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The article is sparse on what a detailed map for mathematics would look like and merely points out that some topics have related techniques for solving them. I

The article is sparse on what a detailed map for mathematics would look like and merely points out that some topics have related techniques for solving them. I don't think a map for math techniques is feasible, but a map relating topics via mathematical steps is possible in Physics [1]. (Disclaimer: I'm the author of that map for Physics.) I think the reason that a map in Physics is feasible is Physicists do not use math techniques in the way mathematicians do, and the objectives are different. https://derivationmap.net/ https://derivationmap.net/

knzhou 6y ago
It seems to me that your map is far too detailed to use practically. You spell out every algebraic step, including stuff as simple as "divide both sides by T", so that deriving f = 1/T from T = 1/f takes about 10 nodes. This is like building a model train to a larger scale than an actual train -- what is the use? Education research tells us that what you actually want to do is the exact opposite: chunk as much as possible. You should learn algebra separately, and then use your preexisting knowledge of algebra to group f = 1/T and T = 1/f into one conceptual node. If you need 10 nodes every time something that basic is done, then your map will contain a vast amount of redundancy and be too large to use to get anywhere...
- physicsgraph 6y ago
  I agree that navigating a map of Physics at the very lowest level would not enlighten any student or researcher. My expectation in mapping atomic steps for a wide swath of the domain might enable insights not otherwise accessible. The chunking of atomic steps is what enables leaps in understanding. The mapping process starts with understanding each step.
  - knzhou 6y ago
    Well, I recommend doing a concrete, nontrivial derivation from start to finish just to see how this approach scales. As a basic example that is typically covered in about half a page in books, try doing a full derivation of the wave equation for a wave on a string. I would bet that once you set up the 1000 nodes required to do this, you'll be completely exhausted, and moreover will have gotten no new insight! If you're not tired yet, try deriving the equation describing waves on a stiff rod -- it'll take at least 1500 nodes, most of which will be exactly the same as the ones for the wave equation. Furthermore, this excessive mathematical structure hides the physical assumptions that really drive the validity of these equations. A real string doesn't actually obey the wave equation perfectly. The reason has to do with physical aspects of the string itself, not minutiae in the mathematical derivation of the wave equation. I can't think of an example where progress in physics was stalled because somebody tried to divide both sides of an equation by T and failed...
    - ZenOfTheArt 6y ago
      A Fitch derivation of the existence of the intersection of all members of a nonempty set is a better place to start because it can be done in less than ten sheets of paper longhand. The ratio of triviality to pages consumed is quite shocking when you finally confront it. It is at that point that you realize intuition has no formal translation but is vital since the level of detail seems to blur and darken intuition when holding a proof to the standard of formal derivation rather than the ordinary informal standard. So far, I’ve seen relatively little interest in mathematical intuition or even honest appraisal of what it is or how mathematicians should develop it. Rather the trend seems to be pretending that mathematical intuition doesn’t exist and treating formalization as a no-op. I think this is due to an anti-intellectual atmosphere that views mathematics as a source of problems for the military as opposed to pastimes for civilians.
    - Koshkin 6y ago
      This has reminded of the Two Capacitor Paradox [0]. (The moral of the story is that you have to know the limits of your model.) [0] https://en.wikipedia.org/wiki/Two_capacitor_paradox https://en.wikipedia.org/wiki/Two_capacitor_paradox
- 7373737373 6y ago
  Here's 2+2=4: https://twitter.com/dd4ta/status/1050433711416721408 https://twitter.com/dd4ta/status/1050433711416721408 I recently asked a related question regarding proof maps and quantifying their similarity/distance, but didn't get any answers: https://math.stackexchange.com/questions/3482135/are-there-projects-that-visualize-how-proofs-relate-to-each-other-similar-to-wh https://math.stackexchange.com/questions/3482135/are-there-p...

knzhou 6y ago

It seems to me that your map is far too detailed to use practically. You spell out every algebraic step, including stuff as simple as "divide both sides by T", so that deriving f = 1/T from T = 1/f takes about 10 nodes. This is like building a model train to a larger scale than an actual train -- what is the use? Education research tells us that what you actually want to do is the exact opposite: chunk as much as possible. You should learn algebra separately, and then use your preexisting knowledge of algebra to group f = 1/T and T = 1/f into one conceptual node. If you need 10 nodes every time something that basic is done, then your map will contain a vast amount of redundancy and be too large to use to get anywhere...

physicsgraph 6y ago
I agree that navigating a map of Physics at the very lowest level would not enlighten any student or researcher. My expectation in mapping atomic steps for a wide swath of the domain might enable insights not otherwise accessible. The chunking of atomic steps is what enables leaps in understanding. The mapping process starts with understanding each step.
- knzhou 6y ago
  Well, I recommend doing a concrete, nontrivial derivation from start to finish just to see how this approach scales. As a basic example that is typically covered in about half a page in books, try doing a full derivation of the wave equation for a wave on a string. I would bet that once you set up the 1000 nodes required to do this, you'll be completely exhausted, and moreover will have gotten no new insight! If you're not tired yet, try deriving the equation describing waves on a stiff rod -- it'll take at least 1500 nodes, most of which will be exactly the same as the ones for the wave equation. Furthermore, this excessive mathematical structure hides the physical assumptions that really drive the validity of these equations. A real string doesn't actually obey the wave equation perfectly. The reason has to do with physical aspects of the string itself, not minutiae in the mathematical derivation of the wave equation. I can't think of an example where progress in physics was stalled because somebody tried to divide both sides of an equation by T and failed...
  - ZenOfTheArt 6y ago
    A Fitch derivation of the existence of the intersection of all members of a nonempty set is a better place to start because it can be done in less than ten sheets of paper longhand. The ratio of triviality to pages consumed is quite shocking when you finally confront it. It is at that point that you realize intuition has no formal translation but is vital since the level of detail seems to blur and darken intuition when holding a proof to the standard of formal derivation rather than the ordinary informal standard. So far, I’ve seen relatively little interest in mathematical intuition or even honest appraisal of what it is or how mathematicians should develop it. Rather the trend seems to be pretending that mathematical intuition doesn’t exist and treating formalization as a no-op. I think this is due to an anti-intellectual atmosphere that views mathematics as a source of problems for the military as opposed to pastimes for civilians.
  - Koshkin 6y ago
    This has reminded of the Two Capacitor Paradox [0]. (The moral of the story is that you have to know the limits of your model.) [0] https://en.wikipedia.org/wiki/Two_capacitor_paradox https://en.wikipedia.org/wiki/Two_capacitor_paradox
7373737373 6y ago
Here's 2+2=4: https://twitter.com/dd4ta/status/1050433711416721408 https://twitter.com/dd4ta/status/1050433711416721408 I recently asked a related question regarding proof maps and quantifying their similarity/distance, but didn't get any answers: https://math.stackexchange.com/questions/3482135/are-there-projects-that-visualize-how-proofs-relate-to-each-other-similar-to-wh https://math.stackexchange.com/questions/3482135/are-there-p...