I was scrolling through to see if you explain how you typeset the math (notoriously hard on webpages), but I happened to instead find a small mistake. You write: "Elements of $\mathbb{R}^n$ are sets of real numbers." That is an incorrect definition.
- It is inaccurate because n doesn't enter into the definition (and thus all Euclidean spaces would be the same).
- You want ordered n-tuples, not sets. If you actually defined elements of R^n as sets of n real numbers, you would not be able to represent the diagonal (or, if you fix that, not be able to represent the difference between certain points on the diagonal of R^n and certain points on the diagonal of certain subspaces).
Moreover, figure 5 talks about "overlapping vectors". This is highly non-standard, and definitely would need to be defined.
Further, you're setting your readers up for trouble by defining vector arithmetic in terms of bases.
Totally agree with this post...however, to be extremely pedantic (and thus not suitable for the article in question), a tuple in the foundations of mathematics is typically defined as a set. That is, (x,y) := {x,{x,y}}, where the latter is the set containing the element x and the set {x,y}. That is how one goes from axiomatic set theory to define tuples of numbers.
There is a big difference between your type of pedantry versus the OP... I don't think the OP was pedantic. It is simply not a set of real numbers. He did say that it was a "small mistake".
The first is more an error of understanding (for the lack of a better term), whereas talking about tuples as being de facto sets is truly about being pedantic.
Absolutely! In this sense you are absolutely right, but a reader who knows how to construct such tuples (i.e. fill in the blanks in the article) also in all likelihood knows the content of the article. And I bet readers who don't know that already, will not realize that those are the sets the author means.
After stumbling across the author's twitter where he already complains that people are "being mean on HN", and seeing his responses there, I have some serious doubts about whether he is fit to be teaching people mathematics. I honestly applaud him for writing the article, and don't hold the math mistake against him at all. But the incredible defensiveness when confronted with a small mistake is absurd.
Who goes from axiomatic set theory at all, if their goal is not to be a logician/type theorist? One does not need to know axiomatic set theory to study linear algebra any more than one needs to know about semiconductors in order to program in Python.
To be even more pointlessly pedantic: sure, but that doesn't give you "sets of real numbers", in the sense of sets where every element can meaningfully be interpreted as a real number.
Keep in mind those are just notes I made for myself. I decided to put them out there just in case someone found them useful and to signal my skill... By practicing ML/DS in the last couple of years, I came to realize that what I put there is probably more than what is needed to know for applied ML/DS, and conceptual inaccuracies like those have little to none relevance
I mean no offense by saying that, and it's human nature to be wrong.
However, this is not a "conceptual inaccuracy". It's wrong. Straight up, old-fashioned wrong. And don't tell me it's of "little to no relevance" that your definition of R^2 does not distinguish between (0,1) and (1,0).
- It is inaccurate because n doesn't enter into the definition (and thus all Euclidean spaces would be the same).
- You want ordered n-tuples, not sets. If you actually defined elements of R^n as sets of n real numbers, you would not be able to represent the diagonal (or, if you fix that, not be able to represent the difference between certain points on the diagonal of R^n and certain points on the diagonal of certain subspaces).
Moreover, figure 5 talks about "overlapping vectors". This is highly non-standard, and definitely would need to be defined.
Further, you're setting your readers up for trouble by defining vector arithmetic in terms of bases.