## Sine function and quantum theory

### March 3, 2007

Some thoughts on the simplicity of the definition of sine function and association with quantum theory

## Readings

### March 3, 2007

I have just finished reading

- Richard Dawkins
*The God Delusion* - Marcus Chown
*The Never-ending Days of being Dead (Dispatches from the Front Line of Science)*

* *

*Dawkins*

It is good to see so much material carefully collected together exposing the absurdities of religion. I find it incomprehensible why the idea of a god should be appealing or helpful. Indeed from the point of view of the exciting mysteries of the existence of the universe the postulation of a god, and the associated faith, leads to a definite dead end because no further investigation can be made.

Of course, when Stephen Hawking used to say, as the understanding of the Big Bang continued to develop, that physicists would soon have a theory of everything, this was just as absurd. At least, though, new developments were made and he could change his mind. Disturbingly there will probably always be scientists who think they have reached the end of the search. It has always been my view that there can never be an end to understanding – as more is comprehended more questions will need to be asked.

Chown’s book gives fascinating insights into some aspects of this.

Dawkins exposes a mechanism as to how humans could have evolved to a state where a religious requirement exists in their brains. Clearly, as for many other evolved conditions, some have it and some don’t. Could it evolve away?

He believes the world would be a better place without it but there are many other conditions, such as nationalism, racism or indeed almost all group levels, that would allow conflict to develop between one group and another.

I suspect that on an island where children are brought up without any mention of god, religion would probably eventually develop within the community and that particular kind of group would exist ready, perhaps, for conflict with another on another island or to split into different parts, with the potential, again, for conflict.

Probably conflict is needed for human development, as in any nonlinear complexity problem one needs the balance between cooperation and conflict for greater complexity to arise.

*Chown*

Many exciting ideas from the edge of science here but I just mention two.

### Incompleteness of Mathematics

There was a crisis in mathematics in the early 20^{th} century.

Consider

**The sentence in bold is not true**

or, just the sentence

I am lying

to see a puzzle.

Godel was able to prove that mathematics must contain things that are true but which cannot be proved. This was a bit devastating for mathematicians.

Just a few years later Turing was able to show that there were things which could never be computed.

Currently, Chaitin has suggested that mathematics should be seen as a series of islands floating on a sea. Each island represent a robust mathematical edifice, such asalgebra, geometry, number theory all rising from a sturdy base of axioms. Each island can be connected by strong strands of reasoning. However surrounding these sturdy islands is an infinite, random collection of truths which cannot be proved and must themselves be axioms. They might form the rocks of other islands but, remember, cannot be proved. The axioms that we are used to are self evident to us because, they are simple and can be related to our experiences of the world. The random ones require creative intuition and may be further from the world but as understanding grows there will others to group with them which in turn will form new robust islands.

This is all rather pleasing and certainly matches in with an equivalent startling randomness in physics, in Quantum Theory.

### Inertial Mass

A mystery from Newtonian Mechanics is the idea of inertial mass. From experiment

Newton found that for a particle

*force = m * acceleration** *

where *m* is just a constant of proportionality which he called the inertial mass and is associated with the particle itself.

It then turns out that this inertial mass is the same as the gravitational mass.

Hawkins has shown certain quantum results concerning behaviour at the edge of a Black Hole. This has led recently to a suggestion that inertial mass is not a property of the particle but a resistance set off by passage through the quantum vacuum.

## Geometric Algebra

### January 2, 2007

Over recent years I became interested in Clifford Algebra and this led me to material by Leo Dorst and Stephen Mann on Geometric Algebra. In the latter case the thrust being the scope for global computational algorithms for geometric objects.

In this weblog I plan to expose my growing understanding of GA in a set of fundamental and not very sophisticated observations.

Clifford Algebra introduction that I have used in teaching.

http://members.aol.com:/stuangsam/geoalg/clifford1/clifford1.html

Geometric Algebra 1

An umbrella for scalar and vector prduct in 2D

http://members.aol.com:/stuangsam/geoalg/ga1/ga1.html

Geometric Algebra 2

3D and Reflection

http://members.aol.com:/stuangsam/geoalg/ga2/ga2.html

Geometric Algebra 3

Planes and their Dual

http://members.aol.com:/stuangsam/geoalg/ga3/ga3.html

Geometric Algebra 4

Rotations