Lately I have been preoccupied with the rather philosophical question of why we are able to discover physical laws through mathematics; or put another way, why does the universe follow mathematical laws and a consistent line of reasoning. One could also go on to ask if mathematics is a human invention or an independent entity that is being discovered bit by bit by humans
Some say mathematics is a form of language invented for the purpose of formulating and describing the world around us and thus there is no sense in asking why it is so effective. I do not agree with this. Firstly, mathematics is not a language, it is a logic system. Furthermore, what is so interesting and astonishing about the power of math is not that it can describe the world around us, but that it can make predictions. Unlike the other sciences, in physics one can deduce new ideas and principles mathematically. I recommend reading “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” by Eugene Wigner (http://hiperc2.buffalostate.edu/~carbonjo/documents/Wigner.pdf) and its follow-up, “The Unreasonable Effectiveness of Mathematics” by R.W. Hamming (http://nedwww.ipac.caltech.edu/level5/March02/Hamming/Hamming.html) to understand the power of mathematics.
As I was reading different ideas about and nature and relation of mathematics to science, I came across the Mathematical Universe Hypothesis proposed by Max Tegmark. Tegmark’s hypothesis states that our external physical reality is a mathematical structure. Tegmark extends his idea to say every mathematical structure has physical reality and our universe is just one mathematical structure, out of many, that has been complex enough to allow us to evolve in it (anthropic principle). The purpose of this post is not to discuss the details of Tegmark’s ideas (which can be found here:http://arxiv.org/PS_cache/arxiv/pdf/0704/0704.0646v2.pdf) but to discuss my own thoughts in regard to the questions proposed in the beginning of this post. The source and effectiveness of mathematics are no longer puzzling if, as Tegmark says, “our universe is mathematics in a well-defined sense.” After all, the universe follows mathematical logic and physicists follow this line of logic to discover how nature works. Maybe developing mathematics is ultimately a journey of discovery – a unique form of discovery which progresses through creativity in the use of the logic that comes to us from nature.
One of the challenging ideas about this school of thought is the relationship of mathematical equations and their interpretations. I find one good example of this to be the equations of quantum mechanics. Our mathematical model of quantum mechanics seems to accurately describe what happens at the quantum level, and their implications have been consistently proven. Yet, there are many interpretations of quantum mechanics that attempt to give some physical meaning to the mathematics of the theory. If the universe is a purely mathematical construct, then there should not be different candidates of reality arising from a single mathematical theory. If this is true, then surely there is more than mathematics to the reality and truth that physicists attempt to understand. However, I do not believe this to be the case. The important point important point is that each scientific interpretation assumes a different and distinguishable mathematical structure. For example, in quantum mechanics the Many Worlds interpretation introduces a unique mathematical system in which the collapse of the wave function is associated with splitting the universe into mutually unobservable, alternate histories—distinct universes within a greater multiverse.Obviously this is a completely distinct mathematical landscape than the Copenhagen interpretation for instance. Therefore, the existence of many interpretations for the same set of mathematical laws is not because mathematics does not completely define reality, but the fact that we do not have the complete mathematical landscape of reality. In fact, conceptual and non-mathematical ideas are just shortcuts for us to examine possible mathematical structures, which will be proven right or wrong through experimentation or further mathematical reasoning as other laws get discovered. All in all, conceptual interpretations are a starting point to get to the ultimate mathematical reality, not the other way around.