Top 70 Axioms Quotes & Sayings

The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms.

It is not enough merely to memorize and spout vocal axioms good singing is infinitely more than much talk and head-knowledge.

The principles of Jefferson are the definitions and axioms of free society.

Freedom can be manifested only in the void of beliefs, in the absence of axioms, and only where the laws have no more authority than a hypothesis.

It's one of the old show business axioms. No matter how successful you've been, there's always a younger and sexier seal coming along.

Ethical axioms are found and tested not very differently from the axioms of science. Truth is what stands the test of experience.

The progress of science, like an ancient desert trail, is strewn with the bleached skeletons of discarded theories, doctrines, and axioms which seemed to possess eternal life.

Axioms are delightful in theory, but impossible in practice.

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.

Geometry, like arithmetic, requires for its logical development only a small number of simple, fundamental principles. These fundamental principles are called the axioms of geometry.

What are your Axioms, and Categories, and Systems, and Aphorisms? Words, words.... Be not the slave of Words.

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.

If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book ... it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way.

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?

Though moral axioms to guide the conduct of the practitioner have existed since the beginnings of the profession of healing, Western doctors are most likely to view the Hippocratic Oath of approximately two-and-a-half millennia ago as the first codified set of statements to which they can look for guidance.

The once-surprising existence of non-Euclidean models of Euclid's first four axioms can be seen as a sort of mathematical joke.

We can... treat only the geometrical aspects of mathematics and shall be satisfied in having shown that there is no problem of the truth of geometrical axioms and that no special geometrical visualization exists in mathematics.

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.

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.

Those authors are to be read at schools that supply most axioms of prudence.

My Design in this Book is not to explain the Properties of Light by Hypotheses, but to propose and prove them by Reason and Experiments: In order to which, I shall premise the following Definitions and Axioms.

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.

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.

For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems.

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.

For axioms in philosophy are not axioms until they are proved upon our pulses.

I am not a mathematician, but I was aware that for centuries, mathematics was considered the queen of the sciences because it claimed certainty. It was grounded on some fundamental certainties - axioms - that led to others.

I tell you the solemn truth, that the doctrine of the Trinity is not so difficult to accept for a working proposition as any one of the axioms of physics.

We know that there is no absolute knowledge, that there are only theories; but we forget this. The better educated we are, the harder we believe in axioms.

The eye of understanding is like the eye of the sense; for as you may see great objects through small crannies or levels, so you may see great axioms of nature through small and contemptible instances.

In addition, the teaching of theories from axioms, or some close imitation of them such as the basic laws of an algebra, is usually an educational disaster.

Pointed axioms and acute replies fly loose about the world, and are assigned successively to those whom it may be the fashion to celebrate.

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.

I collect axioms, paradoxes, maxims, teaching stories, proverbs, and aphorisms of all sorts, because I love to see complex ideas distilled into a few words.

A few rules include all that is necessary for the perfection of the definitions, the axioms, and the demonstrations, and consequently of the entire method of the geometrical proofs of the art of persuading.

I think being self-taught is the only way of really learning things, because you have to question everybody. You don't simply absorb axioms or so-called truths like they are rules. You have to test them, and you need to reflect on them, to consider the effect they create. So, you have very significant opinions about things, because they come out from your own reflections.

Finally there are simple ideas of which no definition can be given; there are also axioms or postulates, or in a word primary principles, which cannot be proved and have no need of proof.

Yet Gotama's Dhamma is more than just a series of axioms. It is to be lived rather than simply adopted and believed in. It entails that one embrace this world in all its contingency and specificity, with all its ambiguity and flaws.

Underlying the preaching of the Puritans are three basic axioms: 1. The unique place of preaching is to convert, feed and sustain, 2. The life of the preacher must radiate the reality of what he preaches, 3. Prayer and solid Bible study are basic to effective preaching.

There have been several occasions during the course of Fleetwood Mac over the years where we've had to undermine whatever the business axioms might be to sort of keep aspiring as an artist in the long term, and the 'Tusk' album was one of those times.

..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?

A priori one should expect a chaotic world which cannot be grasped by the mind in any way... The kind of order created by Newton's theory of gravitation...is wholly different. Even if the axioms of the theory are proposed by man, the success of such a project presupposes a high degree of ordering of the objective world.... That is the "miracle" which is being constantly reinforced as our knowledge expands.

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.'

A science is something which is constructed from truth on workable axioms. There are 55 axioms in scientology which are very demonstrably true, and on these can be constructed a great deal.

When Keats says: 'Axioms in philosophy are not axioms until they are proved upon our pulses', what he means is that we don't necessarily believe what a poem is saying if it comes out and tells us in an absolutely head-on, in-your-face way; we only believe it to be true if we feel it to be true.

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.

We should never forget the two axioms: 'Jesus is with me' and whatever happens, happens by the will of God.

A dozen more questions occurred to me. Not to mention twenty-two possible solutions to each one, sixteen resulting hypotheses and counter-theorems, eight abstract speculations, a quadrilateral equation, two axioms, and a limerick. That's raw intelligence for you.

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.

You can "prove" anything on the verbal level, just be accepting the necessary axioms at the beginning.

What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. Roughly speaking, people know that it deals with numbers, figures, with relations, operations, and that its formal procedures involving axioms, proofs, lemmas, theorems have not changed since the time of Archimedes.

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.

We do everything by custom, even believe by it; our very axioms, let us boast of free-thinking as we may, are oftenest simply such beliefs as we have never heard questioned.

Philosophy begins when one learns to doubt -- particularly to doubt one's cherished beliefs, one's dogmas and one's axioms.

Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs.

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.

Applying the axioms of physical science to human life has something reprehensible to it.

Life cannot be captured in a few axioms. And that is just what I keep trying to do. But it won't work, for life is full of endless nuances and cannot be captured in just a few formulae.

If there are such things as political axioms, the propriety of the judicial power of a government being co-extensive with its legislative, may be ranked among the number.

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.