The foundations of mathematics are flawed. N J Wildberger

[caffeinemachine]

Well this post is "related" two math but I found the chat room to be the most appropriate place for this.

Now I found a lecture series on youtube by a mathematician N J Wildberger.
He claims that the "Foundations of mathematics" are flawed. The area of mathematics he claims need to be revised thoroughly are:

1)Calculus/Analysis (Including the definition of real numbers- be it infinite decimals or dedekind's cuts or others)
2)Geometry/Topology
3)Algebraic Geometry
etc

Here is a link to lecture 94 where he claims that the cauchy sequence definition of reals is not quite correct.
http://www.youtube.com/watch?v=kcirwIwRIUw&feature=relmfu

He starts challenging the foundations of mathematics from lecture 87 onwards.
Guess you guys will find it interesting.

 

[CaptainBlack]

Well this post is "related" two math but I found the chat room to be the most appropriate place for this.

Now I found a lecture series on youtube by a mathematician N J Wildberger.
He claims that the "Foundations of mathematics" are flawed. The area of mathematics he claims need to be revised thoroughly are:

1)Calculus/Analysis (Including the definition of real numbers- be it infinite decimals or dedekind's cuts or others)
2)Geometry/Topology
3)Algebraic Geometry
etc

Here is a link to lecture 94 where he claims that the cauchy sequence definition of reals is not quite correct.
http://www.youtube.com/watch?v=kcirwIwRIUw&feature=relmfu

He starts challenging the foundations of mathematics from lecture 87 onwards.
Guess you guys will find it interesting.

Anyone familiar with stuff I posted in the other place will know that his view and my own are broadly in agreement at least to the extent of wanting to be able see the objects we are talking about, despite others having claimed that no true mathematician has such views.

CB

 

[caffeinemachine]

Anyone familiar with stuff I posted in the other place will know that his view and my own are broadly in agreement at least to the extent of wanting to be able see the objects we are talking about, despite others having claimed that no true mathematician has such views.

CB

He does complain that the real numbers are very complicated objects. But what he is saying is that the construction is logically incorrect. It might not appeal to our intuition, and we may be unhappy that even something as basic as a real number is so conceptually involved, but Wildberger claims that the real number system is "problematic" on logical grounds and has to be rethought completely.

 

[CaptainBlack]

He does complain that the real numbers are very complicated objects. But what he is saying is that the construction is logically incorrect. It might not appeal to our intuition, and we may be unhappy that even something as basic as a real number is so conceptually involved, but Wildberger claims that the real number system is "problematic" on logical grounds and has to be rethought completely.

In that its construction implicitly uses the axiom of choice, which results in the vast majority of reals being non-computable.

Also Wildberger seems reject all completed infinities, which makes him philosophically an intuitive of some ilk (and I think you cannot argue with that position on the basis of logic but only on the grounds that it excludes a lot of interesting maths)

CB

 

[caffeinemachine]

In that its construction implicitly uses the axiom of choice, which results in the vast majority of reals being non-computable.

Also Wildberger seems reject all completed infinities, which makes him philosophically an intuitive of some ilk (and I think you cannot argue with that position on the basis of logic but only on the grounds that it excludes a lot of interesting maths)

CB

But the use of the axiom of choice doesn't mean that the construction logically incorrect. So if that's Wildberger's point then I think I can't agree with him.
What are "completed infinities"?

 

[chisigma]

I know that most of You probably won't agree with me... in my opinion one of the 'chance' to make 'sure' the 'foundations of Mathematics' is to call the so called 'real number' simply number and suppose, as in the case of the set, its definition is 'property of humans' [and also of 'aliens' if they exist, of course...]. Once we have established that natural numbers, integers, rationals, irrationals, etc... are simply subsets of the set of numbers ...

Kind regards

$\chi$ $\sigma$

 

[CaptainBlack]

But the use of the axiom of choice doesn't mean that the construction logically incorrect. So if that's Wildberger's point then I think I can't agree with him.
What are "completed infinities"?

It does if you do not accept arbitrary non-computable choice functions

CB

 

[CaptainBlack]

What are "completed infinities"?

In this context "complete infinities" referes to regarding non-finite structures as things in themselves, in particular to regard the Natural numbers as a thing rather than a process.

You will see this most clearly in induction where some authors will conclude that they have proven the result for all \(x \in \mathbb{N}\) while others conclude that we have proven the result for any natural number \(x\). This is probably not an important distinction, but it does illurstrate the point, and I suspect the difference in wording is often deliberate.

Another example is to regard a line as a thing initself rather than as a segment which may be produced as far as one needs/likes.

CB