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"these propositions made in general terms, may have them ready "to apply to all particular cases as formed rules and sayings. Not "that, if they be equally weighed, they are more clear and evident axcom" than the instances they are brought to confirm; but that,
"more familiar to the mind, the very naming of them is enough to "satisfy the understanding." In farther illustration of this, he adds very justly and ingeniously, that "although our knowledge begins "in particulars, and so spreads itself by degrees to generals; yet, "afterwards, the mind takes quite the contrary course, and having "drawn its knowledge into as general propositions as it can, makes "them familiar to its thoughts, and accustoms itself to have recourse "to them, as to the standards of truth and falsehood."
But although, in mathematics, some advantage may be gained without the risk of any inconvenience, from this
of use only ment of axioms, it is a very dangerous example to be followed in
other branches of knowledge, where our notions are not equally clear and precise; and where the force of our pretended axioms (to use Mr. Locke's words) "reaching only to the sound, and not to the "signification of the words, serves only to lead us into confusion, "mistakes, and errour." For the illustration of this remark, I must refer to Locke.
Another observation of this profound writer deserves our attention, while examining the nature of axioms;-" that they are "not the foundations on which any of the sciences is built; nor " at all useful in helping men forward to the discovery of unknown
noty foun." truths."* This observation I intend to illustrate afterwards, in dalibus of add, to what Mr. Locke has so well stated, that, even in mathematics, & siling.
of the of the art. At present I shall only
it cannot with any propriety be said, that the axioms are the foundation on which the science rests; or the first principles from which its more recondite truths are deduced. Of this I have little doubt that Locke was perfectly aware; but the mistakes which some of the most acute and enlightened of his disciples have committed in treating of the same subject, convince me, that a further elucidation of it is not altogether superfluous. With this view, I shall here introduce a few remarks on a passage in Dr. Campbell's Philosophy of Rhetoric, in which he has betrayed some misapprehensions on this very point, which a little more attention to the hints already quoted from the Essay on Human Understanding might have prevented. These remarks will, I hope, contribute to place the nature of axioms, more particularly of mathematical axioms, in a different and clearer light than that in which they have been commonly considered.
"Of intuitive evidence (says Dr. Campbell) that of the following "propositions may serve as an illustration: One and four make five. "Things equal to the same thing are equal to one another. The "whole is greater than a part; and, in belief all axioms in arithme
• Book iv, chap. 7, § 11,—2. 3.
propositions made in general terms, may have them ready ly to all particular cases as formed rules and sayings. Not f they be equally weighed, they are more clear and evident he instances they are brought to confirm; but that, being familiar to the mind, the very naming of them is enough to the understanding." In farther illustration of this, he adds tly and ingeniously, that "although our knowledge begins ticulars, and so spreads itself by degrees to generals; yet, ards, the mind takes quite the contrary course, and having its knowledge into as general propositions as it can, makes familiar to its thoughts, and accustoms itself to have recourse m, as to the standards of truth and falsehood." Ithough, in mathematics, some advantage may be gained the risk of any possible inconvenience, from this arrangeaxioms, it is a very dangerous example to be followed in ranches of knowledge, where our notions are not equally d precise; and where the force of our pretended axioms (to Locke's words) "reaching only to the sound, and not to the cation of the words, serves only to lead us into confusion, tes, and errour." For the illustration of this remark, I must Locke.
er observation of this profound writer deserves our attenile examining the nature of axioms;" that they are ? foundations on which any of the sciences is built; nor useful in helping men forward to the discovery of unknown This observation I intend to illustrate afterwards, in of the futility of the syllogistic art. At present I shall only hat Mr. Locke has so well stated, that, even in mathematics, with any propriety be said, that the axioms are the founda hich the science rests; or the first principles from which reçondite truths are deduced. Of this I have little doubt ke was perfectly aware; but the mistakes which some of acute and enlightened of his disciples have committed in of the same subject, convince me, that a further elucidation t altogether superfluous. With this view, I shall here infew remarks on a passage in Dr. Campbell's Philosophy ric, in which he has betrayed some misapprehensions on point, which a little more attention to the hints already om the Essay on Human Understanding might have preThese remarks will, I hope, contribute to place the nature more particularly of mathematical axioms, in a different er light than that in which they have been commonly con
tuitive evidence (says Dr. Campbell) that of the following itions may serve as an illustration: One and four make five. equal to the same thing are equal to one another. The s greater than a part; and, in belief all axioms in arithme
Book iv. chap. 7, § 11,~2. 3;
"tic and geometry. These are, in effect, but so many expositions
"But, in order to prevent mistakes, it will be necessary further "to illustrate this subject. It might be thought that, if axioms were "propositions perfectly identical, it would be impossible to advance
a step by their means, beyond the simple ideas first perceived by "the mind. And it must be owned, if the predicate of the proposi
tion were nothing but a repetition of the subject, under the same
aspect, and in the same or synonymous terms, no conceivable advantage could be made of it for the furtherance of knowledge. "Of such propositions, for instance, as these,-seven are seven, "eight are eight, and ten added to eleven are equal to ten added to
eleven, it is manifest that we could never avail ourselves for the "improvement of science. Nor does the change of the term make 66 any alteration in point of utility. The propositions, twelve are a dozen, twenty are a score, unless considered as explications of the "words dozen and score, are equally insignificant with the former. "But when the thing, though in effect coinciding, is considered under "a different aspect; when what is single in the subject is divided in "the predicate, and conversely; or when what is a whole in the one "is regarded as a part of something else in the other; such propositions lead to the discovery of innumerable and apparently remote "relations. One added to four may be accounted no other than a "definition of the word five, as was remarked above. But when I 'say, Two added to three are equal to five,' I advance a truth "which, though equally clear, is quite distinct from the preceding. "Thus, if one should affirm, That twice fifteen make thirty,' and "again, that thirteen added to seventeen make thirty,' nobody "would pretend that he had repeated the same proposition in other "words. The cases are entirely similar. In both cases, the same "thing is predicated of ideas which, taken severally, are different.
'From these again result other equations, as one added to four "are equal to two added to three, and twice fifteen are equal to "thirteen added to seventeen.'
"Now, it is by the aid of such simple and elementary principles, "that the arithmetician and algebraist proceed to the most astonish"ing discoveries. Nor are the operations of the geometrician es"sentially different.”
I have little to object to these observations of Dr. Campbell, as far as they relate to arithmetic and to algebra; for, in these sciences, all our investigations amount to nothing more than to a comparison of different expressions of the same thing. Our common language indeed frequently supposes the case to be otherwise; as when an equation is defined to be, "A proposition asserting the equa"lity of two quantities." It would, however, be much more correct to define it, "A proposition asserting the equivalence of two expres"sions of the same quantity;" for algebra is merely a universal arithmetic; and the names of numbers are nothing else than collectives, by which we are enabled to express ourselves more concisely than could be done by enumerating all the units that they contain. Of this doctrine, the passage now quoted from Dr. Campbell shews that he entertained a sufficiently just and precise idea.
But, if Dr. Campbell perceived that arithmetical equations, such as "one and four make five," are no other than definitions, why should he have classed them with the axioms he quotes from Euclid, "That the whole is greater than a part," and that "Things equal "to the same thing are equal to one another;" propositions which, however clearly their truth be implied in the meaning of the terms of which they consist, cannot certainly, by any interpretation, be considered in the light of definitions at all analogous to the former? The former, indeed, are only explanations of the relative import of particular names; the latter are universal propositions, applicable alike to an infinite variety of instances.*
Another very obvious consideration might have satisfied Dr. Campbell, that the simple arithmetical equations which he mentions, do not hold the same place in that science which Euclid's axioms hold
* D'Alembert also has confounded these two classes of propositions. "What do the greater part of those axioms on which geometry prides itself amount to, but to an expres. sion, by means of two different words or signs, of the same simple idea? He who says that two and two make four, what more does he know than another, who should content himself with saying, that two and two make two and two?" Here, a simple arithmetical equation (which is obviously a mere definition) is brought to illustrate a remark on the nature of geometrical axioms.-With respect to these last (I mean such axioms as Euclid has prefixed to his Elements) D'Alembert's opinion seems to coincide exactly with that of Locke, already mentioned. "I would not be understood, nevertheless, to condemn the use of them altogether: I wish only to remark, that their utility rises no higher than this, that they render our simple ideas more familiar by means of habit, and better adapted to the different purposes to which we may have occasion to apply them."-" Je ne pretends point cependant en condamner absolument l'usage: je veux seulement faire observer, à quoi il se reduit; c'est à nous rendre les idées simples plus familières par l'habitude, et plus propres aux différens usages auxquels nous pouvons les appliquer."-Discours Préliminaire, &c. &c.
in geometry. What I allude to is, that the greater part of these axioms are equally essential to all the different branches of mathematics. That "the whole is greater than a part," and that "things "equal to the same thing are equal to one another," are propositions as essentially connected with our arithmetical computations, as with our geometrical reasonings; and, therefore, to explain in what manner the mind makes a transition, in the case of numbers, from the more simple to the more complicated equations, throws no light whatever on the question, how the transition is made, either in arithmetic or in geometry, from what are properly called axioms, to the more remote conclusions in these sciences.
The very fruitless attempt thus made by this acute writer to illustrate the importance of axioms as the basis of mathematical truth, was probably suggested to him by a doctrine which has been repeatedly inculcated of late, concerning the grounds of that peculiar evidence which is allowed to accompany mathematical demonstration. "All the sciences (it has been said) rest ultimately on first "principles, which we must take for granted without proof; and "whose evidence determines, both in kind and degree, the evidence "which it is possible to attain in our conclusions. In some of the "sciences, our first principles are intuitively certain; in others, "they are intuitively probable; and such as the evidence of these "principles is, such must that of our conclusions be. If our first "principles are intuitively certain, and if we reason from them con"sequentially, our conclusions will be demonstratively certain: but "if our principles be only intuitively probable, our conclusions will "be only demonstratively probable. In mathematics, the first prin"ciples from which we reason are a set of axioms which are not "only intuitively certain, but of which we find it impossible to con"ceive the contraries to be true: And hence the peculiar evidence "which belongs all the conclusions that follow from these prin"ciples as necessary consequences."
To this view of the subject Dr. Reid has repeatedly given his sanction, at least in the most essential points; more particularly, in controverting an assertion of Locke's, that "no science is, or hath "been built on maxims."—" Surely (says Dr. Reid) Mr. Locke was "not ignorant of geometry, which hath been built upon maxims pre"fixed to the Elements, as far back as we are able to trace it. But "though they had not been prefixed, which was a matter of utility "rather than necessity, yet it must be granted, that every demon"stration in geometry is grounded, either upon propositions formerly "demonstrated, or upon self-evident principles."*
On another occasion, he expresses himself thus: "I take it to be "certain, that whatever can, by just reasoning, be inferred from a "principle that is necessary, must be a necessary truth. Thus, as 66 he axioms in mathematics are all necessary truths, so are all the
* Essay on Intell. Powers, p. 647, 4to edit.
"conclusions drawn from them; that is, the whole body of that "science." 99*
That there is something fundamentally erroneous in these very strong statements with respect to the relation which Euclid's axioms bear to the geometrical theorems which follow, appears sufficiently from a consideration which was long ago mentioned by Locke,—that from these axioms it is not possible for human ingenuity to deduce a single inference. "It was not (says Locke) the influence of those " maxims which are taken for principles in mathematics, that hath "led the masters of that science into those wonderful discoveries "they have made. Let a man of good parts know all the maxims "generally made use of in mathematics never so perfectly, and con66 template their extent and consequences as much as he pleases, he "will, by their assistance, I suppose, scarce ever come to know, "that the square of the hypothenuse in a right angled triangle, is "equal to the squares of the two other sides.' The knowledge that "the whole is equal to all its parts,' and, if you take equals from 66 equals, the remainders will be equal,' helped him not, I presume, "to this demonstration: And a man may, I think, pore long enough "on those axioms, without ever seeing one jot the more of mathe"matical truths." But surely, if this be granted, and if, at the same time, by the first principles of a science, be meant those fundamental propositions from which its remoter truths are derived, the axioms cannot, with any consistency, be called the First Principles of Mathematics. They have not (it will be admitted) the most distant analogy to what are called the first principles of natural philosophy;-to those general facts, for example, of the gravity and elasticity of the air, from which may be deduced, as consequences, the suspension of the mercury in the Torricellian tube, and its fall when The pre-carried up to an eminence. According to this meaning of the word, plu of most, the principles of mathematical science are, not the axioms, but the definitions; which definitions hold, in mathematics, precisely the sieme. same place that is held in natural philosophy by such general facts not axiom, as have now been referred to.‡
Ibid. p. 577. See also pp. 560, 561. 606.
Essay on Human Understanding, Book IV. chap. xii. § 15.
In order to prevent cavil, it may be necessary for me to remark here, that when I speak of mathematical axioms, I have in view only such as are of the same description with the first nine of those which are prefixed to the Elements of Euclid; for, in that list, it is well known, that there are several which belong to a class of propositions altogether different from the others. That "all right angles (for example) are equal to one anc ther;" that "when one straight line falling on two other straight lines makes the two in terior angles on the same side less than two right angles, these two straight lines, if produced, shall meet on the side, where are the two angles less than two right angles;" are manifestly principles which bear no analogy to such barren truisms as these, "Things that are equal to one and the same thing are equal to one another." "If equals be added to equals, the wholes are equal." "If equals be taken from equals, the remainders are equal." Of these propositions, the two former (the 10th and 11th axioms, to wit, in Eu. clid's list) are evidently theorems which, in point of strict logical accuracy, ought to be demonstrated; as may be easily done, with respect to the first, in a single sentence. That the second has not yet been proved in a simple and satisfactory manner, has been long