Axiom

“ But this is a sharp man: he has seen that the Axiom does just as well by itself. Did you ever see one of those conjurers bring a globe of live fish out of a pocket-handkerchief? That's the kind of thing we have in Modern Geometry. A man stands before you with nothing but an Axiom in his hands. ”

“ You must know, doctor, that what you assume to be an old axiom is only an old error. In fact, why do you say that out of nothing nothing can arise? Simply because natural energies can do nothing without pre-existing materials. Hence your argument amounts to this. “Natural energies never make anything out of nothing; therefore out of nothing nothing can be made.” That this conclusion is a great blunder I need not prove, I presume, as logic teaches that no conclusion can be more general than its premises. Where is, then, the ground of your pretended axiom? ”

“ If such an idea be not what we too heedlessly term “an axiom,” it is at least preferable, as a Logical basis, to any axiom ever propounded, or to all imaginable axioms combined:—and such, precisely, is the idea with which my deductive process, so thoroughly corroborated by induction, commences. My particle proper is but absolute Irrelation. ”

“ This last principle, which we may call the axiom of constant result, holds good throughout all our mental operations, not only when we compare, but when we add, divide, class, or infer a given matter in any conceivable way. ”

“ The reviewer conceives that some of the so-called axioms are really definitions. The axiom, that ‘magnitudes which coincide with each other, that is, which fill the same space, are equal,’ is a definition of geometrical equality: the axiom, that ‘the whole is greater than its part,’ is a definition of whole and part. ”

“ This Idea of a First Cause, and the principle involved in the Idea, have been the subject of discussion in another manner. As we have already said, we assume as an axiom that a First Cause must exist; and we assert that God, the First Cause, exists eternal and immutable, by the necessity which the axiom implies. ”

“ The grand principle which we are just beginning to indistinctly perceive, and to recognize as underlying every branch of knowledge and as forming the foundation of all positive science, seems, when stated, to be simply an axiom. ”

“ All cases, then, tend to show that deduction involves generalization from an observed case on a single axiom; the axioms, therefore, of induction are the axioms of reasoning, and the generalization of particulars is the work of reason. ”

“ It is a statement of truth which is below and above all argument—a truth which all men recognize as a part of their mental existence. An axiom is simply a reference to a necessary condition in the framework of the human constitution. ”

“ That effects are proportional to their causes is laid down by some writers as an axiom in the theory of causation; and great use is sometimes made of this principle in reasonings respecting the laws of nature, though it is encumbered with many difficulties and apparent exceptions, which much ingenuity has been expended in showing not to be real ones. ”

“ But questions, and interrogations, and things like these, are neither true nor false, while judgments and propositions are necessarily one or the other.
Now of axioms, some are simple, and others are not simple; as Chrysippus, and Archedemus, and Athenodorus, and Antipater, and Crinis, agree in dividing them. Those are simple, which consist of an axiom or proposition, which is not ambiguous, (or of several axioms, or propositions of the same character,) as for instance the sentence, “It is day.” ”

“ The mode of discovering and proving laws of nature, which we have now examined, proceeds on the following axiom: Whatever circumstances can be excluded, without prejudice to the phenomenon, or can be absent notwithstanding its presence, is not connected with it in the way of causation. ”

“ If there be necessary and universal truths expressible in propositions of axiomatic simplicity and obviousness, and having for their subject-matter the elements of all our experience and all our knowledge, surely these are the truths which, if experience suggest to us any truths at all, it ought to suggest most readily, clearly, and unceasingly. ”

“ If a figure is used, all sorts of things seem obviously to follow, which no formal reasoning can prove from the explicit axioms, and which, as a matter of fact, are only accepted because they are obvious. By banishing the figure, it becomes possible to discover all the axioms that are needed ”

“ Tell a child that things which are equal to the same thing are equal to one another, and the child, lacking a sufficiently abstract notion of equality, and lacking, too, the needful practice in comparing relations, will fail to grasp the axiom. Similarly, a rustic, never having thought much about forces and their results, cannot form a definite conception answering to the axiom that action and reaction are equal and opposite. ”

“ An emotional postulate may become the guiding principle of a new science, a methodological principle may become superfluous and be discarded or be superseded by a better, a primitive desire may die down and cease to nourish a postulate, nay even a full blown axiom may be conceived as becoming otiose under changed conditions of experience. ”

“ The psychological reason why axioms, and indeed many propositions not ordinarily classed as such, may be learned from the idea only without referring to the fact, is that in the process of acquiring the idea we have learned the fact. The proposition is assented to as soon as the terms are understood, because in learning to understand the terms we have acquired the experience which proves the proposition to be true. ”

“ We now approach nearer to the application of these principles as observed in the immediate agency and effects which precede and follow action, and as connected with the verb.
It is an axiom in philosophy which cannot be controverted, that every effect is the product of a prior cause, and that every cause will necessarily produce a corresponding effect. This fact has always existed and will forever remain unchanged. It applies universally in physical, mental, and moral science; to God or man; to angels or to atoms; in time or thro eternity. ”

“ It is a matter of convenience, economy and — last, not least — beauty to make the set of axioms as small and as simple as possible — which means that ordinarily of all the possibilities of choice that one is preferred which makes the set of axioms consist of a minimum of simple propositions. ”

“ The Conceptualist may urge that the Nominalist's forms of statement and argument exist for the sake of their meaning, namely, judgments [Pg 15] and reasonings; and that the Materialist's laws of Nature are only judgments founded upon our conceptions of Nature; that the truth of observations and experiments depends upon our powers of perception; that perception is inseparable from understanding, and that a system of Induction may be constructed upon the axiom of Causation, regarded as a principle of Reason, just as well as by considering it as a law of Nature, and upon much the same lines. ”

“ He announces as the first principle of these Axioms — as the highest and firmest of all Principles — the Maxim of Contradiction:12 The same predicate 426cannot both belong and not belong to the same subject, at the same time and in the same sense; or, You cannot both truly affirm, and truly deny, the same predicate respecting the same subject; or, The same proposition cannot be at once true and false. This Axiom is by nature the beginning or source of all the other Axioms. It stands first in the order of knowledge; and it neither rests upon nor involves any hypothesis.13 ”

“ But when we try to grasp experience as a whole, we must set ourselves above the encumbering abstractions of a psychological classification that has transgressed the limits of its validity. By conceiving the axioms as essentially postulates, made with an ultimately practical end, we bridge the gap that has been artificially constructed between the functions of our nature, and overcome the errors of intellectualism. ”

“ And, first, we may notice those axiomatic beliefs which lie at the foundation of human reasoning, and which, while apparently in harmony with nature, do not admit of verification except by an experience impossible to finite beings. ”

“ In regard to his genius alone—we find what he says true as far as we have experienced, and we can judge no further but by larger experience—for axioms in philosophy are not axioms until they are proved upon our pulses. ”

“ That all such as have sought action and work out of their inquiry have been hasty and pressing to discover some practices for present use, and not to discover Axioms, joining with them the new assignations as their sureties. That the forerunning of the mind to frame recipes upon Axioms at the entrance, is like Atalanta's golden ball that hindereth and interrupteth the course, and is to be inhibited till you have ascended to a certain stage and degree of generalities ”

“ There is no small difference between the idols of the human mind, and the ideas of the divine mind; that is to say, between certain idle dogmas, and the real stamp and impression of created ohjects, as they are found in nature.
24. Axioms determined upon in argument can never assist in the discovery of new effects: for the subtilty of nature is vastly superior to that of argument. But axioms properly and regularly abstracted from particulars, easily point out and define new particulars, and therefore impart activity to the sciences. ”

“ All general propositions, whether called axioms, or definitions, or laws of nature, are merely abridged statements of the particular facts, which, as occasion arises, we either think we may proceed on as proved, or intend to assume. ”

“ As an art logic may be called the navigation of the sea of thought.
§ 25. The laws of thought are all reducible to the three following axioms, which are known as The Three Fundamental Laws of Thought.
(1) The Law of Identity—
Whatever is, is;
or, in a more precise form,
Every A is A.
(2) The Law of Contradiction—
Nothing can both be and not be;
Nothing can be A and not A.
(3) The Law of Excluded Middle—
Everything must either be or not be;
Everything is either A or not A.
§ 26. ”

“ Logic states, and partly explains and applies, certain abstract principles which all other sciences take for granted; namely, the axioms above mentioned—the principles of Contradiction, of the Syllogism and of Causation. By exercising the student in the apprehension of these truths, and in the application of them to particular propositions, it educates the power of abstract thought. Every science is a model of method, a discipline in close and consecutive thinking ”

“ are the axioms of logic adequate to reality, or are they measures and means by which alone we can, create realities, or the concept " reality "? ”

“ For in many cases they retain their hold over our affections only faute de mieux. They are the best assumptions we can work with, but not the best we can conceive. And some one may some day discover a way to work with what are now unsupported postulates, and so raise them to axiomatic rank. Thus whatever axioms we may at any time employ are, and ever remain relative to the nature of our desires and our experience, and so long as changes may occur in either, inexhaustible possibilities of corresponding developments must be admitted in the list of our axiomatic principles. ”

“ The other constructs its axioms from the senses and particulars, by ascending continually and gradually, till it finally arrives at the most general axioms, which is the true but unattempted way.
20. The understanding when left to itself proceeds by the same way as that which it would have adopted under the guidance of logic, namely, the first. For the mind is fond of starting off to generalities, that it may avoid labour, and after dwelling a little on a subject is fatigued by experiment. But these evils are augmented by logic, for the sake of the ostentation of dispute. ”

“ So that the attempt to disprove the necessity and use of axioms, as principles of reasoning, fails even when we take those instances which the opponents consider as the more manifestly favourable to their doctrine. ”

“ Axioms, he shows, are but hypotheses which have proved so useful, and have succeeded so well in displacing all rival hypotheses, that today they seem indispensable: they are merely empiric propositions or teleological conventions which have proved victorious in the struggle for acceptance as truth. ”

“ From the two kinds of axioms above specified, arise the two divisions of philosophy and the sciences, and we will use the commonly adopted terms which approach the nearest to our meaning, in our own sense. ”

“ In everyone of the special branches to which I have referred there are now certain propositions which are accepted as axiomatic; if a man's work conforms to these, it is allowed as scientific—he is advancing on the true path ”

“ Then we shall have, not useless universal axioms, but efficacious mediate axioms, true laws from which we can derive works, and which are the sources of power in the same degree as the sources of light. ”

“ When he describes in the SECOND part his method of finding true terms, or rather the method of his school, when he describes this new method of finding 'ideas,' ideas as they are in nature, powers, causes, the elements of history, or forms, as he more commonly calls them, when he describes this new method of deducing axioms, axioms that are ready for practice, he does, indeed, give us instances ”

“ As axioms and definitions are the preliminaries of abstract mathematical reasonings, so our sensuous perceptions through the five senses are the preliminaries to reasonings about concrete realities in the world of change. ”