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


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 timeinterval, 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. Inversesquare 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
inversesquare 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
