There’s a rule in Russian grammar that every first-grader knows by heart

There’s a rule in Russian grammar that every first-grader knows by heart, “жи-ши пиши через и”. It’s just saying that one should write “и” after “ж” and “ш” even when pronunciation suggests “ы”, and it’s very catchy! This is a rule. Like a lot of grammar rules, it’s in some sense arbitrary, and mostly due to tradition and convention rather anything else. Someone somewhere wrote the Big Book Of Writing Correctly In Russian, and it says that grammatically correct spelling is “жи” and “ши”, even though literally nothing would change or go wrong if we spelled it with “ы” (well, maybe some students would get better grades, and we’d need to reissue a lot of documents, but anyway…)

There’s a rule of “having a cat” that you have to feed your cat. If you don’t feed your cat, then your cat would die. This rule is due to the fact that if cats are not fed, they tend to die. I cannot change this rule by becoming the person who writes the Big Book Of Having A Cat.

A mistake that I see a lot of students is that they think of math “rules” as either the Russian grammar rule, or the feed your cat rule. Often the same student will change how they think about the rule depending on the question asked, and they somehow see math rules as both arbitrary and immutable laws of the universe at the same time! (not their fault, of course, this is pretty much how mathematics is presented in school)

But math “rules” are neither. “You can’t divide by zero” is both arbitrary in the choice of the definitions that we started with and immutable in the consequences of those definitions after the choice has been made. Notice, however, that they are not arbitrary in the choice of consequences, nor immutable in the starting definitions.

But when one says “you can’t divide by zero”, do they know the definitions they are choosing, and more importantly, do they know that the choice of definitions is being made? On some level, sure, I think if properly questioned, most students will have no choice but to come up with a lot of classifiers to hang onto every word of ‘you can’t divide by zero’ to make that statement something approaching correct, because the alternative is “well that’s just the way things are”…and generally students (or at least students I had) are better than that. 

The important point to make here is that there are really no “rules” in math the way people think about math “rules” (like “you cannot divide by zero”, or “a square of something is always positive”). Every single word in math is very precisely defined, that’s what makes it so great. “You cannot divide by zero”, what does “cannot” mean? Does that mean someone from the Bureau Of Doing Math Correctly will scold me? Or does that mean that my math will die because I didn’t feed it? No, that sentence doesn’t mean anything. 

What is usually meant by that is something like “If x is a non-zero real number, then there does not exist a real number y such that x=0*y”. And now we can have a field day. What is a real number? What is a zero? What is multiplication?… Ok, say, we know definitions of those things, good. Can we find some other words to go in those slots so that it is true? if x is a blue cat, can there exist a plaid banana y such that x=(universal velvet teaspoon) ∻y? How the heck do I know? I can always make my choices of italicized words that would make this true. Making a good choice is what math is about, but first and foremost, it is still making that choice.

from Hacker News https://ift.tt/3kmoyBU