I don't get why calculus is always brought as an example.
It wasn't particularly hard, the entire class had to learn it in high school. We all had it (in a slightly harder shape) in every university course (no matter how detached from what we actually needed)
I forgot all my calculus after high school, had to relearn it in uni and then I promptly forgot again.
Exactly as the article says, it was more about proving we had the capacity (proxy iq test?) to learn it.
You don't need calculus in real life and I think the focus on calculus is ridiculous when we could explore other more practical areas, like category theory (which only my lucky friends who did advanced math got to play with)
But what is the general application of category theory, outside computer science.. and even there the average programmer who hasn't some type theory experience will stare at you with huge eyes when you mention it..
I love the wikipedia intro: Calculus is the mathematical study of continous change, the same way geometry is the study of shape and algebra... that's it perfectly. And the most basic application is in everyone's life and also one of the basic physics thing: The relationship between location, speed and acceleration. I find this very essential, vs category theory at least..
Category theory is about connecting the dots between different areas of maths. The "general application" is to allow you to reason over the structure of a problem you're interested in, while throwing away all the superfluous details. It arose when geometers and topologists realised they were working on the same problems, dressed up in different ways. I think the utility for technical people, from this perspective, is pretty clear.
As for the general working person? I think it's just an exercise in learning to do abstractions correctly, which is valuable in any line of work.
There are actually people who advocate that we should base maths education on category theory much earlier (much as New Math was interested in teaching set theory early on, as a foundational topic). CT is an unreasonably effective tool in a large section of pure maths, so this doesn't sound unreasonable to me; it wouldn't be nearly so scary if it were introduced gently much earlier on (in the same way we start to learn about things like induction in the UK in secondary school, long before formalities like ordinals are introduced at uni). Currently only a very specific, highly-specialised section of the population learn CT, but if something like this were to happen, I'm sure we'd see lots of benefits which are hard to identify at the moment.
Don't need to convince me ;), but I mean this is for the very average person who argues like "why do I need more math than adding two numbers"... and even just "allow you to reason over the structure of a problem you're interested in, while throwing away all the superfluous details" and "learning to do abstractions correctly" would not seem more approachable to them than the calculus description I gave above? imo. Is there a simple real-life problem or model everyone should know or has touched in school this can relate to? I'd be curious, because as you frame it I may even need to revisit my faded memory, hm.
I honestly don't know. This might be a skill issue on my part (very much not an educator), but I think of it as a language for thinking about structural abstraction, so to me the question is akin to "is there a real-life problem that German relates to?"; I can certainly think of lots of problems that would be made much much easier by understanding the language (e.g., getting around Germany, i.e. noticing abstractions), but it's tough to point to anything for this explicit question other than "conversing with someone in German".
I guess to try and mirror your calculus example, I'd try and motivate why someone should care about abstraction itself, perhaps with examples like 'calculating my taxes each year is exactly the same problem, except the raw numbers have changed'.
Alternatively it might go over better to say something like: "Imagine you have a map with a bunch of points, and paths which you can walk between them. CT is the study of the paths themselves, the impact of walking down them in various routes: for mathematicians, this means looking at things like turning sentences such as 'think of a number, add 4 to it then divide by 2 then add 6 then subtract 1' into 'think of a number and add 7'. Once you've spotted this shortcut on this silly toy map, you'll recognise the same paths and the same shortcut when you see on your tax form 'take your income, add £400 to it, divide by 2, add £600 and subtract £100"
I've only read pieces of it, but I think this moves in the right direction towards making category theory useful to day-to-day life in non-trivial ways.
I have had plenty of math classes both in high school and later at university but I don't recall any significant distinction that would leave me with some concrete idea of "algebra" vs. "calculus" vs. "whatever" years later.
In the US we'd reserve "mathematical analysis" (or more specifically, Real analysis) to the college level classes which involve writing proofs about the continuity of functions between sets of real numbers. You'd probably end up with a lecture on the mean value theorem here, and leave with the ability to prove it, among other things
"Calculus" is the application of that theory without argument. It's an advanced high school class or an early college one. There you'll integrate or differentiate real valued functions for use in optimisation problems or for determining qualitative features of such a function (e.g. where is it flat, where is it defined, etc).
In the US, you can probably pass calculus without writing a proof, but you can't pass mathematical analysis without at least understanding epsilon/delta proofs.
I'm pretty sure we did proofs in high school. But that was a while ago, don't know what they do now.
Hey now that I think of it, "mathematical analysis" had continuity, limits, some integrals. And then every mathematically inclined uni specialization had "integral and differential calculations (let's shorten it to calculus)" which was more advanced use of integrals :)
A rose by any other name would smell as sweet, but it may be called a thorn in a different locale.
> In the US, you can probably pass calculus without writing a proof, but you can't pass mathematical analysis without at least understanding epsilon/delta proofs.
I forgot all my calculus after high school, had to relearn it in uni and then I promptly forgot again.
Exactly as the article says, it was more about proving we had the capacity (proxy iq test?) to learn it.
You don't need calculus in real life and I think the focus on calculus is ridiculous when we could explore other more practical areas, like category theory (which only my lucky friends who did advanced math got to play with)