Solving Hilbert’s sixth problem (part one of many)
Outside in or inside out?
In 1900 David Hilbert proposed a set of problems to guide mathematics in the 20th century. Among them problem six asks for the axiomatization of physics.
Solving problem six is a huge task and the current consensus is that it is a pseudo-problem but I will attempt to prove otherwise in this and subsequent posts. I will also start formulating the beginning of the answer.
Let’s first try to get a feel for the magnitude of the problem. What does axiomatizing physics mean? Suppose the problem is solved and we have the solution on a piece of paper in front of us. Should we be able to answer any physics question without using experiments? Is the answer supposed to be a Theory of Everything? Let’s pause for a second and reflect on what we just stated: eliminate the need for experiments in physics!!! This is huge.
But what about Gödel Incompleteness Theorem? Because of it mathematics is not axiomatizable and has an infinite landscape. Do the laws of physics have an infinite landscape too?
The biggest roadblock for solving Hilbert’s sixth problem turns out to be Gödel Incompleteness Theorem. Let’s get the gist of it. Start with an antinomy (any antinomy will do): this statement is false. If the statement is true, its content is accurate but its content says that the statement is false. Contradiction. Likewise, if the statement is false, its negation is true, but the negation states that the statement is true. Again we have a contradiction. This was well known a long time before Gödel as the liars’ paradox. But now let’s follow Gödel and replace true and false with provable and unprovable. We get: this statement is unprovable. Suppose the statement is false. Then the statement is provable. Then there exists a proof to a false statement. Therefore the reasoning system is inconsistent. The only way to restore consistency is to have that the statement is true. Hence we just constructed a true but unprovable statement!
Now in a sufficiently powerful axiomatic system Sn suppose we start with axioms: a1, a2, …, a_n (at the minimum Sn must include the natural number arithmetic). Construct a statement P not provable in the axiomatic system (Gödel does this using the diagonal argument). Then we can add P to a1,…,a_n and construct the axiomatic system S_n+1 = a1, a2, …, a_n, P. We can also construct another axiomatic system S’_n+1 = a1, a2, …, a_n, not P. Both S_n+1 and S’_n+1 are consistent systems, but together are incompatible (because P and not P cannot be both true at the same time). The process can be repeated forever, and hence in mathematics there is no “Theory of Everything Mathematical”, no unified axiomatic system, and mathematics has an infinite domain.
So it looks like the goal of axiomatizing physics is hopeless. Mathematics is infinite, and mathematicians seem to be able to keep exploring the mathematical landscape. Since mathematicians are part of nature too, axiomatizing physics seem to demand math axiomatization as well. Case closed, Hilbert sixth problem must be a pseudo-problem, right?
However, it turns out there is another way to do axiomatization. Let’s start by looking at nature. We see that space-time is four dimensional, we see that nature is quantum at core, we see that the Standard Model has definite gauge symmetries. Nature is written in the language of mathematics. But WHY some mathematical structures are preferred by Nature over others? We cannot say that those mathematical structures are unique, all mathematical structures are unique! We can say that some mathematical structures are distinguished.
Solving Hilbert sixth problem demands as a prerequisite finding a mechanism to distinguish a handful of mathematical structures from the infinite world of mathematics.
And in a well known case we know the answer. Consider the special theory of relativity: this is a theory based on a physical principle. Finding essential physical principles is what needs to be done first. Suppose we now have all nature’s physical principles written in front of us. What is the next step? The next step is to use them as filters to select distinguished mathematical structures. If we pick the principles correctly, the accepted mathematical structures will be those and only those which are distinguished by nature as well.
So instead of doing an axiomatization in the traditional way outside in: from axioms derive statements, we use it inside out: from physical principles (axioms) we reject all mathematical structures but a distinguished few. The axioms are like the fence of a domain establishing its boundary. The orientation of the boundary matters. In everyday life, or in engineering this way of using the axioms is well known: they are called requirements. When we buy a car we do not start with the Standard Model Lagrangean to arrive at the make and model we will buy, but we start with requirements: the price range, the acceptable colors, etc. In other words we start with the acceptable features. In special theory of relativity, the relativity principle rejects all Lie groups except the Lorenz and the Galilean group. An additional constant of the speed of light postulate picks the final answer.
Whatever gets selected does not need to be a closed form theory of everything and we bypass the limitation from Gödel incompleteness theorem. Now this program is actually very feasible. Next time I will show how to pick the physical principles, we’ll pick two principles and in subsequent posts I’ll use those principles to derive quantum and classical mechanics step by step in a very rigorous mathematical way (it is rather lengthy to derive quantum mechanics and I don’t know how many posts I’ll need for it). It turns out that quantum and classical mechanics are also theories of nature based on physical principles just like theory of relativity is. The role of the constant of the speed of light postulate will be played in the new case by
theorem. In the process of deriving quantum mechanics we’ll make great progress
towards solving Hilbert’s sixth problem, but we’ll still be far short from a
full “theory of everything”.