Axioms Quotes

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.

As a leader... I have always endeavored to listen to what each and every person in a discussion had to say before venturing my own opinion. Oftentimes, my own opinion will simply represent a consensus of what I heard in the discussion. I always remember the axiom; a leader is like a shepherd. He stays behind the flock, letting the most nimble go out ahead, whereupon the others follow, not realizing that all along they are being directed from behind.

If nature leads us to mathematical forms of great simplicity and beauty - by forms I am referring to coherent systems of hypothesis, 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 relationships which nature suddenly spreads out before us and for which none of us was in the least prepared.

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

An Italian proverb says, In men every mortal sin is venial; in woman every venial sin is mortal. And a German axiom, that There are only two good women in the world: one of them is dead, and the other is not to be found.

You want at least 5% of the population being serious. That five, 6% of the population carries the rest of the people. You've heard that old axiom: 5% of the people pull the wagon; 95% are in it.

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.

Fanon calls his ideology a new humanism, not only in contrast to the elite humanism of the West, but also on the axiom that the wretched of the earth, understood socially, think and thus must be a basis of a new politics. This, of course, is not achieved immediately, but it must become an explicit element of the struggle for liberation.

My morality, the morality of reason, is contained in a single axiom: existence exists - and in a single choice: to live.

It should be remembered, as an axiom of eternal truth in politics, that whatever power in any government is independent, is absolute also.

In a conquered country benevolence is not humanitarianism. It is a general political axiom that a conqueror must not inspire a good opinion of his benevolence until he has demonstrated that he can be severe with malefactors.

Summation of Leviathan: "The axiom, fear; the method, logic; the conclusion, despotism."

Natura nihil agit frustra [Nature does nothing in vain] is the only indisputible axiom in philosophy. There are no grotesques in nature; not any thing framed to fill up empty cantons, and unncecessary spaces.

Be willing to dream, and imagine yourself becoming all that you wish to be. Keep in mind the basic axiom -- all that now exists was once imagined. It follows then that what you want to exist for you in the future must now be imagined.

The fundamental axiom, then, for the study of man is the existence of individual consciousness

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.

Many successful people are no more talented than unsuccessful people. The difference between them lies in the old axiom that successful people do those things that unsuccessful people don't like to do.

Axioms are delightful in theory, but impossible in practice.

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.

To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.

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

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

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

We read fine things but never feel them to the full until we have gone the same steps as the author.

It is practically an axiom in psychiatry that precocious intellect combined with physical weakness can give rise to many unpleasant character traits - avarice, delusions of grandeur , and obsessive masturbation, to name just a few.

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.

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.

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 is accepted as an axiom by all Americans that the civil power ought to be not only neutral and impartial as between different forms of faith, but ought to leave these matters entirely on one side, regarding them no more than it regards the artistic or literary pursuits of the citizens.

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.

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.

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.

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.

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

We may lay it down as an incontestible axiom, that, in all the operations of art and nature, nothing is created; an equal quantity of matter exists both before and after the experiment; the quality and quantity of the elements remain precisely the same; and nothing takes place beyond changes and modifications in the combination of these elements. Upon this principle the whole art of performing chemical experiments depends: We must always suppose an exact equality between the elements of the body examined and those of the products of its analysis.

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.

There is no more fundamental axiom of American freedom than the familiar statement: In a free country we punish men for the crimes they commit but never for the opinions they have.

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.

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.

A crucial difference between lite libertarians and the right kind is that to the former, the idea of liberty is propositional - a deracinated principle, unmoored from the realities of history, hierarchy, biology, tradition, culture, values. Conversely, the paleolibertarian grasps that ordered liberty has a civilizational dimension, stripped of which the libertarian non-aggression axiom, by which we all must live, cannot endure.

I've made movies that were adaptations and I've been kind of frustrated by the process because, you know that old axiom, 'It's never as good as the book'? It's often true because nothing competes with your own imagination. When you're reading a book and you imagine something in your head, nothing's going to compete with that.

..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?

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.

Having a moment of clarity was one thing; I'd had moments like that before. It had to be followed with a dedicated push of daily exercise. It's a trite axiom, but practice DOES make perfect. If you want to be a strong swimmer or an accomplished musician, you have to practice. It's the same with sobriety, though the stakes are higher. If you don't practice your program every day, you're putting yourself in a position where you could fly out of the orbit one more time.

What government supports, government controls. This is an ancient axiom repeatedly ratified by experience. . . . Indeed, as is well known, acceptance of tax aid has led to the secularization of many church-related colleges and universities.

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?

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.

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.

We're taught Lord Acton's axiom: all power corrupts, absolute power corrupts absolutely. I believed that when I started these books, but I don't believe it's always true any more. Power doesn't always corrupt. Power can cleanse. What I believe is always true about power is that power always reveals.

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.