Maybe-Mathematical Musings

In various academic contexts, I keep running across variations of the claim that there's no such thing as good or bad writing, just writing that conforms better or worse to arbitrary social standards.

And I understand where this idea came from! AAVE is a perfectly fine dialect of English and there's nothing "wrong" about writing in that dialect rather than Standard American English. And it makes a lot of sense to me to support, like, middle schoolers writing in their native spoken dialect.

But that somehow metastasized within the pedagogy-of-writing community into the idea that there's no such thing as bad writing. And like, I assure you. There is.

Can you point me to any good writings in AAVE?

I don't generally have any on tap (I'm not an AAVE speaker!) but research for that other post pointed me toward "Should Writers Use They Own English". And while I don't think I agree with the central claim, it's definitely (1) good writing (2) that's not in SAE.

In various academic contexts, I keep running across variations of the claim that there's no such thing as good or bad writing, just writing that conforms better or worse to arbitrary social standards.

And I understand where this idea came from! AAVE is a perfectly fine dialect of English and there's nothing "wrong" about writing in that dialect rather than Standard American English. And it makes a lot of sense to me to support, like, middle schoolers writing in their native spoken dialect.

But that somehow metastasized within the pedagogy-of-writing community into the idea that there's no such thing as bad writing. And like, I assure you. There is.

Yeah, you absolutely haven't seen the form this metastasis takes.

I really meant what I said: the claim is "there is no such thing as bad writing". Not "there are no rules you can't break" but "no writing is bad".

Look at e.g. this piece I found randomly on Medium but which is fairly representative. It's skirting the line between the reasonable version and the crazy version. So we have

Which honestly is fair enough, though I think somewhat playing semantic games: "oh, it's not bad writing, it's just ineffective writing!"

But we also get this:

Which is a defensible but pretty aggressive position in terms of, like, what you should teach people and what you shouldn't.

And the thing is, as you read further into the piece and it gets more detail-oriented and practical, it gets more sensible! It's the version that hasn't metastasized.

But you remove that from context, and especially as the ideology filters through faculty who have thought this through a bit less, and you get decontextualized comments reminding everyone that it's bigoted to claim that any writing is "bad", and that judging writing as good or bad is just reenforcing hierarchical power structures. I've gone to talks that argue it's unethical to assign grades to writing as a professor because that puts us in a position of judging our students' writing.

In various academic contexts, I keep running across variations of the claim that there's no such thing as good or bad writing, just writing that conforms better or worse to arbitrary social standards.

And I understand where this idea came from! AAVE is a perfectly fine dialect of English and there's nothing "wrong" about writing in that dialect rather than Standard American English. And it makes a lot of sense to me to support, like, middle schoolers writing in their native spoken dialect.

But that somehow metastasized within the pedagogy-of-writing community into the idea that there's no such thing as bad writing. And like, I assure you. There is.

If you allow middle schoolers to write assignments in a minority dialect, you are permanently locking them out of power and opportunity. If students cannot write in Standard American English, they need to *practice it*

I think there are a number of different goals you could have with a writing assignment, and allowing the use of non-standard dialects is helpful for some of them and not for others.

But even when you are teaching people to write in Standard American English, it can be helpful to acknowledge that you are teaching them a new dialect; their native dialect isn't "wrong", but it's useful to be able to code-switch. My understanding is that's much more effective pedagogy than "correcting" the "bad grammar" of AAVE.

Are quaternions and octonions of the list?

Ooh yeah, I knew I was forgetting at least one thing. Or...two things? Quaternions and octonions probably deserve their own essay each on some level, because each one is relaxing a different key property. But we'll see how things actually work out in the process of writing.

I also could plausibly do a thing on like vector spaces.  Which I don’t think of as numbers but they do generalize them in an important way and highlight some major questions we could ask.

(Also this is all helpful; knowing which things people want to hear about helps me plan!)

matrices feel numberish in some ways, are they? Also like, wtf are tensors maybe?

Hm, that’s an interesting one.

I tend to think of matrices as functions, not as numbers.  But they can be both!  (And in particular most sets of numbers form groups, and then representation theory asks if you can embed those groups in some sort of matrix group.)  Will have to think if I can come up with a good angle on that.  What natural question do matrices answer?

Tensors can really refer to two very different perspectives.  In the physics perspective, tensors are definitely a function and don’t work well as a number.  In the perspective I learned, which is the tensor product as generating a new ring from a collection of other rings, they are a sort of number; but explaining that might anti-enlighten people who are trying to figure out what the fuck a tensor is, because it’s so unrelated to the why they come up normally.

Are quaternions and octonions of the list?

Ooh yeah, I knew I was forgetting at least one thing. Or...two things? Quaternions and octonions probably deserve their own essay each on some level, because each one is relaxing a different key property. But we'll see how things actually work out in the process of writing.

I also could plausibly do a thing on like vector spaces.  Which I don’t think of as numbers but they do generalize them in an important way and highlight some major questions we could ask.

(Also this is all helpful; knowing which things people want to hear about helps me plan!)

Can anyone come up with a function that always sends algebraic numbers to algebraic numbers, but has a local max or min at a transcendental number?  I think there should be one but I can’t come up with one.  

How smooth do you want this function to be?

As g-g says, big dependency on smoothness. We could say f(x)=x if x≤π and f(x)=-x if x>π but that has a pretty major discontinuity.

My instinct is that if f is differentiable and definable in a “normal way” then since it sends algebrac to algebraic, it has to only really have algebraic parameters. In that case, solving f’(x)=0 will give a roughly algebraic solution. So to find something differentiable that does this, it’ll need to be fairly pathological (and might not be constructive?).

Does that example have a local max or min? I guess if you just define it at π at all, that can be a max or a min.  But yeah, I do at least want local continuity at the max or min.

And yeah, your second paragraph is sort of where I’m getting stuck.  The obvious functions that send algebraics to algebraics are, like, polynomials, but then those have polynomial derivatives and so algebraic critical points.  And you can expand those out to various sorts of rational functions, but those should have the same problem, I think.

And conversely I can imagine a function that like is an upside-down vee whose vertex has π as an x-coordinate, but I’m not sure if you can make both sides of that vee be algebraic and still have them meet in the middle?  Like if the left half is y=x, the right half would have to be y = 2π-x or something and that’s not algebraic any more.  But maybe if they have different slopes I can make it work somehow?

Can anyone come up with a function that always sends algebraic numbers to algebraic numbers, but has a local max or min at a transcendental number?  I think there should be one but I can’t come up with one.  

How smooth do you want this function to be?

In an ideal world, I’d be able to describe it in a reasonable way to non-mathematicians, and maybe even show a graph.

But honestly seeing any example at all would make me a lot more comfortable asserting that it exists, which is all I need in the footnote to a digression I’m working on.

Oh and I guess I do want it to be continuous, at least locally around the max or min. (Looking for a counterexample to the extreme value theorem over the algebraic numbers.)

Can anyone come up with a function that always sends algebraic numbers to algebraic numbers, but has a local max or min at a transcendental number?  I think there should be one but I can’t come up with one.  

How smooth do you want this function to be?

In an ideal world, I’d be able to describe it in a reasonable way to non-mathematicians, and maybe even show a graph.

But honestly seeing any example at all would make me a lot more comfortable asserting that it exists, which is all I need in the footnote to a digression I’m working on.

Huh, apparently the Latin alphabet is ultimately descended from Egyptian hieroglyphs. The evolution is Hieroglyphs -> Proto-Sinaitic -> Phoenician -> Euboean Greek -> Etruscan -> Latin. Delightfully, that means we actually know what pictograms the modern letters are derived from. A was an ox's head, for instance, while N was a snake and K was a palm.

one of my favorite facts is that writing has been independently invented like, three times top

I think last time we talked about this, @necarion and I were debating whether it's three or four. I wanna say Egypt, China, Mesopotamia, and Maya? Plus maybe a couple others that might have been independent but died off.

More interesting to me is that the alphabet has only been invented once. Anyone who grew up on civ 1 like I did, it lied to you: writing is a prereq for the alphabet, not the other way around!

What do speakers of other languages call "diagonalizable". I was thinking in class about how it requires english suffix system which AFAIK is not common? Idk, do other languages have equivalents to -ize and -able? Presumably some (German) but not all. This is a hard question to Google

fun fact "diagonalizable" is literally spelled exactly the same in spanish

There’s a trick for this:

Go to the Wikipedia page, then go to the other-language version of that page (see the righthand side of the screenshot above).

This saved my ass in grad school, in a backwards sort of way.

A lot of the papers I had to read for my thesis project were in French. I don't speak French. (Well, I know enough to communicate basic ideas like "I'm an American can you please point me to someone who speaks English", but that doesn't count.)

This wasn't as big a problem as it sounds like, because math French isn't generally hard to read. Most of the nouns are things like "cohomologie galoisienne" (Galois cohomology) or "une extension cyclique de degré p" (a cyclic extension of degree p) or "anneau de valuation discrète" (discrete valuation ring). The French math words are basically the same as the English ones. So I just needed a list of basic verbs and prepositions, and Google Translate could handle that for me.

But at some point I read a comment about the "droite d'Euler". And Google Translate told* me that's the "right of Euler" but that didn't fucking tell me anything. There's no math concept I know of called the "right of Euler" so I was stuck.

And then my sister suggested the Wikipedia trick. So I looked up droite d'Euler on French Wikipedia, changed the language to English, and found that it was something called the Euler line.

So yes. Good tip.

* It tells me the right thing now; but I promise it didn't in 2011.

Me: Oh, hey, here’s a situation that’s likely to come up from time to time in my game. I should dash off a paragraph or two to briefly address it.

Me, a thousand words, six subheadings, two sidebars and a twelve-entry random lookup table later:

This happens to me every time I go to write a math article.

What was going to be a one-paragraph aside in my piece on the replication crisis somehow became a three-post 30k-word series on hypothesis testing.

I mean my actual position, absent annoyance at various Suicide Of The West takes, is that it’s useful to learn to read cursive in some fields, but there’s not much reason to learn to write it beyond immiserating children, which, y’know, if that’s your thing I guess

The argument for learning cursive is that, when you write quickly, you will start joining up letters. Basically everyone does that unless they’re making a real effort not to. So you should learn a way to join up letters that works.

The argument against learning cursive is that the style of cursive we teach in grade school is not good for that. But there are other styles, that work much better!

I re-taught myself cursive a few years ago, in a modern Italic script, and it’s been so worth it.

I mean I don’t personally find myself writing longhand enough for it to matter, but from what I understand about your line of work you might well find your priorities to be different, in which case fair enough.

Yeah so like

If you never write anything by hand, you don't need to be able to write in cursive. But you also don't need to be able to print!

If you basically only write by hand when you're carefully filling out forms, or maybe writing 1-2 word notes to yourself, then cursive doesn't give much advantages.

But most people will write more than that, even if it's just for, like, a grocery list, or notes on a memo during a meeting, or something. And most people will write in a lazy way that takes printed characters and then joins them up. Which is basically a sloppy cursive.

Now, if that's all you're writing, it's probably not worth putting in active practice time as an adult. It took me about 30 hours of work to retrain my handwriting, which isn't a lot but isn't nothing; it's worth it for me because (1) I write on the board a lot and that's way more legible now, and (2) I find it easier to do rough drafts of my writing longhand. But as you say, that doesn't generalize.

On the other hand, this implies that having a good, usable, legible cursive would be mildly useful to a lot of people in an ongoing way. And that doesn't seem like an unreasonable thing to have as part of a grade school curriculum. (Especially since it would be useful in school, when people do in fact do a lot of longhand writing for tests and such.)

The problem is that the cursive we teach people is not good, usable, and legible, so people don't actually derive benefits from it. And then they hate it, quite reasonably.