5 ms·

I wish my teacher showed me stuff like that in high school instead of just scary numbers. While I do understand that eventually you need to get down to the numb

by Spittie 13y ago

I wish my teacher showed me stuff like that in high school instead of just scary numbers. While I do understand that eventually you need to get down to the numbers, my brain seems to pick up the overall concept much faster seeing nice visual representations like those.

jordigh 13y ago
It's not just your brain. It's everyone's brain. Human reasoning is fundamentally geometric. What happens when you see people who apparently are able to understand "just scary numbers" is that they are building in their own head these same images. They've had previous exposure to such images and know how to build their own. But in the end, everyone ends up reasoning with drawings.
- dllthomas 13y ago
  That doesn't conform very well to my understanding or experience. Citation?
  - EC1 13y ago
    You're a special little snowflake then. Here's your citation: [SOURCE]
    - dllthomas 13y ago
      What?
  - recursive 13y ago
    As a person who is generally considered to be good with scary numbers, I can confirm that this is exactly how it works for me.
    - dllthomas 13y ago
      For the record, I have no trouble at all believing that it works that way for many people. The claim was that this is how it works for everyone, and that I find questionable - in my experience, some people work better verbally than visually. I'm pretty good visually, but it definitely feels like I'm engaging different systems when I am reasoning using words than when I am reasoning using images - and I don't expect this generalizes.
  - Double_Cast 13y ago
    I too believe that math is most easily understood geometrically. I've suspected this since highschool, when I personally derived many of these gifs during trig class. I don't have anything more than anecdotes to support this hypothesis, but it shouldn't be surprising given that an entire brain lobe (occipital) is devoted to visual processing [1]. And remember how our first introduction to numbers was "the number line"? The reals are isomorphic to a line, but defining the reals as a line isn't feasible since a line doesn't differentiate the rationals from the reals (or anything in between). But on the other hand, showing kids a line is easier to grok than defining numbers as "an ordered field", isn't it? Also consider that Newton invented calculus using infinitesimals, which made sense to him spatially but didn't find rigorous footing until Weierstrass [2]. Additionally, the Greeks used to refer to finding a figure's area as "quadrature" [3], i.e. finding the area of an equivalent square. If not universal, I'd say geometric interpretations were at least pretty widespread. [1] http://en.wikipedia.org/wiki/Occipital_lobe http://en.wikipedia.org/wiki/Occipital_lobe [2] http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_... [3] http://en.wikipedia.org/wiki/Quadrature_(mathematics) http://en.wikipedia.org/wiki/Quadrature_(mathematics)
    - dllthomas 13y ago
      "If not universal, I'd say geometric interpretations were at least pretty widespread." I'd have had no objection to that. Clearly, many people do learn visually and clearly many people are good at manipulating mental imagery. Support them! Just be careful you're not ignoring those that don't (or at least know that that's what you're doing). If you're going to make a case that there are no such people (or a negligible number) then that needs support. The story here about imagination seems highly relevant (along with the broader point): http://lesswrong.com/lw/dr/generalizing_from_one_example http://lesswrong.com/lw/dr/generalizing_from_one_example
      - Double_Cast 13y ago
        I'm familiar with the typical-mind-fallacy posts and glad to see a LW reference on HN. Just the other day, I read Eliezer's article on Algernon's Law [1]. It's tangentially related. Incidentally, it precedes LW by a decade! His thesis sounds reasonable to me. But I thought MBTI test sounded reasonable too, which I learned doesn't have any empirical backing. So my intuitions may be less than reliable. meh. > According to Galton, people incapable of forming images were overrepresented in math and science. I've since heard that this idea has been challenged, but I can't access the study. So yes, I guess some lack a visual imagination. But I've seen several people enlightened by diagrams, and never seen anyone enlightened via plug-&-chug formulas. So I'd be surprised if a visual imagination did'n confer some kind of advantage in math. I highly doubt that a visual imagination confers a disadvantage, given that Euler had photographic memory. But I concede that the null hypothesis is certainly likely. Yvain says he can't access Galton's study. After a few minutes Googling, I gave up too after I hit a paywal. [1] http://web.archive.org/web/200102021712/http://sysopmind.com/algernon.html http://web.archive.org/web/200102021712/http://sysopmind.com...
- Someone 13y ago
  "A formula is worth a thousand pictures." [1] Also, draw a picture of the monster group (http://en.wikipedia.org/wiki/Monster_group http://en.wikipedia.org/wiki/Monster_group), 11 dimensions of space time (http://en.wikipedia.org/wiki/M-theory http://en.wikipedia.org/wiki/M-theory), or explain how a picture of seven infinitely long cylinders can illustrate that each pair of those cylinders touches each other. So, I disagree that "everyone ends up reasoning with drawings". [1] https://www.utexas.edu/faculty/council/2002-2003/memorials/Dijkstra/dijkstra.html https://www.utexas.edu/faculty/council/2002-2003/memorials/D...
  - jordigh 13y ago
    There's a formalism that we manipulate to reason, but we're always aided by geometric and spatial analogies. The Monster is a permutation group, which means it moves things around, and we visualise 11 dimensions by considering 3 dimensional analogues or by projecting the shadow of 11-dimensional objects onto 2 or 3 dimensions.
- NAFV_P 13y ago
  > But in the end, everyone ends up reasoning with drawings. That is a big generalisation, I have solved problems without using visualisation. How would you interpret your comment if someone was congenitally blind?
  - jordigh 13y ago
    The only blind mathematician I ever knew worked with fractals and was able to draw fractals on the computer and then accurately describe what these fractals looked like.
- wisty 13y ago
  Newton actually released most of his work as graphical proofs, not calculus, partly for this reason (and partly because he was an asshole who wanted to keep his methods a trade secret). Before Newton, math was graphical. The problem is, it's hard to come up with graphical proofs. It's a lot easier to reason symbolically. If you want to solve mathematical problems, equations may be better. If you want to explain your reasoning, then diagrams can be better.

by Spittie 13y ago

jordigh 13y ago

It's not just your brain. It's everyone's brain. Human reasoning is fundamentally geometric. What happens when you see people who apparently are able to understand "just scary numbers" is that they are building in their own head these same images. They've had previous exposure to such images and know how to build their own. But in the end, everyone ends up reasoning with drawings.

dllthomas 13y ago
That doesn't conform very well to my understanding or experience. Citation?
- EC1 13y ago
  You're a special little snowflake then. Here's your citation: [SOURCE]
  - dllthomas 13y ago
    What?
- recursive 13y ago
  As a person who is generally considered to be good with scary numbers, I can confirm that this is exactly how it works for me.
  - dllthomas 13y ago
    For the record, I have no trouble at all believing that it works that way for many people. The claim was that this is how it works for everyone, and that I find questionable - in my experience, some people work better verbally than visually. I'm pretty good visually, but it definitely feels like I'm engaging different systems when I am reasoning using words than when I am reasoning using images - and I don't expect this generalizes.
- Double_Cast 13y ago
  I too believe that math is most easily understood geometrically. I've suspected this since highschool, when I personally derived many of these gifs during trig class. I don't have anything more than anecdotes to support this hypothesis, but it shouldn't be surprising given that an entire brain lobe (occipital) is devoted to visual processing [1]. And remember how our first introduction to numbers was "the number line"? The reals are isomorphic to a line, but defining the reals as a line isn't feasible since a line doesn't differentiate the rationals from the reals (or anything in between). But on the other hand, showing kids a line is easier to grok than defining numbers as "an ordered field", isn't it? Also consider that Newton invented calculus using infinitesimals, which made sense to him spatially but didn't find rigorous footing until Weierstrass [2]. Additionally, the Greeks used to refer to finding a figure's area as "quadrature" [3], i.e. finding the area of an equivalent square. If not universal, I'd say geometric interpretations were at least pretty widespread. [1] http://en.wikipedia.org/wiki/Occipital_lobe http://en.wikipedia.org/wiki/Occipital_lobe [2] http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_... [3] http://en.wikipedia.org/wiki/Quadrature_(mathematics) http://en.wikipedia.org/wiki/Quadrature_(mathematics)
  - dllthomas 13y ago
    "If not universal, I'd say geometric interpretations were at least pretty widespread." I'd have had no objection to that. Clearly, many people do learn visually and clearly many people are good at manipulating mental imagery. Support them! Just be careful you're not ignoring those that don't (or at least know that that's what you're doing). If you're going to make a case that there are no such people (or a negligible number) then that needs support. The story here about imagination seems highly relevant (along with the broader point): http://lesswrong.com/lw/dr/generalizing_from_one_example http://lesswrong.com/lw/dr/generalizing_from_one_example
    - Double_Cast 13y ago
      I'm familiar with the typical-mind-fallacy posts and glad to see a LW reference on HN. Just the other day, I read Eliezer's article on Algernon's Law [1]. It's tangentially related. Incidentally, it precedes LW by a decade! His thesis sounds reasonable to me. But I thought MBTI test sounded reasonable too, which I learned doesn't have any empirical backing. So my intuitions may be less than reliable. meh. > According to Galton, people incapable of forming images were overrepresented in math and science. I've since heard that this idea has been challenged, but I can't access the study. So yes, I guess some lack a visual imagination. But I've seen several people enlightened by diagrams, and never seen anyone enlightened via plug-&-chug formulas. So I'd be surprised if a visual imagination did'n confer some kind of advantage in math. I highly doubt that a visual imagination confers a disadvantage, given that Euler had photographic memory. But I concede that the null hypothesis is certainly likely. Yvain says he can't access Galton's study. After a few minutes Googling, I gave up too after I hit a paywal. [1] http://web.archive.org/web/200102021712/http://sysopmind.com/algernon.html http://web.archive.org/web/200102021712/http://sysopmind.com...
Someone 13y ago
"A formula is worth a thousand pictures." [1] Also, draw a picture of the monster group (http://en.wikipedia.org/wiki/Monster_group http://en.wikipedia.org/wiki/Monster_group), 11 dimensions of space time (http://en.wikipedia.org/wiki/M-theory http://en.wikipedia.org/wiki/M-theory), or explain how a picture of seven infinitely long cylinders can illustrate that each pair of those cylinders touches each other. So, I disagree that "everyone ends up reasoning with drawings". [1] https://www.utexas.edu/faculty/council/2002-2003/memorials/Dijkstra/dijkstra.html https://www.utexas.edu/faculty/council/2002-2003/memorials/D...
- jordigh 13y ago
  There's a formalism that we manipulate to reason, but we're always aided by geometric and spatial analogies. The Monster is a permutation group, which means it moves things around, and we visualise 11 dimensions by considering 3 dimensional analogues or by projecting the shadow of 11-dimensional objects onto 2 or 3 dimensions.
NAFV_P 13y ago
> But in the end, everyone ends up reasoning with drawings. That is a big generalisation, I have solved problems without using visualisation. How would you interpret your comment if someone was congenitally blind?
- jordigh 13y ago
  The only blind mathematician I ever knew worked with fractals and was able to draw fractals on the computer and then accurately describe what these fractals looked like.
wisty 13y ago
Newton actually released most of his work as graphical proofs, not calculus, partly for this reason (and partly because he was an asshole who wanted to keep his methods a trade secret). Before Newton, math was graphical. The problem is, it's hard to come up with graphical proofs. It's a lot easier to reason symbolically. If you want to solve mathematical problems, equations may be better. If you want to explain your reasoning, then diagrams can be better.