I think the story is told as a sort of cultural marker of what a Big F-ing Deal this discovery was/is. Mathematics, collectively, struggled for a long time to find a way to make 3-dimensional numbers into an algebra in a way that extends the algebra of complex numbers. (The cross and dot products are unsatisfying, because they don't have division.) The shock that this can be done in four, -- not three -- dimensions is still sort of reverberating, and that's what I think this story marks. It's a short stand-in for the longer story I've just summarized, and it evokes (or is meant to evoke) the mind-shattering thrill of discovery.
I don't necessarily think the story accomplishes this -- your question is but one piece of evidence that it doesn't -- but I think for those who spend a good amount of time with these kinds of algebra questions, it comes to take on that role, and that's why I think it's repeated.
(Teaser -- if you want to know more about these kinds of questions, Google for "real division algebras". There are not very many, and they way they are organized is not, I think, something one would expect.)
Also, in Hamilton's time they didn't have the view of mathematics as axiom systems that could be played with. Math was a way of finding Truth, so the idea of making up multiplication rules at your convenience probably seemed like an extremely non-obvious move.
The discovery of complex numbers and quaternions probably played a big part in getting people to question what math is, leading to Hilbert's program to study Foundations etc. Hamilton's story is a nice, rare single instance we can point to, symbolizing this discovery.
To find a way to make 3-dimensional numbers into an algebra in a way that extends the algebra of complex numbers
What does that mean? My understanding was that Hamilton was searching for a way to make the manipulation of points in space easier, such as rotation, and noticed that the imaginary part of the complex numbers could be manipulated in the way he wanted. He then created a rather artificial tool in the form of the quaternions that allowed this.
By "extends" here we formally mean "normed division algebra". Basically, we want any 2D slice of our space to be equivalent to the complex numbers. This is analogous to how the reals embed into the complex plane or how slices of vector spaces are still vector spaces. We don't want things to depend on a particular basis (i.e. implementation).
Anyway, it's pretty easy to make up some multiplication on 3D vectors, like multiplying their components. However, in general, it won't play nicely with such arbitrary 2D slices. As it turns out, this slicing property is equivalent to having multiplication play nicely with vector norms:
|ab| = |a| |b|.
that is, multiplication of vectors multiplies their lengths. Getting a multiplication with this property is the hard part, per se, and is only possible in dimensions 1, 2, 4, and 8.
Comments
I think the story is told as a sort of cultural marker of what a Big F-ing Deal this discovery was/is. Mathematics, collectively, struggled for a long time to find a way to make 3-dimensional numbers into an algebra in a way that extends the algebra of complex numbers. (The cross and dot products are unsatisfying, because they don't have division.) The shock that this can be done in four, -- not three -- dimensions is still sort of reverberating, and that's what I think this story marks. It's a short stand-in for the longer story I've just summarized, and it evokes (or is meant to evoke) the mind-shattering thrill of discovery.
I don't necessarily think the story accomplishes this -- your question is but one piece of evidence that it doesn't -- but I think for those who spend a good amount of time with these kinds of algebra questions, it comes to take on that role, and that's why I think it's repeated.
(Teaser -- if you want to know more about these kinds of questions, Google for "real division algebras". There are not very many, and they way they are organized is not, I think, something one would expect.)
Also, in Hamilton's time they didn't have the view of mathematics as axiom systems that could be played with. Math was a way of finding Truth, so the idea of making up multiplication rules at your convenience probably seemed like an extremely non-obvious move.
The discovery of complex numbers and quaternions probably played a big part in getting people to question what math is, leading to Hilbert's program to study Foundations etc. Hamilton's story is a nice, rare single instance we can point to, symbolizing this discovery.
Fair point, but hadn’t complex numbers been around for a while by the time quaternions were discovered?
I think it's more like a meme. Countless other mathematical insights in the same time period have been at least as important and counterintuitive.
What does that mean? My understanding was that Hamilton was searching for a way to make the manipulation of points in space easier, such as rotation, and noticed that the imaginary part of the complex numbers could be manipulated in the way he wanted. He then created a rather artificial tool in the form of the quaternions that allowed this.
By "extends" here we formally mean "normed division algebra". Basically, we want any 2D slice of our space to be equivalent to the complex numbers. This is analogous to how the reals embed into the complex plane or how slices of vector spaces are still vector spaces. We don't want things to depend on a particular basis (i.e. implementation).
Anyway, it's pretty easy to make up some multiplication on 3D vectors, like multiplying their components. However, in general, it won't play nicely with such arbitrary 2D slices. As it turns out, this slicing property is equivalent to having multiplication play nicely with vector norms:
that is, multiplication of vectors multiplies their lengths. Getting a multiplication with this property is the hard part, per se, and is only possible in dimensions 1, 2, 4, and 8.