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Mathematicians still have no proper definition of number

And you say this on the basis of what?

Those whom the Gods wish to destroy They first make mad. — Euripides

by Carrie (migeru at eurotrib dot com) on Sat Sep 16th, 2006 at 07:56:20 AM EST
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If you read Bertrand Russell who devoted his life for search of proper definitions of basic terms of science you would know. He could not find them - his definition of number boiled down to so-called paradox of Russell - description of object is called normal when one don't use this object as its element.
Take defintion of number - it is multitude of multitudes which is equivalent to some exemplary multitude (Sorry for bad English, it is in back translation). But one cannot define this exemplary multitude. So it means mathematicians have no definition of number without using tautology.
by FarEasterner on Sat Sep 16th, 2006 at 08:23:49 AM EST
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Bertrand Russell is 100 years out of date in mathematical logic.

I suggest Naive Set Theory by Paul Halmos and On Numbers and Games by John Conway.

Those whom the Gods wish to destroy They first make mad. — Euripides

by Carrie (migeru at eurotrib dot com) on Sat Sep 16th, 2006 at 08:26:58 AM EST
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Surprise - they found something revolutionary?
Like perpetual engine? They could break chains of tautology?
by FarEasterner on Sat Sep 16th, 2006 at 09:13:43 AM EST
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Look, this is like telling me that Set Theory doesn't work and bringing up Frege as an example.

As you know, the internal consistency of a logical system complex enough to contain the arithmetic of natural numbers [without tautology] cannot be proved from within the system. That is, the internal consistency of any theory of the logic of numbers cannot be proved without appeal to external principles.

Russel and Whitehead did their work on the definition of number 30 years before this fact was discoverd by Gödel. Not only Russell's philosophical approach to the logic of Mathematics, but also David Hilbert's were shattered by Gödel, Turing, Post [hey, I can even quote a Russian mathematician here] and others in teh 1930's and 40's.

I don't know what you mean by tautology.

Now, if you think that the fact that mathematics is based on unproven principles is a new insight, I suggest you take a look at Euclid and Aristotle, for whom it was clear already 2000+ years ago that axioms and postulates were accepted without proof and were not ashamed of it.

Those whom the Gods wish to destroy They first make mad. — Euripides

by Carrie (migeru at eurotrib dot com) on Sat Sep 16th, 2006 at 09:25:40 AM EST
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