|[Still from the 1956 movie The Ten Commandments]|
No one has any idea why mathematics works so well to describe nature, but it is arguably an empirical fact that it works. A corollary of this is that you can formulate theories in terms of mathematical axioms and derive consequences from this. This is not how theories in physics have historically been developed, but it’s a good way to think about the relation between our theories and mathematics.
All modern theories of physics are formulated in mathematical terms. To have a physically meaningful theory, however, mathematics alone is not sufficient. One also needs to have an identification of mathematical structures with observable properties of the universe.
The maybe most important lesson physicists have learned over the past centuries is that if a theory has internal inconsistencies, it is wrong. By internal inconsistencies, I mean that the theory’s axioms lead to statements that contradict each other. A typical example is that a quantity defined as a probability turns out to take on values larger than 1. That’s mathematical rubbish; something is wrong.
Of course a theory can also be wrong if it makes predictions that simply disagree with observations, but that is not what I am talking about today. Today, I am writing about the nonsense idea that the laws of nature are somehow “inevitable” just because you can derive consequences from postulated axioms.
It is easy to see that this idea is wrong even if you have never heard the word epistemology. Consequences which you can derive from axioms are exactly as “inevitable” as postulating the axioms, which means the consequences are not inevitable. But that this idea is wrong isn’t the interesting part. The interesting part is that it remains popular among physicists and science writers who seem to believe that physics is somehow magically able to explain itself.
But where do we get the axioms for our theories from? We use the ones that, according to present knowledge, do the best job to describe our observations. Sure, once you have written down some axioms, then anything you can derive from these axioms can be said to be an inevitable consequence. This is just the requirement of internal consistency.
But the axioms themselves can never be proved to be the right ones and hence will never be inevitable themselves. You can say they are “right” only to the extent that they give rise to predictions that agree with observations.
This means not only that we may find tomorrow that a different set of axioms describes our observations better. It means more importantly that any statement about the inevitability of the laws of nature is really a statement about our inability to find a better explanation for our observations.
This confusion between the inevitability of conclusions given certain axioms, and the inevitability of the laws of nature themselves, is not an innocuous one. It is the mistake behind string theorists’ conviction that they must be on the right track just because they have managed to create a mostly consistent mathematical structure. That this structure is consistent is of course necessary for it to be a correct description of nature. But it is not sufficient. Consistency tells you nothing whatsoever about whether the axioms you postulated will do a good job to describe observations.
Similar remarks apply to the Followers of Loop Quantum Gravity who hold background independence to be a self-evident truth, or to everybody who believes that statistical independence is sacred scripture, rather than being what it really is: A mathematical axiom, that may or may not continue to be useful.
Another unfortunate consequence of physicists’ misunderstanding of the role of mathematics in science are multiverse theories.
This comes about as follows. If your theory gives rise to internal contradictions, it means that at least one of your axioms is wrong. But one way to remove internal inconsistencies is to simply discard axioms until the contradiction vanishes.
Dropping axioms is not a scientifically fruitful strategy because you then end up with a theory that is ambiguous and hence unpredictive. But it is a convenient, low-effort solution to get rid of mathematical problems and has therefore become fashionable in physics. And this is in a nutshell where multiverse theories come from: These are theories which lack sufficiently many axioms to describe our universe.
Somehow an increasing number of physicists has managed to convince themselves that multiverse ideas are good scientific theories instead of what they de facto are: Useless.
There are infinitely many sets of axioms that are mathematically consistent but do not describe our universe. The only rationale scientists have to choose one over the other is that the axioms give rise to correct predictions. But there is no way to ever prove that a particular set of axioms is inevitably the correct one. Science has its limits. This is one of them.