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TimChan

Three crisises in the math foundations

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Today I've been told the first crisis in the foundations of math, that is about the discovering of irrational number. Then i searched keywords in google to find more about these the second and third crisis, but it seems those searched articles are either not concerned or written in languages i don't know. can anyone help me, plz?

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I have never really heard of "crises" in math, but I haven't been following math history a lot.

I guess Gödel's uncertainty theorem might count as a crisis.

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IIRC, the Pythagoreans believed the length of the diagonal of a unit square to be a rational number. (It isn't; it's the square root of 2, irrational.)

They tried again and again to determine the diagonal's length as a rational number, getting closer but never exact. They hid this as a secret, and supposedly even killed one of their own who revealed this information.

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If the first was the existance of irrational numbers then the second would seem to be infinity. More precisely limits at infinity. It basically caused the same problems with the unit circle that the lack of irrational numbers did with the unit square. The diagonal of the square has to have a length. The circle has to have a circumferance. They could estimate it, but they lacked the concepts to state it.

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Quote:
Original post by LilBudyWizer
They could estimate it, but they lacked the concepts to state it.

These numbers will be never 'computed' otherwise they were not 'irrational' [smile]

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as I recall, they can prove the existance of irrational numbers by proving that there is exactly one number that lies between successive rational number approximations to that irrational number(increasing and decreasing)

anyone else heard of this?

btw thanks for the article, it's interesting..

has anyone ever read the book "what is mathematics?"

http://www.amazon.com/exec/obidos/tg/detail/-/0195105192/qid=1126140746/sr=8-1/ref=pd_bbs_1/102-7151216-7568122?v=glance&s=books&n=507846

it's really great, if anyone is interested..

Tim

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