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Theory.. cerning Math: Eid
Continuation of the Subject.-How far it is true that all Mathematical Evidence is resolvable into Identical 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
As this account of mathematical evidence appears to me 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 perSome have ception of identity, has been considered by some as a consequence of the commonly received doctrine, which represents the axioms of infind it from Euclid as the first principles of all our subsequent reasonings in geoy dostum & metry. 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; principle and, at the same time, involves an admixture of important truth, of reasoning in math:
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, L.L.D. Published by G. and W. Nicol, London, 1812.
*It is more than probable, that this theory was suggested to Leibnitz by some very curious observations in Aristotle's Metaphysics, Book IV. chap. iii. and iv.
+ I must here observe, in justice to my friend M. Prevost, that the two doctrines which I have represented in the above paragraph as quite irreconcilable, seem to be regarded by him as not only consistent with each other, but as little more than different modes of stating the same proposition. The remarks with which he has favoured me on this point will be found in the Appendix annexed to this volume. At present, it may suffice to mention, that none of the following reasonings apply to that particular view of the ques tion which he has taken. Indeed, I consider the difference of opinion between us, as to the subject now under consideration, as chiefly verbal. On the subject of the preceding article, our opinions are exactly the same. See Appendix.
which contributes not a little to the specious verisimilitude of the In what conclusion. It is founded on this simple consideration, that the geometrical notions of equality and of coincidence are the same; and that, founded 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 identity the two triangles together, in every circumstance connected both with magnitude and figure.*
Of the justness of the assumption on which this argument pro- Assumpceeds, I do not entertain the slightest doubt. Whoever has the cu- tion just
riosity to examine any one theorem in the elements of plane geometry, in which different spaces are compared together, will easily perceive, that the demonstration, when traced back to its first principles, terminates in the fourth proposition of Euclid's first book: a proposition of which the proof rests entirely on a supposed application of the one triangle to the other. In the case of equal triangles which differ in figure, this expedient of ideal superposition cannot be directly and immediately employed to evince their equality; but the demonstration will nevertheless be found to rest at bottom on the same species of evidence. In illustration of this doctrine, I shall only appeal to the thirty-seventh proposition of the first book, in which it is proved that triangles on the same base, and between the same parallels, are equal; a theorem which appears, from a very simple construction, to be only a few steps removed from the fourth of the same book, in which the supposed application of the one triangle to the other, is the only medium of comparison from which their equality is inferred.
In general, it seems to be almost self-evident, that the equality of Equality, of two spaces can be demonstrated only by showing, either that the
one might be applied to the other, so that their boundaries should spa hon exactly coincide; or that it is possible, by a geometrical construc-howtion, to divide them into compartments, in such a manner, that the sum of parts in the one may be proved to be equal to the sum of parts in the other, upon the principle of superposition. To devise the easiest and simplest constructions for attaining this end, is the object to which the skill and invention of the geometer is chiefly directed.
Nor is it the geometer alone who reasons upon this principle. If How com you wish to convince a person of plain understanding, who is quite
It was probably with a view to the establishment of this doctrine, that some foreign ignorant of
a person 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 be tween 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 triangle, in the contemplation of the logician.
This idea is very explicitly stated, more than once, by Aristotle: ŵN TO TODOV ÈV. "Those things are equal whose quantity is the same; (Met. iv. c. 16.) and still more precisely in these remarkable words, εν τούτοις ή ισοτης ενοτης; "In mathematical quanti ties, equality is identity." (Met. x. c. 3.)
For some remarks on this last passage, see note (F.)
unacquainted with mathematics, of the truth of one of Euclid's theorems, it can only be done by exhibiting to his eye, operations, exactly analogous to those which the geometer presents to the understanding. A good example of this occurs in the sensible or experi mental illustration which is sometimes given of the forty-seventh proposition of Euclid's first book. For this purpose, a card is cut into the form of a right angled triangle, and square pieces of card are adapted to the different sides; after which, by a simple and ingenious contrivance, the different squares are so dissected, that those of the two sides are made to cover the same space with the square of the hypothenuse. In truth, this mode of comparison by a superposition, actual or ideal, is the only test of equality which it is possible to appeal to; and it is from this (as seems from a passage in Proclus to have been the opinion of Apollonius) that, in point of logical rigour, the definition of geometrical equality should have been taken.* The subject is discussed at great length, and with much acuteness, as well as learning, in one of the mathematical lectures of Dr. Barrow; to which I must refer those readers who may wish to see it more fully illustrated.
I am strongly inclined to suspect, that most of the writers who have maintained that all mathematical evidence resolves ultimately into the perception of identity, have had a secret reference, in their minds, to the doctrine just stated; and that they have im
Identikposed on themselves, by using the words identity and equality as lite
rally synonymous and convertible terms. This does not seem to be at all consistent, either in point of expression or of fact, with sound synon logic. When it is affirmed (for instance) that "if two straight lines term." in a circle intersect each other, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other;" can it with any propriety be said, that the relation between these rectangles may be expressed by the formula a = a? Or, to take a case yet stronger, when it is affirmed, that "the area of a circle is equal to that of a triangle having the circumference for its base, and the radius for its altitude;" would it
*I do not think, however, that it would be fair, on this account, to censure Euclid for the arrangement which he has adopted, as he has thereby most ingeniously and dexterously contrived to keep out of the view of the student some very puzzling questions, to which it is not possible to give a satisfactory answer till a considerable progress has been made in the elements. When it is stated in the form of a self-evident truth, that magnitudes which coincide, or which exactly fill the same space, are equal to one another; the beginner readily yields his assent to the proposition; and this assent, without going any farther, is all that is required in any of the demonstrations of the first six books: whereas, if the proposition were converted into a definition, by saying, 'Equal magnitudes are those which coincide, or which exactly fill the same space;" the question would immediately occur, Are no magnitudes equal, but those to which this test of equality can be applied? Can the relation of equality not subsist between magnitudes which differ from each other in figure? In reply to this question, it would be necessary to explain the defini. tion, by adding That those magnitudes likewise are said to be equal, which are capable of being divided or dissected in such a manner that the parts of the one may severally coincide with the parts of the other;-a conception much too refined and complicated for the generality of students at their first outset; and which, if it were fully and clearly apprehended, would plunge them at once into the profound speculation concerning the comparison of rectilinear with curvilinear figures.
not be an obvious paralogism to infer from this proposition, that the triangle and the circle are one and the same thing? In this last instance, Dr. Barrow himself has thought it necessary, in order to reconcile the language of Archimedes with that of Euclid, to have recourse to a scholastic distinction between actual and potential coincidence; and, therefore, if we are to avail ourselves of the principle. of superposition, in defence of the fashionable theory concerning mathematical evidence, we must, I apprehend, introduce a correspondent distinction between actual and potential identity.*
That I may not be accused, however, of misrepresenting the opinion which I am anxious to refute, I shall state it in the words of an author, who has made it the subject of a particular dissertation; and who appears to me to have done as much justice to his argument as any of its other defenders.
"(1) Omnes mathematicorum propositiones sunt identicae, et re"praesentantur hac formula, a = a. Sunt veritates identicae, sub"varia forma expressae, imo ipsum, quod dicitur contradictionis prin
* "Gum demonstravit Archimedes circulum aequari rectangulo triangulo cujus basis radio circuli, catbetus peripheriae exaequetur, nil ille, siquis propius attendat, aliud quic quam quam aream circuli ceu polygoni regularis indefinite multa latera habentis, in tot dividi posse minutissima triangula, quae totidem exilissimis dicti trianguli trigonis aequen. tur; eorum verò triangulorum aequalitas è sola congruentia demonstratur in elementis. Unde consequenter Archimedes circuli cum triangulo (sibi quantumvis dissimili) congruen. tiam demonstravit.-Ita congruentiae nihil obstat figurarum dissimilitudo; verùm seu similes sive dissimiles sint, modò aequales, semper poterunt semper posse debebunt congru. ere. Igitur octavum axioma vel nullo modo conversum valet, aut universaliter converti potest; nullo modo, si quae isthic habetur congruentia designet actualem congruentiam ; universim, si de potentiali tantùm accipiatur."-Lectiones Mathematicae Lect. V.
[When Archimedes demonstrated that a circle is equal to a right-angled triangle whose base is equal to the radius and its altitude to the periphery of the circle, he proved nothing more, if properly understood, than that the area of a circle, or of a regular poly. gon having an indefinite number of sides, may be divided into so many minute triangles as may be equal to as many infinitely small three sided figures in the said triangle. But the equality of these triangles is demonstrated in the elements from their coincidence alone. Hence Archimedes proved the coincidence of the circle with the triangle, how unlike soever to itself. So the want of similitude in the figures, hinders not their coinci. dence; but however like or unlike, so that they be equal, they always ought and will coincide. The converse, therefore, of the eighth axiom, is either never possible, or uni. versally; never, if the coincidence spoken of means an actual coincidence, or universally, if only a potential coincidence.]
(1) [All the propositions of Mathematicians are identical, and are expressed by this for mula, a a. They are identical truths, expressed under a different form; nay, the same thing which is called the beginning of the series, differently stated and involved: for in truth, all the propositions of this class are contained in it. As far as I understand it, the only difference in the propositions is, that they are reduced, some by a longer and soine by a shorter method, to the first principle of all, and resolved into that. So, for instance, the proposition 2 + 2 = 4 immediately resolves itself into this, !+1+1+1=1+1+1+1; that is, the same is the same, and in speak ing with strict propriety, it ought to be so expressed. If it happen that four exist, then four exist: for geometers do not treat of existence, which is only understood hypotheti cally. Thence he who traces the steps of reasoning acquires the most complete certainty, for he observes the identity of the ideas. And this evidence, which we call mathematical or geometrical, immediately compels assent. Yet the nature of geometry is not pecu liar and confined to that science, for it arises from the perception of identity, which may happen, though the ideas do not represent extension.] 13
a thing conforme
cipium, vario modo enunciatum et involutum ; siquidem omnes hujus 66 generis propositiones reverâ in eo continentur. Secundum nos"tram autem intelligendi facultatem, ea est propositionum differen"tia, quod quaedam longa ratiociniorum serie, alia autem breviore via, ad primum omnium principium reducantur, et in illud resol"vantur. Sic. v. g. propositio 2 + 2 = 4 statim huc cedit 1 + 1 “+1+1=1+1 +1 + 1; i. e. idem est idem; et proprie lo"quendo, hoc modo enunciari debet.—Si contingat, adesse vel ex"istere quatuor entia, tum existunt quatuor entia; nam de existen"tia non agunt geometrae, sed ea hypothetice tantum subintelligi"tur. Inde summa oritur certitudo ratiocinia perspicienti; obser"vat nempe idearum identitatem; et haec est evidentia assensum "immediate cogens, quam mathematicam aut geometricam voca"mus. Mathesi tamen sua natura priva non est et propria; oritur "etenim ex identitatis perceptione, quae locum habere potest, eti"amsi ideae non repraesentent extensum."*
With respect to this passage, I have only to remark, that the author confounds two things essentially different;-the nature of the truths which are the objects of a science, and the nature of the evidence by which these truths are established. Granting, for the sake of argushare es-ment, that all mathematical propositions may be represented by the formula a = a, it would not therefore follow, that every step of the reasoning leading to these conclusions, was a proposition of the same nature; and that, to feel the full force of a mathematical demonstration, it is sufficient to be convinced of this maxim, that every thing may be truly predicated of itself; or, in plain English, that the same is the same. A paper written in cypher, and the interpretation of that paper by a skilful decypherer, may, in like manner, be considered as, to all intents and purposes, one and the same thing. They are so, in fact, just as much as one side of an algebraical equation is the same thing with the other. But does it therefore follow, that the whole evidence upon which the art of decyphering proceeds, resolves into the perception of identity?
It may It may be fairly questioned too, whether it can, with strict correctness, be said even of the simple arithmetical equation 2 + 2 = questioned 4, that it may be represented by the formula a = a. The one is
a proposition asserting the equivalence of two different expressions; -to ascertain which equivalence may, in numberless cases, be an object of the highest importance. The other is altogether unmeaning and nugatory, and cannot, by any possible supposition, admit of the slightest application of a practical nature. What opinion then shall we form of the proposition a = a, when considered as the
The above extract (from a dissertation printed at Berlin in 1764, has long had a very extensive circulation in this country, in consequence of its being quoted by By Dr. Beattie, in his Essay on Truth, (see p. 221. 2d edit.) As the learned author of the es say has not given the slightest intimation of his own opinion on the subject, the doctrine in question has, I suspect, been considered as in some measure sanctioned by his authority. It is only in this way that I can account for the facility with which it has been admitted by so many of our northern logicians.