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Physics and Mathematics

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by A. C. Sturt

 

 

 

 

 



Summary

A. Background

B. Mathematics and change

C. The Physics

D. Conclusions

References

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Summary

A. Background

B. Mathematics and change

C. The Physics

D. Conclusions

 

References

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Summary

A. Background

B. Mathematics and change

C. The Physics

D. Conclusions

References

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Summary

A. Background

B. Mathematics and change

C. The Physics

D. Conclusions

 

References

 

 

 

 

Summary

 

Mathematical physics is shown to be quite different from the algebra of mathematics in some fundamental respects. The difference began with Newton in his formulation of the Second Law of Motion and the Law of Universal Gravitation. It cannot be glossed over by the terms “applied” or “pure” mathematics.

 

The analysis considers the development of algebra from its beginning in number. Algebra is an extension of the concept of number as an abstraction. It is this abstraction which gives number the properties which allow us to add, subtract, multiply and divide. This is the basis on which algebra develops the concept of equations and its various operations, the difference between 2 multiplied by 3 which is 6, and 2 apples multiplied by 3 apples which is impossible.

 

Newton ignored all this in his mathematical approach to physics. He based his entire argument on equations containing three variables which are certainly not abstractions: force, length and time-interval, the dimensions of physics. They were chosen because they are different in kind and orthogonal i.e. they vary independently. They are linked by the parameter which he called mass. Thus he made use of all the basic operations of algebra, while ignoring the constraints on which their development depended. He extended it into infinitesimal calculus, the mathematics of change, which was his prime consideration. He used his equations to predict the motion of planets around the Sun i.e. he tested it on the Solar System. It has since been confirmed in countless applications.

 

All subsequent physics derives from the same methodology. The value of the Universal Gravitation Constant was measured in the laboratory. Inverse-square laws were confirmed for electrostatic charges and magnetic poles, reconciling the equations by the device of assigning dimensions to constants, so that they conform to the dimensions of force. Thus all the inverse-square laws for force at a distance were confirmed in static experiments in spite of its definition as mass times acceleration contained in the Second Law.

 

The result of Newton’s methodology is a fundamental difference between mathematical physics and mathematics in the treatment of equations, which has profound consequences for the development of physics. Algebraic equations do not depend on location, because of their quality of abstraction. All algebraic variables are ultimately numbers, and if numbers do not vary in time and space, neither do the algebraic equations which result from them. They are as general as the reasoning which produced them.

 

By contrast, the equations of physics have to be verified by measurements or they are no more than conjecture. The procedure of physics is to formulate a physical model, express the model in the form of an equation, to use the equation to make predictions and to test them by making measurements. If the measurements match the predictions, the physical model is compatible with the real operations of nature in that area. The equation serves to generalise, or physics would have to proceed on a case by case  basis. Computing facilitates the testing of such generalisation.

 

However, the equation is no more general than the physical model which it describes. If the physical model relates to phenomena observed in the locality in which measurements can be made, so does the equation. If there is a broader phenomenon which is giving rise to specific effects in the locality being observed, this cannot be captured by the equation. As the field of observation increases and the underlying phenomenon comes into view, a new physical model is required, and a new equation to describe it. Under these conditions it cannot be assumed that an equation of physics applies to the entire Universe without qualification, simply because an algebraic equation would. This is exactly the situation which is emerging as the scope of physics increases through the use of new technology, with new instruments and new measurements, not only on Earth but out into the Universe and down into the components of the atom. It may turn out that equations developed for the Solar System may not be generally applicable in the enlarged area of interest.

 

In fact this has already happened with Relativity. Relativistic models and equations were developed to account for observed phenomena near the speed of light. It also features in the dual theories of the nature of light. This sort of parallel physics cannot last indefinitely, because it suggests that the real underlying phenomena have not yet been observed or understood. These physical models too will be tested by measurement, overtaken and replaced by the procedure above.

 

The corollary of the argument is that physics cannot advance through ever more sophisticated mathematical reasoning. It will develop as a result of new measurements. It will not necessarily all happen here on Earth. Space technology offers the opportunity to step out outside the Solar System, which may be necessary to detect how ‘local’ our own environment is. The scope for discovery is endless.

 

 

A.     Background

 

It was Newton who linked physics so closely to mathematics. In the very first paragraphs of his Principia he set out the “definitions” (in Latin, of course!) which became the “Laws of Motion” that have guided man to the Moon and back. The paragraphs themselves contain no equations, not least because of the state of algebra at the time, let alone his and Leibniz’s infinitesimal calculus. The Principia develops its arguments verbally with geometric illustrations. The equations which are now used, and even the process of translation, serve to obscure nuances of his thinking and the origin of the assumptions which have come back to haunt later generations in spite of the astonishing success of physics, as of all sciences. And if you need convincing of that success, then to coin another famous phrase in translation, look around you!

 

Physicists of course do not usually see it like that. The equations appear on page 2 of any physics textbook, and their qualifications and assumptions permeate everything that follows. The thought processes which gave rise to them are ignored, perhaps understandably for teaching purposes. But Newton lived in age of alchemy, and many of the phenomena into which his thinking has permeated were not even discovered when he wrote his book. There was no chemistry or electricity, there was no geology or evolution, there were no atoms with their nuclei. Since his time the observed Universe has become immeasurably larger and smaller. Newton quotes the lodestone as a simile for his concept of gravity.

 

His first “Definition” states the hypothesis of mass. It may come as a shock to physicists to hear mass called a hypothesis, because it is now considered the absolute bedrock of analysis, even though Relativity later proposed that it might really be variable. Newton’s reasoning is as follows, using his own examples. If you take four times as much air, you have four times as much matter. The same is true of snow, dust or fine powders, or of “all bodies that are by any causes whatever differently condensed”. He discounts the space between the “parts of such bodies”. These parts constitute the “mass” of the bodies, and that is the basis of the analysis which follows. Masses are additive, as he has shown from his examples. Mass is proportional to weight; he has “checked it by experiments on pendulums, very accurately made”.

 

The analysis continues with parameters which are observations rather than hypotheses. The first parameter is length. You can see length and even touch it. It can be, and has been at various stages of history, a standard set in stone. The second parameter is time. Time can also be observed in the sense of being measured by a clock. The caveat here is not to forget the clock. What you see as the passage of time depends on the sort of clock, witness the havoc caused to early railway timetables by the use of diurnal clocks, and the move to mechanical clocks which it necessitated. Newton was two hundred years before railways. Finally there is the parameter of force. Force is sensed rather than visible, though its agent may be, because you certainly know when someone pushes you or you drop something on your toe. On the basis of one hypothesis and these three direct observations Newton devised what he called Universal laws.

 

There is no doubt at all that there are fundamental laws of physics which apply everywhere at all times. The alternative would be mysticism and anarchy, which does not accord with our observations that the processes of the Universe are regular and repeatable. The Universe is a system. However, it may be that what are now enunciated as laws are not quite as universal as first thought. They may be local manifestations of deeper, more fundamental laws, which is why they may not apply in exactly the same way under all conditions. This may come to light when the growth of physics and science generally causes them to rub up against the wider Universe outside man’s current experience, and it may lead to all sorts of ifs and buts and correction factors, even if only at the margin. All of which is to say that it may be a more complicated Universe than the original interpretations envisage, however successful in their own spheres.

 

This concept is totally foreign to present thinking, and it bears on the relationship between mathematics and physics. To explain it, we need to follow the developments and assumptions inherent in both disciplines from the very beginning.

 

continued


 

 



 


mathematical physics is not algebra



 
development of algebra





Newton’s three variables, the dimensions of physics

applied to Solar System


 
all physics uses his methodology


 

 

 

equations no longer Universal

but depend on location

 






verified by measurement






only as applicable as the physical model

 


need qualification



Relativity

dual theories of light

 

underlying phenomena not understood

 

advance through measurement not mathematical reasoning

 

 

 

 

 

 

Newton’s laws

 

arguments developed geometrically

 

assumptions lost

 

 

 

 

 

 

 

 

 

 

 

 

 

 

hypothesis of mass

 

 

derivation

 

 

 

 

 

 

 

length

 

time –remember the clock

 

 

sorce sensed

 

 

 

 

 

Universe is a system with fundamental laws

 

 

these are phenomena not equations

 

 

highlights relationship between mathematics and physics

 

 

Copyright A. C. Sturt 2005

continued on Page 2

 

 

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