6 ms·

>> the author’s wish to present ... mathematics, as intuitively and informally as possible, without compromising logical rigor The books in the West in general

by zkmon 11mo ago

>> the author’s wish to present ... mathematics, as intuitively and informally as possible, without compromising logical rigor 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?

mjburgess 11mo ago
Nope, but mathematics research is one of the most rarefied fields being extremely difficlt to get into, hard to get money, etc. -- (this is my understanding, at least). Progress is made here by people who, aged 10 are already showing signs of capability. 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
- deleted 11mo ago
  [deleted]
kardianos 11mo ago
I agree with this. But I don't see the students rejecting this, but the education degreed peoples who choose texts and the publishers want to make all learning for all people. This is foolish. Most people don't need to know calculus. And if you do learn it, do so with rigor so you actually learn it and not just the appearance of it, which is much much worse.
- DrSAR 11mo ago
  Not sure I agree with 'appearance [...] is much worse'. 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.
- godelski 11mo ago
  > Most people don't need to know calculus I don't like this line of argument. It applies to many things, many of which we'd laugh at for suggesting. 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! [1] https://easings.net/ https://easings.net/ [2] https://en.wikipedia.org/wiki/Jerk_%28physics%29 https://en.wikipedia.org/wiki/Jerk_%28physics%29
  - altmanaltman 11mo ago
    I get the argument you're making but that's a bit like saying cavemen used to do calculus as they hunt, which is a valid way of looking at this maybe but they didn't really "use calculus" just intuition. Simillarly, when learning calculus, most people do not do so at a driving course, they do it in the classroom. 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!
- rramadass 11mo ago
  > Most people don't need to know calculus. 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/everyday-calculus 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.
  - wing-_-nuts 11mo ago
    Thank you for the book link! One major problem I always had with calculus was that we didn't focus nearly enough on applications of it, outside of some optimization problems, which I loved. 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)
elcapitan 11mo ago
This may be a stupid question, but what do people usually mean when they refer to a mathematical text as being "rigorous"? Does it mean that everything is strictly proof-based rather than application-oriented?
- actinium226 11mo ago
  Generally that's what it means. And also when proofs are presented, a rigorous book will go through it fully, whereas a less rigorous one might just sketch out the main ideas of the proof and leave out some of the nitty gritty details (i.e. it's less rigorous to talk about "continuity" as "you can draw it without lifting the pen" as compared to the epsilon-delta definition, but epsilon-delta is pretty detailed and for intro calculus for non-mathematicians you don't really need it).
  - mike50 11mo ago
    This is the reason that everyone at my university said to just take the Applied version of Calculus 1 and 2 t avoid the proofs.
- layer8 11mo ago
  Not necessarily proof-heavy, but at least with formally rigorous definitions and theorems.
- zorked 11mo ago
  Rigorous = a pain in the ass to learn from, but you gain imaginary points for the pain.
  - godelski 11mo ago
    It certainly is more painful, but it is more beneficial. It is also harder to teach, but I stand by my claim. I'll quote Poincare: Math is not about the study of numbers, but the relationships between them. The difficulty and benefit of the rigor is the abstraction. Math is all about abstraction. 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. ---- Let's take the Fundamental Theorem of Calculus as an example[0]: f'(x) = lim_{h->0} {f(x + h) - f(x)} / {h} Take a moment here and think about it's form. Are there equivalent ones? What do each of these symbols mean? 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} You may also find the symmetric derivative too! f'(x) = lim_{h->0} {f(x + h) - f(x - h)} / {2h} In fact, you see these in computational programs all the time! The symmetric derivative even has the added advantage of error converging at an O(n^2) rate instead of O(n)! Yet, are these the same? (hint: no) Or tell me about the general case of f'(x) = lim_{h->0} {f(x + ah) - f(x + bh)}/{(a-b)h} I'm betting that most classes that went through deriving the derivative did not answer these questions for you (or you don't remember). Yet, had you, you would have instantly known how to do numerical differentiation and understand the limits, pitfalls, and other subjects like FEM (Finite-Element Methods) or Computational Methods would be much easier for those who take them. ---- Yet, I still will say that this is much harder to teach. Math is about abstraction, and abstraction is simply not that easy. But abstraction is incredibly powerful, as I hope every programmer can intuitively understand. After all, all we do is deal with abstractions. One can definitely be overly abstract and it will make a program uninterpretable for most, but one also can make a program have too little abstraction, which in that case we end up writing a million variations of the same thing, taking far more lines to write/read, and making the program too complex. There is a balance, but I'd argue that if one is able to understand abstraction that it is far easier to reduce abstraction than it is to abstract. 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 https://www.youtube.com/watch?v=oIhdrMh3UJw
nabla9 11mo ago
> Russian books doing this. 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 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.
- wirrbel 11mo ago
  is that like a Landau Lifshitz for Math?
  - nextos 11mo ago
    No, it's just a gentle overview.
  - rramadass 11mo ago
    It is very good and has a succinct coverage of a broad range of topics from Mathematics; just the right understandable level of rigor without being overwhelming. Published by Dover Publications and hence quite affordable. See ToC at https://store.doverpublications.com/products/9780486409160 https://store.doverpublications.com/products/9780486409160
- lioeters 11mo ago
  > A.N. Kolomogorov 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.
actinium226 11mo ago
> The books in the West in general kept getting less rigorous, with time. 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.
- deleted 11mo ago
  [deleted]
- psunavy03 11mo ago
  Non-STEM curricula do not necessarily require calculus.
zozbot234 11mo ago
If you care about getting all the nitty gritty details of a "rigorous" proof, maybe the quicker approach is to install Lean on your computer and step through a machine-checked proof from Mathlib. What you get from even the most heavyweight math books is still quite far from showing you all the steps involved.
bmitc 11mo ago
While world class, Russian mathematics is not known for rigor, formality, or detail, so I'm dubious.
cyberax 11mo ago
There are less formal math books in Russian. My absolute favorite calculus textbook is Fikhtenholts's "A Course of Differential and Integral Calculus". It is a bit less formal than many modern texts, but somehow much more approachable. 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.
matheusmoreira 11mo ago
There's an old blog that addresses this topic: https://professorconfess.blogspot.com/ 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.

by zkmon 11mo ago

mjburgess 11mo ago

Nope, but mathematics research is one of the most rarefied fields being extremely difficlt to get into, hard to get money, etc. -- (this is my understanding, at least). Progress is made here by people who, aged 10 are already showing signs of capability. 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

deleted 11mo ago
[deleted]

kardianos 11mo ago

I agree with this. But I don't see the students rejecting this, but the education degreed peoples who choose texts and the publishers want to make all learning for all people. This is foolish. Most people don't need to know calculus. And if you do learn it, do so with rigor so you actually learn it and not just the appearance of it, which is much much worse.

DrSAR 11mo ago
Not sure I agree with 'appearance [...] is much worse'. 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.
godelski 11mo ago
> Most people don't need to know calculus I don't like this line of argument. It applies to many things, many of which we'd laugh at for suggesting. 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! [1] https://easings.net/ https://easings.net/ [2] https://en.wikipedia.org/wiki/Jerk_%28physics%29 https://en.wikipedia.org/wiki/Jerk_%28physics%29
- altmanaltman 11mo ago
  I get the argument you're making but that's a bit like saying cavemen used to do calculus as they hunt, which is a valid way of looking at this maybe but they didn't really "use calculus" just intuition. Simillarly, when learning calculus, most people do not do so at a driving course, they do it in the classroom. 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!
rramadass 11mo ago
> Most people don't need to know calculus. 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/everyday-calculus 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.
- wing-_-nuts 11mo ago
  Thank you for the book link! One major problem I always had with calculus was that we didn't focus nearly enough on applications of it, outside of some optimization problems, which I loved. 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)

elcapitan 11mo ago

This may be a stupid question, but what do people usually mean when they refer to a mathematical text as being "rigorous"? Does it mean that everything is strictly proof-based rather than application-oriented?

actinium226 11mo ago
Generally that's what it means. And also when proofs are presented, a rigorous book will go through it fully, whereas a less rigorous one might just sketch out the main ideas of the proof and leave out some of the nitty gritty details (i.e. it's less rigorous to talk about "continuity" as "you can draw it without lifting the pen" as compared to the epsilon-delta definition, but epsilon-delta is pretty detailed and for intro calculus for non-mathematicians you don't really need it).
- mike50 11mo ago
  This is the reason that everyone at my university said to just take the Applied version of Calculus 1 and 2 t avoid the proofs.
layer8 11mo ago
Not necessarily proof-heavy, but at least with formally rigorous definitions and theorems.
zorked 11mo ago
Rigorous = a pain in the ass to learn from, but you gain imaginary points for the pain.
- godelski 11mo ago
  It certainly is more painful, but it is more beneficial. It is also harder to teach, but I stand by my claim. I'll quote Poincare: Math is not about the study of numbers, but the relationships between them. The difficulty and benefit of the rigor is the abstraction. Math is all about abstraction. 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. ---- Let's take the Fundamental Theorem of Calculus as an example[0]: f'(x) = lim_{h->0} {f(x + h) - f(x)} / {h} Take a moment here and think about it's form. Are there equivalent ones? What do each of these symbols mean? 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} You may also find the symmetric derivative too! f'(x) = lim_{h->0} {f(x + h) - f(x - h)} / {2h} In fact, you see these in computational programs all the time! The symmetric derivative even has the added advantage of error converging at an O(n^2) rate instead of O(n)! Yet, are these the same? (hint: no) Or tell me about the general case of f'(x) = lim_{h->0} {f(x + ah) - f(x + bh)}/{(a-b)h} I'm betting that most classes that went through deriving the derivative did not answer these questions for you (or you don't remember). Yet, had you, you would have instantly known how to do numerical differentiation and understand the limits, pitfalls, and other subjects like FEM (Finite-Element Methods) or Computational Methods would be much easier for those who take them. ---- Yet, I still will say that this is much harder to teach. Math is about abstraction, and abstraction is simply not that easy. But abstraction is incredibly powerful, as I hope every programmer can intuitively understand. After all, all we do is deal with abstractions. One can definitely be overly abstract and it will make a program uninterpretable for most, but one also can make a program have too little abstraction, which in that case we end up writing a million variations of the same thing, taking far more lines to write/read, and making the program too complex. There is a balance, but I'd argue that if one is able to understand abstraction that it is far easier to reduce abstraction than it is to abstract. 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 https://www.youtube.com/watch?v=oIhdrMh3UJw

nabla9 11mo ago

> Russian books doing this. 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 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.

wirrbel 11mo ago
is that like a Landau Lifshitz for Math?
- nextos 11mo ago
  No, it's just a gentle overview.
- rramadass 11mo ago
  It is very good and has a succinct coverage of a broad range of topics from Mathematics; just the right understandable level of rigor without being overwhelming. Published by Dover Publications and hence quite affordable. See ToC at https://store.doverpublications.com/products/9780486409160 https://store.doverpublications.com/products/9780486409160
lioeters 11mo ago
> A.N. Kolomogorov 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.

actinium226 11mo ago

> The books in the West in general kept getting less rigorous, with time. 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.

deleted 11mo ago
[deleted]
psunavy03 11mo ago
Non-STEM curricula do not necessarily require calculus.

zozbot234 11mo ago

If you care about getting all the nitty gritty details of a "rigorous" proof, maybe the quicker approach is to install Lean on your computer and step through a machine-checked proof from Mathlib. What you get from even the most heavyweight math books is still quite far from showing you all the steps involved.

bmitc 11mo ago

While world class, Russian mathematics is not known for rigor, formality, or detail, so I'm dubious.

cyberax 11mo ago

There are less formal math books in Russian. My absolute favorite calculus textbook is Fikhtenholts's "A Course of Differential and Integral Calculus". It is a bit less formal than many modern texts, but somehow much more approachable. 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.

matheusmoreira 11mo ago

There's an old blog that addresses this topic: https://professorconfess.blogspot.com/ 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.