Not the person you are responding, but I also used Spivak for single variable. For multivariable calculus I used Marsden and Tromba.
The other Spivak is a great book, but maybe too much for a first approach to the subject.
As others have argued, one can learn mathematics without its history, but it is very helpful to motivate many of its abstractions. A good example of this are some of the textbooks of David Bressoud. In a "Radical approach to Analysis", concepts are introduced in historical order. The book starts with the heat equation and its solution by Fourier series which was rejected at the time, in constrast to most books that start with the real numbers and limits.
"All of statistics" by Larry Wasserman. I took a bunch of courses that taught stats as a bunch of independent tools for different problems. AOS helped me to build some a strong foundation.
"Fundamental university physics Volume 1: Mechanics" by Alonso and Finn. This book seems to be not very well known in the USA, but it is very popular in Spanish and Portuguese speaking countries. It is your classical introductory physics/mechanics course with a very high emphasis on calculus.
"Computational partial differential equations" by Hans Peter Langtagen. A book on numerical methods for solutions of PDEs. It has the right amount of rigour (so you are able to tackle the literature), but it also includes code and plenty of practical advice.
"Nonlinear dynamics and chaos" by Strogatz. I think this book is really well known and I can't add much.
A very formulaic article, start with an anecdote that could fit many other articles (it seems Feynmann always shows up when we are talking about learning), refer to some Twitter conversation, add a formula or graph which provides no insight, sprinkle memes everywhere and that's all.
In the end, it seems the entire article can be summarised to, "it is all good to google for some recipes to solve your problems, but you also should care about some deeper understanding". Is there anything else to it?
Having skimmed through the article, I agree that the insight could have been summarized as a one-liner.
One could raise the same criticism for majority of content on the Internet, or even the field of literature.
On the other hand, starting with a Feynmann anecdote got my attention; the article takes the reader on an exploration of that single insight - at least it can be credited with staying on point - with some amusing illustrations/memes, and concludes by explaining the contrast between top-down/bottom-up approaches. OK, this last point could have been elaborated, what it means to "work from first principles".
Over all, I enjoyed following the author's thinking process, and felt that the article had enough substance to keep the reader's interest and stimulate some reflection and discussion, to re/consider one's approach.
> Is there anything else to it?
Like you, I seek more philosophical and intellectual substance. I think articles like this one are aimed at a general audience, like "popular science" - it's second-hand digested information, with some opinions based on experience. To get to the real meat, one must get closer to the source of the insights, like Feymann's own work.
On the other hand, this article took me 6 months of having a thought I couldn’t put into words until it clicked earlier this week and the Feynman anecdote is just a stylistic choice for structuring the article.
And you can summarize anything in 1 sentence if you don’t care about readers getting it, are trying to be funny, or are cynical enough. The world is not so complex after all. Everything’s just about increasing or decreasing entropy.
As soon as I finished my PhD in computational science, I left to the Bay Area. I did consider some positions in academia, but as others have described, the low impact of the problems I was working on and the path to permanent position didn't really motivate me.
I have been one year in industry and though I am doing something different, filtering and sensor fusion, I would describe my overall experience as very positive. I still get to read a lot of papers, work with very intelligent and motivated people and every now and then I get some praise from my coworkers on my maths knowledge, which is nice.
Deadlines are tighter and the metrics in which my work is judged on are very different.
Only to the undesirables students. Quoting the paper,
"The Mathematics Department of Moscow State University, the most prestigious mathematics school in Russia, was at that time actively trying to keep Jewish students (and other 'undesirables') from enrolling in the department.
One of the methods they used for doing this was to give the unwanted students a different set of problems on their oral exam."
Ah. Then it makes sense. I thought first it was a set of carefully selected problems, created for the purpose of letting one group fail and another pass. Which would have been very interesting (i.e. are there problems which one group solves better than another, or problems that one group must refuse to solve for religious reasons etc).
Giving a set of hard problems to an unwanted group and a separate set of problems (or no problems at all) to another is just simple discrimination. They could just as well have had a sign that told the unwanted group not to apply and it would have been no less discriminating.
Another point of view on why it is not striking that we can model heat conduction well. The heat equation is well-posed, which among other things, means that the solution depends continuously of the input data. And we observe this in many physical systems, that small changes in the cause produces small changes in the effect, therefore, we should expect from good mathematical models of reality to display this property.
It is about triangles. The idea that sine is the solution of this ODE for f(0)=0 and f'(0)=1 is quite modern.
I would say a course on trigonometry usually covers (my experience):
trigonometric functions
exact value of them for the angles 30, 45, 60, 90 ... degrees
Trigonometric formulas for the sum and difference of angles. A formuka for the double and the half angle.
Law of sine and law of cosine
Lots of relations derived from the Pythagoras theorem (sin^2+cos^=1)
how to solve trigonometric equations
With all this, you are equipped to completely determine a triangle, knowing some of its and the length of some its sides. As as application, I was taught, how to measure heights and distances provided you can measure angles.
Thus, without trigonometry, it would be fairly hard to take a course on analytic geometry.
Now, how would the course be enhanced by introducing sine as the solution of an ODE?
