4 ms·

> fancy notation always gets in the way of understanding because people naturally think in words, not in spatial relationships. I'm pretty sure this is false f

by zeven7 10y ago

> fancy notation always gets in the way of understanding because people naturally think in words, not in spatial relationships. I'm pretty sure this is false for me. I think thinking in spacial relationships comes much more naturally to me than words. Words immediately start making my brain hurt. The notation you wrote is incredibly more painful for me to look at and understand than the triangles in the video.

lisper 10y ago
Let me rephrase that: people naturally communicate using language. In your head you may think in symbols and spatial relationships, but if you want to get a concept out of your head and into someone else's head you have to use language. There's a reason that the symbology in mathematical papers is invariably wrapped in natural language.
- lasfter 10y ago
  I still disagree. I wouldn't have understood half of my abstract algebra classes if the profs had only used words to explain the concepts. When you are first exposed to the First Isomorphism Theorem for Rings, a thousand words can't do what that simple triangle diagram can.
- jacobolus 10y ago
  This is entirely backwards. Mathematics is the study of patterns. Humans have amazing powerful visual and spatial reasoning centers in the brain which can process huge amounts of semi-structured information. Writing words down instead forces everything through low-bandwidth serialized language processing apparatus instead. Mathematical notation (and computer code, and music notation, and ...) benefit tremendously from even slightly exploiting spatial reasoning skills. But better is some kind of diagram, interactive simulation, or similar. The fastest mathematical explanation is direct tutorial, person-to-person, where the amount of symbolic formalism can be reduced to a minimum, and hand gestures, chalk drawings, colorful verbal analogies, etc. can convey most of the message. When limited to pure symbolic formalism in a scholarly paper, mathematics teaching and learning is a much harder slog. The reason formal mathematics uses symbolic formal language as a canonical way of recording results is that it’s easier to figure out how to verify it step-by-step and make sure that the logic is correct. * * * I’ll give you an example, from a paragraph a friend of mine wrote several years ago in a blog diary: (I posit that you could decipher and understand this paragraph at least an order of magnitude faster if I drew a 30 second napkin sketch.) > The tea room I went to is an “eight tatami design”, a large square space. Each side is two tatami lengths (four widths, two widths and a length) long. The eight tatami are arranged in the following way: from the entrance—which is in a corner, second forward on the left—two tatami extend into the room to meet the opposite wall on their short sides; the tatami touching the opposite wall is met on its right long side by a third tatami’s short side. This third tatami’s short side in turn meets a fourth tatami’s long side, whose two short sides meet the opposite wall and a fifth tatami’s short side. The fifth tatami’s second short side meets the room’s the fourth corner and its long side meets the sixth tatami’s short side. The sixth tatami’s second short side meets the first tatami’s long side. These six tatami form a square with an square empty middle. The empty middle’s sides’ dimensions are all one tatami length long. The middle, then, is filled by two tatami whose orientation is like that of tatami one, two, four and five—short sides to the opposite wall and the wall with the entrance. I tried out alternative arrangements and I might be wrong but I think that this solution (or a rotation of it) is the only possible tatami arrangement for the room’s dimensions that fulfills the rule “four corners must never converge”. Going in a circle around the room, starting at the door and following the direction in the description there, we can number the outer tatami one through six and the two inner tatami seven (left from the entrance) and eight (right from the entrance).
  - mannykannot 10y ago
    I am not sure that this is the clearest possible written explanation - "Each side is two tatami lengths (four widths, two widths and a length) long"? It is often the case that words + diagrams is the clearest way to explain something, and in most cases where you have a choice, you can choose both. Symbolic expressions fall somewhere between the two, and when used properly, further increase your chances of getting the point across. There's no need to turn every choice into an exclusive dichotomy, regardless of how common that is on HN.
  - dredmorbius 10y ago
    Comparing a less-than-optimal textual description with an optimsed symbolic or digrammatic description proves little. It helps to know that tatami are rectangles with length 2x width (something a casual and/or non-Japanese reader might reasonably not know). What's described is eight tatamis (length 2x their width), arranged in a square. Much of the confusion comes from describing each tatami's relation with its neighbors, rather than the space as a whole, and referencing other tatami rather than the room as a whole. A simpler description: A square room, four units per side, filled with eight mats, each one unit wide and two units long. Looking away from you, down either wall are two mats placed lenthways. On the near and far walls, a single mat between those, long edge to the wall. In the center square, two mats placed long edges parallel to the side walls. Or the 3600x3600 diagram in this image: http://www.tatami.com.my/layout.jpg http://www.tatami.com.my/layout.jpg The problem is your friend's blog description. Not inherent limits of language.
    - jacobolus 10y ago
      I agree my friend’s description is not the best. He was trying to make it clear and unambiguous, and he’s a smart guy, but he hadn’t practiced this before and I’m sure he didn’t think all that deeply about it. It’s just not a priori obvious how to make a clear verbal description of a spatial relationship. A diagram of the same thing takes much less effort to construct. I don’t think it’s possible to construct a textual description or mathematical formula of a collection of tatami patterns which takes less than 40–60 seconds per pattern to unpack into a clear mental image. If I spend about 5–10 seconds looking at each of the 14 tatami mat diagrams in your picture, I’m confident I could reconstruct the patterns from memory a few minutes later. I can’t imagine doing the same without spending a significant amount of time on a textual explanation. Additionally, after examining an image with a bunch of tatami patterns for a couple minutes, I could probably tell you whether additional patterns satisfy the criteria to be acceptable or not, without doing any explicit analysis of those criteria. I’d have to think a lot harder if presented only with prose or formula descriptions.
monk_e_boy 10y ago
I didn't even know that powers, logs and roots were that related. My math is bad :( I struggled through some of the Khan Academy but I find it really hard to remember all the rules. The triangles really helped me. But maybe the video was overly simplistic? These triangles (visual items) can be placed in the mind palace. I can picture them now and could probably figure out where the log2(x) fits in.

by zeven7 10y ago

lisper 10y ago

Let me rephrase that: people naturally communicate using language. In your head you may think in symbols and spatial relationships, but if you want to get a concept out of your head and into someone else's head you have to use language. There's a reason that the symbology in mathematical papers is invariably wrapped in natural language.

lasfter 10y ago
I still disagree. I wouldn't have understood half of my abstract algebra classes if the profs had only used words to explain the concepts. When you are first exposed to the First Isomorphism Theorem for Rings, a thousand words can't do what that simple triangle diagram can.
jacobolus 10y ago
This is entirely backwards. Mathematics is the study of patterns. Humans have amazing powerful visual and spatial reasoning centers in the brain which can process huge amounts of semi-structured information. Writing words down instead forces everything through low-bandwidth serialized language processing apparatus instead. Mathematical notation (and computer code, and music notation, and ...) benefit tremendously from even slightly exploiting spatial reasoning skills. But better is some kind of diagram, interactive simulation, or similar. The fastest mathematical explanation is direct tutorial, person-to-person, where the amount of symbolic formalism can be reduced to a minimum, and hand gestures, chalk drawings, colorful verbal analogies, etc. can convey most of the message. When limited to pure symbolic formalism in a scholarly paper, mathematics teaching and learning is a much harder slog. The reason formal mathematics uses symbolic formal language as a canonical way of recording results is that it’s easier to figure out how to verify it step-by-step and make sure that the logic is correct. * * * I’ll give you an example, from a paragraph a friend of mine wrote several years ago in a blog diary: (I posit that you could decipher and understand this paragraph at least an order of magnitude faster if I drew a 30 second napkin sketch.) > The tea room I went to is an “eight tatami design”, a large square space. Each side is two tatami lengths (four widths, two widths and a length) long. The eight tatami are arranged in the following way: from the entrance—which is in a corner, second forward on the left—two tatami extend into the room to meet the opposite wall on their short sides; the tatami touching the opposite wall is met on its right long side by a third tatami’s short side. This third tatami’s short side in turn meets a fourth tatami’s long side, whose two short sides meet the opposite wall and a fifth tatami’s short side. The fifth tatami’s second short side meets the room’s the fourth corner and its long side meets the sixth tatami’s short side. The sixth tatami’s second short side meets the first tatami’s long side. These six tatami form a square with an square empty middle. The empty middle’s sides’ dimensions are all one tatami length long. The middle, then, is filled by two tatami whose orientation is like that of tatami one, two, four and five—short sides to the opposite wall and the wall with the entrance. I tried out alternative arrangements and I might be wrong but I think that this solution (or a rotation of it) is the only possible tatami arrangement for the room’s dimensions that fulfills the rule “four corners must never converge”. Going in a circle around the room, starting at the door and following the direction in the description there, we can number the outer tatami one through six and the two inner tatami seven (left from the entrance) and eight (right from the entrance).
- mannykannot 10y ago
  I am not sure that this is the clearest possible written explanation - "Each side is two tatami lengths (four widths, two widths and a length) long"? It is often the case that words + diagrams is the clearest way to explain something, and in most cases where you have a choice, you can choose both. Symbolic expressions fall somewhere between the two, and when used properly, further increase your chances of getting the point across. There's no need to turn every choice into an exclusive dichotomy, regardless of how common that is on HN.
- dredmorbius 10y ago
  Comparing a less-than-optimal textual description with an optimsed symbolic or digrammatic description proves little. It helps to know that tatami are rectangles with length 2x width (something a casual and/or non-Japanese reader might reasonably not know). What's described is eight tatamis (length 2x their width), arranged in a square. Much of the confusion comes from describing each tatami's relation with its neighbors, rather than the space as a whole, and referencing other tatami rather than the room as a whole. A simpler description: A square room, four units per side, filled with eight mats, each one unit wide and two units long. Looking away from you, down either wall are two mats placed lenthways. On the near and far walls, a single mat between those, long edge to the wall. In the center square, two mats placed long edges parallel to the side walls. Or the 3600x3600 diagram in this image: http://www.tatami.com.my/layout.jpg http://www.tatami.com.my/layout.jpg The problem is your friend's blog description. Not inherent limits of language.
  - jacobolus 10y ago
    I agree my friend’s description is not the best. He was trying to make it clear and unambiguous, and he’s a smart guy, but he hadn’t practiced this before and I’m sure he didn’t think all that deeply about it. It’s just not a priori obvious how to make a clear verbal description of a spatial relationship. A diagram of the same thing takes much less effort to construct. I don’t think it’s possible to construct a textual description or mathematical formula of a collection of tatami patterns which takes less than 40–60 seconds per pattern to unpack into a clear mental image. If I spend about 5–10 seconds looking at each of the 14 tatami mat diagrams in your picture, I’m confident I could reconstruct the patterns from memory a few minutes later. I can’t imagine doing the same without spending a significant amount of time on a textual explanation. Additionally, after examining an image with a bunch of tatami patterns for a couple minutes, I could probably tell you whether additional patterns satisfy the criteria to be acceptable or not, without doing any explicit analysis of those criteria. I’d have to think a lot harder if presented only with prose or formula descriptions.

monk_e_boy 10y ago

I didn't even know that powers, logs and roots were that related. My math is bad :( I struggled through some of the Khan Academy but I find it really hard to remember all the rules. The triangles really helped me. But maybe the video was overly simplistic? These triangles (visual items) can be placed in the mind palace. I can picture them now and could probably figure out where the log2(x) fits in.