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Axioms And Their Validity

159. THE preceding considerations must be completed by an examination of Axioms, which, owing to a philosophical prejudice often greatly misleading, are supposed to have a higher validity than Theorems, all truths of a wide generality being held to be more certain than particular truths; and from this higher validity there is often deduced the conclusion of a deeper origin. Because an axiom expresses universal experience, is confirmed from all sides, and admits of no doubt whatever, it is said to be " self-evident," and because it is self-evident, self-luminous, needing no reflected light, it is held to be above Experience.

There are many conveniences in the separation of self-evident truths from reflected truths ; unhappily, like most verbal distinctions, it has come to be regarded in the light of a real distinction ; being classed apart, Axioms have come to be considered as due to another origin. This is untenable when we learn that Axioms have no such exclusive certainty, but arise from the general ground of Experience, out of which all truths arise. The logical processes which constitute the group in a general truth are precisely the processes which constitute a particular truth,—the difference lies in the terms, not in the forms ; in the symbols, not in the operations. The widest of all axioms — " whatever is is " — cannot be more certain, more irresistible, than the most fleeting of particular truths, e. g. " I am sad." The axiom " If equals be taken from equals the remainders are equal," may indeed be more rapidly intuited than the particular truth respecting the square of the hypothenuse in the forty-seventh of Euclid, which can only be seen by a mind that has followed the steps of the demonstration ; but this greater ease and rapidity of vision does not endow the seen with greater certitude; and the second truth is equally irresistible with the first, when once the relations are intuited.

160. Since, then, the characteristic of superior certitude must be given up, and the superiority rest upon the ease with which the conclusion is reached, shall we still adopt the common opinion that the distinguishing mark of an axiom is its self-evidence ? Let us first understand what is affirmed. If a truth is self-evident only when it is self-luminous, i. e. when its luminosity is absolutely independent of all reflection from Experience whatever, being d priori not only to this experience and to that, but to all, -- then I assert that the axiom of equality is not more self-evident than the 47th prop. of Euclid ; for to the mind of the infant neither truth is evident. But if a truth is self-evident when the conclusion is evident in the premises, — self-luminous because its rays issue from within, and the mind in the very act of apprehending the terms apprehends the equation of those terms, — this definition may be accepted, and we should all agree to call a truth self-evident when no other evidence is needed outside the terms of its expression, because no other relations are implied beyond the relations specified. But this is no mark of d priori truths, as distinguished from demonstrated truths. To the mind which has once learned the properties of numbers, the proposition 2 + 2 = 4 is self-evident. The terms mean that, and nothing else. But to the mind uninstructed in such properties, the proposition so far from being self-evident is not evident at all; it may be made so, by placing a group of pebbles, naming it four, and then dividing it into two groups each of which is named two, when the mind sees that four is the group made by two and two.

161. Newton has been censured for the laxity with which he uses the term axiom. Technically his practice may be questionable, psychologically it is defensible. I think he is correct in applying the term to the fundamental principles of Dynamics : axiomata sive leges. The laws of Motion have the same certainty and self-evidence, when their terms are apprehended, as the axioms of Geometry; neither have these characters when the terms are imperfectly apprehended ; both demand that the mind should already be in possession through Experience of the specified relations.

Is there, then, no distinction between axioms and particular propositions ? Assuredly. Axioms express truths of universal application ; and some of them inevitably arise in every man's experience, or may be extricated from it ; whereas particular propositions are limited to special experiences. The former are self-evident, i. e: requiring no extraneous proof, because no doubt is suggested by contradictory experiences. Every instant of our lives we have evidence that a thing is what it is ; and this evidence needs no confirmation, because we have never any experience that a thing can be and not be at the same instant. Every instant of our lives we see things' change their positions in space after some other thing has been brought into new relations with them ; and we express this constant experience in the axiom " Every effect has a cause." Such axioms obviously need no confirmation from particular experiences, because being expressions of universal experience they admit of no doubt. It is otherwise with particular propositions, in which the terms express inconspicuous relations, or relations that are hypothetical; though even particular propositions become irresistible when their terms are conspicuous and real. If the proposition be neither self-evident nor illuminated from general Experience, — as, for example, when first the proposition respecting the square of the hypothenuse is presented, — we have to ascertain what are the relations specified -in its terms ; these are shown to us, demonstrated ; and from that moment the particular proposition is no less irresistible than an axiom. The relations are what they are, and cannot be other than what they are ; and we have ascertained what they are. The contingency which existed at the outset has vanished forever. So long as these terms preserve their homogeneity, so long will the proposition preserve its necessity. Every schoolboy who has learned his multiplication, table sees at once that 6 multiplied by 6 gives 36 ; this intuitive judgment is axiomatic ; but although he may not see at once that the cube of 6 is 216, because he cannot at once intuite the relations, yet after rendering these inconspicuous relations conspicuous (after calculating), his discursive judgment becomes axiomatic : he is not less assured that the cube of 6 is 216, than he is that when equals are taken from equals the remainders are equal.

162. Our conclusion therefore is that axioms have a wider application than particular truths, but not a higher validity, not another origin. Having a wider application, they have a higher scientific value. But they had their origin in Experience, and cannot have a wider range than the inductions from Experience, which proceed on the assumed homogeneity of the unknown or unspecified relations with those that are specified.

I do not wish to be understood as adopting the view that Axioms are founded on Induction ; on the contrary, I hold them to be founded on Intuition. They are founded on Experience, because Intuition is empirical. But it is a mistake to present them as founded on any comparison of instances, or as primarily established by Induction. Indeed the very conception of Induction is so far antagonistic to that of Axiom, that it includes the acknowledgment of a contingency which the Axiom excludes. There is an assumption of homogeneity underlying both. The assumption in the case of an intuition is that the relations are what they are seen to be ; in the case of an induction it is that the relations are what they are 'inferred to be. Now the Axiom which universalizes an intuition assumes the homogeneity of the terms it formulates ; and if these are invariant the conclusion is necessary. One act of Intuition establishes an Axiom, for the Axiom is simply the universalization of its terms. That the whole is greater than any one of its parts, is not indeed self-evident, in the rigorous sense of the words ; but when its evidence has been seen, in the intuition of its relations, — when in any one case the meaning of the terms has been apprehended and the Logic of Feeling has passed into the Logic of Signs, so that a sensible mass is shown to be divisible into smaller masses, and the former is now under-stood to be what is called the whole, while the latter are called the parts, — the relations being intuited, the Axiom is complete, now and forever; and every future whole is seen to be greater than any of its parts, no other intuition being possible so long as the terms intuited are unchanged.

It is possible, indeed, to mistake inductions for intuitions and prejudices for axioms ; but that is only when we fail to discriminate between what is seen and what is inferred. Hence the need of Verification.

163. Only one point more needs to be touched on here. Axioms are commonly said to be indemonstrable judgments. This theory of their lying outside Demonstration is another form of the theory of their being self-evident ; but if the views respecting Demonstration put forward in the preceding chapter are correct, the theory is inadmissible. Admitting Demonstration to be the exhibition of the intuited equivalence, — the showing of what may be seen in the terms,— we must admit that it is even easier to demonstrate the axiom, "A whole is greater than any of its parts," than to demonstrate a particular proposition, " Water is composed of oxygen and hydrogen." Nay, even the axiom " Whatever is is " may be demonstrated, for we can exhibit the equivalence of each side of the equation ; indeed the axiom only is irresistible on the assumption of this equivalence, i.e. that what we express by the word is on the one side, we also express by the same word on the other.

( Originally Published 1874 )

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Axioms And Their Validity

Necessary Truths

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