Imagine writing two versions of the same computer program. The first represents its integers as 32-bit binary numbers. The second writes the numbers in base 10, ASCII strings with each byte used to store one digit.
A while ago, I discussed this post with a friend, who I learned also uses music to memorize numbers. He doesn't have perfect pitch, but relative pitch is just as good for encoding.
Yes, absolutely. This is something I think about a lot, and as a mathematician, my capicity to visualise mathematical ideas is exactly where I pin the physical location of my maths talent (metaphorically speaking, of course). Particular agreement with Jadagul on how to read a textbook.
I'm going to throw it into the air that maybe a computer-science way of understanding this is that difficult mathematical ideas are somehow "large" in a memory sense, and talented mathematicians are mentally equipped with very large "mathematics buffers". In my experience of teaching mathematics to those who are struggling with high-school level stuff, the problem seems intuitively that they can't fit an entire mathematical idea into their head at once.
The underlying power of metaphors comes from analogies, which, I claim, is all of intelligence, the whole basis for intelligence - that is to say, all other forms of reasoning (such as deductive/predicate, bayesian/probabilistic etc) are merely special cases of analogies: I claim that analogy formation is beyond the scope of Bayes.
That's a big claim (and remember readers, you heard it explictly publically stated here by me first). Although you and Eliezer no doubt think that Bayesian reasoning is more general than anlogy formation, there is little basis for your beliefs- in fact - he and you have both got it the wrong way around. (Its Bayes that's the special case of analogy formation, not vice versa).
Further, analogy formation is closely related to ontology merging, the ablity to communicate (or 'map') a valid concept from one knowledge domain to another novel knowledge domain. I repeat my big claim: this is all of intelligence; ontology/knowledge representation is all of the AI problem.
Refer to the detailed discussion by Steven Pinker in his new book 'The Stuff of Thought' on the power of metaphor/analogy.
Incidentally, here's a pro tip for you AI wannabes: ontology is the concrete version of pure math (ie ontologies are 'mathematical artifacts', or to use an analogy/metaphor, ontologies are to pure math as physical objects are to physical laws). I realized this myself months ago, but I was very pleased to get confirmation: first from Pinker, then, from references to other philosophers, who had independently realized this:
Mathematical Beauty!
"Twentieth-century French philosopher Alain Badiou claims that ontology is mathematics. Badiou also believes in deep connections between math, poetry and philosophy."
Jadagul, do you have ant references detailing the principles of good mathematical visualization, or providing case studies/examples? It seems that mathematicians have had no interest in documenting this since the closest that comes to mind is cognitive science research, including Lakoff and Nunez's controversial work, Where Mathematics Comes From.
Douglas, different people visualize in different ways. The textbook tries to give you enough details that you can reconstruct the argument, while giving you few enough that you have to actually reconstruct it, which forces you to frame it in a way that makes sense to you. (I avoid the word 'visualize' because there never seems to be much that's actually visual about the way I understand math proofs; this is sort of the point).
I don't see how Jadagul is disagreeing with steven. Math books give too much detail in proofs; rather, they give the wrong details. They should tell you how to reconstruct the proof, not give you touchstones. Visualization is a part of that.
Steven, math texts aren't supposed to be a good way of visualizing the argument. They're supposed to be a skeleton; every prof I've talked to agrees that you're basically supposed to be rewriting any argument you care about as you read it. I know that if I find an argument confusing I can generally figure it out by starting at the top and writing it out as I would notes for a proof I'm working on myself.
The math text isn't supposed to be the understandable version; it's supposed to be the cliff notes so you can make your own understandable version.
It sure complicated the math by dispersing it all over the place.
On the contrary, two-dimensional "boxed" representations of complicated expressions may be more intuitive than conventional math layout. Here is one paper on Citeseer which uses such a notation to analyze the computer proof of the Robbins-Boolean problem.
Of course, we should recognize that the visualization of a problem does not necessarily make things easier for everyone. I for one, have a harder time understanding a problem or it's solution if I have to think through a specified metaphor, much less graphic visualization (it usually makes everything even more blurry for me). I do better if it cuts straight to the point (in universally accepted math symbols and equations, which are unambiguous). Sometimes I do try to pick an example that uniquely helps me understand better (almost never a visual one though). I doubt there is a universal 'native architecture'.
I've always thought that most math texts do awfully at this. Way too much symbol manipulation, way too little visual intuition-building. Either that or it's a learning-style thing.
Marcello: The Löb's Theorem cartoon was drawn on the theory that the brain has native architecture for tracking people's opinions, and would find it easier to visualize the difference between someone's opinion and someone's opinion about someone's opinion, than to make the corresponding distinction in formal systems.
It sure complicated the math by dispersing it all over the place. Having compact symbolic notation allows to imagine graph-like models or type inference processes right in the context of a single sheet of paper where the math is written, while navigating cartoons takes time and bigger shifts of context, making it harder to follow. It helps to have a single binding context for more difficult proofs, or to modularize the proof so that everything doesn't need to be in mind at once. In my experience, most of the time solving a moderately hard problem is spent building up a library of transformations between alternative visualizations and across the freedoms of the problem, so that at some point steps towards the solution just reveal themselves. Build a map, and then walk it.
A while ago, I discussed this post with a friend, who I learned also uses music to memorize numbers. He doesn't have perfect pitch, but relative pitch is just as good for encoding.
Marcello,
I recommend the mural "Math" by Liz Mitchell, an artist and math teacher in Oakland, California.
Yes, absolutely. This is something I think about a lot, and as a mathematician, my capicity to visualise mathematical ideas is exactly where I pin the physical location of my maths talent (metaphorically speaking, of course). Particular agreement with Jadagul on how to read a textbook.
I'm going to throw it into the air that maybe a computer-science way of understanding this is that difficult mathematical ideas are somehow "large" in a memory sense, and talented mathematicians are mentally equipped with very large "mathematics buffers". In my experience of teaching mathematics to those who are struggling with high-school level stuff, the problem seems intuitively that they can't fit an entire mathematical idea into their head at once.
The underlying power of metaphors comes from analogies, which, I claim, is all of intelligence, the whole basis for intelligence - that is to say, all other forms of reasoning (such as deductive/predicate, bayesian/probabilistic etc) are merely special cases of analogies: I claim that analogy formation is beyond the scope of Bayes.
That's a big claim (and remember readers, you heard it explictly publically stated here by me first). Although you and Eliezer no doubt think that Bayesian reasoning is more general than anlogy formation, there is little basis for your beliefs- in fact - he and you have both got it the wrong way around. (Its Bayes that's the special case of analogy formation, not vice versa).
Further, analogy formation is closely related to ontology merging, the ablity to communicate (or 'map') a valid concept from one knowledge domain to another novel knowledge domain. I repeat my big claim: this is all of intelligence; ontology/knowledge representation is all of the AI problem.
Refer to the detailed discussion by Steven Pinker in his new book 'The Stuff of Thought' on the power of metaphor/analogy.
Incidentally, here's a pro tip for you AI wannabes: ontology is the concrete version of pure math (ie ontologies are 'mathematical artifacts', or to use an analogy/metaphor, ontologies are to pure math as physical objects are to physical laws). I realized this myself months ago, but I was very pleased to get confirmation: first from Pinker, then, from references to other philosophers, who had independently realized this:
Mathematical Beauty!
"Twentieth-century French philosopher Alain Badiou claims that ontology is mathematics. Badiou also believes in deep connections between math, poetry and philosophy."
Jadagul, do you have ant references detailing the principles of good mathematical visualization, or providing case studies/examples? It seems that mathematicians have had no interest in documenting this since the closest that comes to mind is cognitive science research, including Lakoff and Nunez's controversial work, Where Mathematics Comes From.
Douglas, different people visualize in different ways. The textbook tries to give you enough details that you can reconstruct the argument, while giving you few enough that you have to actually reconstruct it, which forces you to frame it in a way that makes sense to you. (I avoid the word 'visualize' because there never seems to be much that's actually visual about the way I understand math proofs; this is sort of the point).
I don't see how Jadagul is disagreeing with steven. Math books give too much detail in proofs; rather, they give the wrong details. They should tell you how to reconstruct the proof, not give you touchstones. Visualization is a part of that.
Jadagul: I agree. You aren't really reading a math text if you don't have a pencil in your hand and plenty of scratch paper.
Steven, math texts aren't supposed to be a good way of visualizing the argument. They're supposed to be a skeleton; every prof I've talked to agrees that you're basically supposed to be rewriting any argument you care about as you read it. I know that if I find an argument confusing I can generally figure it out by starting at the top and writing it out as I would notes for a proof I'm working on myself.
The math text isn't supposed to be the understandable version; it's supposed to be the cliff notes so you can make your own understandable version.
Math-U-See seems interesting.http://www.mathusee.com/
It sure complicated the math by dispersing it all over the place.
On the contrary, two-dimensional "boxed" representations of complicated expressions may be more intuitive than conventional math layout. Here is one paper on Citeseer which uses such a notation to analyze the computer proof of the Robbins-Boolean problem.
"when my psychology teacher put up a string of twenty digits on the board and asked us to memorize them, I was able to do it. "
Did you use rhythms or add "lyrics" to the tunes?
Of course, we should recognize that the visualization of a problem does not necessarily make things easier for everyone. I for one, have a harder time understanding a problem or it's solution if I have to think through a specified metaphor, much less graphic visualization (it usually makes everything even more blurry for me). I do better if it cuts straight to the point (in universally accepted math symbols and equations, which are unambiguous). Sometimes I do try to pick an example that uniquely helps me understand better (almost never a visual one though). I doubt there is a universal 'native architecture'.
I've always thought that most math texts do awfully at this. Way too much symbol manipulation, way too little visual intuition-building. Either that or it's a learning-style thing.
Marcello: The Löb's Theorem cartoon was drawn on the theory that the brain has native architecture for tracking people's opinions, and would find it easier to visualize the difference between someone's opinion and someone's opinion about someone's opinion, than to make the corresponding distinction in formal systems.
It sure complicated the math by dispersing it all over the place. Having compact symbolic notation allows to imagine graph-like models or type inference processes right in the context of a single sheet of paper where the math is written, while navigating cartoons takes time and bigger shifts of context, making it harder to follow. It helps to have a single binding context for more difficult proofs, or to modularize the proof so that everything doesn't need to be in mind at once. In my experience, most of the time solving a moderately hard problem is spent building up a library of transformations between alternative visualizations and across the freedoms of the problem, so that at some point steps towards the solution just reveal themselves. Build a map, and then walk it.