Discovering Nature's Language: Mathematics

[BrowncoatsRule]

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,

"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. f²(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?

 

[Dadface]

I think one aspect of maths is that it is a convenient way of summarising the observations we are able to make.

 

[BrowncoatsRule]

I think one aspect of maths is that it is a convenient way of summarising the observations we are able to make.

That it is indeed :) Statistics and probability are built around the idea of summarising observations and predicting future events.

 

[phinds]

That it is indeed :) Statistics and probability are built around the idea of summarising observations and predicting future events.

While completely true, that seems to me to miss the point of "math described nature". That is, forget for a moment about statistics and think about exactitude. If you want to EXACTLY describe how an object falls in a gravitational field, you use an exact equation. This to me does quite a bit more than "summarize" things, it EXACTLY DESCRIBED them. And it IS the language of nature in that very real sense.

 

[Dadface]

While completely true, that seems to me to miss the point of "math described nature". That is, forget for a moment about statistics and think about exactitude. If you want to EXACTLY describe how an object falls in a gravitational field, you use an exact equation. This to me does quite a bit more than "summarize" things, it EXACTLY DESCRIBED them. And it IS the language of nature in that very real sense.

A summary does not imply that answers are not at the observed level of exactitude. Equations of motion, for example, may assume uniform acceleration and in the domain of applicability within which they can be succesfully applied they produce answers which are a good match to the observations, along with the uncertainties of those observations.

 

[Borek]

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

Sorry to be so blunt, but my thought is that you should search the forum before asking. This particular question - is mathematics invented, or discovered - was asked on many occasions, and there are several threads with good sources listed, as the question is not new.

This thread is from 2004: https://www.physicsforums.com/showthread.php?t=25704, check the "Similar discussion" pane at the bottom of the page.

 

[BrowncoatsRule]

While completely true, that seems to me to miss the point of "math described nature". That is, forget for a moment about statistics and think about exactitude. If you want to EXACTLY describe how an object falls in a gravitational field, you use an exact equation. This to me does quite a bit more than "summarize" things, it EXACTLY DESCRIBED them. And it IS the language of nature in that very real sense.

Bear in mind that an exact equation accounts for all possible factors. In your example, how an object falls in a gravitational field, the equation we use for this motion only accounts for four factors: the initial height in some reference frame, as well as initial velocity, the strength of the gravitational acceleration (and whether you treat it as a constant or not depending on how far the object falls in gravitational field), and of course time. In our mathematical model of a falling object, those four factors are the only ones that determine the motion of the object. While this is mathematically exact on paper, this only approximates the true physical path of the falling object. This is because it is impossible to account for the infinite number subtle complexities and variations that also affect the motion of the object, such as variations in the flow, temperature, or density of the medium in which the object is falling through, as well as other factors that we throw into the word "friction".

If you were to take an apple and drop it from your desk in exactly the same fashion for, say, 10 times, each time the apple fell it would do so in a slightly different way than it did in the previous trial. This is because the arrangement of the air molecules around the apple are slightly different every time, and may also have slightly different temperatures and thus different kinetic energies, which can affect how much momentum each air molecule transfers to the apple as it falls. Or even how the molecule was spinning as it hit the apple. All of these things are impossible for us to account for, and even though their net effect on the motion of the object is often considered negligible, they prevent us from being able to right an exact equation. Thus, we say our simplified equation approximately describes how the object will fall. Therefore, we use mathematics to attempt to describe nature by creating a simplified model of it, where the uncertainties of the model are satisfactorily small. In this sense, mathematics is not the language of nature but rather the language of whatever we want it to be (I could write an equation for the thought process of an imaginary dragon in a parallel universe where I invent the physics if I wanted to -- clearly not nature), and how precise we want to be with our measurements will always be limited, and therefore never exact.

 

[phinds]

Yes, I am of course aware of all that and agree with it, but I think you are engaging in an exercise that would be described in the military as "pissing up a rope".

 

[Redbelly98]

Thread locked pending moderation.