**SOURCE:** Barry Mazur, When Is One Thing Equal to Some Other Thing?

Anyone who writes a number in the form **1729** implies a method of calculation: one thousand, plus seven hundreds, plus two tens, plus nine ones.

Different than writing **tick-marks |||||||||||||||||||||||…¹⁷²⁹** which would imply

Different than Roman numerals **MDCCXXIX**,

hexadecimal** 6C1**,

** **

or the most **agnostic** way to write a number, via its **prime factorization** ”the fourth prime ⨯ the sixth prime ⨯ the eighth prime”.

They’re all ways of *calculating* the number, but they’re not the number itself.

**Daphne**

We could agree to call this number some agreed-upon **name**, like ”Daphne” and use a symbol ₯ for shorthand. **1729** is no more her name than is **6C1**.

Or we could refer to Daphne **by property** without implying a particular calculation: “the smallest sum of two cubes, which can be written two different ways”.

Or we could denote Daphne **by equation**:

- ₯ = 9^3 + 10^3, or
- Daphne ₯ is the number that solves the equation 12^3 – ₯ = 1^3.

It’s the same way with **the square root of two**. Its name is no more **√2** than **⨿2** or **¶2**.

Just like **1729**, **√2** is merely a notation. What makes **√2** be **√2** is the *property* it has.

**Numbers aren’t numerals, they’re … uh … ***things*.

*things*.

All of the above is meant to drive a wedge between numbers as written on paper and numbers as they “exist” abstractly.

Numbers don’t need numerals. And **you can talk about numbers without knowing how to write them**. Just agree on some symbol like π and use π whenever you want to talk about the number you don’t know how to write.

It sounds trivial talking about an integer, but the difference between

**properties**of numbers,- ways to
**calculate**numbers, and - the
**numbers**themselves

is good to keep in mind when you’re thinking deeply about

- the real line (measure theory),
- algebraic numbers (Galois theory),
- transcendental numbers,
- p-adic numbers,
- complex numbers or other bodies of numbers.