Mathematics and Nature

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Mathematics and Physical Laws

Are we painting a picture of the universe which is so complex that it can only exist as a mathematical model in the mind of God? Ian Stewart has written a very interesting book called "Does God Play Dice". It describes the progress made in the mathematical fields of Topology and Chaos theory. The history is a fascinating study of the extreme slowness with which man grasps new concepts. He shows how the progress made by Newton in developing the calculus and applying it to classical mechanics is in fact quite limited. The number of cases in nature where it is possible to make accurate predictions are in fact few and far between. Text books only dwell on the successes. While we can write down the equations governing the motion of a planet around its sun, the solution of those equations to give the equation of an elliptical orbit only works while the problem is limited to just one planet and one sun. Add a second planet and any long term solution of the problem is impossible.

Nature does her sums through differential equations. The planet is here, Its got this force on it and its got this velocity. So in a very small amount of time it must have moved the very small distance from here to there and its velocity has also minutely changed. Nature knows only now and the process of getting from one now to the next now. Nature does not need to know that the planet moves in an elliptical orbit and is incapable of telling that it has moved the planet through some perfect mathematical shape.

We can program computers to work in the same way that nature does. To work out changes in position and velocity for billions of small steps in time following the motion of the planet. Now add a second planet to the mathematical model and let the calculations run on for a few years. Brilliant, we can do it. But what happens when a colleague tries to run the same program on his computer. At first there is good agreement, But if I have an Acorn computer and he has a PC, then the results gradually drift apart as the programs are run. Suppose we make the programs more sophisticated, double the number of significant figures we calculate the answers to. Half the length of the time steps. It only delays the inevitable: the results still drift apart.

What we find is that this process of iteration does not solve our differential equations. At best, the mathematics we perform is only homomorphic with the natural process. That is, we are able to say that in this way and that way the results which our mathematical process provides correspond to the natural process. But we do not know that the there is an isomorphism. We do not know that nature will be able to match every imaginative recombination of our equations. These two words homomorphism and isomorphism come from the mathematics of algebraic structures. Isomorphism implies that the systems are identical, while homomorphism implies that one is identical to a part of the other. I have borrowed them and slightly increased their meaning.

Fitting Mathematics to Nature

1,

2,

But the first method to be understood has quite unlikely rules:

3,

The reason for this is that the third one of these corresponds to a natural experience. That of cutting a pie into five slices and eating three of them, then cutting the next pie down the middle and eating half of it. The answer tells me that I have eaten eleven slices of the size I would get by cutting the pies into ten slices each.

The first corresponds to a much simpler physical situation, that of moving around a paved area in search of the slab under which some treasure is buried. This is the mathematics of vector addition. It is hard to understand why mathematicians have been able to handle the addition of fractions for the last four thousand years, while the much simpler vector addition in the first example was discovered only in 1865.

The lesson to be learnt from the way in which the rules for manipulating fraction developed is that mathematics is first and foremost the servant of reality. Of all the ways in which we could combine two pairs of numbers, the one we use for adding fractions is dictated by the experience of the eating of pies and cakes.

Mathematics of Modern Physics

Consider an electric charge. It has associated with it some kind of field which spreads out from it in perfect symmetry. The situation has a geometry in which all points at the same distance from the charge have the same magnitude of the field. At all points in space the direction of the field is parallel to the radius from the charge to the point. At all points in space, the field is directed in the same sense either inward or outward. As we follow the field outwards, it spreads out through space so that at any point, the magnitude will be inversely proportional to the area of the sphere through that point and centred on the charge. In this way nature uses the geometry of the situation to determine the field. We write the equation

Consider now a second charge and somehow anchor it and the first charge in space. It too has a field. Lets call the two fields and . The fields of the two charges exist independently and are both present at every point in space without distorting each other. I call this the principle of superimposition borrowing the name from the mathematics of differential equations and giving it a far more powerful meaning and spelling it with more letters. (Indeed it is this fundamental property of nature which allows its namesake to work in the solution of differential equations by mathematicians.)

Now let us introduce a third charge which is free to move. We notice that it is affected by the presence of the first and second charges and moves with an acceleration revealing that the fields are affecting it. We could calculate this acceleration by calculating and and then adding the two vectors. Multiply this resultant vector by and we obtain the acceleration. Nature is able to perform a vector addition of two electric fields at the locus of a charge which is affected by them.

Here then are the three mechanisms by which nature is able to make things happen. They are GEOMETRY, SUPERIMPOSITION and VECTOR ADDITION. If the universe is a mathematical model, then these are the three processes by which the mathematics is performed throughout space.

It is important to realise that nature is not able to cross multiply equations, make algebraic substitutions or perform feats of differentiation and integration. These are mathematical devises by which we seek to construct mathematical models which are isomorphic to the way in which nature works. The problem is that we do not know when our model is isomorphic and when is is homomorphic. If it is isomorphic, then anything we do in our mathematical model has a direct parallel in nature and vice versa. If on the other hand, it is homomorphic, then we will find an exact parallel in our mathematics to anything which nature can do, but there will be things which we can do in our mathematical model which nature will not be able to parallel. The consequences of this discussion are that we must at all times in our mathematics be aware of the limitations of nature's power to do mathematics. Our mathematics has great limitations because unlike nature we are not able to perform the almost infinite number of calculations corresponding to the action of every charge in the universe on every other charge. Nature performs a summation. We perform an integration to get the same result. The validity lies in understanding the correspondence between the two and not allowing our mathematics to stray into further deductions without establishing the continued correspondence between our calculations and the way in which nature performs her own.

I am uncertain about scalar addition, it all depends on whether or not true scalars exist in nature. Perhaps scalars are only manifestations of vector effects. Mass you may say is an obvious scalar, but then what is mass? A body which we try to accelerate resists us with a force and we invent the concept of mass to explain the inertial force which we experience. Yet all forces are electromagnetic in nature. The reaction between the end of a spring and the mass it is accelerating is one of electrostatic attraction and repulsion between atoms. If we add a bit onto the mass, nature does not add two masses, she adds together the inertial forces which we attribute to mass. Nature in this case is still performing vector addition.

There is also a link between our three mechanisms in that superimposition and vector addition are both geometric in nature. Thus the universe is but one giant geometrical analogue computer running a mathematical model we call reality. I like this, physics, mathematics and metaphysics held together in a single philosophical understanding of, to quote Douglas Adams, "life, the universe and everything".

Notice the similarity with the computer which can only perform binary addition and swap 1s and 0s. Every other mathematical and logical capability is built from these two building blocks.

The mathematical consequence of this relationship between nature and mathematics is enormous. We have discovered the limitations of the solution of differential equations by integration in all but the most simple situations. By using computers to mimic the step by geometric movement of objects, we have discovered chaotic motion. This revolution gives an insight into just how frail our mathematics has been in attempting to understand the secrets of nature. In trying to understand the physics of relativity, electromagnetism and atomic physics, I have had this gut feeling that something is wrong. The mathematics gets more and more complicated and I am left wondering how on earth (or in the heavens) nature is supposed to perform all this mathematics in order to know how she will function.

I was suspicious that errors in our understanding of nature tend to multiply as we base logical deduction upon unsound suppositions. I am now sure that slight errors in our understanding of the nature of mass, charge and the electromagnetic phenomena have lead to a gradual accumulation of errors until we find respectable scientists taking the science fiction writers into undreamt of adventures in alternative quantum universes.