What are your axioms, and categories, and systems, and aphorisms? Words, words. High air-castles are cunningly built of words, the words well bedded in good logic-mortar; wherein, however, no knowledge will come to lodge.
~Understanding.~--The eye of the understanding is like the eye of the sense; for as you may see great objects through small crannies or holes, so you may see great axioms of nature through small and contemptible instances.--_Bacon._
We do everything by custom, even believe by it; our very axioms, let us boast of our Freethinking as we may, are oftenest simply such beliefs as we have never heard questioned.
He alone reads history aright, who, observing how powerfully circumstances influence the feelings and opinions of men, how often vices pass into virtues, and paradoxes into axioms, learns to distinguish what is accidental and transitory in human nature from what is essential and immutable.--_Macaulay._
Consider the following axioms carefully: "Everything's better when it sits on a Ritz." and "Everything's better with Blue Bonnet on it." What happens if one spreads Blue Bonnet margarine on a Ritz cracker? The thought is frightening. Is this how God came into being? Try not to consider the fact that "Things go better with Coke".
Jean Valjean disconcerted him. All the axioms which had served him as points of support all his life long, had crumbled away in the presence of this man. Jean Valjean's generosity towards him, Javert, crushed him. Other facts which he now recalled, and which he had formerly treated as lies and folly, now recurred to him as realities. M. Madeleine re-appeared behind Jean Valjean, and the two figures were superposed in such fashion that they now formed but one, which was venerable. Javert felt that something terrible was penetrating his soul--admiration for a convict. Respect for a galley-slave--is that a possible thing? He shuddered at it, yet could not escape from it. In vain did he struggle, he was reduced to confess, in his inmost heart, the sublimity of that wretch. This was odious.
Complete systems of axioms have been stated by M. Pasch, _loc. cit._; G. Peano, _loc. cit._; M. Pieri, _loc. cit._; B. Russell, _Principles of Mathematics_; O. Veblen, _loc. cit._; and by G. Veronese in his treatise, _Fondamenti di geometria_ (Padua, 1891; German transl. by A. Schepp, _Grundzüge der Geometrie_, Leipzig, 1894). Most of the leading memoirs on special questions involved have been cited in the text; in addition there may be mentioned M. Pieri, "Nuovi principii di geometria projettiva complessa," _Trans. Accad. R. d. Sci._ (Turin, 1905); E.H. Moore, "On the Projective Axioms of Geometry," _Trans. Amer. Math. Soc._, 1902; O. Veblen and W.H. Bussey, "Finite Projective Geometries," _Trans. Amer. Math. Soc._, 1905; A.B. Kempe, "On the Relation between the Logical Theory of Classes and the Geometrical Theory of Points," _Proc. Lond. Math. Soc._, 1890; J. Royce, "The Relation of the Principles of Logic to the Foundations of Geometry," _Trans. of Amer. Math. Soc._, 1905; A. Schoenflies, "Über die Möglichkeit einer projectiven Geometrie bei transfiniter (nichtarchimedischer) Massbestimmung," _Deutsch. M.-V. Jahresb._, 1906. Entry: BIBLIOGRAPHY
Thus, on the one hand, the individualist conception, when carried out to its full extent, leads to the total negation of all real cognition. If the real system of things, to which conscious experience has reference, be regarded as standing in casual relation to this experience there is no conceivable ground for the extension to reality of the notions which somehow are involved in thought. The same result is apparent, on the other hand, when we consider the theory of knowledge implied in the Leibnitzian individualism. The metaphysical conception of the monads, each of which is the universe _in nuce_, presents insuperable difficulties when the connexion or interdependence of the monads is in question, and these difficulties obtrude themselves when the attempt is made to work out a consistent doctrine of cognition. For the whole mass of cognisable fact, the _mundus intelligibilis_, is contained _impliciter_ in each monad, and the several modes of apprehension can only be regarded as so many stages in the developing consciousness of the monad. Sense and understanding, real connexion of facts and analysis of notions, are not, therefore, distinct in kind, but differ only in degree. The same fundamental axioms, the logical principles of identity and sufficient reason, are applicable in explanation of all given propositions. It is true that Leibnitz himself did not work out any complete doctrine of knowledge, but in the hands of his successors the theory took definite shape in the principle that the whole work of cognition is in essence analytical. The process of analysis might be complete or incomplete. For finite intelligences there was an inevitable incompleteness so far as knowledge of matters of fact was concerned. In respect to them, the final result was found in a series of irreducible notions or categories, the _prima possibilia_, the analysis and elucidation of which was specifically the business of philosophy or metaphysics. Entry: 1798
A much more favourable judgment must be given upon the short _Treatise on eternal and immutable Morality_, which deserves to be read by those who are interested in the historical development of British moral philosophy. It is an answer to Hobbes's famous doctrine that moral distinctions are created by the state, an answer from the standpoint of Platonism. Just as knowledge contains a permanent intelligible element over and above the flux of sense-impressions, so there exist eternal and immutable ideas of morality. Cudworth's ideas, like Plato's, have "a constant and never-failing entity of their own," such as we see in geometrical figures; but, unlike Plato's, they exist in the mind of God, whence they are communicated to finite understandings. Hence "it is evident that wisdom, knowledge and understanding are eternal and self-subsistent things, superior to matter and all sensible beings, and independent upon them"; and so also are moral good and evil. At this point Cudworth stops; he does not attempt to give any list of Moral Ideas. It is, indeed, the cardinal weakness of this form of intuitionism that no satisfactory list can be given and that no moral principles have the "constant and never-failing entity," or the definiteness, of the concepts of geometry. Henry More, in his _Enchiridion ethicum_, attempts to enumerate the "_noemata moralia_"; but, so far from being self-evident, most of his moral axioms are open to serious controversy. Entry: A
But what has become of Logic, with which the traditional order of Andronicus begins Aristotle's works (1-148 b 8)? So far from coming first, Logic comes nowhere in his classification of science. Aristotle was the founder of Logic; because, though others, and especially Plato, had made occasional remarks about reason ([Greek: logos]), Aristotle was the first to conceive it as a definite subject of investigation. As he says at the end of the _Sophistical Elenchi_ on the syllogism, he had no predecessor, but took pains and laboured a long time in investigating it. Nobody, not even Plato, had discovered that the process of deduction is a combination of premisses ([Greek: syllogomos]) to produce a new conclusion. Aristotle, who made this great discovery, must have had great difficulty in developing the new investigation of reasoning processes out of dialectic, rhetoric, poetics, grammar, metaphysics, mathematics, physics and ethics; and in disengaging it from other kinds of learning. He got so far as gradually to write short discourses and long treatises, which we, not he, now arrange in the order of the _Categories_ or names; the _De Interpretatione_ on propositions; the _Analytics_, _Prior_ on syllogism, _Posterior_ on scientific syllogism; the _Topics_ on dialectical syllogism; the _Sophistici Elenchi_ on eristical or sophistical syllogism; and, except that he had hardly a logic of induction, he covered the ground. But after all this original research he got no further. First, he did not combine all these works into a system. He may have laid out the sequence of syllogisms from the _Analytics_ onwards; but how about the _Categories_ and the _De Interpretatione_? Secondly, he made no division of logic. In the _Categories_ he distinguished names and propositions for the sake of the classification of names; in the _De Interpretatione_ he distinguished nouns and verbs from sentences with a view to the enunciative sentence: in the _Analytics_ he analysed the syllogism into premisses and premisses into terms and copula, for the purpose of syllogism. But he never called any of these a division of all logic. Thirdly, he had no one name for logic. In the _Posterior Analytics_ (i. 22, 84 a 7-8) he distinguishes two modes of investigation, analytically ([Greek: analytikos]) and logically ([Greek: logikos]). But "analytical" means scientific inference from appropriate principles, and "logical" means dialectical inference from general considerations; and the former gives its name to the _Analytics_, the latter suits the _Topics_, while neither analytic nor logic is a name for all the works afterwards called logic. Fourthly, and consequently, he gave no place to any science embracing the whole of those works in his classification of science, but merely threw out the hint that we should know analytics before questioning the acceptance of the axioms of being (_Met._ [Gamma] 3). Entry: B
In the _Posterior Analytics_ the syllogism is brought into decisive connexion with the real by being set within a system in which its function is that of material implication from principles which are primary, immediate and necessary truths. Hitherto the assumption of the probable as true rather than as what will be conceded in debate[51] has been the main distinction of the standpoint of analytic from that of dialectic. But the true is true only in reference to a coherent system in which it is an immediate ascertainment of [Greek: nous], or to be deduced from a ground which is such. The ideal of science or demonstrative knowledge is to exhibit as flowing from the definitions and postulates of a science, from its special principles, by the help only of axioms or principles common to all knowledge, and these not as premises but as guiding rules, all the properties of the subject-matter, i.e. all the predicates that belong to it in its own nature. In the case of any subject-kind, its definition and its existence being avouched by [Greek: nous], "heavenly body" for example, the problem is, given the fact of a non-self-subsistent characteristic of it, such as the eclipse of the said body, to find a ground, a [Greek: meson] which expressed the [Greek: aition], in virtue of which the adjectival concept can be exhibited as belonging to the subject-concept [Greek: kath auto] in the strictly adequate sense of the phrase in which it means also [Greek: hê auto].[52] We are under the necessity then of revising the point of view of the syllogism of all-ness. We discard the conception of the universal as a predicate applicable to a plurality, or even to all, of the members of a group. To know merely [Greek: kata pantos] is not to know, save accidentally. The exhaustive judgment, if attainable, could not be known to be exhaustive. The universal is the ground of the empirical "all" and not conversely. A formula such as the equality of the interior angles of a triangle to two right angles is only scientifically known when it is not of isosceles or scalene triangle that it is known, nor even of all the several types of triangle collectively, but as a predicate of triangle recognized as the widest class-concept of which it is true, the first stage in the progressive differentiation of figure at which it can be asserted.[53] Entry: A
The final period is characterized by the successful production of exact systems of axioms, and by the final solution of problems which have occupied mathematicians for two thousand years. The successful analysis of the ideas involved in serial continuity is due to R. Dedekind, _Stetigkeit und irrationale Zahlen_ (1872), and to G. Cantor, _Grundlagen einer allgemeinen Mannigfaltigkeitslehre_ (Leipzig, 1883), and _Acta math._ vol. 2. Entry: BIBLIOGRAPHY
The most important of the applications of continuous groups are to the theory of systems of differential equations, both ordinary and partial; in fact, Lie states that it was with a view to systematizing and advancing the general theory of differential equations that he was led to the development of the theory of continuous groups. It is quite impossible here to give any account of all that Lie and his followers have done in this direction. An entirely new mode of regarding the problem of the integration of a differential equation has been opened up, and in the classification that arises from it all those apparently isolated types of equations which in the older sense are said to be integrable take their proper place. It may, for instance, be mentioned that the question as to whether Monge's method will apply to the integration of a partial differential equation of the second order is shown to depend on whether or not a contact-transformation can be found which will reduce the equation to either [Pd]²z/[Pd]x² = 0 or [Pd]²z/[Pd]x[Pd]y = 0. It is in this direction that further advance in the theory of partial differential equations must be looked for. Lastly, it may be remarked that one of the most thorough discussions of the axioms of geometry hitherto undertaken is founded entirely upon the theory of continuous groups. Entry: W
With this definition of addition it can be proved that prospectivities with the same double point satisfy all the axioms of magnitude. Accordingly they can be associated in a one-one correspondence with the positive and negative real numbers. Let E (fig. 70) be any point on l, distinct from O and U. Then the prospectivity (OEU²) is associated with unity, the prospectivity (OOU²) is associated with zero, and (OUU²) with [infinity]. The prospectivities of the type (OPU²), where P is any point on the segment OEU, correspond to the positive numbers; also if P' is the harmonic conjugate of P with respect to O and U, the prospectivity (OP'U²) is associated with the corresponding negative number. (The subjoined figure explains this relation of the positive and negative prospectivities.) Then any point P on l is associated with the same number as is the prospectivity (OPU²). Entry: A
It follows from axioms 1-12 by projection that the Dedekind property is true for all lines. Again the _harmonic system_ ABC, where A, B, C are collinear points, is defined[34] thus: take the harmonic conjugates A', B', C' of each point with respect to the other two, again take the harmonic conjugates of each of the six points A, B, C, A', B', C' with respect to each pair of the remaining five, and proceed in this way by an unending series of steps. The set of points thus obtained is called the harmonic system ABC. It can be proved that a harmonic system is compact, and that every segment of the line containing it possesses members of it. Furthermore, it is easy to prove that the fundamental theorem holds for harmonic systems, in the sense that, if A, B, C are three points on a line l, and A', B', C' are three points on a line l', and if by any two distinct series of projections A, B, C are projected into A', B', C', then any point of the harmonic system ABC corresponds to the same point of the harmonic system A'B'C' according to both the projective relations which are thus established between l and l'. It now follows immediately that the fundamental theorem must hold for all the points on the lines l and l', since (as has been pointed out) harmonic systems are "everywhere dense" on their containing lines. Thus the fundamental theorem follows from the axioms of order. Entry: 12
The life of Cujas was altogether that of a scholar and teacher. In the religious wars which filled all the thoughts of his contemporaries he steadily refused to take any part. _Nihil hoc ad edictum praetoris_, "this has nothing to do with the edict of the praetor," was his usual answer to those who spoke to him on the subject. His surpassing merit as a jurisconsult consisted in the fact that he turned from the ignorant commentators on Roman law to the Roman law itself. He consulted a very large number of manuscripts, of which he had collected more than 500 in his own library; but, unfortunately, he left orders in his will that his library should be divided among a number of purchasers, and his collection was thus scattered, and in great part lost. His emendations, of which a large number were published under the title of _Animadversiones et observationes_, were not confined to lawbooks, but extended to many of the Latin and Greek classical authors. In jurisprudence his study was far from being devoted solely to Justinian; he recovered and gave to the world a part of the Theodosian Code, with explanations; and he procured the manuscript of the _Basilica_, a Greek abridgment of Justinian, afterwards published by Fabrot (see BASILICA). He also composed a commentary on the _Consueludines Feudorum_, and on some books of the Decretals. In the _Paratitla_, or summaries which he made of the Digest, and particularly of the Code of Justinian, he condensed into short axioms the elementary principles of law, and gave definitions remarkable for their admirable clearness and precision. His lessons, which he never dictated, were continuous discourses, for which he made no other preparation than that of profound meditation on the subjects to be discussed. He was impatient of interruption, and upon the least noise he would instantly quit the chair and retire. He was strongly attached to his pupils, and Scaliger affirms that he lost more than 4000 livres by lending money to such of them as were in want. Entry: CUJAS
The second controversy is that between the view that the axioms applicable to space are known only from experience, and the view that in some sense these axioms are given _a priori_. Both these views, thus broadly stated, are capable of various subtle modifications, and a discussion of them would merge into a general treatise on epistemology. The cruder forms of the _a priori_ view have been made quite untenable by the modern mathematical discoveries. Geometers now profess ignorance in many respects of the exact axioms which apply to existent space, and it seems unlikely that a profound study of the question should thus obliterate _a priori_ intuitions. Entry: VII
AXIOM (Gr. [Greek: axiôma]), a general proposition or principle accepted as self-evident, either absolutely or within a particular sphere of thought. Each special science has its own axioms (cf. the Aristotelian [Greek: archai], "first principles") which, however, are sometimes susceptible of proof in another wider science. The Greek word was probably confined by Plato to mathematical axioms, but Aristotle (_Anal. Post._ i. 2) gave it also the wider significance of the ultimate principles of thought which are behind all special sciences (_e.g._ the principle of contradiction). These are apprehended solely by the mind, which may, however, be led to them by an inductive process. After Aristotle, the term was used by the Stoics and the school of Ramus for a proposition simply, and Bacon (_Nov. Organ._ i. 7) used it of any general proposition. The word was reintroduced in modern philosophy probably by René Descartes (or by his followers) who, in the search for a definite self-evident principle as the basis of a new philosophy, naturally turned to the familiar science of mathematics. The axiom of Cartesianism is, therefore, the _Cogito ergo sum_. Kant still further narrowed the meaning to include only self-evident (intuitive) synthetic propositions, _i.e._ of space and time. The nature of axiomatic certainty is part of the fundamental problem of logic and metaphysics. Those who deny the possibility of all non-empirical knowledge naturally hold that every axiom is ultimately based on observation. For the Euclidian axioms see GEOMETRY. Entry: AXIOM
Scarcely less strong than his interest in Rome is his interest in the moral lessons which her history seemed to him so well qualified to teach. This didactic view of history was a prevalent one in antiquity, and it was confirmed no doubt by those rhetorical studies which in Rome as in Greece formed the chief part of education, and which taught men to look on history as little more than a storehouse of illustrations and themes for declamation. But it suited also the practical bent of the Roman mind, with its comparative indifference to abstract speculation or purely scientific research. It is in the highest degree natural that Livy should have sought for the secret of the rise of Rome, not in any large historical causes, but in the moral qualities of the people themselves, and that he should have looked upon the contemplation of these as the best remedy for the vices of his own degenerate days. He dwells with delight on the unselfish patriotism of the old heroes of the republic. In those times children obeyed their parents, the gods were still sincerely worshipped, poverty was no disgrace, sceptical philosophies and foreign fashions in religion and in daily life were unknown. But this ethical interest is closely bound up with his Roman sympathies. His moral ideal is no abstract one, and the virtues he praises are those which in his view made up the truly Roman type of character. The prominence thus given to the moral aspects of the history tends to obscure in some degree the true relations and real importance of the events narrated, but it does so in Livy to a far less extent than in some other writers. He is much too skilful an artist either to resolve his history into a mere bundle of examples, or to overload it, as Tacitus is sometimes inclined to do, with reflections and axioms. The moral he wishes to enforce is usually either conveyed by the story itself, with the aid perhaps of a single sentence of comment, or put as a speech into the mouth of one of his characters (e.g. xxiii. 49; the devotion of Decius, viii. 10, cf. vii. 40; and the speech of Camillus, v. 54); and what little his narrative thus loses in accuracy it gains in dignity and warmth of feeling. In his portraits of the typical Romans of the old style, such as Q. Fabius Maximus, in his descriptions of the unshaken firmness and calm courage shown by the fathers of the state in the hour of trial, Livy is at his best; and he is so largely in virtue of his genuine appreciation of character as a powerful force in the affairs of men. Entry: LIVY
By much the most important member of this early band of Italian writers was undoubtedly Nicolas Steno (1631-1687), who, though born in Copenhagen, ultimately settled in Florence. Having made a European reputation as an anatomist, his attention was drawn to geological problems by finding that the rocks of the north of Italy contained what appeared to be sharks' teeth closely resembling those of a dog-fish, of which he had published the anatomy. Cautiously at first, for fear of offending orthodox opinions, but afterwards more boldly, he proclaimed his conviction that those objects had once been part of living animals, and that they threw light on some of the past history of the earth. He published in 1669 a small tract, _De solido intra solidum naturaliter contento_, in which he developed the ideas he had formed of this history from an attentive study of the rocks. He showed that the stratified formations of the hills and valleys consist of such materials as would be laid down in the form of sediment in turbid water; that where they contain marine productions this water is proved to have been the sea; that diversities in their composition point to commingling of currents, carrying different kinds of sediment of which the heaviest would first sink to the bottom. He made original and important observations on stratification, and laid down some of the fundamental axioms in stratigraphy. He reasoned that as the original position of strata was approximately horizontal, when they are found to be steeply inclined or vertical, or bent into arches, they have been disrupted by subterranean exhalations, or by the falling in of the roofs of underground cavernous spaces. It is to this alteration of the original position of the strata that the inequalities of the earth's surface, such as mountains, are to be ascribed, though some have been formed by the outburst of fire, ashes and stones from inside the earth. Another effect of the dislocation has been to provide fissures, which serve as outlets for springs. Steno's anatomical training peculiarly fitted him for dealing authoritatively with the question of the nature and origin of the fossils contained in the rocks. He had no hesitation in affirming that, even if no shells had ever been found living in the sea, the internal structure of these fossils would demonstrate that they once formed parts of living animals. And not only shells, but teeth, bones and skeletons of many kinds of fishes had been quarried out of the rocks, while some of the strata had skulls, horns and teeth of land-animals. Illustrating his general principles by a sketch of what he supposed to have been the past history of Tuscany, he added a series of diagrams which show how clearly he had conceived the essential elements of stratigraphy. He thought he could perceive the records of six successive phases in the evolution of the framework of that country, and was inclined to believe that a similar chronological sequence would be found all over the world. He anticipated the objections that would be brought against his views on account of the insuperable difficulty in granting the length of time that would be required for all the geographical vicissitudes which his interpretation required. He thought that many of the fossils must be as old as the time of the general deluge, but he was careful not to indulge in any speculation as to the antiquity of the earth. Entry: 2