The books in the West in general kept getting less rigorous, with time. I don't see Asian or Russian books doing this. The audience getting less receptive to rigor and wishing for more visuals and informal talk. When they get to higher studies and research, would they be able to cope with steep curve of more formalism and rigor?
Mathematics: Its Content, Methods and Meaning by A.D. Aleksandrov, A.N. Kolomogorov, M.A. Lavrent’ev,.. https://www.goodreads.com/book/show/405880.Mathematics
It's still a masterpiece. Originally published in 1962 in 3 volumes. The English translation has all in one.
Published by Dover Publications and hence quite affordable. See ToC at https://store.doverpublications.com/products/9780486409160
That wouldn't be who Kolmogorov complexity is named after? ("The Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program that produces the object as output.") ..Oh it is the same mathematician, Andrey Nikolaevich Kolmogorov. I see he was a major figure with many other concepts, equations and theorems named after him.
I wonder if it's because more people are going to college who would have otherwise gone to a vocational or trade school? If the audience expands to include people who might not have studied calculus had they not chosen to go to college, I feel like textbooks have to change to accommodate that.
I'll quote Poincare:
Math is not about the study of numbers, but the relationships between them.
The abstraction makes it harder to understand how to apply these rules, but if one breaks through this barrier one is able to apply the rules far more broadly.
----
f'(x) = lim_{h->0} {f(x + h) - f(x)} / {h}
If you actually study this, you may realize that there are an infinite number of equations that allow us to describe a secant line. So why this one? Is there something special? (hint: yes)
Let's call that the "forward derivative". Do you notice that through the secant line explanation that the "backward derivative" also works? That is
f'(x) = lim_{h->0} {f(x) - f(x - h)} / {h}
f'(x) = lim_{h->0} {f(x + h) - f(x - h)} / {2h}
Or tell me about the general case of
f'(x) = lim_{h->0} {f(x + ah) - f(x + bh)}/{(a-b)h}
----
This is just a tiny taste of what rigor holds. You are absolutely right to be frustrated and annoyed, but I hope you understand your conclusion is wrong. Unless you're Ramanujan, every mathematician has spent hours banging their head against a literal or metaphorical wall (or both!). The frustration and pain is quite real! But it is absolutely worth it.
[0] Linking an EpsilonDelta video that covers this exact example in more detail https://www.youtube.com/watch?v=oIhdrMh3UJw
> had you, you would have instantly known how to do numerical differentiation and understand the limits, pitfalls, and other subjects like FEM
No, you wouldn't. You would also learn things out of order. You would be exposed to things without understanding why you are learning them. People who argue this usually learn things the intuitive way (whether from rigorous material or not - what goes on in their mind isn't rigorous), and then they go back and reassess the rigor in the light of that. Then they pretend that they learned from the rigorous exposition. No, they didn't.
It is totally fine to iterate. Learn non-rigorously. Go back to it and iterate on rigor later. As it becomes necessary, and if it ever becomes necessary for your field.
> Unless you're Ramanujan, every mathematician has spent hours banging their head against a literal or metaphorical wall
Particularly if you are learning from "rigorous" material. But then you go watch some YouTube videos to make up for the absence of didactics in your textbook.
I mean, why don't we just throw Bourbaki books at freshmen and let them sort it out without classes? They are maximally rigorous, therefore maximally great to learn from, right?
> I mean, why don't we just throw Bourbaki books at freshmen and let them sort it out without classes?
So I'm not sure your argument of "out of order" is accurate. The order is what we make. There's no clear optimal way to teach math. Your argument hinges on that. You might argue that the current status quo is working, so why disrupt it, and I'll point around asking if you really think it's so effective when many demonstrate a lack of understanding all around us. That so many struggle with calculus is evidence itself. We need not even acknowledge that there are many children who learn this (and let's certainly not admit that it's far more common for them to learn it in unconventional ways).
> let them sort it out without classes?
The point of Math education isn't to create a clone army of the most pedantic pure Mathematicians in Academia.
> if you really think it's so effective when many demonstrate a lack of understanding all around us. That so many struggle with calculus is evidence itself
Because the first attempt went so well.
There is a certain irony with that too. The major criticisms were about the de-emphasis of arithmetic, arguing that this is the skill people need more. Here we are today though, where everyone has a calculator in their pocket. Plus, people still struggle with extremely basic arithmetic such as calculating a 10% tip... SAT scores are commonly pointed to as evidence, but in the same period of time Verbal skills fell even more[0]! Would you suggest that New Maths was so bad that it made people worse at English? Or maybe there's another explanation like how the number of SAT takers dramatically increased, especially from those in the bottom percentile of high school[1,2].
If you explore the NAEP data[2] there's no decline and there's been progress the entire time for math scores for ages 9, 13, and 17 (the ages that participated).
The Russians famously did very well too.
It would be even a greater irony if we concluded something was a failure because we made mathematical errors in analyzing...
[0] https://www.reddit.com/r/dataisbeautiful/comments/xg6gsz/his...
[1] The Sandia Report and U.S. Achievement: An Assessment: https://www.jstor.org/stable/pdf/27541911.pdf
[2] https://www.nationsreportcard.gov/ndecore/xplore/ltt [2] https://www.reddit.com/r/dataisbeautiful/comments/qz3the/us_...
Let's take the Fundamental Theorem of Calculus as an example[0]:
f'(x) = lim_{h->0} {f(x + h) - f(x)} / {h}This isn't the Fundamental Theorem of Calculus, it's the usual definition of the derivative of a function of a single real variable. The Fundamental Theorem of Calculus establishes the inverse relationship between differentiation and integration [0].
Unless you're Ramanujan, every mathematician has spent hours banging their head against a literal or metaphorical wall (or both!)
Ramanujan was no stranger to banging his head against the wall. My impression from Kanigel's The Man Who Knew Infinity is that his work ethic and mathematical fortitude were as astonishing as his creativity. For much of his career, he couldn't afford paper in quantity and did his hard work on stone slate, only recording the results. This could make it seem like his results were a product of pure inspiration because he left no trace of the furious activity and struggle that was involved.
From The Man Who Knew Infinity:
When he thought hard, his face scrunched up, his eyes narrowed into a squint. When he figured something out, he sometimes seemed to talk to himself, smile, shake his head with pleasure. When he made a mistake, too impatient to lay down his slate pencil, he twisted his forearm toward his body in a single fluid motion and used his elbow, now aimed at the slate, as an eraser. Ramanujan's was no cool, steady Intelligence, solemnly applied to the problem at hand; he was all energy, animation, force.
[0] https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculu...
My pet peeve about calculus books is that they almost always overlook the importance of continuity. In some extreme cases, they even start with infinitely small sequences, with some rather gnarly theorems like Bolzano–Weierstrass theorem about converging subsequences.
I think this is a mistake. It's much easier to start with continuous functions and build from there. Modern readers then can visualize the epsilon-delta formulation of limits as "zooming in" on the function. The "epsilon" is the height of the screen, and the "delta" is the "zoom level" at which the function fragment fits on the screen.
And once you "get" the idea of continuity and function limits, the other limit theorems just fall out naturally.
Given the choice between a class room of first years who believe (in themselves and) an appearance of calculus knowledge or a room of scared undergrads that recoil from any calculus-inspired argument they 'have never learnt it properly', I'll take the former. I can work with that much more easily. Sure, some things might break - but what's the worst that can happen?
We'll sort out the rigour later while we patch the bruises of overextending some analogies.
> Most people don't need to know calculus
Most people don't need to know how to read. Most people don't need to know how to add. Most people don't need to know how to use a computer. The foolishness of these statements are all subjective and based on what one believes one "needs". Yet, I have no doubt all of these things can improve peoples lives.
I'd argue the same with calculus. While I don't compute derivatives and integrals every day[0], I certainly use calculus every day. That likely sounds weird, but it is only because one thinks that math and computation is the same. When I drive I use calculus as I'm thinking about my rates of change, not only my velocity. Understanding different easing functions[1] I am able to create a smoother ride, be safer, drive faster, and save fuel. All at the same time!
The magic of the rigor is often lost, but the magic is abstraction. That's what we've done here with the car example. I don't need to compute numbers to "do math", I only need to have an abstract formulation. To understand that multiple variables are involved and there are relationships between them, and understanding that there are concepts like a rate of change, the rate of change of the rate of change, and even the rate of change of the rate of change of the rate of change! (the jerk!)[2].
That's still math. It may not be as rigorous, but a rigorous foundation gives you a greater ability to be less rigorous at times and take advantage of the lessons.
So yes, most people "don't need calculus" but learning it can give them a lot of power in how to think. This is true for much of mathematics. You may argue that this is not how it is taught, but with that I'll agree. The inefficiency of how it is taught is orthogonal to the utility of its lessons.
[0] Is a physicist not doing math just when they do symbol manipulation? I can tell you with great confidence, and experience, that much of their job is doing math without the use of numbers. It is about deriving formulations. Relationship!
If you're willing to stretch the definition of what "using" maths is then it can apply to everything and that devalues the concept as a whole. I'm not on the toilet, I'm doing calculus!
The difference may be in two different cavemen. One throws his spear on intuition alone. The other is thinking about the speed he throws, how the animal moves, the wind, and so on. There is a formulation, though not as robust as you'd see in a physics book.
> the definition of what "using" maths is then it can apply to everything
Math is a language, or more accurately a class of languages. If you're formulating your toilet activities, then it might be math. But as you might gather, there's nuance here.
I quoted Poincaré in another comment but I'll repeat here as I think it may help reduce confusion (though may add more)
Math is not the study of numbers, but the relationships between them.
Yes, technique is one thing, but being really good at throwing spears doesn't make you really good at math, is my argument. And most people will encounter maths in a formal setting while lacking the broader perspective that everything is technically "math".
Yet, we need to see the argument from the common person's view, if we're talking about calculus and learning in the traditional sense. The view you stated is quite esoteric and doesn't generalize well in this setting imo.
It's like a musician saying they see music in every action, but to most non-musicians (even if the stated thing is kind of true) that doesn't make a lot of sense etc.
> but is the thinking caveman really doing the same?
> It's like a musician saying they see music in every action, but to most non-musicians (even if the stated thing is kind of true) that doesn't make a lot of sense etc.
The musician, like the mathematician, understands that every sound is musical. If you want to see this in action it's quite enlightening[0]. I'm glad you brought up that comparison because I think it can help you understand what I really mean. There is depth here. Every human has access to the sounds but the training is needed to put them together and make these formulations. Benn here isn't exactly being formal writing his music using a keyboard and formalizing it down to musical notes on a sheet (though this is something I know he is capable of).
But maybe I should have quoted Picasso instead of Poincaré
Learn the rules like a pro, so you can break them like an artist.
I know this is confusing and I wish I could explain it better. But at least we can see that regardless of the field of expertise we find similar trains of thought. Maybe a bridge can be created by leveraging your own domain of expertise
Maybe I can put it this way: gibberish is more intelligible when crafted by someone who can already speak.
People should have at the minimum a conceptual idea of Calculus. A good motivation is Everyday Calculus: Discovering the Hidden Math All around Us by Oscar E. Fernandez - https://press.princeton.edu/books/paperback/9780691175751/ev...
> And if you do learn it, do so with rigor so you actually learn it
This is not strictly necessary for everybody. The conceptual ideas are what are important; else you are merely doing "plug-and-chug" Maths without any understanding. You need to focus on rigor only based on your needs and at your own pace. Concepts come first Formalism comes second.
A good example; In the Principia Newton actually uses the phrase Quantity of Motion for what we define today as Momentum. The phrase is evocative and beautifully captures the main concept instead of the bland p = m x v definition which though correct and needed for calculations conveys no mental imagery.
In Mathematics one should always approach a concept/idea from multiple perspectives including (but not limited to) Imagination, Conceptual, Graphical, Symbolic, Relationship, Applications, Definition/Theorem/Proof.
I originally went for engineering, and took cal I and II, then switched to cs and poof all of the math heavy stuff vanished (except maybe some LA for graphics)
There's not much need for a large amount of PhD places, and funding, for pure mathematics research.
Likewise, on the applied side, "calculus" now as a pure thing has been dead alone time. Gradients are computed with algorithms and numerical approximations, that are better taught -- with the formal stuff maintained via intuition.
I'm much more open to the idea that the west has this wrong, and we should be more focused on developing the applied side after spending the last century overly focused on the pure
https://professorconfess.blogspot.com/
It correlates student loans with the destruction of academic integrity. The idea is school administrators want to capture as many student loan dollars as possible, and that means maximizing the number of enrolled students. To that end, complexity, rigor and difficulty are all reduced as much as possible. Students are prevented from failing, since if they fail they might drop out, reducing profits. I even remember one article which draws comparisons with russian education.