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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 pr

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. Can you substantiate "millions of transfer students between thousands of universities every quarter"? That seems like outlandish hyperbole. Irrespective of the precise number, a system with momentum is not sufficient justification for its continuation.

weland 12y ago
> 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. Studying any kind of engine (or for that matter, anything that moves), solid-state electronic device (from transistors to antennae) or power distribution network and doing any kind of design about them is literally impossible without calculus. Even algebraic approaches to these are formulated through discretization of a continuous model (which -- unsurprisingly -- you need calculus to study). We have such a thing as databases, devices that can manipulate graphics and high-complexity economic models that are analyzed through linear algebra because Newton and Leibniz found the mathematical tools we need in order to model (continuous) change. We could mostly get along fine-ish without linear algebra, but the industrial revolution that spawned our society would be impossible without calculus, because without it you cannot design anything non-trivial that moves or runs on electricity.
cushychicken 12y ago
Calculus has been part of the foundations of nearly all modern science since its inception in the late 17th century. Its genesis shaped the course of modern physics, engineering, chemistry, and architecture, and has consistently for the last several centuries. Linear algebra has really only come into its own since humans have had powerful computational systems at their disposal, as a large part of linear algebra's usefulness (including its relevance to the examples you describe) is efficiently organizing and performing calculations in higher order systems. That is not to say it is irrelevant or non-impactful - far from it! I'm just saying linear algebra has taken hundreds of years for its real power to be leveraged, and in the intervening time, calculus has reigned supreme as a system for modeling and describing real systems. (FWIW, I agree with the author and the paper in question, but I think OP you replied to is absolutely correct that teaching both gives perspective that neither alone would impart.)
d_e_solomon 12y ago
FWIW, I have a math degree and work in tech. Both are incredibly important. I would say that linear algebra and discrete methods are more important for computing, but calculus has had a larger impact on science especially physics. Newtonian physics - a revolution in thinking and unification of mathematics and science - is intricately tied to calculus. Even in the example of economics that you cite, calculus is used heavily for the theoretical underpinnings for finding equilibria and deriving min/max.
- FeymanFan78 12y ago
  I agree that Calculus is a more readily applicable mathematics that easily applied to many things. It really bothers me when people downplay the usefulness of Calculus also the person that posted the fundamental concepts of Linear Algebra are generally a little beyond the Grasp of people that haven't even been through the Rigors of Calculus 1 - Calculus 3.
cabinpark 12y ago
>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.

weland 12y ago

> 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. Studying any kind of engine (or for that matter, anything that moves), solid-state electronic device (from transistors to antennae) or power distribution network and doing any kind of design about them is literally impossible without calculus. Even algebraic approaches to these are formulated through discretization of a continuous model (which -- unsurprisingly -- you need calculus to study). We have such a thing as databases, devices that can manipulate graphics and high-complexity economic models that are analyzed through linear algebra because Newton and Leibniz found the mathematical tools we need in order to model (continuous) change. We could mostly get along fine-ish without linear algebra, but the industrial revolution that spawned our society would be impossible without calculus, because without it you cannot design anything non-trivial that moves or runs on electricity.

cushychicken 12y ago

Calculus has been part of the foundations of nearly all modern science since its inception in the late 17th century. Its genesis shaped the course of modern physics, engineering, chemistry, and architecture, and has consistently for the last several centuries. Linear algebra has really only come into its own since humans have had powerful computational systems at their disposal, as a large part of linear algebra's usefulness (including its relevance to the examples you describe) is efficiently organizing and performing calculations in higher order systems. That is not to say it is irrelevant or non-impactful - far from it! I'm just saying linear algebra has taken hundreds of years for its real power to be leveraged, and in the intervening time, calculus has reigned supreme as a system for modeling and describing real systems. (FWIW, I agree with the author and the paper in question, but I think OP you replied to is absolutely correct that teaching both gives perspective that neither alone would impart.)

d_e_solomon 12y ago

FWIW, I have a math degree and work in tech. Both are incredibly important. I would say that linear algebra and discrete methods are more important for computing, but calculus has had a larger impact on science especially physics. Newtonian physics - a revolution in thinking and unification of mathematics and science - is intricately tied to calculus. Even in the example of economics that you cite, calculus is used heavily for the theoretical underpinnings for finding equilibria and deriving min/max.

FeymanFan78 12y ago
I agree that Calculus is a more readily applicable mathematics that easily applied to many things. It really bothers me when people downplay the usefulness of Calculus also the person that posted the fundamental concepts of Linear Algebra are generally a little beyond the Grasp of people that haven't even been through the Rigors of Calculus 1 - Calculus 3.

cabinpark 12y ago

>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.