Many of the most general physical laws are very simple, from Einstein's law of General Relativity to Schrodinger's Equation in quantum mechanics. Why aren't are most basic physical laws incredibly complex?
we can see that a large proportion of physical equations are extremely simple this is because we have developed these for our understanding of all these units and values. Many phenomena can be described by relatively simple, and often linear, equations well enough for most practical purposes.
But, take Newton's 2nd Law, often written F = ma (the vector force is a scalar, the mass, times a vector acceleration) which is very tidy. That's good enough for a lot of purposes, but a more complete statement is that F = DP/DT (the force is the first derivative of momentum w.r.t time). And then if the magnitude of p is very large the we need to take into account effects described by special relativity and the equation becomes much more complex
Firstly, let me concur that GR is very simple. Yes, the maths is complex in the sense of few people can follow it, but the fact is that people can follow it. That means it is very simple in the space of possible theories.
QM is not quite so simple; without getting into it, there is a lot more controversy/ dissatisfaction around Copenhagen etc.
For my money, Quantum Field Theory is hard to call simple, but again it is understood by some people so is 'simple' in some sense.
One might re-express:
Why do we have so much success describing physical phenomena with "simple" laws? (A)
How can a Universe be constructed by simple laws yet be able to generate a complex enough subsystem (people) that can understand it? (B)
Well, one line of thought is that simple processes (think chaos theory, universal computation, universal complexity) can generate highly complex behaviours. Cf. Turing's halting problem or Wolfram showing how he could generate huge complexity with extremely simple systems.
So, given a sufficiently large universe with simple laws one it is reasonable to meet the demands of (B).
Of course the universe does have to be 'large', in two senses:
- Large enough to contain a sufficiently complex subsystem
- Much larger still so as to have some chance of forming such a subsystem
I suspect there is a relationship between having a 'large universe' and a 'simple' one. Think 'Inflation'; feels like that would be harder to achieve in a complex system.
Although we do have a reasonably simple universe at some level, let's be clear that we do not have a complete model of the universe. The reason for that might well be that such a model is too complex for us to have found yet or ever.
So, let me be clear, I suggest that (B) allows (A) without implying that (A) is complete.
Why are physical laws so simple?
Because the underlying physics is simple. Some might say general relativity isn't simple, and nor is quantum mechanics. But underlying them I think there is a simplicity. I also think William Kingdon Clifford nailed it in his 1870 space theory of matter. Clifford said this:
“I hold in fact:
1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.
2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.
3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.
4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity”.
Clifford died of tuberculosis at the age of 33. If he had not, I suspect the world would be a different place today. A better place.
The Physical Laws are not that simple as it appears. In most of the situation physicists ‘idealize’ the bodies and environs, such that the theoretical picture becomes simple to grasp.
An example is “Perfectly Rigid Bodies” which are rare to be found in nature.
Or Perfectly elastic bodies, frictionless surfaces, and a host of other proposition which goes towards idealization.
An Ideal Gas obeying Gas Laws.
The above is due to an approach in which the basic understanding is acquired in an ideal environ and then “real World” situations can be dealt with using approximation methods or perturbations.
Here let us investigate the structure of physical laws in brief.
A physical theory say T consists, among other things, of a group of laws which are formulated in terms of certain concepts.
But an apparent circularity arises when one considers how the laws of T and the concepts acquire their content, because each seems to acquire content from the other — the laws of T acquire their content from the concepts used in the formulation of the laws, while the concepts are often “introduced” or “defined” by the group of laws as a whole.
To be sure, if the concepts can be introduced independently of the theory T, the circularity does not appear.
But typically every physical theory T requires some new concepts which cannot be defined without using T (we call the latter “T-theoretical concepts”). Is the apparent circularity concerning the laws and the T-theoretical concepts a problem?
As an example, consider the theory T of classical particle mechanics.
For simplicity, we will assume that kinematical concepts, such as the positions of particles, their velocities and accelerations are given independently of the theory as functions of time.
A central statement of T is Newton's second law, F=ma, which asserts that the sum F of the forces exerted upon a particle equals its mass m multiplied by its acceleration a.
While we customarily think of F=ma as an empirical assertion, there is a real risk that it turns out merely to be a definition or largely conventional in character.
If we think of a force merely as “that which generates acceleration” then the force F is actually defined by the equation F=ma.
We have a particle undergoing some given acceleration a, then F=ma just defines what F is. The law is not an empirically testable assertation at all since a force so defined cannot fail to satisfy F=ma.
The problem gets worse if we define the (inertial) mass m in the usual manner as the ratio |F|/|a|.
For now, we are using the one equation F=ma to define two quantities F and m. A given acceleration a at best specifies the ratio F/m but does not specify unique values for F and m individually.
In more formal terms, the problem arises because we introduced force F and mass m as T-theoretical terms that are not given by other theories.
That fact also supplies an escape from the problem.
We can add extra laws to the simple dynamics. For example, we might require that all forces are gravitational and that the net force on the mass m be given by the sum F=Σi F i
of all gravitational forces Fi acting on the mass due to the other masses of the universe, in accord with Newton's inverse square law of gravity. (The law asserts that the force Fi due to attracting mass i with gravitational mass mgi is Gmgmgi ri / ri3,
where mg is the gravitational mass of the original body, ri the position vector of mass i originating from the original body, and G the universal constant of gravitation.)
That gives us an independent definition for F. Similarly we can require that the inertial mass m be equal to the gravitational mass mg. Since we now have independent access to each of the terms F, m and a appearing in F=ma, whether the law obtains is contingent and no longer a matter of definition.
Further problems can arise, however, because of another T-theoretical term that is invoked implicitly when F=ma is asserted. The accelerations a are tacitly assumed to be measured in relation to an inertial system. If the acceleration is measured in relation to a different reference system, a different result is obtained.
For example, if it is measured in relation to a system moving with uniform acceleration A, then the measured acceleration will be a′ = (a − A). A body not acted on by gravitational forces in an inertial frame will obey 0=ma so that a=0. The same body in the accelerated frame will have acceleration a′ = −A and be governed by −mA = ma′. The problem is that the term −mA behaves just like a gravitational force; its magnitude is directly proportional to the mass m of the body.
So the case of a gravitation free body in a uniformly accelerated reference system is indistinguishable from a body in free fall in a homogeneous gravitational field.
A theoretical under determination threatens once again. Given just the motions how are we to know which case is presented to us?
Resolving these problems requires a systematic study of the relations between the various T-theoretical concepts, inertial mass, gravitational mass, inertial force, gravitational force, inertial systems, and accelerated systems and how they figure in the relevant laws of the theory T. Similar problems arise in the formulation of almost all fundamental physical theories.
Another aspect is the rôle of reduction within the global picture of the development of physics. Most physicists, but not all, tend to view their science as an enterprise which accumulates knowledge in a continuous manner.
For example, they would not say that classical mechanics has been disproved by relativistic mechanics, but that relativistic mechanics has partly clarified where classical mechanics could be safely applied and where not.
This view of the development of physics has been challenged by some philosophers and historians of science, especially by the writings of T. Kuhn and P. Feyerabend.
These scholars emphasize the conceptual discontinuity or “incommensurability” between reduced theory T and reducing theory T′.
The structuralistic accounts of reduction now open the possibility of discussing these matters on a less informal level.