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representative of such a formula as the binomial theorem of Sir
I should scarcely have been induced to dwell so long on this why dwell.
theory of Leibnitz concerning mathematical evidence, if I had not observed among some late logicians (particularly among the followers of Condillac) a growing disposition to extend it to all the different sorts of evidence resulting from the various employments of our reasoning powers. Condillac himself states his own opinion on this point with the most perfect confidence. (1) "L'évidence de raison "consiste uniquement dans l'identité: c'est ce que nous avons dé"montré. Il faut que cette vérité soit bien simple pour avoir "échappé à tous les philosophes, quoiqu' ils eussent tant d'intérêt "à s'assurer de l'évidence, dont ils avoient continuellement le mot "dans la bouche."t
*The foregoing reasonings are not meant as a refutation of the arguments urged by any one author in support of the doctrine in question; but merely as an examination of those by which I have either heard it defended, or from which I conceived that it might possibly derive its verisimilitude in the judgment of those who have adopted it. The arguments which I have supposed to be alleged by its advocates, are so completely independent of each other, that, instead of being regarded as different premises leading to the same conclusion, they amount only to so many different interpretations of the same verbal proposition;-a circumstance which, I cannot help thinking, affords of itself no slight proof, that this proposition has been commonly stated in terms too general and too ambiguous for a logical principle. What a strange infer ence has been drawn from it by no less a philosopher than Diderot! "Interrogez des "mathématiciens de bonne foi, et ils vous avoueront que leurs propositions sont toutes "identiques, et que tant de volumes sur le cercle, par exemple, se réduisent à "nous répéter en cent mille facons différentes, que c'est une figure ou toutes les lignes "tirées du centre à la circonference sont égales. Nous ne savons donc presque rien." -Lettre sur les Aveugles.
[Interrogate mathematicians who are honest, and they will confess that all their pro positions are identical, and that so many volumes upon the circle, for example, are made up by repeating a hundred thousand times, that it is a figure in which all the lines drawn from the centre to the circumference are equal. We know then almost nothing.]
+ La Logique, Chap. IX.
On another occasion, Condillac expresses himself thus: "Tout le systême des connois. "sances humaines peut être rendu par une expression plus abrégée et tout-a-fait iden"tique: les sensations sont des sensations. Si nous pouvions, dans toutes les sciences, "suivre également la génération des idées, et saisir le vrai systême des choses, nous "verrions d'une vérité naitre toutes les autres, et nous trouverions l'expression abrégée "de tout ce que nous saurions, dans cette proposition identique: le même est le même.
[The whole system of human knowledge may perhaps be characterized by an expression very concise and entirely identical: Sensations are sensations. If we could, in all the sciences, equally follow the generation of ideas, and seize upon the true system of things, we should discover, that from one truth spring all the others, and we should find a short expression of all our knowledge in this identical proposition; the same is the same.]
(1) [The evidence of reason consists entirely in identity. This we have demonstrated. The extreme simplicity of this truth has occasioned it to be overlooked by Philosophers, alt hough they are so highly interested in this evidence, of which they continually teat.]
The demonstration here alluded to is extremely concise; and if we grant the two data on which it proceeds, must be universally acknowledged to be irresistible. The first is, That the evidence "of every mathematical equation is that of identity:" The second, "That what are called, in the other sciences, propositions or judgments, are, at bottom, precisely of the same nature with equa"tions."--But it is proper, on this occasion, to let our author speak for himself.
Mais, dira-t-on, c'est ainsi qu'on raisonne en mathématiques, ou "le raisonnement se fait avec des équations. En sera-t-il de même "dans les autres sciences, où le raisonnement se fait avec des pro"positions? Je réponds qu' équations, propositions, jugemens, sont "au fond la même chose, et que par conséquent on raisonne de la "même manière dans toutes les sciences."*
Upon this demonstration I have no comment to offer. The truth of the first assumption has been already examined at sufficient length; and the second (which is only Locke's very erroneous account of judgment, stated in terms incomparably more exceptionable) is too puerile to admit of refutation. It is melancholy to reflect, that a writer who, in his earlier years, had so admirably unfolded the mighty influence of language upon our speculative conclusions, should have left behind him, in one of his latest publications, so memorable an illustration of his own favourite doctrine.
It was manifestly with a view to the more complete establishment of the same theory, that Condillac undertook a work, which has appeared since his death, under the title of La Language des Calculs; and which, we are told by the editors, was only meant as a prelude to other labours, 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 endeavoured 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, from 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 re
[But, some will say, It is thus we reason in mathematics where the reasoning is carried on by equations. Will it be the same in other sciences, where this is carried on by propositions? I answer, that equations, propositions judgments, are in fact the same thing; and consequently we reason in the same manner in all the sciences.] La Lo gique, Chap. VIII.
flect 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 names of the whole and one part, to the "name of the other part.-These operations are not incident to "numbers only, but to all manner of things that can be added to"gether, and taken one out of another.-In sum, in what matter "soever there is place for addition and substraction, 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."*
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.
"(1) Per ratiocinationem autem intelligo computationem. Compu"tare vero est plurium rerum simul additarum summam colligere, vel “unâ re ab aliâ detractâ, cognoscere residuum. Ratiocinari igitur
Leviathan, Chap. v.
(1) [By reasoning I understand'computation. To compute is to collect the sum of seve ral things added together; or, having taken one thing from another to ascertain the resi duum. To reason therefore is the same as to add and subtract; or if any one chooses to say also, to multiply and divide, I will not object, since multiplication is the same as the addition of equals, and division the subtraction of equals carried as far as possible. All reasoning therefore is reduced to two operations of the mind, addition and subtraction.]
"idem est quod addere et subtrahere, vel si quis adjungat his multi"plicare et dividere, non abnuam, cum multiplicatio idem sit quod "aequalium additio, divisio quod aequalium quoties fieri potest sub"tractio. Recidit itaque ratiocinatio omnis ad duas operationes ani"mi, additionem 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 favour of our conclusion.
From this long digression, into which I have been insensibly led by the errours of some illustrious foreigners concerning the nature of mathematical demonstration, I now return to a further examination of the distinction between sciences which rest ultimately on facts, and those in which definitions or hypotheses are the sole principles of our reasonings.
Continuation of the Subject.—Evidence of the Mechanical Philosophy, not to be confounded with that which is properly called Demonstrative or Mathematical.-Opposite Errour 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 elemenext to Math. tary principles, exhibits every day more and more of that logical simplicity and elegance which we admire in the works of the Greek ů point of mathematicians. It may, I think, be fairly questioned, whether, in cvidence' 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 conti
*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 Elemens d' Ideologie, where it is honoured with the highest eulogies by the ingenious translator. "L'ouvrage en masse (he observes in one passage) mérite d'être regardé com me un produit precieux des méditations de Bacon et de Descartes sur le systême d'Aristote, et comme le germe des progrès ultérieures de la science." (Disc. Prél. p. 117.) [The whole work merits to be esteemed as the precious product of the meditations of Bacon and Descartes upon the system of Aristotle, and as the germ of further progress in science.]
nent, 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 ent classes of propositions; but as this assimilation (beside its obvi- have led to y ous tendency to involve experimental facts in metaphysical mystery) is apt occasionally to lead to very erroneous logical conclusions, it assimolati 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 which it ought to be understood.
bass of proposition
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 mea-. The order sure, inaccessible to those who have not received a regular mathematical education, it commonly happens, that a taste for it is, in the wh. 5m first instance, grafted on a previous attachment to the researches of studied. 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 facts 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 fication which has misled the mind, more or less, in all its research- simplifica es; and which, in natural philosophy, is peculiarly encouraged by tion! 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 this theorem 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 butter to us 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 in particul 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.*
It is observed by Mr. Robison, in his Elements of Mechanical Philosophy, that "Sir Isaac Newton, in the general scholium on the laws of motion, seems to consider the equali