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Topic: An Unheralded Breakthrough: The Rosetta Stone of Mathematics
Replies: 17   Last Post: May 26, 2013 4:37 PM

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 byron Posts: 891 Registered: 3/3/09
Re: An Unheralded Breakthrough: The Rosetta Stone of Mathematics
Posted: May 25, 2013 10:36 AM

how interesting but rubbish
because
All talk about number theory and geometry being related is meaningless as mathematician dont even know what a number is- without circularity
all their definitions about numbers reduce to just this

a number is a number-circularity impredicative

thus we then dont know what a number is

mathematicians give all these proofs about numbers but they dont even know what a number is so their proofs are worthless
as without knowing what a number is they then cant even IDENTIFY what a number is

Australias lead erotic poet colin leslie dean points out Mathematicians cannot define a number with out being impredicative-ie self referential thus mathematicians dont even know what a number is- thus maths is meaningless All mathematicians can say is a number is a number ?thus they don?t know what a number is thus maths is meaningless

http://www.iep.utm.edu/predicat/
http://www.iep.utm.edu/predicat/

In many approaches to the foundations of mathematics, the property N
of being a natural number is defined as follows. An object x has the
property N just in case x has every property F which is had by zero
and is inherited from any number u to its successor u+1. Or in
symbols:
Def-N N(x) ? ?F[F(0) ? ?u(F(u) ? F(u + 1)) ? F(x)]
This definition has the nice feature of entailing the principle of
mathematical induction, which says that any property F which is had by
zero and is inherited from any number u to its successor u+1 is had by
every natural number:
?F{F(0) ? ?u(F(u) ? F(u + 1)) ? ?x(N(x) ? F(x))}
However, Def-N is impredicative because it defines the property N by
generalizing over all arithmetical properties, including the one being
defined.

again impredicative definition
Let n be smallest natural number such that every natural number can be
written as the sum of at most four cubes.
again impredicative definition

http://en.wikipedia.org/wiki/Impredicativity
Concerning mathematics, an example of an impredicative definition is
the smallest number in a set, which is formally defined as: y = min(X)
if and only if for all elements x of X, y is less than or equal to x,
and y is in X.

http://en.wikipedia.org/wiki/Set-theore ... al_numbers
http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

A consequence of Kurt Gödel's work on incompleteness is that in any effectively generated axiomatization of number theory (ie. one containing minimal arithmetic), there will be true statements of number theory which cannot be proven in that system. So trivially it follows that ZFC or any other effectively generated formal system cannot capture entirely what a number is.

Whether this is a problem or not depends on whether you were seeking a formal definition of the concept of number. For people such as Bertrand Russell (who thought number theory, and hence mathematics, was a branch of logic and number was something to be defined in terms of formal logic) it was an insurmountable problem. But if you take the concept of number as an absolutely fundamental and irreducible one, it is to be expected. After all, if any concept is to be left formally undefined in mathematics, it might as well be one which everyone understands.

Poincaré, amongst others (Bernays, Wittgenstein), held that any attempt to define natural number as it is endeavoured to do so above is doomed to failure by circularity. Informally, Gödel's theorem shows that a formal axiomatic definition is impossible (incompleteness), Poincaré claims that no definition, formal or informal, is possible (circularity). As such, they give two separate reasons why purported definitions of number must fail to define number. A quote from Poincaré: "The definitions of number are very numerous and of great variety, and I will not attempt to enumerate their names and their authors. We must not be surprised that there are so many. If any of them were satisfactory we should not get any new ones." A quote from Wittgenstein: "This is not a definition. This is nothing but the arithmetical calculus with frills tacked on." A quote from Bernays: "Thus in spite of the possibility of incorporating arithmetic into logistic, arithmetic constitutes the more abstract ('purer') schema; and this appears paradoxical only because of a traditional, but on closer examination unjustified view according to which logical generality is in every respect the highest generality."