I’ll be blunt, my friends. I think I’ve discovered the limits of my mathematical abilities.
There are three things I’ll explain in this post. First is why I feel this way, second is how I’ve incorporated what I’ve learned into my worldview, and third is what I’m planning to do from here on out.
What lead me to this conclusion? Well, the course I was taking this semester could be fairly summarized as “normal calculus.” At my uni, there are three main 100 level courses, each two semesters long: ‘Easy’ calculus, aimed towards folks majoring outside the sciences, ‘normal’ calculus, for both neophytes and those aiming for a career in math and sciences, and honors calculus, which is the advanced course for people who really want to get into advanced math. I took the normal course–it’s almost entirely centered around the concept of derivatives and by extension, anti-derivatives. The first thing we covered was the idea of limits–that a certain function (equation, pretty much) can approach a certain number but never reach it. For instance, let’s say we have an equation that’s 1/(x-1) squared. So we can replace x with nearly any number–5, -3, 2.573, whatever, and get something. However, when we replace X with 1, we get an error, because 1-1 is zero, zero squared is still zero, and you can’t divide by zero. Now, when we replace X with numbers VERY CLOSE to 1, like 1.000001, or .999999, the equation gives us very big numbers, like 1000 or 100000 (and since it’s x-1 squared, the answer will always be positive). So we say the limit “as x approaches 1” is positive infinity, because as X goes closer to 1, you can essentially get an infinitely huge answer in the equation, but you can never reach exactly one because you’ll get 1-1 squared which is zero squared which is zero.
That stuff wasn’t so bad simply because the math wasn’t too hard even if the concept was hard to grasp–just stuff like subtracting .9999 from 1, and then using a calculator to get a result. But then came derivatives, and that was tougher.
I don’t wanna get into it too much, but just a basic overview: When you’re asked to “differentiate” or “find the derivative” of a function, you just have to apply some rules to it to get a new function. For instance, if you want to derive the function 5 times x, you apply the constant rule, which just says that anytime you see a number with x next to it, you just remove the x and that’s your answer. So again, if our function is f(x) and its derivative (what we want to find) is f'(x), it’d go like this:
f(x) = 5x
f'(x) = 5
There we go! Easy-peasy, right? Nothing to it. The problem is, there are tons of other derivative rules, and they get really hard to memorize. For instance, the quotient rule, when you want to find the derivative of a function that’s being divided by another function, is SUPER complex. Like, let’s have our first function be f(x) and our second function be g(x). So if a question asks us to “find the derivative of f(x)/g(x),” we have to do this really complicated mess–for some reason, the answer is going to be g(x) times f’x – f(x) times g'(x), and all that is divided by g(x) squared! Pretty weird, right? We have to take the derivatives of the little functions to get the right derivative for the function as a whole. It’s not *too* hard when it’s simple functions like the constant one I described above, but there are also other rules like power rules (for things like 5x squared), logarithmic and trigonometry rules, and more! It gets pretty hard to handle when you have to use all those rules *simultaneously*, and that’s what you’ll have to do with more advanced questions. Needless to say, antidifferentiation, which involves pretty much reversing the rules I described above, is even more complex in its own way!
You have to memorize all those rules to succeed in calc–there’s just no way around it. I coulda cheated if I wanted–the webcam software they ask us to use is pretty easy to game and maybe I could have gotten a few extra points that way. But really, it wouldn’t have done me a whole lot of good, as I wouldn’t be able to cheat my way through actual classes if things opened up in fall, and in any case, a big reason I went back to college was to challenge myself, so cheating would have defeated the whole point. But the point is that all the memorization was more than a sufficient challenge for me, arguably almost more than I could handle.
The other thing that made normal calc has was how many little details you had to keep track of, and I suppose this is true for math (and programming) in general. Remember the notation I used, with f(x) being the function and f'(x) being the derivative? That little apostrophe is super important–if you forget it, you’ll start working with the original function rather than the one you derived, which can REALLY mess you up, even though it’s such a small thing and easy to overlook! And I’m not really good at paying attention to very small details like that. When you’re writing it’s no big deal–misspell a word, autocorrect can catch it, and in any case, it’s very rare that one misspelled word or wrong date can wreck your entire argument. If you don’t pay really close attention to the apostrophes, and other things like negative signs in math, though, you’re in really big trouble.
Finally, some of the methodology was just gettin’ way too abstract for me by the end. For most of the class we had to do derivatives, as I said, but remember how I mentioned antiderivatives, or reverse-derivatives (this is technically called “integration)? That stuff is REALLY complex. There’s a whole thing with substitution (like, representing parts of a given equation as the letter “u” and then working with that) that just lost me. I could handle most of the earlier stuff, but there were so many steps involved with advanced integration that I found it really tough to keep track of. Higher-level math is even MORE abstract and complex, and it’s almost certain I wouldn’t be able to make it.
A brief aside–you’ll note that all this would also apply to computer science. In programming, don’t you also have to be super cautious about indentation, commas and parentheses in the right place, etc.? Yeah, but strangely enough, it’s a lot easier to keep track of that these days. See, my professor had us all use repl.it, and that coding platform actually does a lot of that stuff *for* you. Like, it’ll tell you if you have an indent wrong or if you’re missing a parenthesis somewhere. So I ended up blasting through most of the course with very little difficulty. Now, though, in future classes I might have to do everything by hand rather than with repl.it. THAT might be really hard. But right now things seem good, so we’ll get to that when we come to it.
So, as I said in the title, I think this is as far as I can get, at least in terms of math. As you’ll remember from a couple of my previous entries, when I first went thought about going back to college, it was partially because I was feeling sorta like One-Punch Man, in an odd way. The idea behind that superhero was that he’d grown so powerful that absolutely nothing challenged him anymore–that he was at the very top, and the only solution to his ennui was to find an opponent that could actually give him a good fight. At that time in my life, I felt sort of the same way–I’d accomplished everything I ever wanted, but the end result was that without anything to really challenge me, I started feeling bored. Well, I certainly have found a challenge now, and more importantly, I think it’s one that’s a match for me. There’s nothing and nobody One-Punch Man couldn’t beat, he’s the strongest being in his universe. I could never kid myself into thinking I was like that (intellectually, not physically, obviously), but it does seem like I won’t be able to beat this sort of enemy (higher math).
There’s no shame in admitting and accepting this, as far as I’m concerned. I wanted to do better than I did in the past, and better than I thought I could do, and I succeeded at that. I was never under any impression that I’d be the next Leibniz or Descartes. We can’t all be geniuses, and there’s fundamentally nothing wrong, and certainly nothing shameful, with rationally appraising your own abilities and acknowledging that you’ve done the best you can. I *could* move on to second part of normal calculus rather than the second part of ‘easy’ calculus–I got a B- in my course as a whole, and as long as you get above a C, so I’ve been told, you have a shot at passing Normal Calc II. Still, courses beyond Normal Calc are considerably harder, and I’d have to take a couple of those if I was continuing with a Bachelor’s in Compsci. If I couldn’t get above a B- in Normal Calc I, I’d get bodied by the more advanced stuff.
So, what am I gonna do? I contacted my advisor and we agreed I should probably go for a B.A in compsci rather than a B.Sc. Yeah, a B.A isn’t as impressive, but the thing is, it would be a lot easier in terms of the difficulty of courses, *and* it would require less coursework generally–so that means I could finish up my studies much earlier, which means I’d spend much less money! And while a B.A doesn’t look quite as good to employers as a B.Sc, it doesn’t look too bad either–the math is lighter, but I still have to take a few pretty tough Compsci courses. So I think it should benefit me in the long run, at least a little bit. Thus, after this, my schedule will look like this:
Summer: Intro to Compsci Part 2, Easy Calculus Part 2
After that, I’ll have the basic stuff down, and for a B.A, I’ll just have to take some courses in logic, software programming, and database programming, and then 4 relatively advanced (300 level) compsci courses. Aside from that, I just need to take a couple more easy, low-level science or math courses and I should be set.
Now, a brief word to those who *might* mock me for this. Nobody I know, of course–I’ve not many enemies these days, and pretty much everyone who reads this blog is a friend. But there are elitists out there who’d probably proclaim me a failure for going back down to the easier calc courses rather than sticking with the normal ones.
I don’t see any point in paying attention to such people, and not just because, as I said, none have actually shown up around me. See, for those kinds of elitists would probably call me a failure *no matter what I did.* Yeah, I guess I just wasn’t smart (or driven) enough to cut it in normal calc. But what if I was? Such people would make fun of me for not being good enough for advanced (Honors) calc. And what if, somehow, I managed to make the grade in honors calc? Then they’d STILL say I was a loser for not going on to graduate-level math, or whatever, on and on without end unless I ended up being the next Einstein or Leibniz.
That’s the thing with those folks–typically STEM nerds who think every other field of study is “useless”. There’s no real point in trying to impress them because they’re fundamentally playing a no-win game, and as I’ve repeatedly said, nothing is more pointless than no-win games. The only measure of ability you should care about is putting your skills to good use, whether or not you
Now, it’s true I wasn’t even thinking of “proving the haters wrong” when I decided to give math another shot during my second runthrough of college. As y’all know, my primary concern was to challenge *myself,* to see if I could surpass my own limits. And on that front, I succeeded–if you’d told me in high school I was even capable of doing beginner-level calculus, I never would have believed you (and I *did* pass the normal calc course, it’s not like I failed). But I’m just saying this here not only to ward off such elitists if they should come by, but also to give support to anyone else who either thinks or knows they’re not much for math. If you’re someone who’s not that great at the subject and you come across this entry, just keep in mind what I said above–that the sort of people who look down on you for “not being good at math” would look down on you no matter what happens. So don’t pay them the least amount of mind. If you wanna get better at math, give it a shot, and you may succeed far more than your detractors expect. And if you just don’t care about math, that’s cool too. Find something you do care about and get good at that, whether it’s history or art or whatever. STEM dweebs can mock such subjects for being “insufficiently rigorous” or “not very lucrative,” but if you can make a living doing them, that’s all you need.
And that’s it for this entry. Next up: Some general advice that helped me do a little better in the ‘normal’ calc course–even if it couldn’t get me an A, it saved me from a C- or lower!