What is the Crisis in Mathematics?

[semel]

Mathematics is meant to be a rigorous deductive discipline based upon
sound principles

but
colin leslie dean showing that godels theorem - what godel did- is invalid
as it is based on invalid axioms


throws maths into crisis
because it now turns out that maths is not based upon sound principles

and ad hoc principles can be used if they apparently give the right
result

take the axiom of reducibility used by godel
it is ad hoc and unjustifiable as the The Stanford Dictionary of
Philosophy

The Stanford Dictionary of Philosophy states that ",

many critics
concluded that the axiom of reducibility was simply too ad hoc to be
justified philosophically."

with this admission and the fact that godel used an ad hoc principle
the foundations of maths have been destroyed for anyone can now use any
ad hoc principle to prove anything
take Fermats last theorem
any one can now create an ad hoc principle which will prove the theorem

colin leslie dean has thrown mathematics into crisis by shattering its
logical foundations
and by showing that truth can be arrived at by any ad hoc avenue
thus showing the myth of mathematics as a rigorous deductive discipline
based upon sound principles


to reiterate Godel does use the axiom of reducibility in his proof of HIS
incompleteness theorem

it is is his axiom 1v
and he uses it in his formula 40

Godel uses the axiom of reducibility axiom 1V of his system is the axiom
of reducibility “As Godel says “this axiom represents the axiom of
reducibility (comprehension axiom of set theory)” (K Godel , On formally
undecidable propositions of principia mathematica and related systems in
The undecidable , M, Davis, Raven Press, 1965,p.12-13)
.
Godel uses axiom 1V the axiom of reducibility in his formula 40 where he
states “x is a formula arising from the axiom schema 1V.1 ((K Godel , On
formally undecidable propositions of principia mathematica and related
systems in The undecidable , M, Davis, Raven Press, 1965,p.21

“ [40. R-Ax(x) ≡ (∃u,v,y,n)[u, v, y, n <= x & n Var v & (n+1) Var u
& u Fr y & Form(y) & x = u ∃x {v Gen [[R(u)*E(R(v))] Aeq y]}]

x is a formula derived from the axiom-schema IV, 1 by substitution “

ramsey says of the axiom

Such an axiom has no place in mathematics, and anything which cannot be
proved without using it cannot be regarded as proved at all.

This axiom there is no reason to suppose true; and if it were true, this
would be a happy accident and not a logical necessity, for it is not a
tautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY

The Stanford Dictionary of Philosophy states that

", many critics
concluded that the axiom of reducibility was simply too ad hoc to be
justified philosophically."

 

[yasiru89]

Has anyone noticed how these supposed 'arguments' are simply verbal? I'd rather stick with Godel's theorem. To be verbal about it, it provides a pleasing absolution in one's study of mathematics!

 

[Gib Z]

Your first few lines tells me that your a crackpot who knows nothing about mathematics. Axioms can not be invalid, by definition.

 

[semel]

you say

Axioms can not be invalid, by definition.

but
euclids 5th axiom is invalid in non-eucluidian geometry

 

[Gib Z]

Axioms only apply in the field that they define and are the foundations of. Perhaps if Euclid knew of non-euclidean geometry he would have been more specific, to say that his axioms only apply in Euclidean geometry.

Euclid's axioms are the basis for Euclidean geometry, and within this geometry there are no contradictions. In non-Euclidean geometry, that axiom is not there. We have shown we can still make a mathematically consistent object though, but that doesn't make Euclidean geometry wrong.

 

[HallsofIvy]

Since the OP is citing "colin leslie dean", I googled that name and found this on "Yahoo Answers" http://answers.yahoo.com/question/index?qid=20070618223613AAouaH9:

"Who is this colin leslie dean.
I see his name all over the net for philosophy erotic poetry science religion literary criticism. I see members post here for views on his books. So anyone now anything about this colin leslie dean 6 months ago."

"What we know about Colin Leslie Dean is that he is a self-promoting wanna-be poet from Australia who posts queries here on YA (using fictitious profiles) about his own non-celebrity."

"Somebody who posts questions all over Y/A in the hope of being noticed. But nobody cares."

I particularly liked this from sci.logic on Yahoo Groups:

"Colin Leslie Dean is the only person I know of who actually has proven
that his OWN words are meaningless.
Dean says that words are meaningless. Yet for a man who believes words to
be meaningless he certainly uses a lot of them.
To make his point that words are meaningless, he commits the fallacy of
the stolen concept. i.e. he relies on the concept that words have meaning
to say that they DON'T have meaning.
Hence we can conclude that Dean's words are in fact meaningless! "

Makes me suspect that "semel" is colin leslie dean.

 

[semel]

first you say

Axioms can not be invalid, by definition.

when proven wrong
you now qualify that
you say now

Axioms only apply in the field that they define and are the foundations of. Perhaps if Euclid knew of non-euclidean geometry he would have been more specific, to say that his axioms only apply in Euclidean geometry.

Euclid's axioms are the basis for Euclidean geometry, and within this geometry there are no contradictions.

now you are backtracking adding qualifications to your iniatal statement

-goal post changing it is called when proven wrong on a point just change the point

 

[semel]

The Stanford Dictionary of Philosophy states that ", many critics concluded that the axiom of reducibility was simply too ad hoc to be justified philosophically.
with this admission and the fact that godel used an ad hoc principle
the foundations of maths have been destroyed
for
any one can now use any ad hoc principle to prove anything
take Fermats last theorem
any one can now create an ad hoc principle which will prove the theorem

colin leslie dean has thrown mathematics into crisis by shattering its logical foundations
and by showing that truth can be arrived at by any ad hoc avenue
thus showing the myth of mathematics as a
rigorous deductive discipline based upon sound principles