One of my complaints about math writing is that definitions are hardly ever subtractive, even if that’s how people think of them.
For example, a monoid is a group except without inverses. But that’s not how you’ll see it defined. Instead you’ll read that it’s a set with an associative binary operation and an identity element. A module is a vector space, except the scalars come from a ring instead of a field. But the definition from scratch is more than I want to write here. Any time you have a sub-widget or a pre-widget or a semi-widget, it’s probably best to define the widget first.
I understand the logical tidiness of saying what a thing is rather than what it is not. But it makes more pedagogical sense to describe the difference between a new concept and the most similar familiar concept. And the nearest familiar concept may have more structure rather than less.
Suppose you wanted to describe where a smaller city is by giving directions from larger, presumably more well known city, but you could only move east. Then instead of saying Ft. Worth is 30 miles west of Dallas, you’d have to say it’s 1,000 miles east of Phoenix.
Writers don’t have to chose between crisp logic and good pedagogy. They can do both. They can say, for example, that a pre-thingy is a thingy without some property, then say “That is, a pre-thingy satisfies the following axioms: …”
3 thoughts on “It’s like this other thing except …”
“Then instead of saying Ft. Worth is 30 miles west of Dallas, you’d have to say it’s 1,000 miles east of Phoenix.”
No. One says Dallas is 30 miles east of Fort Worth.
I love your blog, in general, but this post made me yell, “Yes!” out loud.
When I first learned about modules, I had already worked with vector spaces for years. Examples like a Z-module over Z aren’t super helpful in separating them from rings.
I asked my professor, “…so, modules seem to be just like vector spaces except they use rings for scalars, right?”
My professor looked at me annoyed and said something to the effect that modules were different and I should re-read their axioms.
Math pedagogy is so often about building up from nothing rather than taking the familiar and stripping it down to its essence.
On a guitar, moving up the neck to a note is not the same as moving down the neck to the same note. The path is encoded in the sound.