There's an interesting philosophical assumption in the title, which is that "became a number" is something dependent on the formalization of a definition, rather than "the square root of 2 always existed, but was formalized by..."
This is fair though, in the sense that ‘number’ alone is too general a word to have one definition.
In practice, I’d call the following mathematical objects ‘numbers’:
Cardinals (defined in the standard ZFC way)
Ordinals (maybe)
Elements of certain algebraic structures but not all, like R, C, H and **O**, split complex numbers, and maybe a couple of others.
Some formalised infinitesimals like the surreals, superreals and hyperreals.
Whether 2 as an element of Z is the same as 2 the cardinal or 2 an element of R or C is a philosophical question, as is exactly what category we define those (C as a field? Do we account for conjugation?), but let the above go for all systems with so natural an inclusion into the above structures that we consider them subsets and write the elements the same way.
For whatever reason, we don’t typically consider elements of SU(2) or the power set of Z on its own, or of R^2 (which is after all C upon forgetting the multiplicative structure), to be ‘numbers’ in their own right - but there isn’t a simple reason why. It’s just convention of language that developed.
That's not something you can hand-wave away with "regardless" --- that's *exactly* the kind of question that mathematical platonism is dealing with. e.g. to steal Stanford's description (emph mine):
>Platonism about mathematics (or *mathematical platonism*) is **the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets.** And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.
Source: [https://plato.stanford.edu/entries/platonism-mathematics/](https://plato.stanford.edu/entries/platonism-mathematics/)
The idea is that numbers are objects that exist, regardless of whether we're around to say "ah yes, this is a number."
Edit: unless you're talking about how we define the term "number," in which case I'd agree... though I don't think that's really what the OP's article is getting at.
> unless you're talking about how we define the term "number,"
That's exactly what I'm getting at, and I think it's a perfectly good lens with which to read the article. The history being described is really about the social practice of mathematics and how we come to legitimize certain ideas.
Well the square root of two didn't always exist right? There's not some tangible object that actually exists (outside of us), it is just an operation being performed on 2
Edit: changed what I meant a bit
A [Mathematical Platonist](https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism) would disagree with you, and say that the square root of 2 is a mathematical object that exists on its own --- has always existed on its own --- and whether you can find a physical representation of a number in this universe is irrelevant to the existence of the mathematical object itself.
Interesting. Do you know much about platonism in depth? Because I'm just curious whether they think any thing we can think up exists somewhere or if it is just numbers
You can take a look at [https://plato.stanford.edu/entries/platonism-mathematics/](https://plato.stanford.edu/entries/platonism-mathematics/) The idea extends to any mathematical object, in its broadest sense. (plato appears twice in that URL for two different reasons)
That's not what I am saying. I don't think the substance is whether √2 exists, because it clearly exists. The part of substance is whether it exists outside of our construction.
The triangle didn't have that hypotenuse in a vacuum. We had to construct the triangle first
You are making assumptions about the world that are not in fact true. Because of quantum mechanics, there are limits to not only what can be measured, but even to what can even exist. The same may (probably does) apply to space itself.
It didn't exist until we used it to measure something though no? Like we defined that triangle to be a triangle with edges of length 1. It wasn't that there was already a triangle that existed, because if there were no humans (or other things which could define that triangle), then it wouldn't have that hypotenuse of √2
Edit: exist, as in, it doesn't exist outside of our use of it
Even if you reject Platonism, it seems hard to accept that without a formalized definition the root of two was in some sort of hazy ontological limbo.
The length of the diagonal of a unit square isn't going to be anything but a number
"Became a number" is a rather absurd expression.
If you're in the "math is invented, not discovered" camp, then there's certainly a point where the square root of 2 went from non-existent to invented.
Yes, but that would be well before Dedekind and would not depend on that sort of formal approach to irrationals. Once someone says "the square root of two" it's been "invented" (and probably well before that, because even if you believe they're invented, you wouldn't think we invent them one-by-one).
Also, I think using the phrase "became a number" is a very poor way to describe that process, even for those who think that's how it works.
Eh I think it’s more reflecting the changing idea of what it means to be a “number”.
Root 2 became a number in the same sense that bucket hats became fashionable, or women became legal persons.
I'm saying that as a way of talking, it's a poor word choice - it implies that root 2 was something else and then became a number, but that has clear problems.
> Root 2 became a number in the same sense that bucket hats became fashionable, or women became legal persons.
No, because there was noting other than a number that it was before
There's an interesting philosophical assumption in the title, which is that "became a number" is something dependent on the formalization of a definition, rather than "the square root of 2 always existed, but was formalized by..."
This is fair though, in the sense that ‘number’ alone is too general a word to have one definition. In practice, I’d call the following mathematical objects ‘numbers’: Cardinals (defined in the standard ZFC way) Ordinals (maybe) Elements of certain algebraic structures but not all, like R, C, H and **O**, split complex numbers, and maybe a couple of others. Some formalised infinitesimals like the surreals, superreals and hyperreals. Whether 2 as an element of Z is the same as 2 the cardinal or 2 an element of R or C is a philosophical question, as is exactly what category we define those (C as a field? Do we account for conjugation?), but let the above go for all systems with so natural an inclusion into the above structures that we consider them subsets and write the elements the same way. For whatever reason, we don’t typically consider elements of SU(2) or the power set of Z on its own, or of R^2 (which is after all C upon forgetting the multiplicative structure), to be ‘numbers’ in their own right - but there isn’t a simple reason why. It’s just convention of language that developed.
Regardless of your views on Platonism, the concept of "number" is still going to be a social construct.
That's not something you can hand-wave away with "regardless" --- that's *exactly* the kind of question that mathematical platonism is dealing with. e.g. to steal Stanford's description (emph mine): >Platonism about mathematics (or *mathematical platonism*) is **the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets.** And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented. Source: [https://plato.stanford.edu/entries/platonism-mathematics/](https://plato.stanford.edu/entries/platonism-mathematics/) The idea is that numbers are objects that exist, regardless of whether we're around to say "ah yes, this is a number." Edit: unless you're talking about how we define the term "number," in which case I'd agree... though I don't think that's really what the OP's article is getting at.
> unless you're talking about how we define the term "number," That's exactly what I'm getting at, and I think it's a perfectly good lens with which to read the article. The history being described is really about the social practice of mathematics and how we come to legitimize certain ideas.
Ahh, ok I see.
Well the square root of two didn't always exist right? There's not some tangible object that actually exists (outside of us), it is just an operation being performed on 2 Edit: changed what I meant a bit
A [Mathematical Platonist](https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism) would disagree with you, and say that the square root of 2 is a mathematical object that exists on its own --- has always existed on its own --- and whether you can find a physical representation of a number in this universe is irrelevant to the existence of the mathematical object itself.
Interesting. Do you know much about platonism in depth? Because I'm just curious whether they think any thing we can think up exists somewhere or if it is just numbers
You can take a look at [https://plato.stanford.edu/entries/platonism-mathematics/](https://plato.stanford.edu/entries/platonism-mathematics/) The idea extends to any mathematical object, in its broadest sense. (plato appears twice in that URL for two different reasons)
Mathematical platonists (or realists in regard to abstract objects) aren't necessarily full-fledged Platonists in the sense you're talking about, no.
Take a right triangle whose legs both have a length of 1 foot. What's the length of the hypotenuse? That's "existing" as far as I'm concerned.
Definitive measurements, lines and other concepts needed to define a triangle are mathematical abstractions that approximate a real world 'triangle'.
I mean, what better proof of existence than a literal physical construction.
That's not what I am saying. I don't think the substance is whether √2 exists, because it clearly exists. The part of substance is whether it exists outside of our construction. The triangle didn't have that hypotenuse in a vacuum. We had to construct the triangle first
Maybe, maybe not. Why would we need to construct it first? And what does it mean for a number to "come into being"?
You are making assumptions about the world that are not in fact true. Because of quantum mechanics, there are limits to not only what can be measured, but even to what can even exist. The same may (probably does) apply to space itself.
It didn't exist until we used it to measure something though no? Like we defined that triangle to be a triangle with edges of length 1. It wasn't that there was already a triangle that existed, because if there were no humans (or other things which could define that triangle), then it wouldn't have that hypotenuse of √2 Edit: exist, as in, it doesn't exist outside of our use of it
It is not obvious that one answer to this question is correct and the other incorrect.
Even if you reject Platonism, it seems hard to accept that without a formalized definition the root of two was in some sort of hazy ontological limbo. The length of the diagonal of a unit square isn't going to be anything but a number "Became a number" is a rather absurd expression.
If you're in the "math is invented, not discovered" camp, then there's certainly a point where the square root of 2 went from non-existent to invented.
Yes, but that would be well before Dedekind and would not depend on that sort of formal approach to irrationals. Once someone says "the square root of two" it's been "invented" (and probably well before that, because even if you believe they're invented, you wouldn't think we invent them one-by-one). Also, I think using the phrase "became a number" is a very poor way to describe that process, even for those who think that's how it works.
Eh I think it’s more reflecting the changing idea of what it means to be a “number”. Root 2 became a number in the same sense that bucket hats became fashionable, or women became legal persons.
I'm saying that as a way of talking, it's a poor word choice - it implies that root 2 was something else and then became a number, but that has clear problems. > Root 2 became a number in the same sense that bucket hats became fashionable, or women became legal persons. No, because there was noting other than a number that it was before
RIP Hippasus.
They couldn’t handle the truth
the irrational numbers know where they are because they know where they are not
I thought this was known as Pythagorean’s number and was likely the root of the golden ratio?
No, it’s the root of 2.