as have been carefully defined. Various remarks to this purpose occur in the writings both of Mr. Locke and of Dr. Reid. But the example of mathematicians is by no means applicable to the sciences in which these eininent philosophers propose that it should be followed; and, indeed, if it were copied as a model in any other branch of human knowledge, it would lead to errors fully as dangerous as any which result from the imperfections of language. The real fact is, that it has been copied much more than it ought to have been, or than would have been attempted, if the peculiarities of mathematical evidence had been attentively considered. That in mathematics there is no such thing as an ambiguous word, and that it is to the proper use of definitions we are indebted for this advantage, must unquestionably be granted. But this is an advantage easily secured, in consequence of the very limited vocabulary of mathematicians, and the distinctness of the ideas about which their reasonings are employed. The difference, besides, in this respect, between mathematics and the other sciences, however great, is yet only a difference in degree ; and is by no means sufficient to account for the essential distinction which every person must perceive between the irresistible cogency of a mathematical demonstration, and that of any other process of reasoning. From the foregoing considerations it appears, that in mathematics definitions answer two purposes ; first, to prevent ambiguities of language; and, secondly, to serve as the principles of our reasoning. It appears further, that it is to the latter of these circumstances (I mean to the employment of hypotheses instead of facts, as the data on which we proceed) that the peculiar force of demonstrative evidence is to be ascribed. It is, however, only in the former use of definitions, that any parallel can be drawn between mathematics and those branches of knowledge which relates to facts; and, therefore, it is not a fair argument in proof of their general utility, to appeal to the unrivalled certainty of mathematical science,-a pre-eminence which that science derives from a source altogether different, though comprehended under the same name, and which she will forever claim as her own exclusive prerogative. Nor ought it to be forgotten, that it is in pure mathematics alone, that definitions can be attempted with propriety at the outset of our investigations. In most other instances, some previous discussion is necessary to show, that the definitions which we lay down correspond with facts; and, in many cases, the formation of a just definition is the end to which our inquiries are directed. It is very judiciously observed by Mr. Burke, in his Essay on Taste, that “when we define, we are in danger of circumscribing nature within * These two classes of definitions are very generally confounded by logicians ; among others, by the Abbé de Condillac. See La Logique, ou les premiers développemens de l'Art de Penser, chap. vi. the bounds of our own notions, which we often take up by hazard, or embrace on trust, or form out of a limited and partial consideration of the object before us, instead of extending our ideas to take in all that nature comprehends, according to her manner of combining. We are limited in our inquiry by the strict laws to which we have submitted at our setting out.” The same author adds, that “ a definition may be very exact, and yet go but a very little way towards informing us of the nature of the thing defined ;” and that," in the order of things, a definition, let its virtue be what it will, ought rather to follow than to precede our inquiries, of which it ought to be considered as the result.” From a want of attention to these circumstances, and from a blind imitation of the mathematical arrangement, in speculations where facts are involved among the principles of our reasonings, numberless errors in the writings of philosophers might be easily traced. The subject is of too great extent to be pursued any farther here; but it is well entitled to the examination of all who may turn their thoughts to the reformation of logic. That the idea of Aristotle himself, with respect to it, were not very precise, must, I think, be granted, if the following statement of his ingenious commentator be admitted as correct. “Every general term," says Dr. Gillies, “is considered by Aristotle as the abridgment of a definition, and every definition is denominated by him a collection, because it is the result always of observation and comparison, and often of many observations and of many comparisons.” (Gillies's Aristotle, vol. i. p. 92.. Second edition.) These two propositions will be found, upon examination, not very consistent with each other. The first, “That every general term is the abridgment of a definition," applies indeed admirably to mathematics ; and touches with singular precision on the very circumstance which constitutes, in my opinion, the peculiar cogency of mathematical reasoning. But it is to mathematics that it applies exclusively. If adopted as a logical maxim in other branches of knowledge, it would prove an endless source of sophistry and error. -The second proposition, on the other hand, “That every definition is the result of observation and comparison, and often of many observations and many comparisons ;" however applicable to the definitions of natural history, and of other sciences which relate to facts, cannot in one single instance apply to the definitions of geometry; inasmuch as these definitions are neither the result of observations nor of comparisons, but the hypotheses, or first principles, on which the whole science rests. If the foregoing account of demonstrative evidence be just, it follows, that no chain of reasoning whatever can deserve the name of a demonstration (at least in the mathematical sense of that word) which is not ultimately resolvable into hypotheses or defi nitions.* It has been already shown, that this is the case with geometry. And it is also manifestly the case with arithmetic, another science to which, in common with geometry, we apply the word mathematical. The simple arithmetical equations 2 add 2=4; 2 add 3=5, and other elementary propositions of the same sort, are, as was formerly observed, mere definitions, (see page 356,) perfectly analogous, in this respect, to those at the beginning of Euclid; and it is from a few fundamental principles which are essentially of the same description, that all the more complicated results in the science are derived. To this general conclusion, with respect to the nature of mathematical demonstration, an exception may perhaps be, at first sight, apprehended to occur, in our reasonings concerning geometrical problems; all of these reasonings, as is well known, resting ultimately upon a particular class of principles called postulates, which are commonly understood to be so very nearly akin to axioms, that both might, without impropriety, be comprehended under the same name. “ The definition of a postulate,” says the learned and ingenious Dr. Hutton, “will nearly agree also to an axiom, which is a self-evident theorein, as a postulate is a self-evident problem." (Mathematical Dictionary, art. Postulate.) The same author, in another part of his work, quotes a remark from Dr. Barrow, that “there is the same affinity between postulates and problems, as between axioms and theorems.” (Ibid. art. Hypothesis.) Dr. Wallis, too, appears, from the following passage, to have had a decided leaning to this opinion. “According to some, the difference between axioms and postulates is analogous to that between theorems and problems; the former expressing truths which are self-evident, and from which other propositions may be deduced ; the latter, operations which may be easily performed, and by the help of which more difficult constructions may be effected." He afterwards adds, “This account of the distinction between postulates and axioms seems not ill-adapted to the division of mathematical propositions into problems and theorems. And indeed, if both postulates and axioms were to be comprehended under either of these names, the innovation would not, in my opinion, afford much ground for censure. (Wallisii Opera, vol. ii. pp. 667, 668.) In opposition to these very high authorities, I have no hesitation Although the account given by Locke of what constitutes a demonstration, be different from that which I have here proposed, he admits the converse of this doctrine as manifest; viz. That if we reason accurately from our own definitions, our conclusions will possess demonstrative evidence; and “hence" he observes with great truth, “it comes to pass, that one may often meet with very clear and coherent discourses, that amount yet to nothing.” He afterwards remarks, that, " one may make demonstrations and undoubted propositions in words, and yet thereby advance not one jot in the knowledge of the truth of things.” “Of this sort," he adds, “a man may find an infinite number of propositions, reasonings, and conclusions, in books of metaphysics, school-divinity, and some sort of natural philosophy; and, after all, know as little of God, spirits, or bodies, as he did be. fore he set out."-Essay on Human Understanding, book, iv. chap. viii. to assert, that it is with the definitions of Euclid and not with the axioms that the postulates ought to be compared, in respect of their logical character and importance ;-inasmuch as all the demonstrations in plane geometry are ultimately founded on the former, and all the constructions which it recognises as legitimate, may be resolved ultimately into the latter. To this remark it may be added, that, according to Euclid's view of the subject, the problems of geometry are not less hypothetical and speculative, (or, to adopt the phraseology of some late writers, not less objects of pure reason) than the theorems; the possibility of drawing a mathematical straight line, and of describing a mathematical circle, being assumed in the construction of every problem, in a way quite analogous to that in which the enunciation of a theorem assumes the existence of straight lines and of circles corresponding to their mathematical definitions. The reasoning, therefore, on which the solution of a problem rests, is not less demonstrative than that which is employed in proof of a theorem. Grant the possibility of three operations described in the postulates, and the correctness of the solution is as mathematically certain, as the truth of any property of the triangle or of the circle. The three postulates of Euclid are, indeed, nothing more than the definitions of a circle and a straight line thrown into a form somewhat different: and a similar remark may be extended to the corresponding distribution of propositions into theorems and problems. Notwithstanding the many conveniences with which this distribution is attended, it was evidently a matter of choice rather than that of necessity; all the truths of geometry easily admitting of being moulded into either shape, according to the fancy of the mathematician. As to the axioms, there cannot be a doubt, whatever opinion may be entertained of their utility or of their insignificance, that they stand precisely in the same relation to both classes of propositions.* * In farther illustration of what is said above, on the subject of postulates and of problems, I transcribe, with pleasure, a short passage from a learned and interesting memoir, just published, by an author intimately and critically conversant with the classical remains of Greek geometry. “ The description of any geometrical line from the data by which it is defined, must always be assumed as possible, and is admitted as the legitimate means of a geometrical construction : it is therefore properly regarded as a postulate. Thus, the description of a straight line and of a circle are the postulates of plane geometry assumed by Euclid. The description of the three conic sections, according to the definitions of them, must also be regarded as postulates; and though not for: mally stated like those of Euclid, are in truth admitted as such by Apollonius, and all other writers on this branch of geometry. The same principle must be extended to all superior lines. “ It is true, however, that the properties of such superior lines may be treated of, and the description of them may be assumed in the solution of problems, without an actual delineation of them. For it must be observed, that no lines whatever, not even the straight line or circle, can be truly represented to the senses according to the strict mathematical definitions; but this by no means affects the theorelical conclusions which are logically deduced from such definitions. It is only when geometry is applied to practice, either in mensuration, or in the arts connected with geometrical principles, that accuracy of delineation becomes important.”—See an Account of the Life and Writings of Robert Simson, M. D. By the Rev. William Trail, LL. D. Published by G. and W. Nicol, Lond. 1812. * It was probably with a view to the establishment of this doctrine, that some foreign elementary writers have lately given the name of identical triangles to such as agree with each other, both in sides, in angles, and in area. The differences which may exist between them in respect of place, and of relative position (differences which do not at all enter into the reasonings of the geometer) seem to have been considered as of so little account in discriminating them as separate objects of thought, that it has been concluded they only form one and the same iriangle, in the contemplation of the logician. II.- Continuation of the Subject.-How far it is true that all Mathe matical Evidence is resolvable into Indentical Propositions. I had occasion to take notice, in the first section of the preceding chapter, of a theory with respect to the nature of mathematical evidence, very different from that which I have been now attempting to explain. According to this theory (originally, I believe, proposed by Leibnitz) we are taught, that all mathematical evidence ultimately resolves into the perception of identity ; the innumerable variety of propositions which have been discovered, or which remain to be discovered in the science, being only diversified expressions of the simple formula, a=a. A writer of great eminence, both as a mathematician and a philosopher, has lately given his sanction, in the strongest terms, to this doctrine; asserting, that all the prodigies performed by the geometrician are accomplished by the constant repetition of these words,—the same is the same. "Le géomêtre avance de supposition en supposition. Et rétournant sa pensée sous mille formes, c'est en répétant sans cesse, le même est le même, qu'il opère tous ses prodiges.” As this account of mathematical evidence is quite irreconcilable with the scope of the foregoing observations, it is necessary, before proceeding farther, to examine its real import and amount; and what the circumstances are from which it derives that plausibility which it has been so generally supposed to possess. That all mathematical evidence resolves ultimately into the perception of identity, has been considered by some as a consequence of the commonly received doctrine, which represents the axioms of Euclid as the first principles of all our subsequent reasonings in geometry. Upon this view of the subject I have nothing to offer in addition to what I have already stated. The argument which I mean to combat at present, is of a more subtile and refined nature ; and, at the same time, involves an admixture of important truth, which contributes not a little to the specious verisimilitude of the conclusion. It is founded on this simple consideration, that the geometrical notions of equality and of coincidence are the same ; and that, even in comparing together spaces of different figures, all our conclusions ultimately lean with their whole weight on the imaginary application of one triangle to another ;—the object of which imaginary application is merely to identify the two triangles together in every circumstance connected both with magnitude and figure.* |