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| Photo by Roman Mager on Unsplash |
Calculus, like other forms of mathematics, is much more than a language; it’s also an incredibly powerful system of reasoning. It lets us transform one equation into another by performing various symbolic operations on them, operations subject to certain rules. Those rules are deeply rooted in logic, so even though it might seem like we’re just shuffling symbols around, we’re actually constructing long chains of logical inference. The symbol shuffling is useful shorthand, a convenient way to build arguments too intricate to hold in our heads.
If we’re lucky and skillful enough — if we transform the equations in just the right way — we can get them to reveal their hidden implications. To a mathematician, the process feels almost palpable. It’s as if we’re manipulating the equations, massaging them, trying to relax them enough so that they’ll spill their secrets. We want them to open up and talk to us.
In a nutshell, calculus wants to make hard problems simpler. It is utterly obsessed with simplicity. That might come as a surprise to you, given that calculus has a reputation for being complicated. And there’s no denying that some of its leading textbooks exceed a thousand pages and weigh as much as bricks. But let’s not be judgmental. Calculus can’t help how it looks. Its bulkiness is unavoidable. It looks complicated because it’s trying to tackle complicated problems. In fact, it has tackled and solved some of the most difficult and important problems our species has ever faced.
Calculus succeeds by breaking complicated problems down into simpler parts. That strategy, of course, is not unique to calculus. All good problem-solvers know that hard problems become easier when they’re split into chunks. The truly radical and distinctive move of calculus is that it takes this divide-and-conquer strategy to its utmost extreme — all the way out to infinity. Instead of cutting a big problem into a handful of bite-size pieces, it keeps cutting and cutting relentlessly until the problem has been chopped and pulverized into its tiniest conceivable parts, leaving infinitely many of them. Once that’s done, it solves the original problem for all the tiny parts, which is usually a much easier task than solving the initial giant problem. The remaining challenge at that point is to put all the tiny answers back together again. That tends to be a much harder step, but at least it’s not as difficult as the original problem was.
Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus.
