All 13 Uses of
axiom
in
Zen and the Art of Motorcycle Maintenance
- Lateral truths point to the falseness of axioms and postulates underlying one's existing system of getting at truth.†
Part 2 *
- It had long been sought in vain, he said, to demonstrate the axiom known as Euclid's fifth postulate and this search was the start of the crisis.†
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- All the other axioms seemed so obvious as to be unquestionable, but this one did not.†
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- They did this by reasoning that if there were any way to reduce Euclid's postulate to other, surer axioms, another effect would also be noticeable: a reversal of Euclid's postulate would create logical contradictions in the geometry.†
Part 3
- And he retains besides all Euclid's other axioms.†
Part 3
- Thus by his failure to find any contradictions he proves that the fifth postulate is irreducible to simpler axioms.†
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- A German named Riemann appeared with another unshakable system of geometry which throws overboard not only Euclid's postulate, but also the first axiom, which states that only one straight line can pass through two points.†
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- To solve the problem of what is mathematical truth, Poincaré said, we should first ask ourselves what is the nature of geometric axioms.†
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- Should we therefore conclude that the axioms of geometry are experimental verities?†
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- Poincaré concluded that the axioms of geometry are conventions, our choice among all possible conventions is guided by experimental facts, but it remains free and is limited only by the necessity of avoiding all contradiction.†
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- The axioms of geometry, in other words, are merely disguised definitions.†
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- Then, having identified the nature of geometric axioms, he turned to the question, Is Euclidian geometry true or is Riemann geometry true?†
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- His judgment that the scientist selects facts, hypotheses and axioms on the basis of harmony, also left the rough serrated edge of a puzzle incomplete.†
Part 3
Definition:
something assumed to be self-evident