The Secret Talking of Math

June 14, 2025

First, let me grumble. Then I get humble...

Forget calculations. The secret to Math is to learn the private language of the Guild. It is jargon, through and through, right from the beginning. One must memorize it. It took me, oh, only about seventy years to absorb this truth.

I will keep this short: one example to explain everything.

How many different kinds of numbers are there. Stick with the familiar digits here, 0 through 9. How many kinds of numbers are there?

In the private language of Math there are six, at least. Math gives distinct, different definitions for:

• natural,
• whole,
• integer,
• rational,
• irrational, and
• real numbers.

One secret about Math language is its precision. In the list above, natural numbers come first because they are considered the most fundamental kind. These are the numbers we human beings devised for counting, beginning with 1 and going up: 2, 3, 4, and so forth. Math accepts that kind of number as given by Nature.

Whole numbers come next on the list because they include the natural numbers together with zero. Zero is a number that mathematicians invented.

Integers include the whole numbers plus another invention, the negative whole numbers, −1, −2, −3 and so forth. Rational numbers are an entirely made-up category, numbers you get by putting integers together in a certain way.

It goes on from there. Irrational numbers are all of the other numbers that are not of the rational kind. Real numbers comprise the whole kit and kaboodle.

The different kinds of numbers have certain properties and possess identities. Various rules follow from those identities and properties.

It goes on and on from there. Everything in Math language has been defined by building new ideas on top of previous ideas, like the different kinds of numbers. The new ideas have to pass a special kind of math test, called a proof. All of the proofs have names. Math is like a big family tree of proofs.

The speakers of Math use the names of these things in their writings and conversations. One must memorize them and exercise the memories to become fluent in the private language of Math.

Me? I felt no use for all that stuff. I performed calculations well enough to get good grades in high school Math without learning the language. I failed to see why it mattered.

The deficiency in my education caught up with me in college. Flunked me right out of there, in fact.

If I could talk sense into the youngster I used to be, I would say to him what I tell myself today: take the trouble to learn the language of Math the same way as you would learn any other language, including the one you grew up speaking at home. Memorize the vocabulary and its definitions, the parts of speech (properties, identities), the grammar and sentence structure (rules of operation and methods of proving insights with them). Exercise the language until you dream in it.

Calculations are to the art of Math what the fish are to the art of Fishing; which is to say, they are almost beside the point. We pursue the arts out of desire to delve beneath the surface of the world. We take interest in the nature of the problems, while the knowledge we must acquire to approach them satisfies a certain hunger for competence.