APPENDIX.

THE

ARTICLE I. (See page 94

HE following article relates entirely to the question,-"How far it is true, that "all mathematical evidence is resolvable into identical propositions?" The discussion may, in one point of view, be regarded as chiefly verbal; but that it is not, on that account, of so trifling importance, as might at first be imagined, appears from the humiliating inference to which it has been supposed to lead concerning the narrow limits of human knowledge. "Put the question," says Diderot, "to any candid mathematician, and he will acknowledge, that all mathematical "propositions are merely identical; and that the numberless volumes written (for "example) on the circle, only repeat over in a hundred thousand forms, that it is "a figure in which all the straight lines drawn from the centre to the circumference "are equal. The whole amount of our knowledge, therefore, is next to nothing.”—That Diderot has, in this very paradoxical conclusion, stated his own real opinion, will not be easily believed by those who reflect on his extensive acquaintance with mathematical and physical science; but I have little doubt, that he has expressed the amount of the doctrine in question, agreeably to the interpretation put on it, by the great majority of readers.

As the view of this subject which I have taken in the text, has not been thought satisfactory by my friend M. Prevost, I have thought it a duty, both to him and to myself, to annex to the foregoing pages, in his own words, the remarks subjoined to the excellent and faithful translation with which he has honoured this part of my work, in the Bibliothèque Britannique. Among these remarks, there is scarcely a proposition to which I do not give my complete assent. The only difference between us turns on the propriety of the language in which some of them are expressed; and on this point it is not surprising, if our judgments should be soine what biassed by the phraseology to which we have been accustomed in our earlier years. The few sentences to which I am inclined to object, I have distinguished from the rest, by printing them in small capitals. Such explanations of my own argument as appear to be necessary, I have thrown into the form of notes, at the foot of the page.

In the course of M. Prevost's observations on the point in question, he has introduced various original and happy illustrations of the important distinction between conditional and absolute truths;-a subject on which I have the pleasure to find, that all our views coincide exactly.

(1) "A la fin de l'article que l'on vient de lire, l'ingénieux auteur renvoie à ce qu'il a dit au commencement. Il pense y avoir suffisamment prouvé que l'évidence

(1) At the conclusion of the article which we have just read, the ingenious author refers to what he had said at the commencement. He thinks he has there sufficient

Chap. II. Sect. 3. Art. II. of this Volume.

particulière qui accompagne le raisonnement mathématique ne peut pas se résoudre dans la perception de l'identité. Recourons donc à cette preuve. Elle se trouve consister toute entière en réfutation.

"I. L'auteur commence par remarquer, que quelques personnes fondent l'opinion qu'il rejette sur celle qui prend les axiomes pour premiers principes. Et comme il a combattu celle-ci, il en conclut que sa conséquence doit être fausse. Un tel argument a en effet beaucoup de force pour ceux qui sont partis d'une certaine theorie sur les axiomes pour en conclure l'assertion contestée; mais il n'en a point pour les autres. Le rédacteur de cet article se range parmi ces derniers. II a dit et il pense encore, que le mathématicien avance de supposition en supposition; que c'est en retournant sa pensée sous diverses formes, qu'il arrive à d'utiles résultats; QUE C'EST LA RECONNOISSANCE DE QUELQUE IDENTITE qui auTORISE CHACUNE DE SES CONCLUSIONS; et toutefois il a dit et il persiste à croire, que les axiomes mathématiques ne font que tenir la place où de définitions ou de théorêmes, ; et que les définitions sont les seuls principes des sciences de la nature de la géométrie. Voici ses propres expressions. "J'observe "que de bonnes définitions initiales sont les seuls principes rigoureusement suffisans dans les sciences de raisonnement pur....... C'est dans les définitions que sont vé"ritablement contenues les hypothèses dont ces sciences partent......... On pourroit "concevoir [toujours dans ces mêmes sciences], que les principes fussent si nette"ment posés, que l'on n'y trouvât autre chose que de bonnes définitions. De ces "definitions retournées, résulteroient toutes les propositions subséquentes. LES "DIVERSES PROPRIETES DU CERCLE QUE SONT-ELLES AUTRE CHOSE, QUE "DIVERSES FACES DE LA PROPOSITION QUI DEFINET CETTE COURBE ?— "C'est donc l'imperfection (peutêtre inévitable) de nos conceptions, qui a "engagé à faire entrer les axiomes pour quelque chose dans les principes des sci"ences de raisonnement pur. Et ils y font un double office. Les uns remplacent "des définitions. Les autres remplacent des propositions susceptibles d'être "démontrées."

"Il est manifeste que celui qui a tenu de tout temps ce langage n'a pas fondé son opinion, vraie ou fausse, relativement à l'evidence mathematique, sur une opinion

ly proved, that the particular evidence which accompanies mathematical reasoning cannot be resolved into the perception of identity. Let us recur to this proof. It will be found to consist wholly in refutation.

I. The author commences by remarking, that some persons lay the foundation of the opinion which he opposes, upon the principle which considers axioms as first principles; and as he has denied this, he concludes that its consequence is false. This argument may be a good one against those who deduce the opinion which he combats, from a certain theory of axioms, but is of no force whatever against others. The Reviewer of this article ranks himself amongst these last. He has said and he still thinks, that a mathematician advances from one supposition to another; that it is by exhibiting his position in different forms that he is enabled to arrive at useful results; that it is the recognition of some Identity which authorises each of his conclusions; and he has always said, and continues to believe, that mathematical axioms serve no other purpose, but to supply the place either of definitions or of theorems. These are his words. "I observe, that good elementary definitions are, strictly speaking, the only solid principles in the sciences of pure reasoning. It is in definitions that are really contained the hypotheses whence "these sciences spring. It is possible to conceive, (always speaking of the same "sciences,) that their principles have been laid down so distinctly, that we find in "them only good definitions. From these definitions frequently examined, result "all the subsequent propositions. The different properties of the circle, what are "they, but different phases of the proposition which defines this curve? It is then "the imperfection, perhaps inevitable, of our conceptions which has occasioned the "use of axioms for certain purposes in the principles of the sciences of pure reason"ing. They there perform a double office. Some supply the place of definitions. "Others of proportions susceptible of demonstration."

66

It is clear that the present Reviewer, who has at all times held this language, has not founded his opinion, whether true or false, relative to mathematical evidence,

⚫ Essais de Philos. Tom. II. p. 29, à Genève chez Paschoud, 1804.

fausse relativement aux axiomes; ou du moins, qu'étant si parfaitement d'accord avec Mr Dugald Stewart en ce qui concerne les premiers principes des mathematiques, ce n'est point de la que dérive l'apparente discordance de ses expressions et de celles de son ami, sur ce qui concerne le principe de l'évidence mathématique dans la déduction démonstrative. Dès lors il est évident que ce premier argument de l'auteur reste pour lui comme nul.

"II. Passons au second. Celui-ci est encore purement négatif et personnel. Il s'adresse à ceux qui derivent, d'un principe propre à la géométrie, l'assertion que l'auteur combat. De ce que l'égalité en géométrie se démontre par la congruence, ces philosophes se pressént de conclure, que, dans toutes les mathématiques, les vérités reposent sur l'identité. Ceux donc qui n'ont jamais songé à donner un tel appui à l'assertion contestée ne peuvent absolument pas se rendre à l'attaque dirigée contre cet appui. Il est probable qu'un très-grand nombre de partisans du principe de l'identité, considéré comme base de la démonstration, se trouvent (comme le rédacteur peut ici le dire de lui-même) tout à fait étrangers à la manière de raisonner que l'auteur réfute; et n'ont point formé leur opinion relativement à l'évidence mathématique d'après la congruence (réelle ou potentielle) de deux espaces. C'est ce que le rédacteur affirme ici, quant à lui, de la manière la plus positive; et de là résulte que l'argument personnel,* dirigé contre ceux qui ont été menés d'une de ces opinions à l'autre, ne l'atteint point.

"Il est un peu plus difficile de prouver cette affirmation, que quand il étoit question des axiomes, parce que ceux-ci ne peuvent pas manquer de s'offrir aux recherches du logicien, au lieu qu'il n'est pas appelé à prévoir l'application inconsidérée du principe de superposition à toute espèce de démonstration. Si cependant il fait voir que son opinion sur la démonstration dérive de principes universels et tout differens de celui qu'on a en vue, il aura fait, je pense, tout ce qu'il est possible d'attendre de lui.

"Qu'il soit maintenant permis au rédacteur de quitter la tierce personne, et pour éviter quelques longueurs et quelques expressions indirectes, d'établir nettement son opinion et la marche qu'il a tenue en l'exposant.

upon a false opinion relative to axioms. Or at least, being so perfectly in agreement with Mr. Dugald Stewart, as to what concerns the first principles of mathematics, it is not from this is derived the apparent disagreement of his expressions with those of his friend, as to what concerns the principle of mathematical evidence in demonstrative induction. Hence it is evident, that this first argument of the author remains, as respects the present writer, of no force whatever.

II. Let us pass to the second. This is still purely negative and personal. It is addressed to those who derive from a principle confined to geometry, the assertion which the author combats. From the circumstance, that equality in geometry is demonstrated by coincidence, these philosophers hasten to conclude, that in all the branches of Mathematics, truth depends upon identity. Those, then, who have never dreamed of calling in such a proof of the contested point, are not absolutely obliged to expose themselves to the attack which is directed against this proof. It is probable, that a very large number of the partisans of the principle of identity, considered as the basis of demonstration, are, as the Reviewer confesses himself to be, altogether strangers to the manner of reasoning which the author refutes, and have not formed their opinion relative to mathematical evidence, by the coincidence, (real or potential,) of two spaces. This the Reviewer affirms as respects himself, in the most positive manner, and does not feel the force of the argumentum ad hominem, directed against those who are led by one of those opinions to the other. It is rather more difficult to prove this affirmation, than when it was a question of Axioms, because these last do not fail to offer themselves to the researches of the logician, whereas he is not expected to foresee the inconsiderate application of the principle of superposition to every species of demonstration. If, nevertheless, he makes it appear, that his opinion upon demonstration is drawn from principles which are universal, and altogether different from those in view, he will have done, I think, all that can be expected from him.

May it now be permitted the Reviewer to drop the third person, and in order to avoid some prolixity and indirect modes of expression, to state plainly his opinjon, and the course he has taken in supporting it.

* Ad hominem.

"Dès les premières pages de ma logique, je pars de la distinction à faire entre les deux genres de vérité; la conditionnelle et l'absolue. Puis j'ajoute :

"LE MOYEN UNIQUE, PAR LEQUEL NOUS CONNOISSONS SI UNE PROPOSI"TION CONDITIONNELLE EST VRAIE, OU LE CARACTERE D'UNE TELLE VE"RITE, EST L'IDENTITE BIEN ETABLIE ENTRE LE PRINCIPE ET LA CONSE"QUENCE. CETTE IDENTITE N'EST PAS COMPLETE SANS DOUTE; MAIS "ELLE EST TELLE A QUELQUE EGARD, QUE LA CONSEQUENCE DOIT ETRE "TOUTE ENTIERE COMPRISE DANS LE PRINCIPE."*

"Traitant ensuite des sciences selon leur genre, j'appelle sciences de raisonnement pur celles qui ne s'occupent que de la vérité conditionnelle. Je cherche, d'une manière générale et abstraite, les caractères de ces sciences. J'en fais ensuite l'application aux mathématiques dans les deux branches qu'elles comprennent; et c'est par cette voie, que je me trouve avoir déterminé la nature de la démonstration. J'ai soin du reste de faire remarquer que la nature du raisonnement pur, ou proprement dit, ne dépend nullement du sujet, et qu'il n'est propre aux mathématiques qu'en ce sens que ces dernières s'ocuppent de raisonnement d'une manière exclusive et n'y mêlent point des propositions de vérité absolue, comme font les sciences de fait et d'expérience. En voilà assez, je crois, pour faire voir que ce n'est pas témérairement que j'affirme n'avoir en aucune façon conçu la nature de la démonstration d'après le point de vue borné de la superposition. Je ne puis donc,

In the first pages of my Logic, I speak of the distinction between the two kinds of Truth; conditional and absolute; afterwards I add :

The only means by which we know if a conditional proposition is true, or the character of such a certain truth, is the well established Identity between the principle and the consequence. This Identity, doubtless, is not complete, but it is, nevertheless, so far complete, that the consequence must always be entirely comprised in the principle.*

In afterwards treating of the sciences according to their kinds, I call those, sciences of pure reasoning, which are occupied with conditional truth. I seek, in a general and abstract manner, the characters of these sciences. I afterwards make the application to mathematics in the two branches which they comprise; and by this inethod, I find I have determined the nature of demonstration. I have been careful to remark, that the nature of pure reasoning, properly so called, does not depend upon the subject, and that it is not peculiar to mathematics, except in the sense that these are taken up exclusively with reasoning, and are not mingled with propositions of absolute truth, as are the sciences which relate to fact and experience. I have said enough, I think, to make it clear that I do not speak rashly, when I affirm, that I have never in any way considered the nature of demonstration in the point of view bounded by superposition. I cannot, then, as respects myself, give

* Essais de Phil. Tom. II. P. 2. "Le lecteur équitable voudra bien se rappeler que l'ouvrage, dont ce passage est tiré, n'est que l'esquisse d'un cours fort étendu, dans lequel se trouvent développés, par des exemples et de toute manière, les simples énoncés du texte. A peine est-il nécessaire de dire ici en explication ce que j'entends par l'identité complète ou non complète entre le principe et sa conséquence. Si je conclus, par exam. ple, du genre à l'espèce, il y a identité incomplète; comme lorsqu'ayant prouvé une vérité de tout polygone, je l'affirme du triangle en particulier. Il y a identité complète dans une équation. Et on entend bien que l'identité dont il s'agit est celle de la quantité (du nombre des unités), et non de toute autre. Ces deux exemples me semblent suffire pour prévenir toute équivoque."t

+[The candid reader will recollect that the work from which this passage is taken, is only a sketch of a very extended course, in which are developed by examples and in every way, the simple propositions enunciated in the text. It is hardly necessary to speak here in explanation of what I mean by Identity, complete and incomplete, between the premises and the consequence. If I reason, for instance, from the genus to the species, it is an incomplete identity; as when having proved a truth of every polygon. I affirm it of a triangle in particular. In an equation the identity is complete; and we well understand, that the identity of which we treat is that of quantity, (of a number of units,) and not of every other. These two examples appear to me sufficient to prevent all mistake.]

quant à moi, donner mon assentiment à un argument qui n'attaque que ceux dont l'opinion a cette base.

"III. On est toujours long quand on réfute une réfutation. J'aurois donc tort de m'étendre au-delà de ce qui est strictement nécessaire pour établir nettement l'état de la question. Je ne discuterai pas des opinions qui me sont étrangères, telles que celles de Leibnitz, de l'auteur d'une Dissertation latine imprimée à Berlin en 1764, de Barrow, Condillac, Destutt-Tracy. Il me suffit d'avoir répondu, pour moi et pour ceux qui pensent comme moi, aux deux seuls argumens de l'auteur, contre l'opinion que j'ai dès long-temps adoptée.

"J'ajouterai cependant un mot au sujet d'une remarque, que l'auteur introduit en disant, qu'elle est applicable à toutes les tentatives que l'on a faites pour établir l'opinion dont il s'agit." Accordant, dit-il, que toutes les propositions mathema"tiques puissent être représentées par la formule a=a, il ne s'ensuivroit nullement "que chaque pas du raisonnement, qui conduit à ces conclusions, soit une propo"sition de même nature." Je prie l'auteur de cette objection de vouloir bien refléchir un instant sur le sens du mot pas ramené à son expression propre et non figurée. Certainement un pas du raisonnement n'est autre chose qu'une proposition. Si donc on accorde que toute proposition est représentée par a=a, il faudra bien que tout pas soit de même nature.*

"Quant à la lettre chiffrée, certainement elle diffère de la non-chiffrée quant aux signes écrits; comme aussi les plus exagérés partisans du principe de l'identite ne nieront pas que l'expression deux plus deux ne soit differente de l'expression quatre. Dans l'un et l'autre cas le signe diffère, le sens que l'on a en vue est le même.

my assent to an argument which bears upon those only whose opinion rests upon this foundation.

III. In refuting a refutation we are necessarily prolix. I should then be wrong to go farther than is strictly necessary to establish clearly the state of the question. I will not discuss opinions which are foreign to me; such as those of Leibnitz, of the author of the latin Dissertation printed at Berlin in 1764, of Barrow, Condillac, Destutt-Tracy. It is sufficient that I answer, for myself and for those who think with me, the only two arguments of the author against the opinion which I have for a long time adopted.

I shall add, nevertheless, one word on the subject of a remark, which the author introduces by saying, that it is applicable to all the attempts which have been made to establish the opinion of which he treats. "Granting," says he, "for "the sake of argument, that all mathematical propositions may be represented by "the formula a= a, it would not therefore follow, that every step of the reason"ing leading to these conclusions, is a proposition of the same nature." I beg the author of this objection, for a moment to reflect upon the meaning of the word step, when recalled to its literal expression. Certainly, a step of reasoning differs in no respect from a proposition. If then we grant that every proposition is represented by a= a, it will follow, that every step is of the same nature.*

As to the letter in cyphers, certainly it differs from one not in cyphers as respects written signs; as also the most zealous partizans of identity will not deny that the expression, two plus two, is different from the expression four. In both cases the sign is different, the sense which we see it in, is the same.

*That the word pass or step is a figurative expression, when applied to a process of rea soning, cannot be disputed; and the same remark may be extended to the word proposition, and to almost every other term employed in discussions connected with the Human Mind. It may be doubted, however, whether it can be correctly asserted, that a step of reasoning differs in no respect from a proposition. In our language, at least, the word step properly denotes, not a proposition, but the transition to a new proposition from others already known. Thus, when I say, "the area of a triangle, having the circum"ference of a circle for its base, and the radius for its altitude, is greater than the area of "any polygon inscribed in the circle," I enunciate a proposition. When I say that "the area of the same triangle is less than that of any circumscribed polygon," I enunciate another proposition. But when I infer from these two propositions, that the areas of the triangle and circie are equal, I obtain possession of a new truth distinct from either; nor is it easy to imagine a more significant metaphor for expressing this acquisition, than to say, that I have advanced or gained a step in the study of geometry.