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The top-rated answer is either defeatist, or just rationalization for the sentiment "I had to go through this and figure out everything myself, so you should to

The top-rated answer is either defeatist, or just rationalization for the sentiment "I had to go through this and figure out everything myself, so you should too". There is a huge amount of information encoded in the choice of exactly how to define thing, and which theorems people care about. This reflects a long process of trial-and-error as the field was constructed. For a famous philosophical treatise on this using the Euler characteristic as an example, see "Proofs and Refutations” by Imre Lakatos https://en.wikipedia.org/wiki/Proofs_and_Refutations https://en.wikipedia.org/wiki/Proofs_and_Refutations Part 1: https://math.berkeley.edu/~kpmann/Lakatos.pdf https://math.berkeley.edu/~kpmann/Lakatos.pdf Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject. It's a huge problem, it applies to physics textbooks too, and it doesn't have to be this way. Unfortunately, the problem has been known for decades and there's not much reason to expect things to change. (Lakatos wrote the above in 1976.)

derefr 10y ago
> This reflects a long process of trial-and-error as the field was constructed. A way to think about this process: "math" isn't quite a tower of abstract concepts; those only exist in any given mathematician's head, and die with them. Instead, "math" is a name for the novel pieces of language we invent, with the purpose of using them to talk about and (hopefully) share those internal abstract concepts, exposing to the rest of the world concepts that were entirely inaccessible (to everyone but the originator) before that language was introduced. Math is the attempt to communicate never-before-communicated epiphanies; to describe the shapes of never-before-pondered abstractions, with properties only describable using never-before-spoken words. One thing that this implies, is that the first piece of language that lets us even vaguely point at some particular idea so that we can get to work on analyzing it further, might stick around long after we come up with some clearer or more coherent language, because the former one now has the momentum of historical use behind it. Math, when done this way, then becomes a precarious pile of "just good enough to survive" conceptualizations, rather than a precise tower of "best-tool-for-the-job" conceptualizations. And that's no good for teaching.
- mrout 10y ago
  That's a nice sounding, appealing-to-read philosophy. But I'm not sure that it's actually true at all.
- libeclipse 10y ago
  That is a beautiful definition, thank you.
- DonaldFoss 10y ago
  This applies to _most_ disciplines, not just math or math related fields.
  - derefr 10y ago
    I'm not so sure. In most fields, we certainly discover new things (concepts, relationships, laws.) But we can usually describe those new things by analogy to existing things, because—given enough abstraction and analogy—they have recognizable "shapes." There are existing mental tools in our civilization that fit the concepts, and let us toss them around and look at them from all the angles. Math is where we invent language to refer to new forms of abstraction themselves: novel possible shapes for our thoughts to take when we think about other things. You can't talk about a new shape by analogy to existing shapes. Nor can you abstract an abstraction in a way that gives you anything more familiar. (Instead, you'll usually get more novel, ontologically-primitive abstractions, like going from monads to arrows, or going from numbers to fields to rings.) Sometimes disciplines like physics will find a concept that we don't have any mental tools in our toolkit for yet. Then we build some. But we still refer to the process of doing that as doing mathematics—and then we apply that new mathematics back in the problem domain to talk about the new concept. (For a good example, the formalization of quantum theory in physics, required the creation of infinite dimensional analysis in mathematics. Physicists did most of that work, but the work itself was still mathematics, not physics.) --- Now, other fields do still have a similar problem to mathematics, of historical momentum carrying forward old "things" (again: concepts, relationships, laws) when there are better, clearer "things" that could be used in their place. But when we're not working with pure abstractions—ways of thinking—we can make the effort to compare and contrast old and new "things", and decide that some might be more edifying than others. It is possible for an especially-gifted physics teacher to write a very accessible physics textbook, because they need only pick all the clearest "things" to demonstrate. That teacher will still be stuck in a given paradigm—a way of thinking, a belief in the worth of some "things" over others, popular in the culture of their discipline at the time and place they worked. But they might be able to (barely) rise above it, if they think hard about the history of their discipline and the paradigms it has gone through in the past, compare-and-contrast those, and synthesize something that isn't quite just the paradigm they're immersed in. Mathematics is uniquely problematic because it is entirely paradigm. It is a tree of paradigms—each new abstraction only making sense assuming the paradigm it was created in, and then becoming the paradigm for further abstractions still. Every mathematician, all the time, is trying to discover what a particular paradigm—their specialization—can be twisted to accomplish. Not one of us has the brain power to know the total space of things that one of these ways of thinking can be used to express—the problem domains the tool is applicable to—in order to know which tools show more or less promise at being "powerful." We know what we've discovered so far, but we have such an infinitesimal idea of the "space of all possible abstractions" that we could be totally missing some of the best, and using ones that are barely satisfactory. Point a hypercomputer AI at "solving physics", and it'll spit out a description of the universe that will certainly have more "things" in it than we know about today—but which will still also contain a subset of the "things" we do have. (The most "carving nature at its joints"ing ones, presumably.) Those "things" that get carried over will, of course, be defined much more precisely; their concept-boundaries will be adjusted to include or exclude edge-cases we weren't aware of. But they'll be the same "things." Point a hypercomputer AI at "solving mathematics"—of giving us the most powerful abstractions possible to solve problems with—and the result might be entirely unlike anything we've invented so far. It might not share any resemblance to current mathematics, beyond a shared fondness for sets. (For a silly hypothetical: it might turn out that there's a better abstraction for thinking about relationships between things than "functions", and that all the rest of mathematics and physics and whatever other problem domains simplify greatly if we think about the relationships in them in terms of that abstraction, instead of trying to specify relationships using the "function" abstraction.)
deleted 10y ago
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superobserver 10y ago
Highly recommended book by Lakatos. The history and meta-mathematical aspects of proof-theoretic constructs are so often neglected that one can only imagine that there's a latent desire to attain a priestly purity to the proof-theoretic process that serves only to hamper true mathematical maturity in the field and individual learning.
watwut 10y ago
"I had to go through this and figure out everything myself, so you should too". Some of us like it and don't see value in the long winded motivational style. Things like group theory are targeted on people who are interested in math itself and like the mathematics for what it really is. It is interesting to learn history once in a while and there are good books about that. However, most of the time you want to move on faster.
vog 10y ago
> The top-rated answer is ... Please use absolute references (a link to the specific answer) instead of fuzzy references that change over time. Currently, your answer is the top-rated, so which one did you mean? (And even if your answer was not at the top, the top will almost certainly change over time, leading to all kinds of misunderstandings.) EDIT: It seems that this referred to the top-rated answer on MO, not HN: https://mathoverflow.net/a/13149/66043 https://mathoverflow.net/a/13149/66043 This demonstrates my point even more: Please use absolute references, as fuzzy references make for misunderstandings! (And as my comment received quit a lot of upvotes in the beginning, I'm pretty sure I'm not the only one who thought this was about the top-rated HN answer.)
- Tistron 10y ago
  I'm pretty sure he means the top rated answer in the linked article. Still valid that rating ratios can change though.
- stormbrew 10y ago
  I took their post to be referring to the top reply on the math exchange site, which may actually be a stable thing at this point as the question is not fresh. I may be wrong though.
- triplesec 10y ago
  I took it to mean the (then) top rated answerhere on HN, so I'm glad of clarification.
  - vog 10y ago
    Why all these downvotes? What's wrong with criticizing ambiguous references?
    - triplesec 10y ago
      Humans are strange. Perhaps, though it's not certain, if your original comment were couched as a softer request, then that softer approach might not pique certain small subsets so much?
barking 10y ago
I can remember feeling really intimidated by the proofs of theorems. What I now think is that at least some proofs are analogous to code you struggle when writing to get something working. Later, once you'd gotten the thing done you'd realised that there were steps that could be eliminated and still others that could be cleverly combined. Then later again when you wanted to show it off you tidied it up some more and what you then presented to others is much more elegant but perhaps harder to understand.
- balabaster 10y ago
  It's becoming more obvious to me with time that math and its corresponding proofs are very similar in concept to our code and unit tests. Theorems like our software are built from rudimentary building blocks, each needing to be understood in order to build the software and then the tests are the proofs that each of the building blocks are correct... What's missing for a lot of people is "What does this theorem apply to? Why does it apply? How do I break this theorem down into its component pieces to understand it?" Indeed if your math text started off with a concept of a real world thing you're trying to figure out and then break it down to a series of paradigms that the student understands before applying each of the mathematical components that applies to each of these paradigms before finally combining them all to the final theorem, people would get it far more easily and be much less intimidated by it. Then over time students will spot paradigms that combine more elegantly to the final theorem and understand how to substitute them. I think Math is being taught backwards in schools and this is why so many people are intimidated by and shy away from math... and I was a student who both struggled and enjoyed Math... and I still struggle with it because we were taught to think about it backwards. We started out well with the basic building blocks, but somewhere between basic algebra and calculus, this all got turn around ass backwards. Where does the motivation to understand derivatives and integration come from if one doesn't understand the implications of their application?
  - gjulianm 10y ago
    That's a somewhat naive way to say it. Math is not code, it's not programming. You can code a Python script without knowing how does a operating system work. You cannot understand nor use a theorem without knowing the definitions involved. I can tell people, for example, that the concept of derivatives and integration comes very handily when solving differential equations that model real world phenomena (for example, fluid motion). But how can I explain what is a differential equation in a manner that's useful to someone who doesn't know what a derivative is? Math, for better or worse, tends to not have direct applications (on the other side, much like other subjects in their basic stages) and you need to advance before getting to a point where you learn something you really need and then see the usefulness of it all. It's like everything: you don't know what you need and how strongly you need it (despite how many times people will tell you that you need it) until you really have that necessity. Starting off with the general concept and breaking down into paradigms is not always possible. For example, say you want to teach people the Stokes' theorem of integration of manifolds [1]. It is the generalization of the divergence, Kelvin-Stokes and Green theorems that are very useful in electromagnetis, fluid mechanics and probably more. The theorem says, broadly, that if you are measuring some quantity in a certain space, you only need to know how does it "accumulate" on the boundaries of that space. But that will not be useful to an engineer or physicist. To really understand that theorem and break it down you need an entire course on differential geometry, with requisites on topology and calculus. Therefore, in the analysis courses for engineers/physicists the most probable situation is that either they present you the theorem and tell you how to perform the required operations on differential forms (which, if you don't know what they are, is a complete mistery/dark magic), or they go to the specific theorems (Gauss, Kelvin-Stokes, Green) which seem clunky and completely magic again. The summary is that to be able to know what does a theorem apply to, why does it and how to break it, you literally need to study mathematics: that's what mathematics is. And the way to study it is always backwards, it's not like physics or programming where they can show you the high-level stuff, you can see what's happening and then you start to understand it. In math, most of the time, if you learn some subject starting from the high level stuff, it is difficult to even know what on earth are they talking about. 1: https://en.wikipedia.org/wiki/Stokes%27_theorem https://en.wikipedia.org/wiki/Stokes%27_theorem
audunw 10y ago
> It's a huge problem, it applies to physics textbooks too, and it doesn't have to be this way. I've noticed this too. I've tried to learn some quantum physics for fun, and it seems to me that most textbooks have surprisingly little description of the actual physical experiments behind the physics. Maybe this is something they usually cover more in lectures? Like, I haven't found much details about the actual physical experiments around the spin of electrons (and certainly not of other particles). Sometimes there's an abstract description of a basic experimental setup. I read just a few days ago that Feynman would redo old experiments that other physicist weren't doing anymore because they were established science. This is a great learning experience if you have access to the equipment, but most people don't. There should be books that describe how physics was developed, experiment by experiment, with lots of details.
- wayn3 10y ago
  People severely underestimate how much detail is going into science. If you read a 1000 page textbook on experimental physics, thats just the cliffnotes. The very highest level "here are the results of 400 years of serious research". There can't be any detail. If you want the details on the stern gerlach experiment, for example, read the research papers. They will contain all the detail necessary to reproduce the experiment. Lectures are just like those books. They just teach a very broad overview of what has been discovered. The books you think should exist do exist. They're just not written for casuals.
  - moron4hire 10y ago
    The vast majority of research papers are barely readable by people in the field, say nothing about a beginner.
    - wayn3 10y ago
      research papers can all be read by "people in the field". its just that someone who is doing environmental physics doesn't necessarily understand papers on string theory. science is specialized, yea. if you want to understand research papers, get an education. or just do something else. cutting edge research is not for everyone. not everyone is supposed to understand it. whats next? people demanding research papers be dumbed down to adhere to the freedom of information act? science is hard. big deal. you cant have all the things at the same time. read the hawking books written for casuals if you want a cute little story about science.
infinity0 10y ago
> Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject. From "A Mathematician's Apology", G. H. Hardy: > Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. From another angle: when developing new theories or models, your thoughts are all over the place and frankly it's boring to go over your own crappy notes afterwards and try to reconstruct them in a way that others can understand. And much of the time you forget exactly what happened along the way as well, so any story you reconstruct is going to have some hindsight bias, which defeats the purpose of trying to "teach the story". Really, from the first answer: > Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle". This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something.
- jessriedel 10y ago
  >> Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle". > This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something. My point is not just that most textbooks make it too hard/inefficient to learn this stuff. My point is that most of it is never learned. The Legendre transform connects the Lagrangian and Hamiltonian mechanics, the two fundamental formulations of both quantum and classical physics, and yet most physicist cannot tell you why the transform is defined as it is. The reason is they don't take seriously the possibility that we'll find non-Lagrangian phenomena, and so they have not been forced to consider what observational and theoretical evidence led to it's identification in the first place.
- cousin_it 10y ago
  I learned math through guided struggle. My high school had a kind of macho attitude about it, we were all about math competitions and pushing through university level stuff etc. Today I feel that was a good way to study, but much of the struggle could've been avoided. You must solve problems to progress, but they don't have to be hard problems. They just need to be formulated at level n but require solutions at level n+1. Devising such problems is hard and many teachers don't bother, instead they give you definitions at level n+1 right away and make you solve problems about those. That's the root of the problem IMO.
marcosscriven 10y ago
In case anyone else is looking to download this: 1) Search JSTOR: https://www.jstor.org/action/doBasicSearch?Query=proofs+and+refutations&filter= https://www.jstor.org/action/doBasicSearch?Query=proofs+and+... 2) Use Sci-Hub to download it (handy bookmarklet): javascript:document.location.hostname=document.location.hostname + ".sci-hub.cc";
andrepd 10y ago
Or maybe, just maybe, some things really are complicated and there is no quick and easy way to learn them. You really have to struggle and hit your head several times until enlightment comes to you. Books and teachers can only do so much, they can help you and guide you, but often there simply does not exist any magical way to convey stuff to make you instantly understand it.
- FabHK 10y ago
  While that is true, there is really no need to make it (deliberately?) harder by withholding relevant examples, applications, and historical background.
  - watwut 10y ago
    No one is deliberately withholding them out of malice. Practical applications are often convoluted and complicated or too forced. It is like wanting to do triple flips on trampoline before you can jump to sit on it. The sport analogy hold for historical background too - it sometimes help or makes it more interesting, but most often helps as much as knowing history of flips helps to learn flips. Contrary to what you say, practical applications are easy to be found where they are simple and helpful. They make for fun math hobbyist like to play with and that works nice in terms of making kids interested in math. Unfortunately they do not lead to cutting edge science nor math needed for physics and engineering.
mebassett 10y ago
The top-rated answer states that "the goal is to learn how to do mathematics, not to 'know' it". I don't think that's a defeatist sentiment or a rationalization for the state-of-things. It's certainly true that one would gain more insight that shows how modern-maths-as-we-know-it evolved from a centuries' long struggle to understand basic patterns (exempli gratia, Cox's "Primes of the Form x^2 + ny^2"). But it is also true that, when you are doing mathematics, such a book is not nearly as helpful as an impenetrable tome like Neukirch's Algebraic Number Theory. From my own experience (which, I'll admit, is not much to extrapolate from), the former was great for its "insight" and the latter just had too much unnecessary "information". Until I got stuck in an actual maths problem. Then suddenly my opinion switched, and the latter became a trusted map through strange lands. You're right about the amount of information that's encoded in a definition or theorem. And it is very difficult to portray why that information is important (never mind how to access it, see other's comments about scaffolding and cathedrals and the like). I fear that trying to describe why groups are defined like they are would just lead to more impenetrable tomes - but instead of being maps to the lost traveler, they'd be more like tourist adverts. But the questions the source was asking is "why are books like this?" and "how do others [learn maths] in this situation?" Well, maths books are like this because they are guides to help people doing maths. And you learn maths by doing it. I don't think its defeatist to say that no book or lecture about riding a bike would compare to the experience of actually riding a bike. To paraphrase the top-rated answer, why would you expect anything different from maths? But to answer the underlying question of both yourself and the source: how do we improve this? I think pedagogy should focus on getting to "do maths" faster. You really learn by getting stuck in a problem.
- jessriedel 10y ago
  > I don't think its defeatist to say that no book or lecture about riding a bike would compare to the experience of actually riding a bike. To paraphrase the top-rated answer, why would you expect anything different from maths?] This it the wrong analogy and does not capture the MO answer. The correct analogy would be training vehicle designers by teaching them all the parts necessary to build a modern bicycle without teaching them any of the ways early bicycles were designed and why those designs were discarded.
ethan_g 10y ago
> The top-rated answer is either defeatist, or just rationalization for the sentiment "I had to go through this and figure out everything myself, so you should too". Yes, I agree this MO answer (and several others) seem to be rationalizing the situation rather than acknowledging how suboptimal it is. I think I'm a good example of the system's failure. Coming in to college, I was something of an ideal candidate for becoming a mathematician. I had some success in Olympiads and already had decided I wanted to study math. My goal in life was to be a math professor. I enrolled in a top college and took many graduate-level classes. However, by my senior year, when it was time for me to decide the next step in life, grad school or industry, I had become somewhat disenchanted with theoretical math. Math was so abstract I started losing interest: all this commutative algrebra (for example) I learned wasn't making me feel like I had any new insights into solving math problems, outside commutative algebra problem sets. And so I went into industry. However, I can't help but think that if I had more knowledge of the motivation behind all the abstract math, I wouldn't have lost interest. All that machinery of commutative algebra was invented for specific reasons, such as solving polynomial equations in the rationals through algebraic geometry. Years later, through casually reading math on the internet, I've been getting hints as to what power these highly abstract frameworks give you for solving concrete problems. But without seeing the end goal, and having some idea why I should be learning this in the first place, I felt like I was just getting lost in abstract nonsense. >Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject. This is exactly right. I personally failed to reconstruct enough to keep myself interested in the subject.

derefr 10y ago

> This reflects a long process of trial-and-error as the field was constructed. A way to think about this process: "math" isn't quite a tower of abstract concepts; those only exist in any given mathematician's head, and die with them. Instead, "math" is a name for the novel pieces of language we invent, with the purpose of using them to talk about and (hopefully) share those internal abstract concepts, exposing to the rest of the world concepts that were entirely inaccessible (to everyone but the originator) before that language was introduced. Math is the attempt to communicate never-before-communicated epiphanies; to describe the shapes of never-before-pondered abstractions, with properties only describable using never-before-spoken words. One thing that this implies, is that the first piece of language that lets us even vaguely point at some particular idea so that we can get to work on analyzing it further, might stick around long after we come up with some clearer or more coherent language, because the former one now has the momentum of historical use behind it. Math, when done this way, then becomes a precarious pile of "just good enough to survive" conceptualizations, rather than a precise tower of "best-tool-for-the-job" conceptualizations. And that's no good for teaching.

mrout 10y ago
That's a nice sounding, appealing-to-read philosophy. But I'm not sure that it's actually true at all.
libeclipse 10y ago
That is a beautiful definition, thank you.
DonaldFoss 10y ago
This applies to _most_ disciplines, not just math or math related fields.
- derefr 10y ago
  I'm not so sure. In most fields, we certainly discover new things (concepts, relationships, laws.) But we can usually describe those new things by analogy to existing things, because—given enough abstraction and analogy—they have recognizable "shapes." There are existing mental tools in our civilization that fit the concepts, and let us toss them around and look at them from all the angles. Math is where we invent language to refer to new forms of abstraction themselves: novel possible shapes for our thoughts to take when we think about other things. You can't talk about a new shape by analogy to existing shapes. Nor can you abstract an abstraction in a way that gives you anything more familiar. (Instead, you'll usually get more novel, ontologically-primitive abstractions, like going from monads to arrows, or going from numbers to fields to rings.) Sometimes disciplines like physics will find a concept that we don't have any mental tools in our toolkit for yet. Then we build some. But we still refer to the process of doing that as doing mathematics—and then we apply that new mathematics back in the problem domain to talk about the new concept. (For a good example, the formalization of quantum theory in physics, required the creation of infinite dimensional analysis in mathematics. Physicists did most of that work, but the work itself was still mathematics, not physics.) --- Now, other fields do still have a similar problem to mathematics, of historical momentum carrying forward old "things" (again: concepts, relationships, laws) when there are better, clearer "things" that could be used in their place. But when we're not working with pure abstractions—ways of thinking—we can make the effort to compare and contrast old and new "things", and decide that some might be more edifying than others. It is possible for an especially-gifted physics teacher to write a very accessible physics textbook, because they need only pick all the clearest "things" to demonstrate. That teacher will still be stuck in a given paradigm—a way of thinking, a belief in the worth of some "things" over others, popular in the culture of their discipline at the time and place they worked. But they might be able to (barely) rise above it, if they think hard about the history of their discipline and the paradigms it has gone through in the past, compare-and-contrast those, and synthesize something that isn't quite just the paradigm they're immersed in. Mathematics is uniquely problematic because it is entirely paradigm. It is a tree of paradigms—each new abstraction only making sense assuming the paradigm it was created in, and then becoming the paradigm for further abstractions still. Every mathematician, all the time, is trying to discover what a particular paradigm—their specialization—can be twisted to accomplish. Not one of us has the brain power to know the total space of things that one of these ways of thinking can be used to express—the problem domains the tool is applicable to—in order to know which tools show more or less promise at being "powerful." We know what we've discovered so far, but we have such an infinitesimal idea of the "space of all possible abstractions" that we could be totally missing some of the best, and using ones that are barely satisfactory. Point a hypercomputer AI at "solving physics", and it'll spit out a description of the universe that will certainly have more "things" in it than we know about today—but which will still also contain a subset of the "things" we do have. (The most "carving nature at its joints"ing ones, presumably.) Those "things" that get carried over will, of course, be defined much more precisely; their concept-boundaries will be adjusted to include or exclude edge-cases we weren't aware of. But they'll be the same "things." Point a hypercomputer AI at "solving mathematics"—of giving us the most powerful abstractions possible to solve problems with—and the result might be entirely unlike anything we've invented so far. It might not share any resemblance to current mathematics, beyond a shared fondness for sets. (For a silly hypothetical: it might turn out that there's a better abstraction for thinking about relationships between things than "functions", and that all the rest of mathematics and physics and whatever other problem domains simplify greatly if we think about the relationships in them in terms of that abstraction, instead of trying to specify relationships using the "function" abstraction.)

deleted 10y ago

[deleted]

superobserver 10y ago

Highly recommended book by Lakatos. The history and meta-mathematical aspects of proof-theoretic constructs are so often neglected that one can only imagine that there's a latent desire to attain a priestly purity to the proof-theoretic process that serves only to hamper true mathematical maturity in the field and individual learning.

watwut 10y ago

"I had to go through this and figure out everything myself, so you should too". Some of us like it and don't see value in the long winded motivational style. Things like group theory are targeted on people who are interested in math itself and like the mathematics for what it really is. It is interesting to learn history once in a while and there are good books about that. However, most of the time you want to move on faster.

vog 10y ago

> The top-rated answer is ... Please use absolute references (a link to the specific answer) instead of fuzzy references that change over time. Currently, your answer is the top-rated, so which one did you mean? (And even if your answer was not at the top, the top will almost certainly change over time, leading to all kinds of misunderstandings.) EDIT: It seems that this referred to the top-rated answer on MO, not HN: https://mathoverflow.net/a/13149/66043 https://mathoverflow.net/a/13149/66043 This demonstrates my point even more: Please use absolute references, as fuzzy references make for misunderstandings! (And as my comment received quit a lot of upvotes in the beginning, I'm pretty sure I'm not the only one who thought this was about the top-rated HN answer.)

Tistron 10y ago
I'm pretty sure he means the top rated answer in the linked article. Still valid that rating ratios can change though.
stormbrew 10y ago
I took their post to be referring to the top reply on the math exchange site, which may actually be a stable thing at this point as the question is not fresh. I may be wrong though.
triplesec 10y ago
I took it to mean the (then) top rated answerhere on HN, so I'm glad of clarification.
- vog 10y ago
  Why all these downvotes? What's wrong with criticizing ambiguous references?
  - triplesec 10y ago
    Humans are strange. Perhaps, though it's not certain, if your original comment were couched as a softer request, then that softer approach might not pique certain small subsets so much?

barking 10y ago

I can remember feeling really intimidated by the proofs of theorems. What I now think is that at least some proofs are analogous to code you struggle when writing to get something working. Later, once you'd gotten the thing done you'd realised that there were steps that could be eliminated and still others that could be cleverly combined. Then later again when you wanted to show it off you tidied it up some more and what you then presented to others is much more elegant but perhaps harder to understand.

balabaster 10y ago
It's becoming more obvious to me with time that math and its corresponding proofs are very similar in concept to our code and unit tests. Theorems like our software are built from rudimentary building blocks, each needing to be understood in order to build the software and then the tests are the proofs that each of the building blocks are correct... What's missing for a lot of people is "What does this theorem apply to? Why does it apply? How do I break this theorem down into its component pieces to understand it?" Indeed if your math text started off with a concept of a real world thing you're trying to figure out and then break it down to a series of paradigms that the student understands before applying each of the mathematical components that applies to each of these paradigms before finally combining them all to the final theorem, people would get it far more easily and be much less intimidated by it. Then over time students will spot paradigms that combine more elegantly to the final theorem and understand how to substitute them. I think Math is being taught backwards in schools and this is why so many people are intimidated by and shy away from math... and I was a student who both struggled and enjoyed Math... and I still struggle with it because we were taught to think about it backwards. We started out well with the basic building blocks, but somewhere between basic algebra and calculus, this all got turn around ass backwards. Where does the motivation to understand derivatives and integration come from if one doesn't understand the implications of their application?
- gjulianm 10y ago
  That's a somewhat naive way to say it. Math is not code, it's not programming. You can code a Python script without knowing how does a operating system work. You cannot understand nor use a theorem without knowing the definitions involved. I can tell people, for example, that the concept of derivatives and integration comes very handily when solving differential equations that model real world phenomena (for example, fluid motion). But how can I explain what is a differential equation in a manner that's useful to someone who doesn't know what a derivative is? Math, for better or worse, tends to not have direct applications (on the other side, much like other subjects in their basic stages) and you need to advance before getting to a point where you learn something you really need and then see the usefulness of it all. It's like everything: you don't know what you need and how strongly you need it (despite how many times people will tell you that you need it) until you really have that necessity. Starting off with the general concept and breaking down into paradigms is not always possible. For example, say you want to teach people the Stokes' theorem of integration of manifolds [1]. It is the generalization of the divergence, Kelvin-Stokes and Green theorems that are very useful in electromagnetis, fluid mechanics and probably more. The theorem says, broadly, that if you are measuring some quantity in a certain space, you only need to know how does it "accumulate" on the boundaries of that space. But that will not be useful to an engineer or physicist. To really understand that theorem and break it down you need an entire course on differential geometry, with requisites on topology and calculus. Therefore, in the analysis courses for engineers/physicists the most probable situation is that either they present you the theorem and tell you how to perform the required operations on differential forms (which, if you don't know what they are, is a complete mistery/dark magic), or they go to the specific theorems (Gauss, Kelvin-Stokes, Green) which seem clunky and completely magic again. The summary is that to be able to know what does a theorem apply to, why does it and how to break it, you literally need to study mathematics: that's what mathematics is. And the way to study it is always backwards, it's not like physics or programming where they can show you the high-level stuff, you can see what's happening and then you start to understand it. In math, most of the time, if you learn some subject starting from the high level stuff, it is difficult to even know what on earth are they talking about. 1: https://en.wikipedia.org/wiki/Stokes%27_theorem https://en.wikipedia.org/wiki/Stokes%27_theorem

audunw 10y ago

> It's a huge problem, it applies to physics textbooks too, and it doesn't have to be this way. I've noticed this too. I've tried to learn some quantum physics for fun, and it seems to me that most textbooks have surprisingly little description of the actual physical experiments behind the physics. Maybe this is something they usually cover more in lectures? Like, I haven't found much details about the actual physical experiments around the spin of electrons (and certainly not of other particles). Sometimes there's an abstract description of a basic experimental setup. I read just a few days ago that Feynman would redo old experiments that other physicist weren't doing anymore because they were established science. This is a great learning experience if you have access to the equipment, but most people don't. There should be books that describe how physics was developed, experiment by experiment, with lots of details.

wayn3 10y ago
People severely underestimate how much detail is going into science. If you read a 1000 page textbook on experimental physics, thats just the cliffnotes. The very highest level "here are the results of 400 years of serious research". There can't be any detail. If you want the details on the stern gerlach experiment, for example, read the research papers. They will contain all the detail necessary to reproduce the experiment. Lectures are just like those books. They just teach a very broad overview of what has been discovered. The books you think should exist do exist. They're just not written for casuals.
- moron4hire 10y ago
  The vast majority of research papers are barely readable by people in the field, say nothing about a beginner.
  - wayn3 10y ago
    research papers can all be read by "people in the field". its just that someone who is doing environmental physics doesn't necessarily understand papers on string theory. science is specialized, yea. if you want to understand research papers, get an education. or just do something else. cutting edge research is not for everyone. not everyone is supposed to understand it. whats next? people demanding research papers be dumbed down to adhere to the freedom of information act? science is hard. big deal. you cant have all the things at the same time. read the hawking books written for casuals if you want a cute little story about science.

infinity0 10y ago

> Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject. From "A Mathematician's Apology", G. H. Hardy: > Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. From another angle: when developing new theories or models, your thoughts are all over the place and frankly it's boring to go over your own crappy notes afterwards and try to reconstruct them in a way that others can understand. And much of the time you forget exactly what happened along the way as well, so any story you reconstruct is going to have some hindsight bias, which defeats the purpose of trying to "teach the story". Really, from the first answer: > Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle". This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something.

jessriedel 10y ago
>> Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle". > This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something. My point is not just that most textbooks make it too hard/inefficient to learn this stuff. My point is that most of it is never learned. The Legendre transform connects the Lagrangian and Hamiltonian mechanics, the two fundamental formulations of both quantum and classical physics, and yet most physicist cannot tell you why the transform is defined as it is. The reason is they don't take seriously the possibility that we'll find non-Lagrangian phenomena, and so they have not been forced to consider what observational and theoretical evidence led to it's identification in the first place.
cousin_it 10y ago
I learned math through guided struggle. My high school had a kind of macho attitude about it, we were all about math competitions and pushing through university level stuff etc. Today I feel that was a good way to study, but much of the struggle could've been avoided. You must solve problems to progress, but they don't have to be hard problems. They just need to be formulated at level n but require solutions at level n+1. Devising such problems is hard and many teachers don't bother, instead they give you definitions at level n+1 right away and make you solve problems about those. That's the root of the problem IMO.

marcosscriven 10y ago

In case anyone else is looking to download this: 1) Search JSTOR: https://www.jstor.org/action/doBasicSearch?Query=proofs+and+refutations&filter= https://www.jstor.org/action/doBasicSearch?Query=proofs+and+... 2) Use Sci-Hub to download it (handy bookmarklet): javascript:document.location.hostname=document.location.hostname + ".sci-hub.cc";

andrepd 10y ago

Or maybe, just maybe, some things really are complicated and there is no quick and easy way to learn them. You really have to struggle and hit your head several times until enlightment comes to you. Books and teachers can only do so much, they can help you and guide you, but often there simply does not exist any magical way to convey stuff to make you instantly understand it.

FabHK 10y ago
While that is true, there is really no need to make it (deliberately?) harder by withholding relevant examples, applications, and historical background.
- watwut 10y ago
  No one is deliberately withholding them out of malice. Practical applications are often convoluted and complicated or too forced. It is like wanting to do triple flips on trampoline before you can jump to sit on it. The sport analogy hold for historical background too - it sometimes help or makes it more interesting, but most often helps as much as knowing history of flips helps to learn flips. Contrary to what you say, practical applications are easy to be found where they are simple and helpful. They make for fun math hobbyist like to play with and that works nice in terms of making kids interested in math. Unfortunately they do not lead to cutting edge science nor math needed for physics and engineering.

mebassett 10y ago

The top-rated answer states that "the goal is to learn how to do mathematics, not to 'know' it". I don't think that's a defeatist sentiment or a rationalization for the state-of-things. It's certainly true that one would gain more insight that shows how modern-maths-as-we-know-it evolved from a centuries' long struggle to understand basic patterns (exempli gratia, Cox's "Primes of the Form x^2 + ny^2"). But it is also true that, when you are doing mathematics, such a book is not nearly as helpful as an impenetrable tome like Neukirch's Algebraic Number Theory. From my own experience (which, I'll admit, is not much to extrapolate from), the former was great for its "insight" and the latter just had too much unnecessary "information". Until I got stuck in an actual maths problem. Then suddenly my opinion switched, and the latter became a trusted map through strange lands. You're right about the amount of information that's encoded in a definition or theorem. And it is very difficult to portray why that information is important (never mind how to access it, see other's comments about scaffolding and cathedrals and the like). I fear that trying to describe why groups are defined like they are would just lead to more impenetrable tomes - but instead of being maps to the lost traveler, they'd be more like tourist adverts. But the questions the source was asking is "why are books like this?" and "how do others [learn maths] in this situation?" Well, maths books are like this because they are guides to help people doing maths. And you learn maths by doing it. I don't think its defeatist to say that no book or lecture about riding a bike would compare to the experience of actually riding a bike. To paraphrase the top-rated answer, why would you expect anything different from maths? But to answer the underlying question of both yourself and the source: how do we improve this? I think pedagogy should focus on getting to "do maths" faster. You really learn by getting stuck in a problem.

jessriedel 10y ago
> I don't think its defeatist to say that no book or lecture about riding a bike would compare to the experience of actually riding a bike. To paraphrase the top-rated answer, why would you expect anything different from maths?] This it the wrong analogy and does not capture the MO answer. The correct analogy would be training vehicle designers by teaching them all the parts necessary to build a modern bicycle without teaching them any of the ways early bicycles were designed and why those designs were discarded.

ethan_g 10y ago

> The top-rated answer is either defeatist, or just rationalization for the sentiment "I had to go through this and figure out everything myself, so you should too". Yes, I agree this MO answer (and several others) seem to be rationalizing the situation rather than acknowledging how suboptimal it is. I think I'm a good example of the system's failure. Coming in to college, I was something of an ideal candidate for becoming a mathematician. I had some success in Olympiads and already had decided I wanted to study math. My goal in life was to be a math professor. I enrolled in a top college and took many graduate-level classes. However, by my senior year, when it was time for me to decide the next step in life, grad school or industry, I had become somewhat disenchanted with theoretical math. Math was so abstract I started losing interest: all this commutative algrebra (for example) I learned wasn't making me feel like I had any new insights into solving math problems, outside commutative algebra problem sets. And so I went into industry. However, I can't help but think that if I had more knowledge of the motivation behind all the abstract math, I wouldn't have lost interest. All that machinery of commutative algebra was invented for specific reasons, such as solving polynomial equations in the rationals through algebraic geometry. Years later, through casually reading math on the internet, I've been getting hints as to what power these highly abstract frameworks give you for solving concrete problems. But without seeing the end goal, and having some idea why I should be learning this in the first place, I felt like I was just getting lost in abstract nonsense. >Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject. This is exactly right. I personally failed to reconstruct enough to keep myself interested in the subject.