Books and publications on the
interaction of systems in real time by A. C. Sturt
Mathematical physics is shown to be quite different from
the algebra of mathematics in some fundamental respects. The difference began
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.
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
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.
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.
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.
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.
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.
equations no longer Universal
but depend on location
dual theories of light
underlying phenomena not understood
advance through measurement not mathematical reasoning
arguments developed geometrically
hypothesis of mass
time –remember the clock
Universe is a system with fundamental laws
these are phenomena not equations
highlights relationship between mathematics and physics
Copyright A. C. Sturt 2005