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I don't think this is inherent in the idea. You can absolutely make an interface somewhat like this that could arrive at general solutions. Indeed, I think Bret

by asolove 13y ago

I don't think this is inherent in the idea. You can absolutely make an interface somewhat like this that could arrive at general solutions. Indeed, I think Bret would consider this most of the point. He covers this in "Up and down the ladder of abstraction" [0] If by algebra you mean "the transformation of one written notation to another by rule," that is only one way of finding general solutions to mathematical problems. There is a lot of writing about "how" people doing original math think, and many of them work in ways far divorced from algebraic notation. Some seem to develop graphical models they picture in their head, others report using other senses like sound and smell. Freeman Dyson on Feynman: "The reason Dick's physics was so hard for ordinary people to grasp was that he did not use equations... Dick just wrote down the solutions out of his head without ever writing down the equations. He had a physical picture of the way things happen, and the pictures gave him the solutions directly with a minimum of calculation." Einstein on his thinking: "The words of language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements of thought are certain signs ( symbols ) and more or less clear images which can be voluntarily reproduced and combined... The above mentioned elements are, in my case, of visual and some muscular type." More interesting discussion here: http://www.sv.vt.edu/classes/ESM4714/methods/vizthink.html http://www.sv.vt.edu/classes/ESM4714/methods/vizthink.html The idea that "real math is bare symbols" isn't true. Not only isn't it true for doing groundbreaking math, it isn't true even for easy math. I would challenge you to teach children mathematics as a purely symbolic game, never showing them that numbers correspond to physical entities, or that derivatives correspond to tangents on a curve. Those of us who are "good" at algebra are good at it because we understand it in a way deeper than the symbols. Yes, symbolic notation is a great form of communication and necessary for proof, but it is usually not the way to create the initial intuitive spark underlying a solution. [0] http://worrydream.com/LadderOfAbstraction/ http://worrydream.com/LadderOfAbstraction/

RogerL 13y ago
I'm not trying to be waspish, but are you suggesting we just be like Feynman? I will say I'm rather dubious we can achieve that with a simple interface change. There is certainly merit and utility to graphical representation; we all use it via matlab, python+scientific packages, and so on. And, of course, the faster we can change something and see the result the better. It is not always an incremental change, but a pretty fundamental change, to see something change in real time. Yet, this only really works in tiny cases. I recall Victor demonstrating using scrubbing variables to change how a game reacts. Okay, but the vastest amount of the programming I do (and I do fairly numerical things in the computer vision line) I either don't see the utility, or practicality. I can't variable scrub my way to a better algorithm. I've seen, for example, graphical demonstrations of things like binary tree algorithms, sorting, and the like. It's momentarily interesting, but I can't claim that it really made any difference in my understanding of the subject. I think this is a case where people vary. I am not terribly visual, and most of Victor's arguments go over like a lead balloon with me. Yet, I cannot deny the excitement of others. I find it far more commodious to work with, for example, linear algebra. Certainly there are times when I am drawing 3D arrows and the like to work out a problem, but by-and-large the symbolic representation serves far better. There is a reason we use text for communication on HN, and not pictures. It is not because we lack vision or foresight (no pun intended), but because it is the most expressive and flexible way to communicate our ideas. And, of course, sometimes we labor away, trying to communicate in words what would be so easy to illustrate with a single image. Neither is 'best'; we just use the best tool for the job. None of that is to argue against one of your points - that math is more than symbolic representation. Once you are reduced to applying symbolic rules to an equation you are often quite lost, and that is often not taught in schools. Yet so much of math really cannot be done graphically at all, yet we succeed. A lot of people stall out in math (I think) at after the algebra/calc I,II,III level because beyond that there is often no recourse to graphics and physical representations; graphics can be as much a crutch as a help. Anyway, visual thinking is certainly a form of thinking that should not be downplayed, but it is not how some to many brains work, nor is it even possible in many forms of important thought (I leave savants to one side, we can not make decisions based on exceptional/unique people).
leephillips 13y ago
I would add to your references Hadamard's book[0], where he talks about how ideas form in various mathematicians' heads. I don't really disagree with you, it's just that Victor seems to be claiming too much for his new interfaces, and selling symbolic manipulation short. Of course all kinds of things need to be happening inside our heads before we start manipulating symbols. But Feynman's papers and books are full of equations and derivations, not just answers. Most of Victor's examples are actually counter-examples, because they are examples of problems where a little algebra would not only be faster and easier but would yield a simple formula that you could use later, the next time the problem comes up. Take the book design example. What happens the next time you need to fit a similar graph on the page? If you had used algebra you could have written a little calculator in Python in which you could plug in the new numbers and which would return the solution. Victor's way requires you to treat each graph-fitting problem as a special case, even if they're all the same. [0]http://www.amazon.com/The-Psychology-Invention-Mathematical-Field/dp/0486201074 http://www.amazon.com/The-Psychology-Invention-Mathematical-...

by asolove 13y ago

RogerL 13y ago

I'm not trying to be waspish, but are you suggesting we just be like Feynman? I will say I'm rather dubious we can achieve that with a simple interface change. There is certainly merit and utility to graphical representation; we all use it via matlab, python+scientific packages, and so on. And, of course, the faster we can change something and see the result the better. It is not always an incremental change, but a pretty fundamental change, to see something change in real time. Yet, this only really works in tiny cases. I recall Victor demonstrating using scrubbing variables to change how a game reacts. Okay, but the vastest amount of the programming I do (and I do fairly numerical things in the computer vision line) I either don't see the utility, or practicality. I can't variable scrub my way to a better algorithm. I've seen, for example, graphical demonstrations of things like binary tree algorithms, sorting, and the like. It's momentarily interesting, but I can't claim that it really made any difference in my understanding of the subject. I think this is a case where people vary. I am not terribly visual, and most of Victor's arguments go over like a lead balloon with me. Yet, I cannot deny the excitement of others. I find it far more commodious to work with, for example, linear algebra. Certainly there are times when I am drawing 3D arrows and the like to work out a problem, but by-and-large the symbolic representation serves far better. There is a reason we use text for communication on HN, and not pictures. It is not because we lack vision or foresight (no pun intended), but because it is the most expressive and flexible way to communicate our ideas. And, of course, sometimes we labor away, trying to communicate in words what would be so easy to illustrate with a single image. Neither is 'best'; we just use the best tool for the job. None of that is to argue against one of your points - that math is more than symbolic representation. Once you are reduced to applying symbolic rules to an equation you are often quite lost, and that is often not taught in schools. Yet so much of math really cannot be done graphically at all, yet we succeed. A lot of people stall out in math (I think) at after the algebra/calc I,II,III level because beyond that there is often no recourse to graphics and physical representations; graphics can be as much a crutch as a help. Anyway, visual thinking is certainly a form of thinking that should not be downplayed, but it is not how some to many brains work, nor is it even possible in many forms of important thought (I leave savants to one side, we can not make decisions based on exceptional/unique people).

leephillips 13y ago

I would add to your references Hadamard's book[0], where he talks about how ideas form in various mathematicians' heads. I don't really disagree with you, it's just that Victor seems to be claiming too much for his new interfaces, and selling symbolic manipulation short. Of course all kinds of things need to be happening inside our heads before we start manipulating symbols. But Feynman's papers and books are full of equations and derivations, not just answers. Most of Victor's examples are actually counter-examples, because they are examples of problems where a little algebra would not only be faster and easier but would yield a simple formula that you could use later, the next time the problem comes up. Take the book design example. What happens the next time you need to fit a similar graph on the page? If you had used algebra you could have written a little calculator in Python in which you could plug in the new numbers and which would return the solution. Victor's way requires you to treat each graph-fitting problem as a special case, even if they're all the same. [0]http://www.amazon.com/The-Psychology-Invention-Mathematical-Field/dp/0486201074 http://www.amazon.com/The-Psychology-Invention-Mathematical-...