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Too Much Calculus – Gilbert Strang (2001) [pdf]

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Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#71

Post by cpitman »

Statistics are wildly important for both business and understanding the world around you, and without solid calculus skills you cannot truly work with statistics.

Its not something that comes up every day at work, but it is still pretty often. For example, I was on a large team developing a complicated distributed system. The performance requirements were uncertain (like 500000 clients +- 200000, sending 10 +- 30 messages per second, etc). The system itself was in early development, hadn't been tuned, and we had only rough performance numbers for it.

Given only that, I needed to purchase the correct amount of hardware for the test lab, keeping in mind that the lead time for hardware was several months. I needed both statistics and some basic calculus to build a model based on what we knew, calculating variances for each intermediate and output, and then getting 50%, 90%, 99% confidence level estimates.

In the end, we got pretty close to a bullseye. On the other hand, I learned the important lesson to hide all the math from the client, lest they get awfully confused.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#72

Post by edanm »

vowelless wrote: post #67
ekidd wrote: post #7

As a programmer, I've found far more uses for linear algebra than for calculus. It shows up in robotics, in machine learning, in game development, and in many other cool subfields. I didn't get very much out of my university linear algebra course. (It was pure theory with nearly zero applications.) I don't think I really developed a good intuition for linear algebra—and why it's so useful—until I read Jim Hefferon's…

I agree although I work in robotics / computer vision. What I wish I had done a lot more of is statistics. Being able to reason about probabilities clearly, getting an idea of how a system performs given various variables, and most importantly, being able to communicate this to others has been very useful to me. I think anyone (front end, backend, etc) can benefit from learning about statistics.

Do you have a recommendation for a good statistics textbook, for someone who is mathematically literate, but doesn't know almost any statistics?

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#74

Post by jeffreyrogers »

cbgb wrote: post #66
jeffreyrogers wrote: post #46

I think Strang has some good points, and his advice is true for the majority of students who won't go on to use substantial amounts of mathematics or need to learn much math in the future. However, if you do want to learn more math beyond calculus and linear algebra, then a thorough understanding of calculus is absolutely a prerequisite and you honestly probably haven't learned it well enough taking the typical seque…

This may be true for mathematics in the analytical tradition (things like Topology, Measure Theory, Real/Complex Analysis), but Linear Algebra is far more important in, unsurprisingly, algebraic disciplines. These include Number Theory, Field Theory, and Mathematical Logic. While studying mathematics in college, once I had finished my Real Analysis requirement, I jumped headfirst into the algebraic side of things and…

That's a good point, my own studies have been biased towards the analytical side of mathematics, but I can definitely see how calculus could be less useful in other fields, e.g. I don't remember ever needing to do an integral when learning algorithms. And the opinions of my professors are of course biased as well, since they've spent their entire careers on analysis.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#75

Post by brendano »

edanm wrote: post #72
vowelless wrote: post #67

Earlier quoted context omitted.

I agree although I work in robotics / computer vision. What I wish I had done a lot more of is statistics. Being able to reason about probabilities clearly, getting an idea of how a system performs given various variables, and most importantly, being able to communicate this to others has been very useful to me. I think anyone (front end, backend, etc) can benefit from learning about statistics.

Do you have a recommendation for a good statistics textbook, for someone who is mathematically literate, but doesn't know almost any statistics?

This book is great and designed for exactly this type of audience: http://www.stat.cmu.edu/~larry/all-of-statistics/

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#76

Post by samatman »

dcosson wrote: post #55
ekidd wrote: post #7

As a programmer, I've found far more uses for linear algebra than for calculus. It shows up in robotics, in machine learning, in game development, and in many other cool subfields. I didn't get very much out of my university linear algebra course. (It was pure theory with nearly zero applications.) I don't think I really developed a good intuition for linear algebra—and why it's so useful—until I read Jim Hefferon's…

Thanks for the link to the book. I had a similar experience where my college linear algebra class was not good, taught by a grad student who seemed to have no enthusiasm for the material or for teaching, plus his accent was hard to understand at times. It was also the first exposure to proofs in the math curriculum so from the beginning I remember being overwhelmed by those and totally losing sight of the forest for…

As an aside, I get really confused and disheartened when I hear that someone's first exposure to proof was in college in a linear algebra class.

I had a somewhat old-fashioned education for an American, and learned proof via Euclid when I was 12. And again, but poorly, when I was 16, but that's another story. Proof is supposed to be learned a) thoroughly b) early and c) as a part of geometry. What's the point in even studying Euclid if you don't learn the method of proof?

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#77

Post by laurentoget »

tel wrote: post #53
laurentoget wrote: post #51

14 years later, the popularity of machine learning and "big data", which requires quite a bit of linear algebra, makes this only truer.

It also requires quite a bit of calculus though!

Calculus is indeed a required tool if you want to design new machine learning algorithm. But without linear algebra, good luck implementing an algorithm or even using a library to apply it on a concrete case.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#78

Post by mturmon »

xamuel wrote: post #10

(Speaking from my experience as a calc teacher at a big U.S. state U.; mileage may vary elsewhere) Most non-STEM students won't see the 'Calculus I, Calculus II, Calculus III' that starts the essay so dramatically. Increasingly, liberal arts students are able to get by with no more than a "math appreciation" course and maybe what amounts to high school algebra. The "math appreciation" course might actually touch on v…

You sound like exactly the constituency Gil Strang's corrective essay is addressed to.

I don't see any of the points you mention as particularly strong. There would be nothing wrong with presenting a more hands-on linear algebra course as a first course for freshmen. (Calling it "matrix arithmetic" does not do the notion service.) The material seems very immediate, and you can use a REPL like octave or matlab to reinforce concepts. You could deepen it as you wished by introducing more abstract concepts and by doing proofs, but I don't think that's necessary. I learned a lot from Golub and van Loan, for instance.

Second, the notion that calculus has had more impact on history/society than linear algebra is very hard to defend. Think of all the numerical modeling that has been done using matrices. Think of all the large-scale computations/optimizations that are done first by finding a linear system, or approximating one. I'm not saying there is a clear answer to this question, just that it can't be the foundation of an argument, because it's so undecidable. Another thing that makes it undecidable is cases where you need both (e.g., weather forecasting).

And saying it would be hard to teach the subject in a uniform way -- really, this is a very bureaucratic objection.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#79

Post by EdwardCoffin »

I don't disagree that linear algebra isn't given enough emphasis. However, the first thing I thought of when I read this was something attributed to Richard Hamming, who is known for the transcript of his talk that pops up on here frequently, You and your Research [1]. I got this from a review of his book on Amazon [2]:

> Hamming, in discussion, concluded his life thinking "the best tool to teach thinking was to teach the calculus."

I'm just thinking here that Strang is saying Linear Algebra is more useful for _applying_, where Hamming would say Calculus is more useful for reasons that go beyond applications (not that it doesn't also have applications).

[1] http://www.cs.virginia.edu/~robins/YouAndYourResearch.html

[2] http://www.amazon.com/review/RWPF0EL3E7GJZ/ref=cm_cr_pr_perm...

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#80

Post by bewuethr »

dfan wrote: post #52
hzhou321 wrote: post #44

I wish Gilbert Strang would teach calculus and write a book as he did for linear algebra.

You are in luck! Strang, Calculus, 2nd edition: http://www.amazon.com/Calculus-Second-Edition-Gilbert-Strang...

And the first edition is freely available at Open Courseware: http://ocw.mit.edu/resources/res-18-001-calculus-online-text...
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