ellauri042.html on line 804: Euclid’s fifth proposition in the first book of his Elements (that the base angles in an isosceles triangle are equal) may have been named the Bridge of Asses (Latin: Pons Asinorum) for medieval students who, clearly not destined to cross over into more abstract mathematics, had difficulty understanding the proof—or even the need for the proof. An alternative name for this famous theorem was Elefuga, which Roger Bacon, writing circa ad 1250, derived from Greek words indicating “escape from misery.” Medieval schoolboys did not usually go beyond the Bridge of Asses, which thus marked their last obstruction before liberation from the Elements.
ellauri071.html on line 403: Kvaternionit jäivät pitkäksi aikaa vektorilaskennan varjoon, ja vasta viime aikoina Hamiltonin elämäntyö on löydetty uudelleen. Nykyisin kvaternionien algebraa käytetään mm. avaruustekniikassa, robotiikassa, ydinfysiikassa ja tietokonegrafikassa. Kvaternionit ovat vektoriavaruuxia yleisempiä struktuureita. Ne eivät ole enää kuntia, vaan normitettuja jakoalbegroja (normed division algebra, en ole varma suomennoksesta). Niin sanottu karvapallolause (hairy ball theorem) sanoo, että ei ole olemassa kolmiuloitteista reaalinormista algebraa.
ellauri096.html on line 186: But the skeptic should not lose his nerve. Proof does not always yield knowledge. Consider a student who correctly guesses that a step in his proof is valid. The student does not know the conclusion but did prove the theorem. His instructor might have trouble getting the student to understand why his answer constitutes a valid proof. The intransigence may stem from the prover’s intelligence rather than his stupidity. L. E. J. Brouwer is best known in mathematics for his brilliant fixed point theorem. But Brouwer regarded his proof as dubious. He had philosophical doubts about the Axiom of Choice and Law of Excluded Middle. Brouwer persuaded a minority of mathematicians and philosophers, known as intuitionists, to emulate his inability to be educated by non-constructive proofs.
ellauri096.html on line 191: Yes, there are infinitely many. Kurt Gödel’s incompleteness theorem demonstrated that any system that is strong enough to express arithmetic is also strong enough to express a formal counterpart of the self-referential proposition in the surprise test example ‘This statement cannot be proved in this system’. If the system cannot prove its “Gödel sentence”, then this sentence is true. If the system can prove its Gödel sentence, the system is inconsistent. So either the system is incomplete or inconsistent. (See the entry on Kurt Gödel.)
ellauri096.html on line 195: J. R. Lucas (1964) claims that this reveals human beings are not machines. A computer is a concrete instantiation of a formal system. Hence, its “knowledge” is restricted to what it can prove. By Gödel’s theorem, the computer will be either inconsistent or incomplete. However, a human being with a full command of arithmetic can be consistent (even if he is actually inconsistent due to inattention or wishful thinking).
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