Encyclopædia Britannica, Ninth Edition/Axiom

AXIOM, from the Greek [Greek], is a word of great import both in general philosophy and in special science ; it also has passed into the language of common life, being applied to any assertion of the truth of which the speaker happens to have a strong conviction, or which is put forward as beyond question. The scientific use of the word is most familiar in mathematics, where it is customary to lay down, under the name of axioms, a number of propositions of which no proof is given or considered necessary, though the reason for such procedure may not be the same in every case, and in the same case may be vari ously understood by different minds. Thus scientific axioms, mathematical or other, are sometimes held to carry with them an inherent authority or to be self-evident, wherein it is, strictly speaking, implied that they cannot be made the subject of formal proof ; sometimes they are held to admit of proof, but not within the particular science in which they are advanced as principles ; while, again, some times the name of axiom is given to propositions that admit of proof within the science, but so evidently that they may be straightway assumed. Axioms that are genuine principles, though raised above discussion within the science, are not therefore raised above discussion alto gether. From the time of Aristotle it has been claimed for general or first philosophy to deal with the principles of special science, and hence have arisen the questions concerning the nature and origin of axioms so much debated among the philosophic schools. Besides, the general philo sopher himself, having to treat of human knowledge and its conditions as his particular subject-matter, is called to determine the principles of certitude, which, as there can be none higher, must have in a peculiar sense that character of ultimate authority (however explicable) that is ascribed to axioms; and by this name, accordingly, such highest principles of knowledge have long been called. In the case of a word so variously employed there is, perhaps, no batter way of understanding its proper signification than by considering it first in the historical light not to say that there hangs about the origin and early use of the name an obscurity which it is of importance to dispell. The earliest use of the word in a logical sense appears in the works of Aristotle, though, as will presently be shown, it had probably acquired such a meaning before his time, find only received from him a more exact determination. In his theory of demonstration, set forth in the Posterior Analytics, he gives the name of axiom to that immediate principle of syllogistic reasoning which a learner must bring with him (i. 2, 6) ; again, axioms are said to be the common principles from which all demonstration takes place com mon to all demonstrative sciences, but varying in expression according to the subject-matter of each (i. 10, 4). The principle of all other axioms the surest of all principles is that called later the principle of Contradiction, in demonstrable itself, and thus fitted to be the ground of all demonstration (Metaph., iii. 2, iv. 3). Aristotle s fol lowers, aud, later on, the commentators, with glosses of their own, repeat his statements. Thus, according to Themistius (ad Post. Anal.), two species of axioms were distinguished by Theophrastus one species holding of all things absolutely, as the principle (later known by the name) of Excluded Middle, the other of all things of the same kind, as that the remainders of equals are equal. These, adds Themistius himself, are, as it were, connate and com mon to all, and hence their name Axiom ; " for what is put over either all things absolutely or things of one sort universally, we consider to have precedence with respect to them. ; The same view of the origin of the name reappears in Boethius s Latin substitutes for it dignitas zudmaxima (propositio), the latter preserved in the word Maxim, which is often used interchangeably with Axiom. In Aristotle, however, there is no suggestion of such a meaning. As the verb diow changes its original meaning of deem worthy into think fit, think simply, and also claim or require, it might as well be maintained that do/m which Aristotle himself employs in its original ethical sense of worth, also in the secondary senses of opinion or dictum (Metaph., iii. 4), and of simple proposition (Topics, viii. 1) was conferred upon the highest principles of reasoning and science because the teacher might require them to be granted by the learner. In point of fact, later writers, like Proclus and others quoted by him, did attach to Axiom this particular meaning, bringing it into relation with Postulate (atrr//xa), as defined by Aristotle in the Posterior Analytics, or as understood by Euclid in his Elements. It may here be added that the word was used regularly in the sense of bare proposition by the Stoics (Diog. Laert., vii. 65, though Simplicius curiously asserts the contrary, ad Epict. Ench., c. 58), herein followed in later times by the Ilamist logicians, and also, in effect, by Bacon. That Aristotle did not originate the use of the term axiom in the sense of scientific first principle, is the natural conclusion to be drawn from the reference he makes to " what are called axioms in mathematics " (Metapli., iv. 3). Sir William Hamilton (Note A, Reid s Works, p. 765) would have it that the reference is to mathematical works of his own now lost, but there is no real ground for such a supposition. True though it be, as Hamilton urges, that the so-called axioms standing at the head of Euclid s Elements acquired the name through the influence of the Aristotelian philosophy, evidence is not wanting that by the time of Aristotle, a generation or more before Euclid, it was already the habit of geometricians to give definite expression to certain fixed principles as the basis of their science. Aristotle himself is the authority for this asser tion, when, in his treatise De Coelo, iii. 4, he speaks of the advantage of having definite principles of demonstration, and these as few as possible, such as are postulated by mathematicians (xaddirep dio>cri KOL ol ev TCHS {jiaO^ao-iv}, who always have their principles limited in kind or num ber. The passage is decisive on the point of general mathematical usage, and so distinctly suggests the very word axiom in the sense of a principle assumed or postulated, that Aristotle s repeated instance of what he himself calls by the name If equals be taken from equals, the remainders are equal can hardly be regarded otherwise than as a citation from recognised mathematical treatises. The conclusion, if warranted, is of no small interest, in view of the famous list of principles set out by Euclid, which has come to be regarded in modern times as the typical specimen of axiomatic foundation for a science. Euclid, giving systematic form to the elements of geome trical science in the generation after the death of Aristotle, propounded, at the beginning of his treatise, under the name of opoi, the definitions with which modern readers are familiar j under the name of atr^/xara, the three principles of construction now called postulates, together with the three theoretic principles, specially geometrical, now printed as the tenth, eleventh, and twelfth axioms ; finally, under the name of KOWXI ewoiat, or common notions, the series of general assertions concerning equality and inequality, having an application to discrete as well as continuous quantity, now printed as the first nine axioms. Now, throughout the Elements, there are numerous indications that Euclid could not have been acquainted with the logical doctrines of Aristotle : a most important one has been signalised in the article ANALYSIS, and, in general, it may suffice to point out that Euclid, who is said to have flour ished at Alexandria from 323 (the year of Aristotle s death) to 283 B.C., lived too early to be affected by Aristotle s work all the more that he was, by philosophical profession, a Platonist. Yet, although Euclid s disposition of geome trical principles at the beginning of his Elements is itself one among the signs of his ignorance of Aristotle s logic, it would seem that he had in view a distinction between his postulates and common notions not unlike the Aristotelian distinction between cu-n^ia-a and diuyxaTa. All the postulates of Euclid (including the last three so-called axioms) may be brought under Aristotle s description of atr^ftara principles concerning which the learner has, to begin with, neither belief nor disbelief, Post. Anal., i. 10, G) ; being (as De Morgan interprets Euclid s meaning) such as the " reader must grant or seek another system, whatever be his opinion as to the propriety of the assumption." Still closer to the Aristotelian conception of axioms come Euclid s common notions, as principles " which there is no question every one will grant" (De Morgan). From this point of view, the composition of Euclid s two lists, as they originally stood, becomes intelligible : be this, however, as it may, there is evidence that his enumeration and division of principles were very early subjected to criticism by his followers with more or less reference to Aristotle s doctrine. Apollonius (250-220 B.C.) is mentioned by Proclus (Com. in EucL, i.) as having sought to give demonstrations of the common notions under the name of axioms. Further, according to Proclus, Geminus made the distinction between postulates and axioms which has become the familiar one, that they are indemonstrable principles of construction and demonstration respectively. Proclus himself (412-485 A.D.) practically comes to rest in this distinction, and accordingly extrudes from the list of postulates all but the three received in modern times. The list of axioms he reduces to five, striking out as derivative the two that assert in equality (4th and 5th), also the two that assert equality between the doubles and halves of the same respectively (Gth and 7th). Euclid s postulate regarding the equality of right angles and the other assumed in the doctrine of parallel lines, now printed as the llth and 12th axioms, he holds to be demonstrable : the 10th axiom (regarded as an axiom, not a postulate, by some ancient authorities, and so cited by Proclus himself) Two straight lines cannot enclose a space he refuses to print with the others, as being a special principle of geometry. Thus he restricts the name axiom to such principles of demonstration as are common to the science of quantity generally. These, he then declares, are principles immediate and self-manifest untaught anticipations whose truth is darkened rather than cleared by attempts to demonstrate them. The question as to the axiomatic principles, whether of knowledge in general or of special science, remained where it had thus been left by the ancients till modern times, when new advances began to be made in positive scientific inquiry and a new philosophy took the place of the peri patetic system, as it had been continued through the Middle Ages. It was characteristic alike of the philosophic and of the various scientific movements begun by Des cartes to be guided by a consideration of mathematical method that method which had led in ancient times to special conclusions of exceptional certainty, and which showed itself, as soon as it was seriously taken up again, more fruitful than ever in new results. To establish philo sophical and all special truth after the model of mathe matics became the direct object of the new school of thought and inquiry, and the first step thither consisted in positing principles of immediate certainty whence deduction might proceed. Descartes accordingly devised his criterion of perfect clearness and distinctness of thought for the determination of ultimate objective truth, and his followers, if not himself, adopted the ancient word axiom for the principles which, with the help of the criterion, they proceeded freely to excogitate. About the same time the authority of all general principles began to be considered more explicitly in the light of their origin. Not that ever such consideration had been wholly over looked, for, on the contrary, Aristotle, in pronouncing the principles of demonstration to be themselves indemon strable, had suggested, however obscurely, a theory of their development, and his followers, having obscure sayings to interpret, had been left free to take different sides on the question ; but, as undoubtedly the philosophic investiga tion of knowledge has in the modern period become more and more an inquiry into its genesis, it was inevitable that principles claiming to be axiomatic should have their pretensions scanned from this point of view with closer vision than ever before. Locke it was who, when the Cartesian movement was well advanced, more especially gave this direction to modern philosophic thought, turn ing attention in particular upon the character of axioms ; nor was his original impulse weakened rather it was greatly strengthened by his followers substitution of positive psychological research for his method of general criticism. The expressly critical inquiry undertaken by Kant, at however different a level, had a like bearing on the question as to the nature of axiomatic principles ; and thus it has come to pass that the chief philosophic interest now attached to them turns upon the point whether or not they have their origin in experience. It is maintained, on the one hand, that axioms, like other general propositions, result from an elaboration of particular experiences, and that, if they possess an excep tional certainty, the ground of this is to be sought in the character of the experiences, as that they are exceptionally simple, frequent, and uniform. On the other hand, it is held that the special certainty, amounting, as it does, to positive necessity, is what no experience, under any circum stances, can explain, but is conditioned by the nature of human reason. More it is hardly possible to assert gene rally concerning the position of the rival schools of thought, for on each side the representative thinkers differ greatly in the details of their explanation, and there is, moreover, on both sides much difference of opinion as to the scope of the question. Thus Kant would limit the application of the name axiom to principles of mathematical science, denying that in philosophy (whether metaphysical or natural), which works with discursive concepts, not with intuitions, there can be any principles immediately certain ; and, as a matter of fact, it is to mathematical principles only that the name is universally accorded in the language of special science not generally, in spite of Newton s lead, to the laws of motion, and hardly ever to scientific principles of more special range like the atomic theory. Other thinkers, however, notably Leibnitz, lay stress on the ultimate prin ciples of all thinking as the only true axioms, and would contend for the possibility of reducing to these (with the help of definitions) the special principles of mathematics, commonly allowed to pass and do duty as axiomatic. Still others apply the name equally and in the same sense to the general principles of thought and to some principles of special science. In view of such differences of opinion as to the actual matter in question, it is not to be expected that there should be agreement as to the marks character istic of axioms, nor surprising that agreement, where it appears to exist, should often be only verbal. The charac ter of necessity, for example, so much relied upon for ex cluding the possibility of an experiential origin, may either, as by Kant, be carefully limited to that which can be claimed for propositions that are at the same time syn thetic, or may be vaguely taken (as too frequently by Leibnitz) to cover necessity of mere logical implication the necessity of analytic, including identical, propositions which Kant allowed to be quite consistent with origin in experience. The question being so perplexed, uo other course seems open than to try to determine the nature of axioms mainly upon such instances as are, at least practically, admitted by all, and these are mathematical principles. That propositions with an exceptional character of certainty are assumed in mathematical science is notorious ; that such propositions must be assumed as principles of the science, if it is to be at once general and demonstrative, is now conceded even by extreme experientialists ; while it is, farther, universally held that it is the exceptional character of the subject-matter of mathematics that renders possible such determinate assumption. What the actual principles to be assumed are, has, indeed, always been more or less disputed ; but this is a point of secondary importance, since it is possible from different sets of assumption to arrive at results practically the same. The particular list of proposi tions passing current in modern times as Euclid s axioms, like his original list of common notions, is open to objection, not so much for mixing up assertions not equally underiva- tive (as the ancient critics remarked), but for including two the 8th and 9th which are unlike all the others in being mere definitions (viz., of equals and of whole or part). Being intended as a body of principles of geometry in particular within the general science of mathematics, the modern list is not open to exception in that it adds to the pro positions of general mathematical import, forming Euclid s original list, others specially geometrical, provided the addi tions made are sufficient for the purpose. It does, in any case, contain what may be taken as good representative in stances of mathematical axioms both general and special; for example, the 1st. Things equal to the same are equal to one another, applicable to all quantity; and the 10th, Two straight lines cannot enclose a space, specially geometrical. (The latter has been regarded by some writers as either a mere definition of straight lines, or as contained by direct implication in the definition ; but incorrectly. If it is held to be a definition, nothing is too complex to be so called, and the very meaning of a definition as a principle of science is abandoned ; while, if it is held to be a logical implication of the definition, the whole science of geometry may as well be pronounced a congeries of analytic propositions. When straight line is strictly defined, the assertion is clearly seen to be synthetic.) Now of such propositions as the two just quoted it is commonly said that they are self-evident, that they are seen to be true as soon as stated, that their opposites are inconceivable; and the expressions are not too strong as descriptive of the peculiar certainty pertaining to them. Nothing, however, is thereby settled as to the ground of the certainty, which is the real point in dispute between the experiential and rational schools, as these have become determinately opposed since the time and mainly through the influence of Kant. Such axioms, according to Kant, being necessary as well as synthetic, cannot be got from experience, but depend on the nature of the knowing faculty ; being immediately synthetic, they are not thought discursively but apprehended by way of direct intuition. According to the experientialists, as represented by J. S. Mill, they are, for all their certainty, inductive generalisations from particular experiences ; only the experiences are peculiar (as already said) in being extremely simple and uniform, while the experience of space Mill does not urge the like point as regards number is farther to be distinguished from common physical experience in that it supplies matter for induction no less in the imaginative (representative) than in the presentative form. Mill thus agrees with Kant on a vital point in holding the axioms to be synthetic propositions, but takes little or no account of that which, in Kant s eyes, is their distinctive characteristic their validity as universal truths in the guise of direct intuitions or singular acts of percep tion, presentative or representative. The synthesis of subject and predicate, thus universally valid though, imme diately effected, Kant explains by supposing the singular presentation or representation to be wholly determined from within through the mind s spontaneous act, instead of being received as sensible experience from without ; to speak more precisely, he refers the apprehension of quantity, whether continuous or discrete, to " productive imagina tion," and regards it always as a pure mental construction. Mill, who supposes all experience alike to be passively received, or, at all events, makes no distinction in point of original apprehension between quantity and physical quali ties, fails to explain what must be allowed as the specific character of mathematical axioms. Our conviction of their truth cannot be said to depend upon the amount of support ing experience, for increased experience (which is all that Mill secures and secures only for figured magnitude, without psychological reason given) does not make it stronger ; and, if they are conceded on being merely stated, which, unless they are held to be analytic proposi tions, amounts to their being granted upon direct inspection of a particular case, it can be only because the case, so decisive, is made and not found is constituted or con structed by ourselves, as Kant maintains, with the guarantee for uniformity and adequacy which direct construction alone gives. Still it does not therefore follow that the construc tion whereby synthesis of subject and predicate is directly made is of the nature described by Kant due to the activity of the pure ego, opposed to the very notion of sensible experience, and absolutely a priori. As we have a natural psychological experience of sensations passively received through bodily organs, we also have what is not less a natural psychological experience of motor activity exerted through the muscular system. Only by muscular movements, of which we are conscious in the act of perform ing them, have we perception of objects as extended and figured, and in itself the activity of the describing and circumscribing movements is as much matter of experience as is the accompanying content of passive sensation. At the same time, the conditions of the active exertion -and of the passive affection are profoundly different. While, in objective perception, within the same or similar movements, the content of passive sensation may indefinitely vary beyond any control of ours, it is at all times in our power to describe forms by actual movement with or without a content of sensation, still more by represented or imagined movement. Our knowledge of the physical qualities of objects thus becomes a reproduction of our mani fold sensible experience, as this in its variety can alone be reproduced, by way of general concepts ; our knowledge of their mathematical attributes is, first and last, an act of conscious production or construction. It is manifestly so, as movement actual or imaginary, in the case of magnitude or continuous quantity ; nor is it otherwise in the case of number or discrete quantity, when the units are objects (points or anything else) standing apart from each other in space. When the units are not objects presented to the senses or represented as coexistent in space, but are mere subjective occurrences succeeding each other in time, the numerical synthesis, doubtless, proceeds differently, but it is still an act of construction, dependent on the power we have of voluntarily determining the flow of subjective consciousness. Thus acting constructively in our experience both of number and form, we, in a manner, make the ultimate relations of both to be what for us they must be in all circumstances, and such relations when expressed are truly axiomatic in every sense that has been ascribed to the name. Beyond the mathematical principles which may be thus accounted for, there are, as was before remarked, no other principles of special science to which the name of axiom is uniformly applied. It may now be understood why the name should be withheld from such a fundamental generalisation as the atomic theory in chemistry, even when we have become so familiar with the facts as to seem to see clearly that the various kinds of matter must combine with each other regularly in definite proportions : the proposition answers to no intuition or direct apprehen sion. At most could it be called axiomatic in the sense, of course applicable to mathematical principles also, that it is assumed as true in the body of science compacted by means of it. The laws of motion, however, formulated by Newton as principles of general physics, not only were called by him axiomatic in this latter sense, but have been given out by others since his time as propositions intuitively certain ; and, though it cannot seriously be pretended that there is the same case for ascribing to them the character of a priori truths, there must be some reason why the name of axiom in the full sense has been claimed for them alone by the side of the mathematical principles. The a priori character, it is clear, can only in a peculiar sense be claimed for truths which all the genius of the ancients failed to grasp, and which were established in far later times as inductions from actual experiments ; Newton, certainly, in calling them axioms, by no means claimed for them aught but an experiential origin. On the other hand, it must be conceded that motion as an experience has in it a character of simplicity, like that belonging to number and form, consisting mainly in a clear apprehension of the circumstances under which the phenomenon varies, while, again, such apprehension is conditioned by the psychological nature of the experience, namely, that it is one depending on activity of our own which we can control, and does not corne to us as bare passive affection which we must take as we find it. "We do in truth make or constitute motion, as we construct number and space ; moving, as we please, without external occasion, and, when apprehending objective movements, following these with conscious motions of our members. Notwithstanding, our proper motions far less adequately correspond to the reality of external motions than do our subjective constructions of space and number answer to the reality of things figured and numbered. With limited store of nervous energy and muscles of con fined sweep, we cannot execute at all such continued unvarying movements as occur, at least approximately, in nature ; we cannot, by any such combinations of movements as we are able to make, determine beforehand the result of such complex motions as nature in endless variety exhibits ; nor, again, can we with any accuracy appreciate the relation between action and reaction by opposing our muscular organs to one another. We must wait long upon experience that comes to us, or rather, in face of the objective complexity presented by nature, sally forth to make varied experiments with moving things, and there upon generalise, before anything can be determined posi tively respecting motion. This is precisely what inquirers, until about the time of Galileo, were by no means content to do, and they had accordingly laws of motion which were, indeed, devised a priori, but which were not objectively true. Since the time of Galileo true, or at least effective, laws of motion have been established inductively, like all other physical laws ; only it is more easy than in the case of the others, which are less simple, to come near to an adequate subjective construction of them, and hence the claim sometimes set up for them to be in fact a priori and in the full sense axiomatic. It remains to inquire in what sense the general principles of all knowledge or principles of certitude may be called, as they often are called, axioms. The laws of Contradiction and of Excluded Middle, noted though not named by Aristotle, together with that formulated as the law of Identity, presupposed as they are in all consistent thinking, have, with a character of widest generality, also a character of extreme simplicity, and may fitly be denominated axioms in the sense of immediate principles. They stand, however, as pure logical principles, apart from all others, being wholly formal, without a shade of material content. There can be no question, therefore, of their certainty being- guaranteed by a direct intuition, valid for all cases because fully representative of all ; as little does there appear valid ground for calling them, in the proper sense, inductive generalisations from experience. They may rather be held to admit only of the kind of proof that Aristotle calls dialectical : whoever denies them will find that he cannot argue at all or be argued with ; he cuts himself off from all part in rational discourse, and is no better, as Aristotle forcibly expresses it, than a plant. The like position of being postulated as the condition of making progress belongs to the very different principle or principles (which may, however, be called logical, in the wider sense) implied in the establishment of truth of fact, more particularly the inductive investigation of nature. Whether expressed in the form of a principle of Sufficient Reason, as by Leibnitz, or, as is now more common, in the form of a principle of Uniformity of Nature, with or without a pendant principle of Causality for the special class of uniformities of succession, some assumption is indispensable for knitting together into general truths the discrete and particular elements of experience. Such postulates must be declared to have an experiential origin rather than to be a priori principles, but experience may more truly be said to suggest them than to be their ground or foundation, since they are themselves the ground, express or implied, of all ordered experience. Their case is perhaps best met by pronouncing them hypothetical principles, and as there are no axioms not even those of mathematics that are thought of without reference to their proved efficiency as principles leading to definite conclusions, they may be called axiomatic on account of their extreme generality, however little they possess the character of immediacy. The name axiom, at the end of the inquiry, is thus left undeniably equivocal, and it clearly behoves those whe- employ it, whether in philosophy or science, always to make plain in what sense it is meant to be taken. Before closing, it is, perhaps, necessary to add why, in dealing with the question of origin, no account has been taken of the doctrine of evolution which has become so promi nent in the latest scientific and philosophical speculation. From the point of view of the present article, that doc trine has only an indirect bearing on the inquiry. If the conditions of experience as they are found in the individual suffice to explain the different assurance with which general assertions are made in different de partments of knowlege, there is no need to carry the psychological consideration farther back. The effect of such difference in the conditions of experience may, of course, be accumulated in the life of the race, and the accumulation may go far to determine the psychological history of the individual, but the question, as a rational one, must be decided upon analysis of the conditions as they are. (G. c. E.)