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"is actually a better impedance match to people's natural mental processes, which involve language" This is false when applied to all people. And this is the r

"is actually a better impedance match to people's natural mental processes, which involve language" This is false when applied to all people. And this is the reason you disagree with the demonstration. Not all people think in language. I would wager that only half the population thinks in language. I think visually. It's work for me to translate my thoughts into words. I have to do a translation process to write this comment. It's easy work, but my natural thoughts are pictures of concepts. Richard Feynmann liked to talk about this a lot, but the internal mechanisms of thought vary drastically from person to person. Ironically, it seems common for people who think in words to suppose everyone else is that way. I am not sure why.

lisper 10y ago
> Not all people think in language Yes, you're right. But everyone communicates in language. I've updated the original comment.
jordigh 10y ago
> I think visually. I'm pretty sure everyone thinks visually or geometrically or spatially or something like that. Most certainly visually if you're not blind. Few people would find this to be an unsatisfactory proof that the nth triangular number is equal to the total number of pairings of n+1 objects: http://i.imgur.com/HCfGOYp.gif http://i.imgur.com/HCfGOYp.gif Better notation is certainly not worthless, and having a certain degree of symmetry or geometry to our notation is a good thing. I insist, nevertheless, that notation is not the biggest impedance towards conveying ideas.
- yorwba 10y ago
  > Few people would find this to be an unsatisfactory proof that the nth triangular number is equal to the total number of pairings of n+1 objects: I accept it as a proof for the specific case, but it took me quite a while to understand what was going on. I definitely can't tell whether it generalizes to arbitrary sizes this way. Maybe if I had a look at all frames at once I could notice a pattern, but the constant movement is really distracting.
- wongarsu 10y ago
  >I'm pretty sure everyone thinks visually or geometrically or spatially or something like that. Most certainly visually if you're not blind. I don't think visually or geometrically or spatially or something like that. I'm naturally thinking in terms of words and abstract concepts. Visualizing something in my mind is actual work; you can describe a person to me in minute detail without me ever visualizing what they might look like. My sense of direction sucks, and I typically don't solve problems visually (or geometrically or spatially). Of course I can still convert between thoughts and images (though conversion between thoughts and words is often easier), and sometimes a problem is easier to explain visually. I understand your proof and find it satisfactory, but it took me some time and I would have prefered a proof by induction.
  - jordigh 10y ago
    What about being convinced that local extrema of smooth functions occur when the derivative is zero? You find it easier to work through an epsilon-delta algebraic proof than to look at a graph and see that it's flat wherever there's a local extremum? Do you find graphs to represent data useless and you would rather look at long lists of numbers?
    - wongarsu 10y ago
      The derivative is the rate of change. If I chose any two points to the left and right to the local extremum, they are non-zero and have different signs. Since there has to be a zero between those two points, the local extremum must have a derivative of zero. That's the line of reasoning I would go. Looking at a graph, searching for exceptions and concluding that there aren't any is probably as easy in this case. Like probably most people I have a harder time thinking about epsilon-delta proofs. >Do you find graphs to represent data useless and you would rather look at long lists of numbers? No, graphs are great. They sacrifice precision to show simple relations between large amounts of numbers, and are really good at that. Numbers a row of numbers can't do that (the closest equivalent to a line graph is to replace each but the first number with their difference to the previous number, but that's still inferior). Natural language is lacking precision at reasonable density for the job. So graphs are pretty much the only decent tool for their job we have invented so far.
- nkurz 10y ago
  Few people would find this to be an unsatisfactory proof that the nth triangular number is equal to the total number of pairings of n+1 objects: http://i.imgur.com/HCfGOYp.gif http://i.imgur.com/HCfGOYp.gif It would be interesting to test this. I forced myself to stare at that image longer than was comfortable, and I got nothing from it. Then I came back and read your words, then went back to the diagram, and still couldn't figure out what relevance the diagram had. Possibly I'm part of a small minority, possibly you are, or possibly it's 50/50. Without testing, I wouldn't wager a guess which of these is most likely. You are likely better than me at visually assembling the individual frames into a coherent whole. The sequence forms a 3-dimensional image, but with 2 dimensions of space and 1 dimension of time. I feel my brain struggling (and failing) to project it into 3 spatial dimensions. The frenetic chaos leaves me longing for the serenity of words.
mannykannot 10y ago
>It seems common for people who think in words to suppose everyone else is that way. I am not sure why. Perhaps because that's how you get to know what someone else is thinking? Even when you draw pictures (which is quite common in technical fields) they usually need some explanation.

lisper 10y ago

> Not all people think in language Yes, you're right. But everyone communicates in language. I've updated the original comment.

jordigh 10y ago

> I think visually. I'm pretty sure everyone thinks visually or geometrically or spatially or something like that. Most certainly visually if you're not blind. Few people would find this to be an unsatisfactory proof that the nth triangular number is equal to the total number of pairings of n+1 objects: http://i.imgur.com/HCfGOYp.gif http://i.imgur.com/HCfGOYp.gif Better notation is certainly not worthless, and having a certain degree of symmetry or geometry to our notation is a good thing. I insist, nevertheless, that notation is not the biggest impedance towards conveying ideas.

yorwba 10y ago
> Few people would find this to be an unsatisfactory proof that the nth triangular number is equal to the total number of pairings of n+1 objects: I accept it as a proof for the specific case, but it took me quite a while to understand what was going on. I definitely can't tell whether it generalizes to arbitrary sizes this way. Maybe if I had a look at all frames at once I could notice a pattern, but the constant movement is really distracting.
wongarsu 10y ago
>I'm pretty sure everyone thinks visually or geometrically or spatially or something like that. Most certainly visually if you're not blind. I don't think visually or geometrically or spatially or something like that. I'm naturally thinking in terms of words and abstract concepts. Visualizing something in my mind is actual work; you can describe a person to me in minute detail without me ever visualizing what they might look like. My sense of direction sucks, and I typically don't solve problems visually (or geometrically or spatially). Of course I can still convert between thoughts and images (though conversion between thoughts and words is often easier), and sometimes a problem is easier to explain visually. I understand your proof and find it satisfactory, but it took me some time and I would have prefered a proof by induction.
- jordigh 10y ago
  What about being convinced that local extrema of smooth functions occur when the derivative is zero? You find it easier to work through an epsilon-delta algebraic proof than to look at a graph and see that it's flat wherever there's a local extremum? Do you find graphs to represent data useless and you would rather look at long lists of numbers?
  - wongarsu 10y ago
    The derivative is the rate of change. If I chose any two points to the left and right to the local extremum, they are non-zero and have different signs. Since there has to be a zero between those two points, the local extremum must have a derivative of zero. That's the line of reasoning I would go. Looking at a graph, searching for exceptions and concluding that there aren't any is probably as easy in this case. Like probably most people I have a harder time thinking about epsilon-delta proofs. >Do you find graphs to represent data useless and you would rather look at long lists of numbers? No, graphs are great. They sacrifice precision to show simple relations between large amounts of numbers, and are really good at that. Numbers a row of numbers can't do that (the closest equivalent to a line graph is to replace each but the first number with their difference to the previous number, but that's still inferior). Natural language is lacking precision at reasonable density for the job. So graphs are pretty much the only decent tool for their job we have invented so far.
nkurz 10y ago
Few people would find this to be an unsatisfactory proof that the nth triangular number is equal to the total number of pairings of n+1 objects: http://i.imgur.com/HCfGOYp.gif http://i.imgur.com/HCfGOYp.gif It would be interesting to test this. I forced myself to stare at that image longer than was comfortable, and I got nothing from it. Then I came back and read your words, then went back to the diagram, and still couldn't figure out what relevance the diagram had. Possibly I'm part of a small minority, possibly you are, or possibly it's 50/50. Without testing, I wouldn't wager a guess which of these is most likely. You are likely better than me at visually assembling the individual frames into a coherent whole. The sequence forms a 3-dimensional image, but with 2 dimensions of space and 1 dimension of time. I feel my brain struggling (and failing) to project it into 3 spatial dimensions. The frenetic chaos leaves me longing for the serenity of words.

mannykannot 10y ago

>It seems common for people who think in words to suppose everyone else is that way. I am not sure why. Perhaps because that's how you get to know what someone else is thinking? Even when you draw pictures (which is quite common in technical fields) they usually need some explanation.