For quite some time now, I have had a very fundamental question about Nature and how we perceive Her. More specifically, I have had a very fundamental question about the tool we use to investigate Nature. There seems to be only one valid way of understanding how the world works, and that is through mathematics. As Richard Feynman put it,(adsbygoogle = window.adsbygoogle || []).push({});

"If you wish to learn about nature, to appreciate nature, it is necessary to understand the language she speaks in. She offers her information only in one form; we are not so unhumble as to demand that she change before we pay attention" -- The Character of Physical Law (1982).

Indeed, going through your undergraduate curriculum, it is easy to see how one can get the impression that it is impossible to obtain a deep understanding of Nature with anything but mathematics. It seems for just about every physical phenomenon, there is an area of mathematics that describes it. In this sense, it would appear that the physical phenomenon and the mathematics come “packaged” together.

My question is this:

We use mathematics to learn more about the universe we live in, but where does the mathematics itself come from? Did it come as an inherent characteristic of our world that we discover through investigation, much like how the physical laws of Nature are discovered through experimentation? Or is mathematics invented, completely abstract from the fundamental characteristics of Nature, created by humans as we need it to describe a physical phenomenon? This is a question about mathematics itself, and is aimed primarily at mathematicians who have a deep understanding of where mathematics comes from.

People often credit Newton (and/or Leibniz) for “inventing calculus”. Did Newton invent it, or discover it as he investigated the quality of Nature in question? Similarly, there are mathematicians and mathematical physicists today who are developing new areas of mathematics to tackle unsolved problems. Are they creating it out of thin air or discovering it as they investigate these unsolved qualities of Nature?

Here is my stance on the matter as well as my arguments:

Mathematics is completely abstract from the fundamental characteristics of our universe. It is created/invented by humans through logical arguments and operations in order to make sense of what we perceive our universe to be. Thus, because we are human and prone to error, we create mathematics that incorrectly describes or fails to describe a phenomenon. Therefore, all physical theories based upon mathematics are only approximations of Nature based on what we perceive to be true through experimentation and theory. Furthermore, there are areas of mathematics that do not describe any physical system at all. Linear algebra, for example, does not itself describe a physical system, but rather provides a tool that allows us to solve systems of equations that do describe a physical system. Additionally, there are mathematical operations that do not have (or are not known to have) any physical meaning, such as a fractional or real-numbered derivative (e.g. f^{2/3}(x) or f^{PI}(x), as opposed to the integer derivatives (e.g. f^{2}(x)) we use for velocity, acceleration, potential gradients, etc.

If mathematics were an inherent quality of Nature to be discovered, then all mathematical knowledge we obtain through investigation should correctly describe Nature without error (here, as in all science, we assume Nature only presents herself the way she is, and would not deceive us through any false manifestation or by changing herself in either space or time). Additionally, if mathematics were an inherent characteristic of Nature, then mathematics itself should be unique to natural systems and not describe any artificial systems. This is not the case. Financial markets, economic theory, artificial intelligence and other computerized systems (all human constructs) are not natural systems, yet they use much of the same mathematics we use to describe physical phenomena.

Thus, in conclusion, I would have to disagree with Mr. Feynman (even though he is my favorite scientist and all-time hero) and say that mathematics is not the language of Nature, nor is it the only way she offers her information. Mathematics is simply the only way that we humans, in our limited capacity, know how to understand Nature.

So, is mathematics invented or discovered? What are your thoughts?

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# Food for thought

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