There are and can be only two ways of searching into and discovering truth. The one flies from the senses and particulars to the most general axioms, and from these principles, the truth of which it takes for settled and immovable, proceeds to judgment and to the discovery of middle axioms. And this way is now in fashion. The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried.
It cannot be that axioms established by argumentation should avail for the discovery of new works, since the subtlety of nature is greater many times over than the subtlety of argument. But axioms duly and orderly formed from particulars easily discover the way to new particulars, and thus render sciences active.
While the game of deadlocks and bottle-necks goes on, another more serious game is also being played. It is governed by two axioms. One is that there can be no peace without a general surrender of sovereignty: the other is that no country capable of defending its sovereignty ever surrenders it. If one keeps these axioms in mind one can generally see the relevant facts in international affairs through the smoke-screen with which the newspapers surround them.
The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms.
An axiomatic system comprises axioms and theorems and requires a certain amount of hand-eye coordination before it works. A formal system comprises an explicit list of symbols, an explicit set of rules governing their cohabitation, an explicit list of axioms, and, above all, an explicit list of rules explicitly governing the steps that the mathematician may take in going from assumptions to conclusions. No appeal to meaning nor to intuition. Symbols lose their referential powers; inferences become mechanical.
One of the most important axioms is, that as the quantity of any commodity, for instance, plain food, which a man has to consume, increases, so the utility or benefit derived from the last portion used decreases in degree. The decrease in enjoyment between the beginning and the end of a meal may be taken as an example.
William Stanley Jevons
We must start with scientific fundamentals, and that means with the data of experiments and not with assumed axioms predicated only upon the misleading nature of that which only superficially seems to be obvious. It is the consensus of great scientists that science is the attempt to set in order the facts of experience.
R. Buckminster Fuller
When a truth is necessary, the reason for it can be found by analysis, that is, by resolving it into simpler ideas and truths until the primary ones are reached. It is this way that in mathematics speculative theorems and practical canons are reduced by analysis to definitions, axioms and postulates.
..why is it that in problematic situations almost everyone resorts to axioms and societal remedies that in actuality almost nobody believes in?...ask yourself, have you ever known anyone whose marriage was saved by a marriage counselor, whose drinking was cured by a psychiatrist, whose son was kept out of reform school by a social worker?
James Lee Burke
Those reliable axioms about the taste and expectations of the mass movie audience are not so much laws of nature as artifacts of corporate strategy. And the lessons derived from them conveniently serve to strengthen a status quo that increasingly marginalizes risk, originality and intelligence.
A. O. Scott
In the Principia Mathematica, Bertrand Russell and Alfred Whitehead attempted to give a rigorous foundation to mathematics using formal logic as their basis. They began with what they considered to be axioms, and used those to derive theorems of increasing complexity. By page 362, they had established enough to prove "1 + 1 = 2.
Whatever the degree of your knowledge, these two-existence and consciousness-are axioms you cannot escape, these two are the irreducible primaries implied in any action you undertake, in any part of your knowledge and in its sum, from the first ray of light you perceive at the start of your life to the widest erudition you might acquire at its end.
Not only the individual experience slowly acquired, but the accumulated experience of the race, organized in language, condensed in instruments and axioms, and in what may be called the inherited intuitions--these form the multiple unity which is expressed in the abstract term "experience.
George Henry Lewes
The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry [as did Clifford], for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.
Eric Temple Bell
Can the difficulty of an exam be measured by how many bits of information a student would need to pass it? This may not be so absurd in the encyclopedic subjects but in mathematics it doesn't make any sense since things follow from each other and, in principle, whoever knows the bases knows everything. All of the results of a mathematical theorem are in the axioms of mathematics in embryonic form, aren't they?
You can go through your whole life telling yourself that life is logical, life is prosaic, life is sane. Above all sane. And I think it is. I've had a lot of time to think about that. And what I keep coming back to is [her] dying declaration: 'So you understand that when we increase the number of variables, the axioms themselves never change.'
I do not want to presuppose anything as known. I see in my explanation in section 1 the definition of the concepts point, straight line and plane, if one adds to these all the axioms of groups i-v as characteristics. If one is looking for other definitions of point, perhaps by means of paraphrase in terms of extensionless, etc., then, of course, I would most decidedly have to oppose such an enterprise. One is then looking for something that can never be found, for there is nothing there, and everything gets lost, becomes confused and vague, and degenerates into a game of hide and seek.
Art, to me is the interpretation of the impression which nature makes upon the eye and brain. The word 'Impressionism' as applied to art has been abused, and in the general acceptance of the term has become perverted. [...] The true impressionism is realism. So many people do not observe. They take the ready-made axioms laid down by others, and walk blindly in a rut without trying to see for themselves.
Based on the considerations of history, ancient history, and international axioms, the logic of following up a citizen with his shadow for the purpose of the demarcation of political frontiers of any state has been discounted for international conventions. For example the Arabs cannot ask Spain just because they were there some time in the past nor can they ask for any other area outside the frontiers of the Arab homeland
An axiomatic system establishes a reverberating relationship between what a mathematician assumes (the axioms) and what he or she can derive (the theorems). In the best of circumstances, the relationship is clear enough so that the mathematician can submit his or her reasoning to an informal checklist, passing from step to step with the easy confidence the steps are small enough so that he cannot be embarrassed nor she tripped up.
We cannot save everyone, can we?' I said to her as we continued walking on our way. I turned my head back only to find the spot the beggar had occupied empty. 'Not everyone, ' she said. She took my hand once more in hers, kissed the back of it, and finished one of the sincerest axioms I had heard in sometime. 'We must save, ' she said, 'only the ones we can while we can.' Leila Bakr, in A TIME TO LOVE IN TEHRAN, and speaking to her love, John Lockwood
Blaise Pascal used to mark with charcoal the walls of his playroom, seeking a means of making a circle perfectly round and a triangle whose sides and angle were all equal. He discovered these things for himself and then began to seek the relationship which existed between them. He did not know any mathematical terms and so he made up his own. Using these names he made axioms and finally developed perfect demonstrations, until he had come to the thirty-second proposition of Euclid.
Catharine Cox Miles
It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.
How odd that Americans, and not just their presidents, have come to think of their Constitution as something separable from the government it's supposed to constitute. In theory, it should be as binding on rulers as the laws of physics are on engineers who design bridges; in practice, its axioms have become mere options. Of course engineers don't have to take oaths to respect the law of gravity; reality gives them no choice. Politics, as we see, makes all human laws optional for politicians.
The axioms of physics translate the laws of ethics. Thus, "the whole is greater than its part;" "reaction is equal to action;" "the smallest weight may be made to lift the greatest, the difference of weight being compensated by time;" and many the like propositions, which have an ethical as well as physical sense. These propositions have a much more extensive and universal sense when applied to human life, than when confined to technical use.
Ralph Waldo Emerson
If nature leads us to mathematical forms of great simplicity and beauty""by forms, I am referring to coherent systems of hypotheses, axioms, etc.""to forms that no one has previously encountered, we cannot help thinking that they are "true," that they reveal a genuine feature of nature... . You must have felt this too: the almost frightening simplicity and wholeness of the relationships which nature suddenly spreads out before us and for which none of us was in the least prepared.
There are two principles on which all men of intellectual integrity and good will can agree, as a 'basic minimum,' as a precondition of any discussion, co-operation or movement toward an intellectual Renaissance. . . . They are not axioms, but until a man has proved them to himself and has accepted them, he is not fit for an intellectual discussion. These two principles are: a. that emotions are not tools of cognition; b. that no man has the right to initiate the use of physical force against others.
The constructs of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth.
We find sects and parties in most branches of science; and disputes which are carried on from age to age, without being brought to an issue. Sophistry has been more effectually excluded from mathematics and natural philosophy than from other sciences. In mathematics it had no place from the beginning; mathematicians having had the wisdom to define accurately the terms they use, and to lay down, as axioms, the first principles on which their reasoning is grounded. Accordingly, we find no parties among mathematicians, and hardly any disputes.
As soon as science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement. Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms. We call such a system of thought a theory. The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the 'truth' of the theory lies.
Too often, contemporary continental philosophers take the 'other' of philosophy to mean literature, but not religion, which is for them just a little too wholly other, a little beyond their much heralded tolerance of alterity. They retain an antagonism to religious texts inherited straight from the Enlightenment, even though they pride themselves on having made the axioms and dogmas of the Enlightenment questionable. But the truth is that contemporary continental philosophy is marked by the language of the call and the response, of the gift, of hospitality to the other, of the widow, the orphan and the stranger, and by the very idea of the 'wholly other, ' a discourse that any with the ears to hear knows has a Scriptural provenance and a Scriptural resonance. ("A Prologue", Journal of Philosophy and Scripture 1.1, Fall 2003, p. 1).
John D. Caputo
In brief, the teaching process, as commonly observed, has nothing to do with the investigation and establishment of facts, assuming that actual facts may ever be determined. Its sole purpose is to cram the pupils, as rapidly and as painlessly as possible, with the largest conceivable outfit of current axioms, in all departments of human thought-to make the pupil a good citizen, which is to say, a citizen differing as little as possible, in positive knowledge and habits of mind, from all other citizens. In other words, it is the mission of the pedagogue, not to make his pupils think, but to make them think right, and the more nearly his own mind pulsates with the great ebbs and flows of popular delusion and emotion, the more admirably he performs his function. He may be an ass, but this is surely no demerit in a man paid to make asses of his customers.
(...)the truths Phaedrus began to pursue were lateral truths; no longer the frontal truths of science, those toward which the discipline pointed, but the kind of truth you see laterally, out of the corner of your eye. In a laboratory situation, when your whole procedure goes haywire, when everything goes wrong or is indeterminate or is so screwed up by unexpected results you can't make head or tail out of anything, you start looking laterally. That's a word he later used to describe a growth of knowledge that doesn't move forward like an arrow in flight, but expands sideways, like an arrow enlarging in flight, or like the archer, discovering that although he has hit the bull's-eye and won the prize, his head is on a pillow and the sun is coming in the window. Lateral knowledge is knowledge that's from a wholly unexpected direction, from a direction that's not even understood as a direction until the knowledge forces itself upon one. Lateral truths point to the falseness of axioms and postulates underlying one's existing system of getting at truth.
Robert M. Pirsig
For the same reason there is nowhere to begin to trace the sheaf or the graphics of differance. For what is put into question is precisely the quest for a rightful beginning, an absolute point of departure, a principal responsibility. The problematic of writing is opened by putting into question the value of the arkhe. What I will propose here will not be elaborated simply as a philosophical discourse, operating according to principles, postulates, axioms, or definitions, and proceeding along the discursive lines of a linear order of reasons. In the delineation of differance everything is strategic and adventurous. Strategic because no transcendent truth present outside the field of writing can govern theologically the totality of the field. Adventurous because this strategy is a not simple strategy in the sense that strategy orients tactics according to a final goal, a telos or theme of domination, a mastery and ultimate reappropriation of the development of the field. Finally, a strategy without finality, what might be called blind tactics, or empirical wandering if the value of empiricism did not itself acquire its entire meaning in opposition to philosophical responsibility. If there is a certain wandering in the tracing of differance, it no more follows the lines of philosophical-logical discourse than that of its symmetrical and integral inverse, empirical-logical discourse. The concept of play keeps itself beyond this opposition, announcing, on the eve of philosophy and beyond it, the unity of chance and necessity in calculations without end.