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# 1.3 A Mathematical Introduction

The mathematical analysis and modelling of physical situations is a very powerful and vital tool. Mathematical modelling derived from this analysis is an important way of extending the analysis into new areas. However it must always be borne in mind that mathematics is not physics, but a discipline that goes beyond all science and its physical limitations must be recognised and understood.

Firstly because all practical models will have some limits or boundary conditions within which they are confined for reasons of avoiding an excess of complexity. For example one cannot model all the particles in a gas individually but have to think of them collectively in various ways in order to understand the behaviour of gases and the way that they flow. This does not prevent a good understanding of the performance of gases and other fluids but chaos theory will always step in at some point to limit predictions.

Secondly the "mathematical universe" of all possible mathematical expressions is vastly bigger than the physical universe we occupy. This is important if the modelling is an attempt at synthesising an existing system based on some underlying simple principles. This is because the vast mathematical universe will present a vast number of possible solutions without any consideration of practical physical constraints. The mathematical synthesis will produce a vast array of all mathematically possible solutions with very little to point out how to select between them

### The fundamental weakness of mathematical synthesis

If one takes any generalised mathematical model of an interaction process and gives it a great deal of freedom and then attempts to solve the model in a general way the result will almost always be a vast range of possible solutions and unless there are good experimental ways for limiting and selecting the solutions (boundary conditions) it will result in an impasse. This is what I firmly believe is happening with our currently favoured approaches to cosmological theories based on string and other theories aimed at unifying gravitation and quantum mechanics.

The use of Occam's razor and choosing to start with the very simplest versions that will allow the complexity needed to solve any problem is about the only mathematical procedure that can be used to start to run through the vast array of possibilities

It follows that what is needed is some sort of conceptual process or hypothesis based on our physical understanding to create these boundary conditions and it is hoped that this paper may help by going some way to providing one of them.

**Deciding Where To Start**

Although mathematical synthesis in the form of string theory has provided us with too much choice is has come to the conclusions that we need more than the simple three dimensions of space and one of time to come up with solutions that could fit with our universe. Eleven dimensions are the favoured solution although greater numbers have been suggested.

This seems to be at first sight a very large number of spare dimensions however it must be remembered that although we have three dimensions of space one of the important features of many things in our universe are cyclic resonances for example the orbits of planets an stars in galaxies and the resonant orbitals of electrons and quarks these are described in terms of complex numbers as are resonant waves in strings so there are at least six dimensions of particles and strings in space as we know it.

Max Tegmark wrote a paper looking at space time multidimensionality and came to the conclusion there were only two possible places where long lived predictable universes could exist and theres were:-

1 Universes with three (complex) dimensions of space and one (linear) dimension of time

This fits with our own universe as one might expect

2 Universes with three (complex) dimensions of "time" and one (linear) dimension of "space"

This seems rather odd until one considers this represents the later stages of the collapse of a rotating black hole towards the Kerr metric ring singularity where space is becoming s unidirectional time because of frame dragging and time is expanding to become space like

These were largely due to the fact that three dimensional spaces produce long range forces with inverse square law properties and the only situation that allows stable long term complex structures is an inverse square law.

However in the second case all particles were Tachyons, ie travelled faster than light or backwards in time. it is interesting to note that by strict CPT symmetry these could be antiparticles

Also string theories and particle theories have shown some remarkable fits between some special but preferred cases of E8xE8 symmetries and matter antimatter symmetries together with Strict CPT symmetry match evolutionary concepts very well and give a much more positive mathematical foundation to this thinking where there is in effect an interchange between the dimensions of space and time. This also allows an understanding of the "ghost" particles that plagued the mathematicians when they explored these concepts

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Look at 1-1 A Philosophical Introduction 1-2 A Scientific Introduction

on to 2 Our Universe as Observed and Modelled

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