I have long opposed the idea in mathematics that there exist “real numbers,” e.g., that the square root of 2 is a “real number.” Or that pi is a real number (or any kind of number).

Is pi 3.14 … (where the ellipsis is supposed to be part of the number)? Yes, pi is that, if you recognize that 3.14 is an interval.

The key concept is the standard of precision. Anything less than that is effective zero, or “nill.” The concept of “nill” is the key to everything in mathematics. Nill is all those intervals shorter than the interval that you take as the minimum in your context.

For example, if your minimum for length is 1 mm, then nill is some but any interval shorter than that. E.g., .9 is nill, so is .09, and .000212 and any decimal number smaller than 1 mm.

Epsilon (ε) is often used for a very small amount so I use ε to denote half of the minimum. so we can say “± ε” to cover all existing but dispense-with intervals.

ε is shrinkable. It is only fixed in a given context, being determined by the minimum interval you can use or care about.

Accordingly, all the following are true:

pi = 3.0 ± ε

pi = 3.1 ± ε

pi = 3.14 ± ε

pi = 3.14159 ± ε

and so on.

Pi is defined as the ratio of the circumference of a circle to its diameter. But there must be some standard of precision to calling any concrete shape a circle, and to measuring its diameter.

And that standard of precision varies with your purpose and with the measuring technology available. Bracketing every decimal value of pi with ± ε acknowledges these facts.

So, you may ask, “What is the difference between saying pi is a real number designated by 3.14 … and saying pi is a range of numbers between 3.14 – ε and 3.14 + ε?”

The difference lies in the whole conception of mathematics. The “real number” doctrine claims that there is a number, but we need to get to infinity to find it. My view holds that there is no single number for pi and the irrationals but there is a range of intervals, adjustable by going to a higher resolution.

The first is mystical, the second is hard-headed engineering.

And by denying that pi (or any irrational number) is a number, I preserve the essence of the concept “number”: a fixed sequence of symbols used to measure quantity.

Pi is a number? Then what is the number after pi in the sequence?

That’s unanswerable.

The concept of “pi” is a mathematical device, and it uses numbers. But “pi” is not a concept of a number. Pi is number-like, it is numeric; you might call it and the irrationals “numerands.”

Note that I am not saying that only integers are numbers: 4.2 is a number, and we know what comes next: 4.3. 5/8 is a number, and we know what comes next: 6/8.

“. . .” is not a part of any number. For that matter, ± ε is not part of any number.

Numbers are concepts for counting—counting either discrete entities (4 apples, 5 apples), discrete parts of entities (2 slices of the whole pizza), or the number of times a continuous quantity exceeds a standard quantity laid off against it.

“The table is 18 inches wide” means that the table’s length exceeds 18 inch-marks on the tape measure, but not the 19th; the 19th inch-mark goes past the edge of the table. So the designation is: the table is between 18 and 19 inches wide. If you measure more precisely, nothing changes. “The table is 18 3/16 inches wide” means its width exceeds 18 3/16 but not 18 and 4/16.

(If it seems to be exactly 18 inches wide, that means its length is between 18 minus ε and 18 + ε, where 2ε is the smallest you can see.)

There are no irrational numbers and therefore no real numbers. That doesn’t mean things like pi and the square root of two have no role in measuring quantity. They do.