N J Wildberger


School of Maths UNSW Sydney 2052 Australia


Tel:61 (02) 9385 7098

Fax:61 (02) 9385 7123

Views and opinions mathematical...

One of the exciting aspects about creating a mathematical website is to have a semi-formal opportunity for expressing views on mathematical matters that otherwise one generally keeps to oneself, or only shares with close colleagues. Is it not a rare privilege to be able to speak one's mind, without regard for the established conventions? I sure think it is, and take up the challenge with some delight. Hope you find something in here interesting or thought provoking.

A) Set Theory: Should you believe? Modern mathematics is slack when it comes to thinking clearly about foundational issues. It's time to get serious about some core problems. Set theorists and analysts be warned: you are not going to like it! (pdf version) POSTED APRIL 2006

Responses: There have been many, most think I am nuts. There doesn't seem to be a great willingness though to actually tackle the challenges in my paper. A few supporters and sympathizers, in particular Wolfgang Mueckenheim has some interesting views, check out for example his paper `Physical constraints of numbers'.

B) Numbers, Infinities and Infinitesimals (pdf) This paper outlines a more concrete and less philosophical approach to infinities and infinitesimals. It promotes the idea of thinking of an infinity not as an infinite set, but more simply as the growth rate of a function defined from natural to natural numbers. An infinitesimal then becomes the reciprocal of an infinity, and it is shown how these concepts allows us to recapture the more useful aspects of ordinal arithmetic. They also let us apply nonstandard analysis to everday calculus in an algebraically simple manner. This paper is a follow up to A) Set Theory: Should you believe?

C) The Wrong Trigonometry (pdf) It may be difficult to accept, but the reality is that the way we currently understand trigonometry represents a major misunderstanding of the subject. This paper tries to outline first of all what is wrong with the current theory, namely its ... too complicated. We unquestioningly accept that the right concept to describe the separation of two lines is an angle, and that the trig functions are some kind of natural functions, despite the difficulties in spelling out precisely what is going on here.

D) A Rational Approach to Trigonometry (pdf) Here is a start on how to think about this important subject the right way. Purely algebraically, using quadrance and spread instead of distance and angle, and replacing the usual Pythagoras' theorem, Sine law, Cosine Law and Sum of angles equal 180 degrees formulas with purely rational analogs. Hence no analysis is needed, computations can proceed by hand, and the whole theory extends to general fields.

But most important is that this approach brings a solid, comprehensible understanding of trigonometry within the grasp of an average student. No longer is memorization of arcane formulas necessary. The sophisticated analysis lurking behind the scenes is replaced by high school algebra accessible to all.

E) Evolution versus Intelligent Design: a mathematician's view The debate has caused both sides to overstate their positions. In this paper I show how mathematics might lead one to consider a mild form of intelligent design as quite reasonable. To support the contention that mathematics reveals a remarkable level of structure, richness and beauty in the intrinsic nature of things, I look at some sensational examples: the Stern Brocot tree, Ford circles, and my own slant on these subjects view what I call `wedges' in the plane. Finally we examine some of the evidence for the idea that free will is an illusion. (pdf version in colour)