Pushing symbols around (or the metaphorical equivalent) might be necessary for building intuitions. Incidentally, they call novice chess players woodpushers.
When I teach piano improv, I show my students a way of making decent-sounding notes that takes about 30 seconds to explain. And then I have them play that way for a long time, only occasionally adding complexity—a new voicing, a different chord order.
Likewise, to train a neural network, you let it make blind guesses, then tell it how it did, until it gets very good at finding the right answer. There seems to be a lot of value in messing around as a form of training.
I get the sense that in your model, blind calculation is a distraction from intuitive problem solving. In my model, blind calculation builds intuition for the problem. In both our models, intuitive problem-solving must be followed by proof, which is the step that Pearson skipped.
I agree with this to some extent. Playing around with the symbols is useful for getting a general intuition for what-sorts-of-results-are-easy. It’s a good way to find things which are “nearby” in the expression-manipulation graph, and to notice patterns in the general structure of that graph. Where it usually doesn’t suffice is harder problems, where you have to go pretty “far away” in the expression-graph or find the right possibility in a very large space. That’s where the exponentially large number of possibilities really kick in, and we need more powerful tools.
So I agree that playing around is often a useful way to build intuition for some aspects of the problem, and sometimes even a necessary step, but it usually isn’t sufficient for harder problems.
We talked about this issue in the comments on this post of yours 9 months ago :)
I get the sense that you’re considering problems where an open-ended search doesn’t tell you if you’re heading in the right direction.
So for example, if we play Marco Polo, when I shout “Marco” and everybody else shouts “Polo,” this gives me some information about where they are. They might move, but they can’t necessarily move fast enough to avoid me.
If we changed the game so people only have to whisper “Polo,” I rarely, if ever, will gain information from shouting “Marco,” and will mostly be stumbling around in the dark.
I might need some pretty sophisticated thinking to tag somebody. Perhaps I’d consider the shape of the swimming pool, who’s likely to try sneaking up behind me just for fun, who’s likely to get bored and forget that I’m still out to tag them, my energy level, and perhaps shift to quietly swimming around while listening for the splashes of the other players.
And these are not considerations that are immediately suggested by the “rules of the game,” which encourage you to think in terms of the actions: swimming, shouting “Marco,” and tagging. A kid who operated by just experimenting with when they should shout “Marco” and swimming around randomly is extremely unlikely to arrive at the precise manner of playing the game that might let them tag somebody in whisper Marco Polo.
A problem statement can prime us to think in terms of misleading similarities (i.e. using a correlation coefficient as a proxy for information), or non-strategic movement (i.e. making a lot of noise splashing around and shouting “Marco” in a way that makes it hard to hear where others are).
Shifting our focus from babbling within constraints to babbling about constraints seems to be a useful move.
Babbling within constraints examples:
Pushing chess pieces around according to the rules of the game
Tracing random routes through a maze
Playing arbitrary notes over a I-V-vi-IV chord progression
Swimming around and shouting “Marco” as fast as you can
Shifting materials in such a way as to form a platform crossing the river
Babbling about constraints examples:
Imagining principles that might be helpful (“move my pieces closer to the other king,” “put pieces where they have the largest number of possible moves,” “check to make sure they can’t take your piece for free on the next move”).
Look for areas of the maze that you definitely don’t need to move through.
Periodically changing the register, scrambling the order of the chords every 8 measures
Trying a tactic of standing still, listening for swimmers, and then surprise-lunging; not shouting “Marco” until you hear the others close by
Observing the height of the river over time, in order to determine how high and wide it gets when it floods.
I notice that I tend to alternate between these two modes. It’s often quite useless for me to force myself to come up with constraints for a problem I haven’t tried messing around with yet. Likewise, at a certain point, messing around becomes obviously fruitless, and imposing constraints becomes more productive.
Pushing symbols around (or the metaphorical equivalent) might be necessary for building intuitions. Incidentally, they call novice chess players woodpushers.
When I teach piano improv, I show my students a way of making decent-sounding notes that takes about 30 seconds to explain. And then I have them play that way for a long time, only occasionally adding complexity—a new voicing, a different chord order.
Likewise, to train a neural network, you let it make blind guesses, then tell it how it did, until it gets very good at finding the right answer. There seems to be a lot of value in messing around as a form of training.
I get the sense that in your model, blind calculation is a distraction from intuitive problem solving. In my model, blind calculation builds intuition for the problem. In both our models, intuitive problem-solving must be followed by proof, which is the step that Pearson skipped.
I agree with this to some extent. Playing around with the symbols is useful for getting a general intuition for what-sorts-of-results-are-easy. It’s a good way to find things which are “nearby” in the expression-manipulation graph, and to notice patterns in the general structure of that graph. Where it usually doesn’t suffice is harder problems, where you have to go pretty “far away” in the expression-graph or find the right possibility in a very large space. That’s where the exponentially large number of possibilities really kick in, and we need more powerful tools.
So I agree that playing around is often a useful way to build intuition for some aspects of the problem, and sometimes even a necessary step, but it usually isn’t sufficient for harder problems.
We talked about this issue in the comments on this post of yours 9 months ago :)
I get the sense that you’re considering problems where an open-ended search doesn’t tell you if you’re heading in the right direction.
So for example, if we play Marco Polo, when I shout “Marco” and everybody else shouts “Polo,” this gives me some information about where they are. They might move, but they can’t necessarily move fast enough to avoid me.
If we changed the game so people only have to whisper “Polo,” I rarely, if ever, will gain information from shouting “Marco,” and will mostly be stumbling around in the dark.
I might need some pretty sophisticated thinking to tag somebody. Perhaps I’d consider the shape of the swimming pool, who’s likely to try sneaking up behind me just for fun, who’s likely to get bored and forget that I’m still out to tag them, my energy level, and perhaps shift to quietly swimming around while listening for the splashes of the other players.
And these are not considerations that are immediately suggested by the “rules of the game,” which encourage you to think in terms of the actions: swimming, shouting “Marco,” and tagging. A kid who operated by just experimenting with when they should shout “Marco” and swimming around randomly is extremely unlikely to arrive at the precise manner of playing the game that might let them tag somebody in whisper Marco Polo.
A problem statement can prime us to think in terms of misleading similarities (i.e. using a correlation coefficient as a proxy for information), or non-strategic movement (i.e. making a lot of noise splashing around and shouting “Marco” in a way that makes it hard to hear where others are).
Shifting our focus from babbling within constraints to babbling about constraints seems to be a useful move.
Babbling within constraints examples:
Pushing chess pieces around according to the rules of the game
Tracing random routes through a maze
Playing arbitrary notes over a I-V-vi-IV chord progression
Swimming around and shouting “Marco” as fast as you can
Shifting materials in such a way as to form a platform crossing the river
Babbling about constraints examples:
Imagining principles that might be helpful (“move my pieces closer to the other king,” “put pieces where they have the largest number of possible moves,” “check to make sure they can’t take your piece for free on the next move”).
Look for areas of the maze that you definitely don’t need to move through.
Periodically changing the register, scrambling the order of the chords every 8 measures
Trying a tactic of standing still, listening for swimmers, and then surprise-lunging; not shouting “Marco” until you hear the others close by
Observing the height of the river over time, in order to determine how high and wide it gets when it floods.
I notice that I tend to alternate between these two modes. It’s often quite useless for me to force myself to come up with constraints for a problem I haven’t tried messing around with yet. Likewise, at a certain point, messing around becomes obviously fruitless, and imposing constraints becomes more productive.
That’s an awesome analogy/example. Well done.
Also, this comment would only need a little more flesh on it to be a great post in its own right.
Thanks, John :) Your post has been thought-provoking for me (three top-level comments so far), so that might have to happen!