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.
Quotes about axioms (16 Quotes)
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.
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.
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.
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.
After I had addressed myself to this very difficult and almost insoluble problem, the suggestion at length came to me how it could be solved . . . if some assumptions (which are called axioms) were granted me.
... There can be no doubt about faith and not reason being the ultima ratio. Even Euclid, who has laid himself as little open to the charge of credulity as any writer who ever lived, cannot get beyond this. He has no demonstrable first premise. He requires postulates and axioms which transcend demonstration, and without which he can do nothing. His superstructure indeed is demonstration, but his ground his faith. Nor again can he get further than telling a man he is a fool if he persists in differing from him. He says 'which is absurd,' and declines to discuss the matter further. Faith and authority, therefore, prove to be as necessary for him as for anyone else.
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.
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.
The law embodies the story of a nations development through many centuries, and it cannot be dealt with as if it contained only the axioms and corollaries of a book of mathematics.
Knowledge is convertible into power, and axioms into rules of utility and duty. But knowledge itself is not Power. Wisdom is Power and her Prime Minister is Justice, which is the perfected law of Truth.
And of all axioms this shall win the prize, Tis better to be fortunate than wise.
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. Everybody who looks at the spectacle of modern algebra will be struck by this complementarity of freedom and necessity.
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.
Axioms in philosophy are not axioms until they are proved upon our pulses we read fine things but never feel them to the full until we have gone the same steps as the author.
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.