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Chapter 1 is brilliant. I've been shouting from the rooftops for years that math[0] courses need more context. We can prove X, Y, and Z, and this class will te

Chapter 1 is brilliant. I've been shouting from the rooftops for years that math[0] courses need more context. We can prove X, Y, and Z, and this class will teach you that, but the motivating problem that led to our ability to do X, Y, and Z is mentioned only in passing. We can work something out, and then come back and rework it in more generality, but then that reworking becomes a thing in and of itself. And this is great! Further advances come from doing just this. But for pedagogical purposes, stuff sticks in the human brain so much better if we teach the journey, and not just the destination. I found teaching Calculus I was able to draw in students so much more if I worked in what problems Newton was trying to solve and why. It gave them a story to follow, a reason to learn this stuff. Kudos to the author for chapter 1 (and probably the rest, but chapter 1 is all I've had time to skim). [0] And honestly, nearly every subject.

gowld 2y ago
Hot take: "18th Century" mathematics was intiuitive and informal, to the point that it was inconsistent. The 19th and 20th Centuries added rigor and formalism (and elitism) and devalued intuition, to the point that it begame uninterpretable to most. The 21st Century's major contribution to mathematics (including YouTube! and conversational style writing) was to bring back intuition, with the backing of formal foundations.
- Agingcoder 2y ago
  I like your idea. I was schooled in abstract 20th century math - indeed YouTube is the opposite, and it’s a good thing. One of my math teachers was once talking to Jean Dieudonné https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9 https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9 who was part of the Bourbaki group and asked him why on earth he insisted on inflicting raw dry theory to the world with no intuition , when his day job involved drawing ideas all day long ! Edit: interestingly enough, one of my colleagues thinks very strongly that intuition should not be shared, and the path to intuition should be walked by everyone so that they ´ Make their own mental images ´. I guess that there’s a tradeoff between making things accessible, and deeply understood, but I don’t know what to make of his opinion.
  - wging 2y ago
    Do you know how Dieudonné answered?
    - Koshkin 2y ago
      On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say 'mathematics is just a combination of meaningless symbols,'... Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient.
  - stoneman24 2y ago
    The ‘make thier own mental models’ vs sharing/providing full information is difficult. ‘Make thier own model’ of the domain can lead to deeper understanding but takes time and may lead to different (possibly incorrect) understanding of the issues and complexities. If not reviewed with others. Providing full information upfront to a person can be quicker but lead to a superficial knowledge. I think that it comes down to whether that deeper knowledge is directly needed for the main task. Can I get by with an superficial (leaky) abstraction and concentrate on the main job.
  - fredilo 2y ago
    > interestingly enough, one of my colleagues thinks very strongly that intuition should not be shared, and the path to intuition should be walked by everyone so that they ´ Make their own mental images ´. I guess that there’s a tradeoff between making things accessible, and deeply understood, but I don’t know what to make of his opinion. If the objective is to advance mathematics instead of making it accessible, then this is a somewhat reasonable position. The mathematical statements that a person can come up with is often a direct product of their mental image. If everyone has the same image, everyone comes up with similar mathematical statements. For this reason you want to avoid that everyone has the same picture. Forcing everyone to start with a clean canvas increases the chance that there is diversity in the images. Maybe someone finds a new image, that leads to new mathematical statements. At least that's the idea. One could also argue that it just leads to blank canvases everywhere.
- paulddraper 2y ago
  Elements is formal, right?
  - hollerith 2y ago
    Pretty much. According to someone on Stack Exchange, "The most serious difficulties with Euclid from the modern point of view is that he did not realize that an axiom was needed for congruence of triangles, Euclids proof by superposition is not considered as a valid proof." But making mistakes in a formal treatment of a subject does not negate the fact that it is a formal treatment of the subject IMHO, and AFAICR none of the theorems in Elements is wrong; i.e., Elements is formal enough to have avoided a mistake even though the axioms listed weren't all the axiom that are actually needed to support the theorems. 18th Century European math was much more potent than ancient Greek math, and although parts of it like algebra and geometry were, for a long time, most of it was not understood at a formal or rigorous level for a long time even if we accept the level of rigor found in Elements.
    - koolala 2y ago
      Isn't how formal or rigorous something is just a social convention? Grammer Nazi's used to make online speech be formal with perfect rigor. Isn't it all relative to what your society defines?
      - chongli 2y ago
        No. That's the colloquial definition of formal. In mathematics, the word formal refers to something more specific: one or more statements written using a set of symbols which have fully-defined rules for mechanically transforming them into another form. A formal proof is then one which proceeds by a series of these mechanical steps beginning with one or more premises and ending with a conclusion (or goal).
        koolala 2y ago
        Both are a formality based on whats in fashion. I like the Axiom of Choice and not taking Math or words as literal or biblical truth.
dataflow 2y ago
Do they explain anywhere why we consider radicals special and ask these questions about them in the first place? Nobody ever explains that. To my programmer brain, whether I'm solving t^3 = 2 or t^100 + t + 1 = 7 numerically, I have to use an iterative method like Newton's either way. ("But there's a button for Nth root" isn't an argument here - they could've added a more generic button for Newton.) They don't fundamentally seem any different. Why do I care whether they're solvable in terms of radicals? It almost feels as arbitrary as asking whether something is solvable without using the digit 2. I would've thought I'd be more interested in whether they're solvable in some other respect (via iteration, via Newton, with quadratic convergence, or whatever)?
- Joker_vD 2y ago
  I imagine it has something to do with the fact that there is a somewhat simple pen-and-paper procedure, quite similar to long division, for calculating square/cubic/etc. roots to arbitrary precision, digit by digit. So finding roots is, in a sense, an arithmetic operation; and of calculus one better not speak in a polite society.
- phs 2y ago
  Radicals are a "natural" extension in a certain sense, just as subtraction and division are. They invert an algebraic operation we often encounter when trying to solve equations handed to us by e.g. physics. I find it understandable to want to give them a name. Why not something like "those things we can solve with Newton"? As you note Newton is broadly applicable; one would hope, given how popular the need to invert an exponent is, that something better (faster, more stable) if more specific than Newton might be created. It is hard to study a desired hypothetical operation without giving it a name. On a related note, how come we don't all already have the names of the 4th order iterative operation (iterated exponents) and its inverse in our heads? Don't they deserve consideration? Perhaps, but nature doesn't seem to hand us instances of those operations very often. We seemingly don't need them to build a bridge or solve some other common practical engineering problem. I imagine that is why they fail to appear in high school algebra courses.
- bubble12345 2y ago
  You are right in the sense that solvability by radicals has no practical importance, especially when it comes to calculations. It is just a very classical pure math question, dating back hundreds of years ago. Its solution led to the development of group theory and Galois theory. Group theory and Galois theory then are foundational in all kinds of areas. Anyway, so why care about solvability by radicals? To me the only real reason is that it's an interesting and a natural question in mathematics. Is there a general formula to solve polynomials, like the quadratic formula? The answer is no - why? When can we solve a polynomial in radicals and how? And so on. If you like pure math, you might find solvability by radicals interesting. It's also a good starting point and motivation for learning Galois theory.
atribecalledqst 2y ago
On the subject of more context in math, I've always wondered if having a grasp of the history of math would be helpful in getting better at solving mathematical problems. i.e. would learning more about how math developed over time, and how people solved important problems in the past, help me in trying to solve some other problem today? Years ago I bought the 3-volume set "Mathematical Thought from Ancient to Modern Times", but never had the time to get past the first few chapters. I'd be interested in any recommendations for math history tomes like that.
- mbivert 2y ago
  > I'd be interested in any recommendations for math history tomes like that. Not a book, but FWIW, I've enjoyed a few videos from Norman Wildberger's "Math History" playlist[0]. Interestingly, he has a unconventional view of infinite processes in mathematics, a point of view that used to be common about a century ago or so. I'm sure knowing some amount of history is useful, but there must be a limit to how much of it is practically useful though. https://www.youtube.com/watch?v=dW8Cy6WrO94&list=PL55C7C83781CF4316 https://www.youtube.com/watch?v=dW8Cy6WrO94&list=PL55C7C8378...
  - mindcrime 2y ago
    I've been going through that very video lecture series the last couple of weeks. Good stuff. And in the lectures he mentions a number of books. I looked a few up on Amazon, and then looked at the associated Amazon recommendations, and so far have this small list of books related to Maths history that look worth reading: Mathematics and Its History (Undergraduate Texts in Mathematics) 3rd ed. 2010 Edition by John Stillwell The History of the Calculus and Its Conceptual Development (Dover Books on Mathematics) by Carl B. Boyer A History of Mathematics by Carl B. Boyer A Concise History of Mathematics: Fourth Revised Edition (Dover Books on Mathematics)A Concise History of Mathematics: Fourth Revised Edition (Dover Books on Mathematics) by Dirk J. Struik Introduction to the Foundations of Mathematics: Second Edition (Dover Books on Mathematics) Second Edition by Raymond L. Wilder Mathematical Thought from Ancient to Modern Times by Morris Kline (3 volume set) The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time by Jason Socrates Bardi There is also a "thing" in mathematics that is sometimes called the "genetic approach" where "genetic" is roughly equivalent to "historical" or maybe "developmental". IOW, a "genetic approach" book teaches a subject by tracing the development of the subject over its history. One popular book in this mold is: The Calculus: A Genetic Approach by Otto Toeplitz
deleted 2y ago
[deleted]
BeetleB 2y ago
> I found teaching Calculus I was able to draw in students so much more if I worked in what problems Newton was trying to solve and why. It gave them a story to follow, a reason to learn this stuff. Be careful. In undergrad, I worked as a math tutor at their tutoring institute. We had two calculus classes - one for engineers/science and the other for business/economics. If you're into Newtonian stuff or it's relevant to your degree, then your approach is all great. What I consistently saw is people in degrees like biology (or even industrial engineering) had a harder time learning because of those physics applications in the book. They came in with one problem: Having trouble with the calculus. And then they discovered they had two problems: To understand calculus they suddenly had to understand physics as well. You'd get students who were totally adept at differentiating and integrating, but would struggle with the problems that involved physics. Yes, it is important to be able to translate real world problems to math ones, but it's a bigger problem when you don't care about that particular field. Same problem with the business students. Since all the tutors were engineering folks, they had trouble tutoring the business students because their textbook was full of examples that required basic finance knowledge (and it really doesn't help that financial quantities are discrete and not continuous, adding to the tutors' confusion). If you can find an application the student is interested in, then by all means, use that approach. For a general purpose textbook, though, it hinders learning for many students who don't care for the particular choice of application the book decided to use.
- dgacmu 2y ago
  I got to see that yesterday with my daughter: her algebra book used PV=nRT as an example of joint proportion and asked some questions about it. She, having never seen the ideal gas law, was quite thrown by it. That said, after we went through it and had a brief physics lesson, it worked quite well and I'm glad they used the example instead of just making something up -- but it required having a tutor (me) on hand to help make the context make sense.
  - HiroshiSan 2y ago
    Ah yes I know that exact problem. AoPS intro to algebra. That question was great for getting an intuitive sense of proportion, though I do remember one of the sub problems gave me some trouble.
chrisweekly 2y ago
Yeah! You're probably already aware of https://betterexplained.com https://betterexplained.com -- an amazing resource that exemplifies this same mindset.
- VyseofArcadia 2y ago
  I was not, thank you for this.
alejohausner 2y ago
When I took real analysis, I didn't get an intuitive sense of limits and all that delta/epsilon stuff. It was too abstract. Strangely enough, when I read David Foster Wallace's "Everything and More" on the history of infinity it all made more sense, because he described all the paradoxes and dead ends that mathematicians had run into before Cauchy and others brought rigor to the problem. Wallace was a postmodern novelist. I don't know why he described infinity so clearly; I didn't enjoy his other work.
- Koshkin 2y ago
  Interesting... This ε-δ stuff is needed when you talk about functions, but when you start instead with the limits of numeric sequences, which are easy to grasp, the function limits come easy as well, because of the analogy between the ways these are defined.
  - alejohausner 2y ago
    It’s not so much that I didn’t understand the principles behind continuity. It’s just that the way my teachers presented the material lacked historical context. After a BSc in pure math I discovered that I enjoyed applied math and CS much more, which told me that I need concrete examples to understand a theory: if you tell me about abstractions like groups and rings, which took years to establish, I lose interest. Tell me that groups express properties of matrix multiplications, or permutations, or modular arithmetic, and I’ll get it right away. It’s the way my mind works, but I’m sure I’m not alone, and mathematical pedagogy would benefit from historical context.
- silasdavis 2y ago
  The maths appendices in Infinite Jest are about half the book. There's a section written by Hal's friend (?), proving something like the intermediate value theorem where he says something like: we'll use epsilon delta because it's mad fun to say.
ForOldHack 2y ago
Chapter X is brilliant. Galois was very brilliant. He is always my #1 choice for ever of who, if anyone from history, I would like to have lunch with. He was introduced to me by my 5th grade calculus teacher Mr Steven Giavant, PhD. Galois bridged several disciplines and invented a new area of math. Most unfortunately he met his end extremely tragically. https://en.wikipedia.org/wiki/%C3%89variste_Galois https://en.wikipedia.org/wiki/%C3%89variste_Galois
- tzs 2y ago
  > He was introduced to me by my 5th grade calculus teacher [...] 5th grade calculus!?
XMPPwocky 2y ago
Absolutely agree. When I was a kid, I took a MOOC (MITx 6.002x) because I was interested in EE. They said calculus was a prerequisite, which I didn't know yet, but I went ahead anyways; every time the course required calculus, I'd go read the relevant chapters in Strang's Calculus. Felt incredibly natural, and I ended up going through the rest of Strang just for fun- probably the best learning experience I've ever had, and I doubt I'd have ever done it without having the MOOC there to motivate the problem.
raegis 2y ago
> I've been shouting from the rooftops for years that math[0] courses need more context. A slightly different perspective: It used to be common for professors to teach a lot of history in graduate algebra courses. (Generation X and older.) I've talked to students from a few other schools who experienced this. None of us cared for it at the time. My teacher spent at least two class periods writing down the history on the board. We were all quite bored. But then again, we were in math grad school, so there was no need to try to motivate us.

gowld 2y ago

Hot take: "18th Century" mathematics was intiuitive and informal, to the point that it was inconsistent. The 19th and 20th Centuries added rigor and formalism (and elitism) and devalued intuition, to the point that it begame uninterpretable to most. The 21st Century's major contribution to mathematics (including YouTube! and conversational style writing) was to bring back intuition, with the backing of formal foundations.

Agingcoder 2y ago
I like your idea. I was schooled in abstract 20th century math - indeed YouTube is the opposite, and it’s a good thing. One of my math teachers was once talking to Jean Dieudonné https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9 https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9 who was part of the Bourbaki group and asked him why on earth he insisted on inflicting raw dry theory to the world with no intuition , when his day job involved drawing ideas all day long ! Edit: interestingly enough, one of my colleagues thinks very strongly that intuition should not be shared, and the path to intuition should be walked by everyone so that they ´ Make their own mental images ´. I guess that there’s a tradeoff between making things accessible, and deeply understood, but I don’t know what to make of his opinion.
- wging 2y ago
  Do you know how Dieudonné answered?
  - Koshkin 2y ago
    On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say 'mathematics is just a combination of meaningless symbols,'... Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient.
- stoneman24 2y ago
  The ‘make thier own mental models’ vs sharing/providing full information is difficult. ‘Make thier own model’ of the domain can lead to deeper understanding but takes time and may lead to different (possibly incorrect) understanding of the issues and complexities. If not reviewed with others. Providing full information upfront to a person can be quicker but lead to a superficial knowledge. I think that it comes down to whether that deeper knowledge is directly needed for the main task. Can I get by with an superficial (leaky) abstraction and concentrate on the main job.
- fredilo 2y ago
  > interestingly enough, one of my colleagues thinks very strongly that intuition should not be shared, and the path to intuition should be walked by everyone so that they ´ Make their own mental images ´. I guess that there’s a tradeoff between making things accessible, and deeply understood, but I don’t know what to make of his opinion. If the objective is to advance mathematics instead of making it accessible, then this is a somewhat reasonable position. The mathematical statements that a person can come up with is often a direct product of their mental image. If everyone has the same image, everyone comes up with similar mathematical statements. For this reason you want to avoid that everyone has the same picture. Forcing everyone to start with a clean canvas increases the chance that there is diversity in the images. Maybe someone finds a new image, that leads to new mathematical statements. At least that's the idea. One could also argue that it just leads to blank canvases everywhere.
paulddraper 2y ago
Elements is formal, right?
- hollerith 2y ago
  Pretty much. According to someone on Stack Exchange, "The most serious difficulties with Euclid from the modern point of view is that he did not realize that an axiom was needed for congruence of triangles, Euclids proof by superposition is not considered as a valid proof." But making mistakes in a formal treatment of a subject does not negate the fact that it is a formal treatment of the subject IMHO, and AFAICR none of the theorems in Elements is wrong; i.e., Elements is formal enough to have avoided a mistake even though the axioms listed weren't all the axiom that are actually needed to support the theorems. 18th Century European math was much more potent than ancient Greek math, and although parts of it like algebra and geometry were, for a long time, most of it was not understood at a formal or rigorous level for a long time even if we accept the level of rigor found in Elements.
  - koolala 2y ago
    Isn't how formal or rigorous something is just a social convention? Grammer Nazi's used to make online speech be formal with perfect rigor. Isn't it all relative to what your society defines?
    - chongli 2y ago
      No. That's the colloquial definition of formal. In mathematics, the word formal refers to something more specific: one or more statements written using a set of symbols which have fully-defined rules for mechanically transforming them into another form. A formal proof is then one which proceeds by a series of these mechanical steps beginning with one or more premises and ending with a conclusion (or goal).
      - koolala 2y ago
        Both are a formality based on whats in fashion. I like the Axiom of Choice and not taking Math or words as literal or biblical truth.

dataflow 2y ago

Do they explain anywhere why we consider radicals special and ask these questions about them in the first place? Nobody ever explains that. To my programmer brain, whether I'm solving t^3 = 2 or t^100 + t + 1 = 7 numerically, I have to use an iterative method like Newton's either way. ("But there's a button for Nth root" isn't an argument here - they could've added a more generic button for Newton.) They don't fundamentally seem any different. Why do I care whether they're solvable in terms of radicals? It almost feels as arbitrary as asking whether something is solvable without using the digit 2. I would've thought I'd be more interested in whether they're solvable in some other respect (via iteration, via Newton, with quadratic convergence, or whatever)?

Joker_vD 2y ago
I imagine it has something to do with the fact that there is a somewhat simple pen-and-paper procedure, quite similar to long division, for calculating square/cubic/etc. roots to arbitrary precision, digit by digit. So finding roots is, in a sense, an arithmetic operation; and of calculus one better not speak in a polite society.
phs 2y ago
Radicals are a "natural" extension in a certain sense, just as subtraction and division are. They invert an algebraic operation we often encounter when trying to solve equations handed to us by e.g. physics. I find it understandable to want to give them a name. Why not something like "those things we can solve with Newton"? As you note Newton is broadly applicable; one would hope, given how popular the need to invert an exponent is, that something better (faster, more stable) if more specific than Newton might be created. It is hard to study a desired hypothetical operation without giving it a name. On a related note, how come we don't all already have the names of the 4th order iterative operation (iterated exponents) and its inverse in our heads? Don't they deserve consideration? Perhaps, but nature doesn't seem to hand us instances of those operations very often. We seemingly don't need them to build a bridge or solve some other common practical engineering problem. I imagine that is why they fail to appear in high school algebra courses.
bubble12345 2y ago
You are right in the sense that solvability by radicals has no practical importance, especially when it comes to calculations. It is just a very classical pure math question, dating back hundreds of years ago. Its solution led to the development of group theory and Galois theory. Group theory and Galois theory then are foundational in all kinds of areas. Anyway, so why care about solvability by radicals? To me the only real reason is that it's an interesting and a natural question in mathematics. Is there a general formula to solve polynomials, like the quadratic formula? The answer is no - why? When can we solve a polynomial in radicals and how? And so on. If you like pure math, you might find solvability by radicals interesting. It's also a good starting point and motivation for learning Galois theory.

atribecalledqst 2y ago

On the subject of more context in math, I've always wondered if having a grasp of the history of math would be helpful in getting better at solving mathematical problems. i.e. would learning more about how math developed over time, and how people solved important problems in the past, help me in trying to solve some other problem today? Years ago I bought the 3-volume set "Mathematical Thought from Ancient to Modern Times", but never had the time to get past the first few chapters. I'd be interested in any recommendations for math history tomes like that.

mbivert 2y ago
> I'd be interested in any recommendations for math history tomes like that. Not a book, but FWIW, I've enjoyed a few videos from Norman Wildberger's "Math History" playlist[0]. Interestingly, he has a unconventional view of infinite processes in mathematics, a point of view that used to be common about a century ago or so. I'm sure knowing some amount of history is useful, but there must be a limit to how much of it is practically useful though. https://www.youtube.com/watch?v=dW8Cy6WrO94&list=PL55C7C83781CF4316 https://www.youtube.com/watch?v=dW8Cy6WrO94&list=PL55C7C8378...
- mindcrime 2y ago
  I've been going through that very video lecture series the last couple of weeks. Good stuff. And in the lectures he mentions a number of books. I looked a few up on Amazon, and then looked at the associated Amazon recommendations, and so far have this small list of books related to Maths history that look worth reading: Mathematics and Its History (Undergraduate Texts in Mathematics) 3rd ed. 2010 Edition by John Stillwell The History of the Calculus and Its Conceptual Development (Dover Books on Mathematics) by Carl B. Boyer A History of Mathematics by Carl B. Boyer A Concise History of Mathematics: Fourth Revised Edition (Dover Books on Mathematics)A Concise History of Mathematics: Fourth Revised Edition (Dover Books on Mathematics) by Dirk J. Struik Introduction to the Foundations of Mathematics: Second Edition (Dover Books on Mathematics) Second Edition by Raymond L. Wilder Mathematical Thought from Ancient to Modern Times by Morris Kline (3 volume set) The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time by Jason Socrates Bardi There is also a "thing" in mathematics that is sometimes called the "genetic approach" where "genetic" is roughly equivalent to "historical" or maybe "developmental". IOW, a "genetic approach" book teaches a subject by tracing the development of the subject over its history. One popular book in this mold is: The Calculus: A Genetic Approach by Otto Toeplitz

deleted 2y ago

[deleted]

BeetleB 2y ago

> I found teaching Calculus I was able to draw in students so much more if I worked in what problems Newton was trying to solve and why. It gave them a story to follow, a reason to learn this stuff. Be careful. In undergrad, I worked as a math tutor at their tutoring institute. We had two calculus classes - one for engineers/science and the other for business/economics. If you're into Newtonian stuff or it's relevant to your degree, then your approach is all great. What I consistently saw is people in degrees like biology (or even industrial engineering) had a harder time learning because of those physics applications in the book. They came in with one problem: Having trouble with the calculus. And then they discovered they had two problems: To understand calculus they suddenly had to understand physics as well. You'd get students who were totally adept at differentiating and integrating, but would struggle with the problems that involved physics. Yes, it is important to be able to translate real world problems to math ones, but it's a bigger problem when you don't care about that particular field. Same problem with the business students. Since all the tutors were engineering folks, they had trouble tutoring the business students because their textbook was full of examples that required basic finance knowledge (and it really doesn't help that financial quantities are discrete and not continuous, adding to the tutors' confusion). If you can find an application the student is interested in, then by all means, use that approach. For a general purpose textbook, though, it hinders learning for many students who don't care for the particular choice of application the book decided to use.

dgacmu 2y ago
I got to see that yesterday with my daughter: her algebra book used PV=nRT as an example of joint proportion and asked some questions about it. She, having never seen the ideal gas law, was quite thrown by it. That said, after we went through it and had a brief physics lesson, it worked quite well and I'm glad they used the example instead of just making something up -- but it required having a tutor (me) on hand to help make the context make sense.
- HiroshiSan 2y ago
  Ah yes I know that exact problem. AoPS intro to algebra. That question was great for getting an intuitive sense of proportion, though I do remember one of the sub problems gave me some trouble.

chrisweekly 2y ago

Yeah! You're probably already aware of https://betterexplained.com https://betterexplained.com -- an amazing resource that exemplifies this same mindset.

VyseofArcadia 2y ago
I was not, thank you for this.

alejohausner 2y ago

When I took real analysis, I didn't get an intuitive sense of limits and all that delta/epsilon stuff. It was too abstract. Strangely enough, when I read David Foster Wallace's "Everything and More" on the history of infinity it all made more sense, because he described all the paradoxes and dead ends that mathematicians had run into before Cauchy and others brought rigor to the problem. Wallace was a postmodern novelist. I don't know why he described infinity so clearly; I didn't enjoy his other work.

Koshkin 2y ago
Interesting... This ε-δ stuff is needed when you talk about functions, but when you start instead with the limits of numeric sequences, which are easy to grasp, the function limits come easy as well, because of the analogy between the ways these are defined.
- alejohausner 2y ago
  It’s not so much that I didn’t understand the principles behind continuity. It’s just that the way my teachers presented the material lacked historical context. After a BSc in pure math I discovered that I enjoyed applied math and CS much more, which told me that I need concrete examples to understand a theory: if you tell me about abstractions like groups and rings, which took years to establish, I lose interest. Tell me that groups express properties of matrix multiplications, or permutations, or modular arithmetic, and I’ll get it right away. It’s the way my mind works, but I’m sure I’m not alone, and mathematical pedagogy would benefit from historical context.
silasdavis 2y ago
The maths appendices in Infinite Jest are about half the book. There's a section written by Hal's friend (?), proving something like the intermediate value theorem where he says something like: we'll use epsilon delta because it's mad fun to say.

ForOldHack 2y ago

Chapter X is brilliant. Galois was very brilliant. He is always my #1 choice for ever of who, if anyone from history, I would like to have lunch with. He was introduced to me by my 5th grade calculus teacher Mr Steven Giavant, PhD. Galois bridged several disciplines and invented a new area of math. Most unfortunately he met his end extremely tragically. https://en.wikipedia.org/wiki/%C3%89variste_Galois https://en.wikipedia.org/wiki/%C3%89variste_Galois

tzs 2y ago
> He was introduced to me by my 5th grade calculus teacher [...] 5th grade calculus!?

XMPPwocky 2y ago

Absolutely agree. When I was a kid, I took a MOOC (MITx 6.002x) because I was interested in EE. They said calculus was a prerequisite, which I didn't know yet, but I went ahead anyways; every time the course required calculus, I'd go read the relevant chapters in Strang's Calculus. Felt incredibly natural, and I ended up going through the rest of Strang just for fun- probably the best learning experience I've ever had, and I doubt I'd have ever done it without having the MOOC there to motivate the problem.

raegis 2y ago

> I've been shouting from the rooftops for years that math[0] courses need more context. A slightly different perspective: It used to be common for professors to teach a lot of history in graduate algebra courses. (Generation X and older.) I've talked to students from a few other schools who experienced this. None of us cared for it at the time. My teacher spent at least two class periods writing down the history on the board. We were all quite bored. But then again, we were in math grad school, so there was no need to try to motivate us.