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Necessary Truths

164. Two errors have been rife in modern Philosophy : 1°, the reliance on Demonstration when the operation has been accurately performed, without regard to the intuitions, — in other words, whether the symbols operated on have, or have not, assignable values; 2°, the reliance on clear ideas as objectively axiomatic, and on axioms as objectively true. The first of these errors may be traced in Mathematics and in Metaphysics ; the second is sometimes avowed, and always implied, in Metaphysics.

In the preceding chapters we have seen that axioms, intuitions, and demonstrations all need critical control, control not only of the operations but of the premises ; verification of the premises consisting in the reduction of every inference. to its corresponding sensation. A little reflection shows that clear ideas may be treacherous grounds of reliance, clearness of conception being no evidence of the existence of any corresponding perception. It is notorious that propositions may be perfectly clear, and even coercive, yet prove on inspection to be illusory. Nothing was clearer and for centuries nothing could be more irresistible, than the conception of the sun revolving round the earth ; it is now rejected from Science. The proposition that mercury is lighter than water is clearer — more readily intelligible --- than that two parallel lines cannot enclose space. The falsehood of the one, and truth of the other, must be proved on quite other grounds than that of clearness ; although when proved the two propositions previously obscure will become transparent, and by this transparency the first proposition will be seen to want objective correspondence. No sooner are the properties of parallel lines and of mercury and water ascertained, than the truth and falsehood of the propositions which formulate these properties become evident.

165. Hume asserted that only the sciences of Quantity admit of demonstration ; "all other inquiries regard only matters of fact and existence, and these are evidently incapable of demonstration. Whatever is may not be." This argument has been urged a thousand times, no one seeming to have suspected its paralogism, namely, that two different propositions are involved in the sentence, " Whatever is may not be." But of this anon. Hume continues : " No negation of a fact can involve a contra-diction. The non-existence of any being, without exception, is as clear and distinct an idea as its existence. The proposition which affirms it not to be, however false, is no less conceivable and intelligible than that which affirms it to be. The case is different with the sciences properly so called. Every proposition which is not true is there confused and unintelligible." * This seems to me false in every respect. It is not true of any proposition ; or is true only by a substitution of terms which would make it equally true of mathematical propositions. When Hume says that the proposition which affirms a thing not to be, however false, is as conceivable as the proposition which affirms it to be, he confounds a verbal with a real proposition. No one can conceive the thing now existing to be now not existing. He can state this verbally, he cannot realize the symbols. He can indeed conceive that, under other conditions, what is now existing might not exist, or might exist differently ; but this change of terms substitutes in thθ place of the one proposition, " The thing exists," another wholly different proposition, " The thing no longer exists." Now by similar changes in the terms it is equally easy to conceive two parallel lines enclosing space, — the lines originally parallel are replaced by lines converging ; and we, preserving the integrity of our proposition in spite of the change in the meaning of terms, say parallel lines may enclose space.

166. Here most of my readers will doubtless consider that I overlook the distinction between the contingency of the proposition in the one case, and the necessity in the other. It is not that I overlook, but that I deny, this celebrated distinction.

The position to be attacked is this : some truths, in-deed most truths, are contingent, general or particular ; others are necessary, and universal. The one class ex-presses facts which we easily conceive might have been otherwise, and for which there is no guaranty that in other times, or in other worlds, they would be what they are at this time, and under these conditions. They are therefore contingent. Contrasted with them are the truths that express facts which are not only seen to be facts now, and under the present conditions, but are seen to be facts which no effort of imagination can figure otherwise. Here and everywhere, now and always, they must preserve their unalterable characters. That acids redden vegetable blues, and that bodies unsupported must fall, are general truths, inductions contingent on certain conditions. We recognize in them no internal necessity why the facts must be so. We easily imagine a state of things in which the results would be different; nor indeed have we any guaranty that in other planets they are not different. But the truths that " Every effect must have a cause," and that " Two parallel lines cannot enclose space," have an internal necessity, - no intellectual ingenuity can conceive a variation in them.

167. Such is the thesis. First, remark the confusion of contingency in a proposition with contingency in a truth. Because there are propositions which express or imply contingency outside the conditions, the mind easily glides into the supposition that there is a contingency inside the conditions ; because a group of phenomena may change, that group itself is held to be not, what it is. A little reflection discloses that a proposition is either a true statement of the facts expressed in it, or a false statement of them; if true, it is necessarily true, and universally true, whenever and wherever those facts recur unchanged; but, of course, if anywhere, at any time, a change occurs in the facts expressed by the proposition, then the old proposition no longer truly expresses the new group of facts. That a body moving under certain conditions as if attracted by a force varying inversely with the square , of the distance will describe an ellipse having the centre of attraction in one of the foci of the ellipse, is a proposition which when demonstrated — found to be a correct expression of the terms — is a truth having no contingency whatever: it is as necessarily true as the axiom respecting parallels. That the earth is a body under approximately similar conditions, and consequently describes what approximates to an ellipse, is also a proposition which having been verified is seen to be true, and will eternally be true so long as the conditions which are the terms of the proposition are unchanged. It is indeed conceivable that under other geometrical conditions —in a space of two, or of four dimensions — neither proposition may be true ; or that even in our own space of three dimensions the second proposition may cease to be applicable because of some slight change in the co-operant factors. But this contingency — that the factors might be otherwise — in no degree affects the necessity of the truths : that the facts are what they are. ingenious geometers have of late years shown that even the much-relied-on axiom respecting parallels is affected with an analogous contingency ; it would not be true in a space of four dimensions ; while Mr. Mill and others have questioned the legitimacy of extending the axiom of causation beyond our world. I am unable to accept either of these positions ; but I certainly admit that if the view of necessary truth which is current in Philosophy is to be accepted at all, it logically forces the acceptance of this contingency in the axioms. In other words, all truths are necessarily true, and all propositions are liable to a double contingency, — first, the contingency of enumeration (i. e. whether all the factors are, or are not, taken into account) ; secondly, the contingency of application (i. e. whether the old formula is applied to the old conditions, or to changed conditions, which would require a new formula). The only necessity is that a thing is what it is, and cannot be other than what it is ; the only contingency is that our proposition may not state what the thing is.

168. The β priori doctrine maintains that only those truths are necessary which formulate facts transcending Experience by their universality, and are therefore incapable of direct verification ; they are seen intuitively to be unchangeable. In opposition to this I maintain that all propositions are contingent which formulate anything transcending Experience, direct or indirect, — in which the co-operant factors cannot be enumerated and verified ; whereas on the contrary, every verified proposition, what-ever its nature, is necessarily true, and universally true, — under the formulated conditions.

Note this final clause. It is the pivot of the question. That a particular acid does redden this vegetable blue is a proposition in no respect contingent ; that hitherto all known acids have been found to redden all vegetable blues, when applied under certain conditions, is also a proposition having no contingency ; but if for these intuitions we substitute an induction, if from these two necessary truths we infer that all acids will under all circumstances redden this or all vegetable blues, the proposition is contingent with a double contingency : it has not been and cannot be verified ; the reddening depends on factors which may or may not be co-operant in any particular case ; and because we are unable to enumerate what will be the factors, our proposition must be contingent ; but if we could enumerate them the contingency would vanish. It is because we can be assured of our factors that most mathematical propositions have no other contingency than that of a possible miscalculation or uncertainty as to the condition. Thus if n be a whole number, the existence of the equivalent series for (1 + x)'' is necessary, because the operation which gives it may be accurately defined. On the contrary, if n be not a whole number but a general symbol, then, because we cannot de-fine the operation by which we pass from (1 + x)" to its equivalent series, a series which exists under such conditions only by virtue of the principle of the permanence of equivalent forms, the connection is contingent ; the series becomes necessary when its existence is assumed ; in other words, " if such an equivalent does exist it must be the series in question, and no other."

169. The arguments which support the a priori view have been ingeniously thrown into this syllogism by Mr. Killick : The necessary truth of a proposition is a mark of its not being derived from Experience. (Experience cannot inform us of what must be :) The inconceivability of the contradictory is the mark of the necessary truth of a proposition : Therefore the inconceivability of its contradictory is a mark of a proposition not being derived from Experience.

This syllogism is perfect in form, but has a radical defect in its terms. The inconceivability of a contradictory results from the entire absence of experiences on which a contradiction could be grounded. If there were any truths independent of Experience, contradictions to them would be conceivable, since there would be no positive obstacle to the conception ; but a contradiction is inconceivable only when all Experience opposes itself to the formation of the contradictory conception.

170. There are truths which can be intuited, seen at a glance, because they express relations simple, constant, familiar; there are other truths which cannot be seen until the complicated relations formulated are unfolded, and presented to Intuition : and there are truths which can be seen at a glance, but which, formulating particular relations seen to exist in the present conjuncture of events, but known not to be constant in recurrence, yield no assurance that they will not be contradicted to-morrow. There can be no objection against a classification of such truths into universal, general, and particular, or into necessary and contingent, if we mean no more by contingency than the impossibility of determining before-hand what will be the co-operant factors. When it is said that a necessary truth is one seen not only to be true, but one which there is no possibility of our conceiving otherwise ; this can only be valid on the assumption that no change be made in the terms formulated : on this assumption, however, all truths are equally necessary ; without this assumption no truth is so.

170a. What is Possibility ? It is the ideal admission as present of absent factors : it states what would be the fact, if the requisite factors were present. What is Contingency ? It is the ideal admission that certain factors now present may be on any other occasion absent ; and when they are absent the result must be different from what it is now. What is Necessity ? It is the intuition of the actual factors, — the perception of adequate relations, — the recognition that what is, must be what it is. All inductions are contingent because they are generalizations of experience under the assumption of homogeneity; and the contingency lies in this, that the unknown cases which we assume to resemble the known cases in all the characters which constitute the terms of our proposition may not resemble them in some of these characters. All identical equations are necessary, and universal, when we universalize the terms.

171. This understood, we may set aside the serious and very common error which asserts that an universal proposition is truer than a general proposition, a general proposition truer than a particular proposition. Nine philosophers in ten will declare the proposition "Every effect must have a cause," to be more certainly true. than the proposition "Sugar is sweet." But the case is really this: the universality of a proposition carries with it the predicate of necessity in virtue of the assumed homogeneity of its terms ; the generality of a proposition carries with it the predicate of constancy in virtue of the same assumption of homogeneity ; the particularity of a proposition carries with it the predicate of contingency in virtue of an assumed heterogeneity in its terms : so long as its terms remain under the limitation of specified conditions, its truth remains unshakable.

In Rule' X. attention is drawn to this assumption of homogeneity, which underlies all Inference, and all Generalization. We construct a triangle, or define its terms.

This done once is done forever. The truths respecting triangles are not generalizations but intuitions, universalized by universalizing the terms, not generalized by comparing all known triangles, and concluding from them to the unknown. We operate on the triangle, not on triangles. When any modification of the terms introduces a new kind, — as for instance a spherical triangle, — there comes a corresponding modification in our propositions, and some that are true of rectilinear triangles are no longer true of spherical triangles. It is here that Verification steps in. What we have to do in any particular ease is not to ascertain whether a proposition is necessary or contingent, but whether it is true, expressing the actual factors of the fact ; and what we have to do in any general case is to ascertain whether all the particulars thus generalized preserve that homogeneity which justifies the extension; and whenever an exception appears, we know that this must be due to some heterogeneity in the terms, — in other words, that for this case a new proposition is needed.

172. It is one thing to state a proposition in terms which, themselves involve no contradiction, another thing to state it in terms which correspond with fact. The objective truth must be verified, i. e. the conceptions must be reduced to perceptions, the inferences to sensations, and, when verified, its certainty is not deepened by assuming an universal expression, nor endangered by a particular expression. When we say that the proposition which is true now and here, may not be true tomorrow and elsewhere, we speak elliptically ; written out in full, the statement would, run thus : to-morrow and elsewhere the circumstances may be so far changed that the result now observable will no longer present itself ; and it is because we do not know whether there will or will not be a change in the circumstances that we call our proposition contingent. There are indeed some propositions which exclude this possibility of change. That the whole is greater than any one of its parts, or that two things equal to a third are equal to each other: these are unassailable, because they are reducible to identical propositions. It is a mistake, however, to class these apart as necessary truths since all truths may be exhibited as propositions of identity ; nor is any proposition verified until this has been effected. To make the argument plain consider the following contrasted propositions. "This bit of iron," says Prof. Bowen, "I find by direct observation melts at a certain temperature; but it may well happen that another piece of iron, quite similar to it in external appearance, may be fusible only at a much higher temperature, owing to the unsuspected 'presence" [Note this clause] "in it of a little more or less carbon in composition. But if the angles at the base of this triangle are equal to each other, I know that a corresponding equality must exist in every figure which conforms to the definition of an isosceles triangle ; for that definition excludes every disturbing element."

Here we have a contingent and a necessary truth accurately indicated. Why is the first contingent ? Simply because one bit of iron may structurally differ from another, although resembling it in external appearance ; and the fusibility does not depend on the external appearance, but on the molecular structure or composition of the iron, i. e. depends on that factor which is assumed to be the same in both, but may really be different in both. The "unsuspected presence of more carbon" is not excluded by the external resemblance. But the presence of any disturbing element is excluded from the isosceles triangle, by the definition of the triangle, and the conclusion that corresponding equality must exist in every figure which conforms to the definition is irresistible; but a similar conclusion may be established by a similar artifice with respect to the iron ; and we may state the identical proposition that not only will this bit of iron always melt at this temperature under these conditions, but every piece of iron having a similar molecular structure and composition will melt at this temperature under these conditions. Our propositions respecting triangles will be not less contingent than our propositions about iron ores, if we admit the element of contingency (§ 170a) and leave undetermined whether the term " triangle " designates a scalene, isosceles, equilateral, or spherical triangle.

173. It was perhaps with surprise that the reader just now saw the statement that the proposition " Sugar is sweet " was a necessary truth ; yet ha may now be prepared to admit this, under the same limitations as apply to all necessary truths, namely, that no change be made in the terms. It simply formulates the fact that a given substance, A, in relation to a given organ of sense, B, has the sensation, C, for its. product. We learn that in the many substances grouped under the general name "Sugar," and in the many sensory organs named "Taste," there are many variations, so that different sugars produce different sensations in the same organ of taste, and the same sugar will produce different sensations in different, organs of taste. The formula " Sugar is sweet" is independent of all these variations ; it is wholly abstract, A + B = C, and is necessarily true in abstraction. If in particular concrete cases we find Ax or Bx we no longer conclude the product will be C, but Cx. This is equally necessary.

The proposition " Two parallel lines do not enclose space and would not meet were they produced indefinitely" is a necessary truth ; but if any one alters the terms, and under the name of parallels includes two lines concave to each other ; or if he admits the contingency that the straight lines may at any point alter their equidistance, then indeed the truth is no longer necessary, — or rather one necessary truth is displaced to make way for another.

174. Those ingenious geometers who have endeavored to show that if our bodies had no thickness and if we lived in the surface of, a sphere, our Geometry would be very different from that now regarded as necessarily true, — that in such a world there would be an infinite number of curved lines between two points which would be shorter than a straight line, and that parallels would end by enclosing space, — may claim to have shaken the serenity of those who rely on the superior necessity of mathematical truths since this imaginary geometry shows that the axioms are not true of every conceivable space, but only true of our known space ; but so far as I understand the argument it fails altogether in throwing doubt on the only correct interpretation of an universal and necessary truth. Every truth is necessary ; every truth is universal, when its conditions are universalized. When we assert that parallel lines do not meet, we stand upon our sensible intuition of the actual fact ; when we assert that they can never meet, we stand on our rational intuition of the virtual fact ; we assume that the lines will be what they are, and cannot be other than what they are, so long as the identity of the terms is preserved. Any change in the conditions which would make the , lines approach each other would require a new proposition to express the changed terms.

Is it not owing to neglect of the need of intuition of a figure that the geometers of fictitious space are able to argue that the angles of a triangle would always more or less exceed two right angles ? Is not their whole argumentation based on the disregard of the psychological principle that symbols are only valid when their sensible values can be assigned?

175. In the next Problem we shall consider more at length the position already indicated that every truth is an identical proposition, or is capable of being reduced to one. A truth is the assertion that something is, and, being what it is, cannot be different, unless the conditions of its existence change. The proof of such an assertion in .every particular case is direct, or dependent. The direct intuition of equality in A = A gives an identical proposition. The indirect demonstration of equality in A = C gives a dependent proposition which is' established through two intuitions, — since A = B and B = C, then A = C. The conclusion here is necessary ; yet there is a contingency, unless the homogeneity of the terms be assumed ; for the equations A = B and B = C, although exact under the defined conditions, may be false under others; we must universalize the terms to make an universal truth.

176. Here one may remark how the common (and useful) distinction between necessary and contingent truths may ,take its place beside the algebraic distinction of identical equations and equations of condition. An identical equation is one of which the two sides are but different expressions of the same number, thus may be assigned to each symbol. An equation of condition is one which is true only in the case specified, there being but one assignable number which will satisfy the equation. Thus a + 7 = 20 is true only when the value of a is 13 ; no other value would satisfy the equation. Although this equation of condition is a new kind, and is particular because confined to the particular number which must be found in the equation itself, yet no sooner is the number found, than the identity is disclosed, and the truth a + 7 20 is necessary, and universal under the stated conditions.

177. Let us take a more familiar illustration. "Fire burns" would be called a contingent truth. It may be so ; it may also be a necessary truth, — an identical proposition. The fact that the conception of Fire is the conception of something which burns combustible things is not rendered dubious, contingent, by the fact that we can conceive Fire placed in relations which would not be those of combustion. When we affirm that fire must burn combustible paper if the requisite conditions are present, our affirmation is simply the expression of certain verified facts; and this expression is not disturbed by the discovery of incombustible paper, is not affected by the substitution of new conditions requiring a new expression. That under particular conditions, the thing we designate Fire will burn the thing we designate Paper, is a contingent proposition, an equation of condition, which must be verified, i. e. the number found; when verified, it is not only a necessary truth from which all contingency has vanished, but easily assumes the universal form, namely, " Fire of this kind under these conditions will always and everywhere burn paper of this kind."

178. There are certain relations which are invariable in our experience, others which are variable. Identical equations, and equations of condition, comprise both orders. Every proposition is contingent which in its expression admits the possibility of a variation in its terms ; every proposition is necessary which excludes such variation ; and whether the contingent or the necessary proposition be true, or. not, depends on its being, or not being, reducible to an identical proposition. The square root of an unknown quantity may be any quantity so long as x has no value assigned ; but given the value of x as a function of y, and the square root is determined forever. If only one black ball be placed in a box with a thousand white balls, there is very little probability of the black being withdrawn on a first trial; but though not probable the withdrawal is possible, — it is therefore, before trial, a contingency ; after the trial there is no contingency whatever. From a box containing nothing but white balls it is absolutely certain that no black ball can be withdrawn. In the former case our ignorance of the position of the black ball and of the direction of the drawer's hand render the withdrawal of a black ball contingent ; if we knew these the contingency would vanish. In the second case, in spite of our ignorance of the positions of the balls, and the direction in which the hand will move, there is no contingency whatever in the conclusion that no black ball will be drawn, for we know one condition which absolutely excludes it, namely, there is no black ball present. Apply this to parallels. That parallel straight lines can never meet, is a necessary truth ; their meeting is excluded by the conditions of the problem, no less than the withdrawal of a black ball is excluded when no black ball is present. Should a black ball appear, we know that these conditions have been violated, and that in a box, assumed to be without a black ball, there was a black ball present. Should the parallel lines deviate in direction, the conditions of the problem have been violated.

179. It is a necessary truth that when several events are equally likely to happen, let one be proved to happen all the others must also happen ; or if one be proved not to have happened this will be proof that none have. Obviously the cogency of this conclusion rests on the assumed homogeneity. Should one of the events happen, and any one of the others be shown not to happen, we conclude that there was some error in the original classification, and that the conditions present in all the other cases were not present in this one. Here as elsewhere it may be said that the necessity of a proposition depends on the transparency of its terms, the contingency on the opacity of the terms ; in other words, whenever we have distinct intuition of all the generating conditions, we know the only possible result ; whenever our vision of the generating conditions is obscure, we do not know the only possible result. In Mathematics we always have an intuition of the generating conditions, and hence the unalterable necessity of the conclusions.

180. It is because philosophers have failed steadily to bear in mind that the truth of a proposition subjectively is rigorously limited to the terms of the proposition, and objectively that the fixity of a result is coexistent with the fixity of its conditions, — that there has arisen this supposition of a class of truths, or class of results, essentially distinct in origin. What I have been in various ways endeavoring to make clear is that all true propositions are necessarily true, their truth when generalized depending on the generalization or assumed homogeneity of their ternis ; whereas, whenever a proposition admitted to be true under the defined conditions presents the character of contingency, and the mind recognizes the possibility of error in generalizing the proposition, and sees that the result now certain might have been uncertain, there has been the unconscious substitution of new terms in place of the old, making in fact a proposition framed to express one set of conditions the expression of another set. This fallacy is common. When we say that what has occurred once will occur again, and will always recur, we mean (or ought to mean) that under precisely similar conditions there must always be similar results. If A = B, or fire burns paper, under any con-junctures, 'it must do so always under these conjunctures. When we say that what has occurred to-day may perhaps never recur, or will recur but seldom, we mean that the conditions are likely to be changed, and with any change in the conditions there must necessarily occur a change in the result : instead of A =B there will be A = C or Ax = Bx. This latter proposition is equally necessary with the former, but is obviously a different proposition.

Those who speak of the Laws of Nature being contingent truths, meaning that a modification or reversal of such Laws is conceivable, and that under changed conditions the propositions would be changed, seem not to be aware of the fallacy. A Law formulates certain specified conditions, and in itself is not at all contingent ; it is either a true formula, or a false formula ; by altering the conditions specified, substituting new conditions, and applying the old formula, we do not disturb the truth of the Law. The contingency lies elsewhere : it lies in our ignorance of the generating conditions.

181. "The belief in the uniformity of Nature," says Mr. Mansel, " is not a necessary truth, however constantly guaranteed by our actual experience. We are not compelled to believe that because A is ascertained to be the cause of B at a particular time, whatever may be meant by that relation, A must therefore inevitably be the cause of B on all future occasions." * This is undeniable, but only by the concealed equivoque lying in the words " on all future occasions." If the co-operant conditions which now determine B to succeed A are preserved unaltered on all future occasions, the result must be then what it is now; but if we are at liberty to suppose, or have any reason to suspect, that on some future occasions the co-operant conditions will be altered, we conclude on the same principles that A will not be followed by B. Get rid of this equivoque by the phrase " on all similar occasions under similar conditions," and the truth that A is the antecedent of B becomes necessary. While every contingent proposition becomes necessary if its terms are made invariable every necessary proposition becomes contingent if its terms are contingent. If we define and thus specify the generating conditions of an equilateral triangle as a triangle having its three sides equal to each other, or define the growth of an organism by specifying the generating conditions, — the simultaneous process of molecular composition and decomposition, — the one proposition is not more necessary than the other ; both express ideal constructions from real intuitions. The mathematician, indeed, who is occupied with ideal figures, is so far at an advantage that he is not like the biologist called upon to regard any possible variation in the objects of the terms of his propositions. The circles and angles of which he treats are not the. figures drawn on paper, but the figures conceived in his mind. But this advantage ceases when he comes to apply his mathematical propositions to real figures. The biologist also when dealing with general principles disregards all variation : it is the ideal organism, the ideal tissue, not the real objects which his truths formulate. The organism is an abstraction. The tissue is an abstraction,— a group of organic elements which approximates to the defined limit. But just as no mathematician ever saw a circle absolutely corresponding with his conception, so no biologist ever saw a tissue absolutely corresponding with the histological definition of it ; but from the complex and variable group of organic elements, he extricates certain elements and names the abstraction, purified of all variation, a tissue. The ulterior questions whether there are in nature objects which approximate to the definition, circle and tissue, and whether these objects have the properties deduced from these definitions or seen by intuition, may be debated ; but once demonstrated, there is no more contingency in a true biological proposition than in a true mathematical proposition. If we have shown to intuition that the circle has the property of comprising the maximum area with the minimum perimeter, or if we have shown that the nerve-tissue has the property of transmitting a stimulus from periphery to centre, from centre to centre, and from centre to periphery, — the certainty of the one is not less absolute than that of the other. Nor let it be urged that the property of the circle is necessarily universal, true of all circles and in all places ; whereas the property of nerve-tissue is contingent, particular, subject to variation, being dependent on variable conditions : this is so ; but the objection rests on a fallacy. The circle is no real figure, but the ideal figure defined by the geometer, and this ideal transported into distant times and places carries with it all its characters unchanged. The nerve - tissue similarly treated shows an equal constancy ; and when we speak of its properties as variable, we draw on our experience of the variable conditions under which real tissues exist; we know that sometimes the nerve is exhausted by action, or by disease; we know that its properties depend on many complex conjunctures ; and since we cannot at any moment be sure of knowing all the generating conditions, we say that the property is contingent. There would be the same contingency respecting circles were our propositions respecting them supposed to refer to real figures. I mean that if the necessity of ΰ truth respecting nerves be denied because in reality nerves are observed under conditions which seem to contradict this truth, on such grounds the necessity of a truth about circles should be denied, since in reality it is never true that a real circle is a figure having every point in its circumference equidistant from the centre. If I define a nerve to be a part of a living organism capable of transmitting stimuli, and define a circle a plane figure having every, point of its circumference equidistant from its centre, — the propositions which are true of either are true necessarily, universally ; whereas if I displace this nerve and substitute for it something else which deviates from the terms of my definition of nerve, — or if I replace the circle by an ellipse, — the old propositions no longer apply, new propositions are needed to express the new terms. That equal forces perpendicularly applied at the opposite ends of equal arms of a straight lever will exactly balance each other, is an absolute truth, and is reducible to a series of identical equations. But that two particular objects, sup-posed to be equal in weight, will exactly balance each other on the arms, supposed to be equal of this particular lever, supposed to be supported at its centre, — this is a contingent truth, comparable to that which says that any given nerve when stimulated will excite the contractility of the muscle in which it terminates. The three suppositions here specified are of generating conditions and it is only by assuming the presence of such conditions that we can apply our abstract proposition. Is it not obvious that if we are allowed to assume the presence of generating conditions in the case of nerve-action, our propositions will have equal necessity ?

182. This will be disputed by the ΰ priorists. They affirm that it is precisely the inability we are under of assuming the presence of the generating conditions which renders physical truths of inferior certainty to mathematical. We do not know, it is argued, why a nerve excites a muscle at all, and we can easily conceive a state of things in which such a property would not belong to nerves ; whereas it is impossible to conceive a state of things wherein ,mathematical truths should not be precisely what we now know them to be.

I admit this fully ; but reject the conclusion founded on it. I admit the contingency which hovers over our application to particular cases of general propositions respecting nerves ; but while admitting the contingency in any particular case, — that is, while ,assuming the possibility or probability that in the particular case there will be other conditions present than those which the general proposition formulates, — I wholly deny that the general proposition is thereby invalidated as a general proposition. We do not indeed know why a nerve excites a muscle, as we know why the three angles of a triangle are equal to two right angles : we have not in the one case a clear intuition of all the generating conditions, as we have in the other. But if we know the fact that a nerve does excite a muscle, under certain conditions, we at the same time know that it will always and everywhere do so under the same conditions. We generalize the fact in generalizing the conditions. And this is all we are enabled to do in Mathematics. We do not there treat of variable but of invariable conditions : it is the triangle, in the abstract, of which we speak. And if we treat of the nerve acting on the muscle, there is a similar certainty. The abstract biological truth is not invalidated because of its failure to embrace all concrete cases, when these cases present relations not expressly included in it. The abstract mathematical truth is not invalidated because of its failure to embrace particular cases when these involve relations not formulated by it; and this is always observed in Applied Mathematics. Fix the terms, specify all the relations formulated, and a biological truth stands on the same level of certainty and universality as a mathematical truth. If nerve and muscle are terms which designate objects partly known, partly unknown, all propositions which include the co-operation of the unknown factors are of course hypothetical, contingent ; but if the terms simply designate the known factors, and the propositions simply formulate these, the contingency vanishes, the propositions become identical; and having been verified once, are necessarily true in all identical cases.

183. The very great importance of the question here discussed must be my excuse for having, with perhaps wearisome iteration, presented my solution of it under various aspects. It is a fundamental question, and of late years all metaphysical discussion may be said to turn on it. More than twenty years have elapsed since I first suggested the solution here reproduced ; but although it has been reargued in the second, third, and fourth editions of my History of Philosophy, I have not observed that any English writer has adopted or refuted it. This silence warrants the suspicion that I had not presented the arguments with sufficient clearness, or else that the view itself is radically defective. Naturally I prefer the former supposition ; and I am confirmed in this conclusion by the gratifying fact that a distinguished foreign thinker, who shows no trace whatever of acquaintance with my writings, has put forth a view substantially similar, in two works to which I wish to express many obligations. Believing, then, that the view is a real contribution to the philosophy of the subject, I have endeavored by a fuller and more varied illustration to carry it home to the conviction of every reader. Those who still hesitate to accept it are referred to the further elucidation which will be reflected from the next chapter.

( Originally Published 1874 )

Problems of Life and Mind:
The Reality Of Abstractions

Ideal Construction In Science

What Are Laws Of Nature?

The Use And Abuse Of Hypothesis

The Passage From The Abstract To The Concrete

Ideal Construction In Metaphysics

The Search After Causes

Intuition And Demonstration

Axioms And Their Validity

Necessary Truths

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