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>I'm amazed that you say that calculus has had a much greater impact on society when virtually every database, graphics manipulation or economic analysis uses principles of linear algebra. I'm not. Consider all of physics. Physics is written down in terms of differential equations. You need calculus to get these equations. In terms of graphics, for example, are you magically creating only static images that never change in time? How do you think people came up with the algorithms to do things like refraction or reflection? These all involve calculus. Calculus is the most important tool in applying mathematics to the real world. It is how we derive the models for pretty much everything I can think of. Any time you want to model something where anything changes, that is calculus. Any time you want to find an optimal solutions, that is calculus. I can't think of a single area of applied mathematics that doesn't use calculus in some form. Even mathematics in general, calculus is used in some form pretty much everywhere in some form, except maybe in the foundations of mathematics like set theory or logic. Solving them, on the other hand, is a different problem. This is where linear algebra is important. When solving an ODE or PDE, when I discretise it, all I am doing is re-writing it as a giant linear algebra problem. Quantum mechanics, for example, is dedicated to finding the eigenvalues and eigenvectors of the Hamiltonian. Linear algebra provides the tools to solve the problems posed by calculus. But it isn't always required. In fact, if you look at the history of linear algebra, it only really became wide-spread when quantum mechanics was developing since there is a deep connection between linear algebra and quantum mechanics. Heisenberg had never heard of a matrix before despite discovering Heisenberg matrix mechanics. He was told by Max Born that what he had been doing is actually this thing mathematicians called "matrices". Now linear algebra is a required course for physicists (incidentally, linear algebra was the first lecture I ever attended at university). So I would argue that learning calculus is the most important thing someone should know mathematically, since it is the tool we use to build models. The second most is linear algebra since it provides us the tools to solve these problems. But in my mind, knowing how to derive these models is a lot more important than being able to solve them. That and linear algebra doesn't give you any insight into why we are solving the problem that way, it is just a tool to solve the problem. Also in Calculus 1,2,3 and so on, you learn increasingly more complicated techniques of calculus to solve more problems. But when you start numerically solving them, linear algebra doesn't care. It doesn't care if your equation came from the Einstein field equations or Newton's second law. It's all the same to it. So in a sense, I can teach you everything you'll need to know about linear algebra in a single course but the same is not true of calculus, which requires multiple courses. |
I think:
Okay, then go back to pre-calculus times and somehow still invent the modern world without it and then get back to me on how irrelevant it is.
Also, you don't get to wear this shirt: http://www.topatoco.com/merchant.mvc?Screen=PROD&Store_Code=...
My feeling is that it would take a few hundred years extra (or maybe more!) to get to the point where you had a computer without calculus. Might even take a thousand. You'd only be able to do discrete stuff, so while Babbage's difference engine might be up for grabs (and mechanical calculators definitely would be) things like the Norden bomb sight or naval firing computers wouldn't be, since they're continuous not discrete systems.
You might not EVER get to computers because things like Nyquist's theorem and stuff like that doesn't even exist in a discrete-only world. Things just sometimes work and sometimes don't and it's sorta related to the frequency, but how is unclear. Aliasing is just something that randomly bites you in the ass and there's no way to develop a rigorous understanding of it and to engineer yourself out of the morass.
I would not volunteer for that mission. I like the modern world.