The role of mathematics in physics
Whether it is the standard model of particle physics, general relativity or string theory, modern physical theory is expressed in mathematical terms. However, to get a theory with physical significance, mathematics alone is not enough, and it needs to be verified by observing nature and the universe.
In the past few centuries, the most important lesson that physicists have learned is that if a theory has internal contradictions, it is wrong, not a problem of the universe. The so-called internal contradiction is the saying that the postulate of theory will lead to contradictions. For example, a physical quantity is defined as probability, and its value is greater than 1, which is obviously wrong.
On the other hand, if the predictions made by a physical theory are completely inconsistent with the observed results, then this theory is also wrong, but this is not what this paper is going to discuss. The point of this paper is, to some extent, what are the laws of nature? Inevitably? Because we can deduce the result from the hypothesis axiom.
Even if you've never heard of it? Epistemology? It is also easy to see that this view is wrong. Are the things we can deduce from axioms the same as the hypothetical axioms? Inevitably? This means that the result is not inevitable. But this idea is wrong, and this is not the interesting part. Interestingly, it is still popular with physicists and science writers, who seem to believe that physics can magically explain themselves.
Where do you get the axioms of physics theory?
According to the existing knowledge, physicists describe the observed phenomena in the most effective way. Of course, once physicists have written down some axioms, anything derived from these axioms can be said to be an inevitable result, which is the requirement of internal consistency.
However, the axiom itself will never be proved to be correct, so the axiom itself will never be inevitable. We can only say that the predictions produced by the axiom are consistent with the observation results. Right? .
This not only means that we may find a different set of axioms, which can better describe the experimental results. More importantly, any statement about the inevitability of natural laws is actually that we can't find a better explanation for the observed phenomena.
Mathematical self-consistency does not mean that the theory is correct.
In fact, this confusion between the inevitability of conclusions given by some axioms and the inevitability of natural laws themselves is not harmless. String theory is a typical example. String theorists firmly believe that they must be on the right track simply because they have successfully constructed a basically consistent mathematical structure. Although the mathematical consistency of string theory is necessary to describe nature correctly, it is not enough. The consistency of mathematical structure does not directly indicate whether the axioms assumed by the theory can describe the observation results well.
A similar situation applies to the theory of loop quantum gravity, and relevant physicists believe that background independence (the geometric structure of space-time is a dynamic quantity, not a static quantity) is a self-evident truth. Similarly, some physicists believe that statistical independence is truth. However, mathematical axioms are not universal truths, which may or may not be useful in reality.
In addition, there is a representative example that physicists misunderstand the role of mathematics in science, that is, the multiverse theory, or the parallel universe theory.
How to choose the correct physical theory?
If a physical theory causes internal contradictions, it means that at least one axiom is wrong. But one way to eliminate internal contradictions is to simply abandon axioms until contradictions disappear.
Abandoning axioms is not a scientifically effective strategy, because the final theory is ambiguous and cannot make effective predictions. But this is a convenient and labor-saving method, which can solve mathematical problems, so it is very popular in physics.
In short, the multiverse theory appears because these theories lack sufficient axioms to describe our universe. However, somehow, more and more physicists have successfully convinced themselves that the multiverse hypothesis is a good scientific theory.
In physics, there are many axioms that are mathematically self-consistent, but they cannot describe our universe completely and accurately. If we want to choose a better theory, physicists must adhere to the principle that these axioms can lead to correct predictions. However, there is no way to prove that a particular set of axioms is necessarily correct because science has its limitations.