A Brief Introduction to Mathematics

What is mathematics?

Mathematics is something which many people consider strange and mysterious. They consider it to be some arcane mystery. Or some incomprehensibly mysterious thing with “unreasonable effectiveness”.

It is true that mathematics can be an incredibly powerful and useful tool. With it, we can perform many feats of measurement. Feats which would be impossible if we simply use a ruler or the like.

We can use mathematics to identify relationships and make integrations which might otherwise be impossible.

But mathematics is not, despite what people such as Plato might think, the mysterious reflection of some mysterious and superior realm.

The reason mathematics seems to be so widely applicable is not because reality is a pale shadow of mathematics. It is because math by its nature applies to virtually everything.

What then is it?

In the words of Ayn Rand:

“Mathematics is a science of method (the science of measurement, i.e., of establishing quantitative relationships), a cognitive method that enables man to perform an unlimited series of integrations.”

Let us examine this definition and see what this tells us about mathematics.

Mathematics is a science of method. What does this mean? It means that mathematics is the study of abstract methods. It is a series of abstractions which we use for a purpose. The purpose of performing measurements.

This is why mathematics is so fundamentally abstract.

Because that is what it needs to be. Mathematics is a series of abstractions which allows us to perform measurements. Which are either impossible or impractical to make via more direct methods, such as using a ruler.

Mathematics, however abstract, describes something in the real world.

Perhaps an example would serve to make this clearer. Let us take the example of an empty Coke bottle. Suppose that I know nothing about this bottle. It has no label or anything else that tells me anything about it.

I want to store jelly beans in the bottle. How many jelly beans can I fit into the bottle?

Now, obviously, this is an attribute of the bottle that I cannot directly observe. This is not a directly observable attribute like the colour of the bottle.

In fact, it is an abstract quality of the bottle. It describes a relationship between the bottle and whatever I might be trying to fit into the bottle.

To do this I need to find the volume of the bottle. This volume is, of course, an abstraction. It allows me to estimate “how much of something” I can fit into the bottle.

I can use mathematics to figure out the volume of this bottle. Integral calculus might be quite useful. But in any case, I can get a very accurate approximation of the volume of this bottle.

How? Well, mathematics is a science of method.

That is, it is a rigorous study of logically validated steps we can take in order to achieve a purpose. In this case, our purpose is to measure things.

Note that these steps have to be validated to actually perform the measurements they purport to. Otherwise, they are not part of the science of mathematics.

We will not go into the issue of how to validate these methods. Except to say that they must be shown to be logically valid. For this, one of two conditions must be met:

  1. The method is in concordance with what we can directly observe.
  2. The method is a logical consequence of what we can directly observe. It might not follow directly from observable facts, but it must be possible to backtrack through a series of logic to show that it logically follows from observable fact.
Sometimes a math proof is nice and simple and easy to visualize like this. Other times … it involves lots of abstract steps.

To reiterate, mathematics is a science of method. That is, it is a study of method for the purposes of measurement.

Why can’t we just perform the measurements and skip all the math?

Have you tried measuring the volume of a Coke bottle? It is not something you can just pick up a ruler and measure. You could pour liquid into the bottle, millilitre by millilitre. Until you fill the bottle and get a good measure of its volume that way.

But what if you cannot do this? Or, just don’t want to? That sounds like it could be a bit impractical!

What if you want to find the distance from Earth to another galaxy?

No amount of liquid or direct observation is going to help you do that! So, then, how are you going to measure that distance?

This is where mathematics comes to the rescue. It allows you to measure things which cannot be directly measured. We can directly observe the colour of something.

But we cannot directly observe the volume of something. Nor can we directly observe its velocity if it is moving.

Mathematics allows us to extend the range of our perception beyond that which is directly observable. It allows us to perceive attributes and relationships, which cannot be directly observed, via the process of measurement.

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