[FOM] Wildberger on Foundations

[joeshipman at aol.com]

Professor Norman Wildberger, of the University of New South Wales, has 
made hundreds of instructional math videos on YouTube. His videos on 
Foundations of Mathematics are very critical of the way mathematics is 
commonly taught (which I agree with, in particular the way High School 
and College classes pass the buck to each other and so don't do 
Calculus rigorously), and also critical of the cavalier postulation of 
infinite sets (which I also agree with, to a point).

However, I think he goes too far in rejecting the standard development 
of analysis as, not only nonrigorous, but incapable of being made 
rigorous. I had a discussion with him in the comments of this video:

http://www.youtube.com/watch?v=Nnp4qZX9H1U

which is entitled "The decline of rigour in modern mathematics".

His introduction reads

***
Rigour means logical validity or accuracy. In this lecture we look at 
this concept in some detail, describe the important role of Euclid's 
Elements, talk about proof, and examine a useful diagram suggesting the 
hierarchy of mathematics. We give some explanation for why rigour has 
declined during the 20th century (there are other reasons too, that we 
will discuss later in this course).

Critical in this picture is the existence of key problematic topics at 
the high school / beginning undergrad level, which form a major 
obstacle to the logical consistent development of mathematics. We list 
some of these topics explicitly, and they will play a major role in 
subsequent videos in this series.

This lecture is part of the MathFoundations series, which tries to lay 
out proper foundations for mathematics, and will not shy away from 
discussing the serious logical difficulties entwined in modern pure 
mathematics. The full playlist is at 
http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all
***

In his discussion with me, he asks for examples of texts where the 
modern framework of Analysis is developed completely rigorously from 
first principles. I tried to discuss the Robertson-Seymour Graph Minor 
Theorem with him in order to pinpoint an example of a theorem which he 
would admit as meaningful but would reject all proofs of, but his 
unfamiliarity with logical concepts like "Second Order Arithmetic" and 
"Primitive Recursive Functions" made this difficult.

Can anyone suggest some source books that might satisfy his request? I 
recommend his videos highly, but want to argue that he goes a little 
too far (not so much in his criticism of mathematics as taught in High 
School and College, but in his criticism of research mathematics as 
professionally practiced).

-- JS