A Quote by Michael Frayn
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.
Related Quotes
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.
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.
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.
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.
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.
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.
Reasoned arguments and suggestions which make allowance for the full difficulties of the state of war that exists may help, and will always be listened to with respect and sympathy.
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.
Men propound mathematical theorems in besieged cities, conduct metaphysical arguments in condemned cells, make jokes on the scaffold, discuss a new poem while advancing to the walls of Quebec, and comb their hair at Thermopylae. This is not panache; it is our nature.
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.
I have heard it remarked that men are not to be reasoned out of an opinion they have not reasoned themselves into.
If you were not reasoned into your beliefs, you cannot be reasoned out of them.
That can never be reasoned down which was not reasoned up.
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 confronted with two courses of action I jot down on a piece of paper all the arguments in favor of each one, then on the opposite side I write the arguments against each one. Then by weighing the arguments pro and con and cancelling them out, one against the other, I take the course indicated by what remains.
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