Solving Hilbert’s sixth problem (part two of many)
Picking the physical principles
We can now try to pick essential physics principles. Suppose we play God and we need to select the building blocks of reality. To avoid infinite regression (who created God?) we need something which is timeless. Outside space and time, the only things which qualify are mathematical relationships. Euclidean geometry existed well before ancient Greeks, and E=mc^2 was valid before Einstein and before the solar system was formed. The names of the mathematical relationships are just historical accidents.
Fine, but the nature of mathematical relationships is very different than the nature of reality. Sticks and stones may break my bones, but when was the last time you heard that someone was killed by Pythagoras’ theorem? If reality is nothing but mathematical relationships arranged in a way to avoid contradictions, we need to look at the essential differences between mathematics and reality (http://arxiv.org/abs/1001.4586).
One key difference is that of “objective reality”. How can we quantify this? Objective reality means that any two observers can agree on statements about nature. In other words, one can define a universal (non-contextual) notion of truth. In mathematics truth is defined as a consequence of the axioms but in nature truth is defined as the agreement with experiment. Between two incompatible axiomatic systems there is no possible concept of true and false and the same statement can be true in one system, and false in another. Take for example the statement p=”two parallel lines do not intersect”. The same p is true in Euclidean geometry and false in non-Euclidean geometry.
If universal truth is to exist, it implies the possibility to reason consistently and to define probabilities. In a more mundane setting we demand positivity: it is what can define a bit. We take positivity as the first physical principle. We are not specifying what kind of bit we are talking about: classical bit, quantum qbit, current probability density (zbit); only that objective reality (it) can generate information such that any two observers can agree.
There is another key difference between the abstract world of math and the concrete real world. In mathematics there is a disjoined set of mathematical structures and the job of a mathematician is to explore this landscape and find bridges between seemingly isolated areas. Nature on the other hand is uniform and the laws of nature are the same (invariant). There are no island universes in our reality (even if the multiverse may exist, we cannot interact with other pockets of reality with different laws of physics). In mathematics two triangles can be combined to form something else than another triangle, but in nature, the laws of physics for system A and the laws of physics for system B are the same with the laws of physics for system A+B. For example, the Newtonian laws of motion for the Earth, are the same with the Newtonian laws of motion for the Sun, and they do not change when we consider the Earth+Sun system. This may look trivial, but it is an extremely powerful observation and from it we will derive three kinds of dynamics: classical mechanics, quantum mechanics, and another type of mechanics not present in nature (which will show that it violates the positivity condition).
The second physical principle we consider is composition: the laws of nature are invariant under tensor composition.
So if you are God, your requirements for the job are: use timeless mathematical structures as your building blocks, do it in such a way that you create objective reality (ability to define a context independent notion of truth) and make sure that the laws of reality (physics) are invariant. If however you are a physicist wanting to solve Hilbert’s sixth problem, your starting physical principles are: positivity and composition. The idea that reality is made of nothing but of mathematical structures is known as the “mathematical universe hypothesis” but Tegmark’s proposal is done incorrectly: it looks at the similarity between mathematics and reality and proposes computability as the physical principle. The right way is to look at the differences and this leads to composition and positivity. Composition (or composability-which is part of the name of this blog) was initially proposed and explored by Emile Grgin and Aage Petersen, while positivity (the objective reality) as a physical principle was first proposed and explored by the author.
Next time I will start using composability (or the invariance of the laws of nature under tensor composition) to start deriving three (and only three) possible dynamics (two of which being classical and quantum mechanics) in the Hamiltonian formalism.