By “video-game” order I mean in an order which makes it increasingly challenging, as opposed to making it increasingly easy because built on more solid foundations.
For instance (as I dimly remember it), calculus was introduced as a collection of rules, of “things to memorize”, rather than worked out from axiomatic principles. It was only later (and as an elective class) that I was introduced to non-standard analysis which provides a rigorous treatment of infinitesimals.
This may be a limitation of mine, but I can only approach math the way I approach coding—I have to know how each layer of abstraction is built atop the underlying one, I’m unable to accept things “on faith” and build upwards from something I don’t understand deeply. I can’t work with expositions that go “now here we need a crucial result that we cannot prove for now, you’ll see the proof next year, but we’re going to use this all through this year”.
Calculus is built on limits, not infinitesimals. At least, that’s how it’s normally defined. They both work, and neither was understood when calculus was discovered.
I think most people are fine using the tools without understanding the rules, and find that easier than learning the rules. Schools are built to teach the way that the majority learns best, as it’s better than teaching the way that the minority learns best.
Can you explain what you mean by this?
By “video-game” order I mean in an order which makes it increasingly challenging, as opposed to making it increasingly easy because built on more solid foundations.
For instance (as I dimly remember it), calculus was introduced as a collection of rules, of “things to memorize”, rather than worked out from axiomatic principles. It was only later (and as an elective class) that I was introduced to non-standard analysis which provides a rigorous treatment of infinitesimals.
This may be a limitation of mine, but I can only approach math the way I approach coding—I have to know how each layer of abstraction is built atop the underlying one, I’m unable to accept things “on faith” and build upwards from something I don’t understand deeply. I can’t work with expositions that go “now here we need a crucial result that we cannot prove for now, you’ll see the proof next year, but we’re going to use this all through this year”.
Calculus is built on limits, not infinitesimals. At least, that’s how it’s normally defined. They both work, and neither was understood when calculus was discovered.
I think most people are fine using the tools without understanding the rules, and find that easier than learning the rules. Schools are built to teach the way that the majority learns best, as it’s better than teaching the way that the minority learns best.