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,
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.
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.
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.