Views and opinions mathematical...
One of the exciting aspects about creating
a mathematical website is to have a semiformal 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)
