mathematics

A Brief Introduction to Mathematics

What is measurement?

In the words of Ayn Rand (Introduction to Objectivist Epistemology, pages 7-8):

“Measurement is the identification of a relationship—a quantitative relationship established by means of a standard that serves as a unit. Entities (and their actions) are measured by their attributes (length, weight, velocity, etc.) and the standard of measurement is a concretely specified unit representing the appropriate attribute. Thus, one measures length in inches, feet and miles—weight in pounds—velocity by means of a given distance traversed in a given time, etc.”

Thus, measurement is the process of identifying quantitative relationships. By using objectively defined standards of measurement. Which allows us to measure the attributes and relationships of things in relation to this standard of measurement.

But why do we need to measure anything?

Why do we need to grasp these quantitative relationships?

Because our consciousness is inherently quantitative. It depends on grasping quantitative relationships. So that we can understand the attributes of entities and the relationships between them. Except for the most directly perceivable attributes of something, we cannot perceive these attributes nor the relationships between things.

We can see that something is red just by looking at it. That is a directly observable attribute of something. But if it is moving, I cannot directly observe its velocity. I am going to have to find some way to measure that.

If I place two differently sized apples next to each other, I can directly observe that one is bigger than the other. That is a relationship I can grasp without having to use mathematics or performing any measurements.

But I cannot directly observe the static charge of a balloon. I am going to have to find some other way to be able to perceive that.

atom mathematics

Good luck measuring the attributes of this atom without mathematics …

Just as with the velocity of moving objects, I am going to have to find some method to measure this attribute of the balloon. That is, I am going to have to use mathematics.

We require mathematics in order to extend our consciousness beyond the range of the directly perceivable. So as to allow us to grasp attributes and relationships which are not otherwise possible to grasp.

Measurement is an important cognitive requirement.

Measurement is the process by which we relate that which is not directly perceivable to that which is directly perceivable. We can directly perceive one meter, but we cannot directly perceive the distance from Los Angeles to Mexico. But if we measure the distance from Los Angeles to Mexico in kilometers and then compare that distance to one meter, we can perceive and comprehend that distance.

So, in short, mathematics is a science of method by which we perform measurements, that is, establish quantitative relationships. So that we can perceive attributes and relationships which are not directly perceivable.

This allows us to extend the range of our consciousness beyond that which is directly perceivable through our senses. And thus, perceive any attribute and relationship we can measure mathematically.

You might say that mathematics is the means by which we can understand attribute or relationship by relating it to perceivable units. Mathematics allows us to make the otherwise non-perceivable perceivable!

Given the huge range of attributes and relationships which are impossible or difficult to directly perceive, it is no wonder mathematics is so widely applicable! Examples abound in every field of science and countless other fields of study. In any field of study requiring us to measure.

So, is it really unreasonable that mathematics would be so universally applicable? Not at all. It should be entirely expected that mathematics should be so universally applicable. Because it is a vital cognitive tool. Not the reflection of some superior Platonic realm.

So much for Plato’s theory …

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