Mathematics: The Language of the Universe

Nature of Reality:
Quasi-Realism
Representational Realism
Existence and Natures
Knowledge of Reality
The Language of Reality

Toolbelt of Knowledge: Concepts
Algorithms
Equivalence
Emergence
Math
The Anthropic Principle
Substrate-Independence
Significance

In our discussions about the nature of reality, we have come to the view that reality is a thing unto itself, independent of perception, belief, or knowledge. Anything we perceive or think we know about reality is not reality itself, but only a representation we have constructed in our minds. A representation is true to the degree that its logic matches with the logic of the real thing it is describing. Today, we are going to talk about that logic, mathematics.

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A lot of people see math as something mysterious that they will never understand. But math is not supernatural. It is not hidden knowledge available only to an elite few. People who know math are not wizards or prophets, they are normal people just like you. I hope that after reading this discussion, you will be convinced that you can learn math too, if you so desire.

To start, let’s forget about numbers and just think about something physical, like air pressure. We know from centuries of experiments that, the pressure in a given volume is proportional to the number of molecules in the volume and the temperature. This may sound complicated, but all it means is if more air is added or the temperature is increased, the pressure increases.

Let’s look at the italicized statement. We have four physical quantities: pressure, volume, number of molecules, and temperature. Let’s shorten each of these to just their first letters: P, V, N, and T. “Is proportional to” means if you change what comes after it, then what comes before it changes by the same percentage. We can represent this by an equals sign and a constant, the letter k. Put this together, and we have,

It’s an equation! We have just done something marvelous; we have taken a fact about reality and written it as a mathematical statement. By doing this, we realize a profound truth: math is not just a tool to work with numbers and get answers to homework problems; it’s a language and a writing system. By becoming math-literate, we break into a higher level of understanding the universe.

Let’s try it again. This time we’ll start with an equation, and figure out what it means.

The first thing we need when trying to read this equation is what the letters mean. In normal languages, letters have mostly the same sounds wherever they appear. In math, it is not so; we must be told what each letter means every time. It is the organization, operations, and numbers that have consistent meaning. So here is what the letters in our new equation mean: capital T stands for temperature, small t stands for time, and k is a constant.

What operations does this equation have? The first thing we notice is d/d. This means, the rate at which the thing on top changes as the thing on the bottom changes. So for us, it would be the rate the temperature changes over time. Next, we notice a triangle before the T on the right. This triangle means the difference between two of what comes after it. So in our case, ΔT means the difference between the temperatures of two objects.

Putting all this together, we can read the equation. It says, “The rate at which temperature flows between two touching objects is proportional to the difference in temperature between the two objects.” This means if two touching objects have very different temperatures, heat will flow quickly between them, but if their temperatures are near each other, the heat will flow slowly.

There is one final piece to the equation, and that is the minus sign. This tells us that the temperatures are changing closer to one another, not running away to extremes. This makes sense. Cold things heat up when they touch hot things, and hot things cool down when they touch cold things. Heat always flows toward equilibrium.

The ability to read equations is only one small part of math. There is also geometry, group theory, set theory, vectors, tensors, and much more. All of these fields of study are called the same thing, math, so what do they all have in common? The answer is that mathematics is the set of all well-defined abstract ideas that follow the principle of non-contradiction. To create math, we must declare one or more axioms, statements that define an imaginary object.

Let’s take an example. "A circle is a shape where every point on its boundary is the same distance from its center." Based on this axiom, we can figure out all kinds of things about lines drawn through circles, intersecting circles, circles in curved space, and more. Everything in math is like this; we start with axioms, and then use logic on them to figure out all that we can about them.

Philosophers and scientists have often wondered at how well math is able to describe the universe. To some, it seems miraculous. However, based on everything we have talked about in the Nature of Reality series, I think it makes perfect sense. Here’s why:

1) A representation is true to the degree that its logic lines up with the logic of the part of reality it is meant to represent.
2) An idea is a representation.
3) Reality is well-defined and always follows the principle of non-contradiction.
4) Every idea that is well-defined and follows non-contradiction is mathematical.
Therefore, everything in reality can be truthfully represented by mathematical ideas.

If we accept the views of reality we have argued for on this blog, this is why Mathematics is the language of the universe.