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Reminds me of how the ancient Greeks thought of mathematics in a purely geometric way. The pythagoreans wouldn't buy a proof that you couldn't demonstrate geome

Reminds me of how the ancient Greeks thought of mathematics in a purely geometric way. The pythagoreans wouldn't buy a proof that you couldn't demonstrate geometrically, with whole numbers. They couldn't conceive of infinite or infinitesimal entities, so they explained them away, making calculus one of those ideas that "the Greeks could have had but didn't".

palish 15y ago
Mm, fair point. Although, "geometric thinking" wasn't a hindrance; check this out, it's pretty interesting: For example we can visualize "taking the derivative of the function f(x)" by "rolling a wheel over the top of its graph": http://dl.dropbox.com/u/315/random_pics/derivative_wheel.png http://dl.dropbox.com/u/315/random_pics/derivative_wheel.png The red circle is a "wheel" which happily rolls back and forth along our f(x) graph. The green line is the derivative of f(x). It's intuitively obvious what that green line is -- if you had to fit a flat piece of paper between the wheel and where it touches the graph, then you'd get that line. That's the derivative, also called the "tangent line" or "slope of the graph" at that point. (The wheel will always touch the graph only once, because we can visualize it as small as we need it to be --- infinitely small, even.) It's common knowledge that a curve's derivative is zero at its maximum / minimum. (In other words, "when the wheel reaches the top or bottom of a 'hill', then its derivative is zero, aka the tangent line is perfectly horizontal".) So right away, we immediately understand where the zeroes of the f'(x) graph must be located. But the real power is, we also have a rough idea of how the f'(x) graph must look, by "watching the wheel's tangent line as we roll it along the graph", in our mind's eye. So that's a neat trick, but why does it matter? Well, here's one example: It's the most natural and obvious thing in the world that the derivative of cos(x) is -sin(x). http://dl.dropbox.com/u/315/random_pics/derivative_wheel_cos.png http://dl.dropbox.com/u/315/random_pics/derivative_wheel_cos... We start at x=0; we see that the wheel is "resting on the very top of the slope", meaning the tangent line is flat, so we know right off the bat that the derivative of cos(0) is 0. From there, the wheel "rolls downhill"; therefore the derivative "goes negative"; and it goes back to zero when the wheel reaches the bottom of the hill (which we see is at x=pi). The wheel then begins climbing uphill, so its derivative "goes positive"; and it goes back to zero when the wheel reaches the top of the slope (at x=2pi). So we see that the graph of the derivative of cos(x) has zeroes at 0, pi, and 2pi; and we also intuitively understand that it's negative from 0 to pi; and we see that it's positive from pi to 2pi. Now all that's left is to think about the graph of -sin(x). It's exactly what we just described: zero at x=0, x=pi, and x=2pi; negative from 0 to pi; and positive from pi to 2pi. So that's why "it's obvious to us that the derivative of cos(x) is -sin(x)", QED. So if you've had to worry about rote memorizing those sorts of formulas, I'd urge you to give these visual techniques a shot. You'll never again need to memorize seemingly-arbitrary equations, which is pretty sweet IMO. Sorry for rambling on; I just get so excited about presenting math "in a visual way". It's a lot of fun, and likely wouldn't hinder the ambitious mathematician at all --- just the opposite, in fact: Feynman, for example, had an arsenal of similar techniques, which probably contributed to his intuitive understanding of the physics behind the formulas and his ability to "see past" the raw equations.
- kalid 15y ago
  Awesome example about the derivative! I think the best part of geometric thinking is running ideas on your "native hardware". Our brains are massively parallel, visual-processing monsters... and we try to think about ideas linearly, like we're a turing machine. No -- use that GPU! :). When I think visually, I can immediately grok the idea (it's running in hardware) and my software layer (my conscious thought process) can work on stitching together deeper ideas. Shameless plug, but I've written about visualizing Euler's Identity (http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ http://betterexplained.com/articles/intuitive-understanding-...): * Radians are distance from the mover's point of view * e is continuous growth * i is a rotation (so e^i*anything is imaginary growth, i.e. growth in a perpendicular direction) Combining ideas like this is just so enjoyable! It brings out the natural beauty in learning.
Dn_Ab 15y ago
I would say it was a matter of interest led by cultural biases, not lack of imagination. It is impossible to look at the ideas of Archimedes and not be dumbfounded by his supreme creativity. Archimedes was able to think of infinitesimals and come up with some of the basics of integral calculus but seemed to think of his methods as an unwholesome but useful tool so he never published them. Who knows where things would have gone if he had. http://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems http://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theore...

palish 15y ago

Mm, fair point. Although, "geometric thinking" wasn't a hindrance; check this out, it's pretty interesting: For example we can visualize "taking the derivative of the function f(x)" by "rolling a wheel over the top of its graph": http://dl.dropbox.com/u/315/random_pics/derivative_wheel.png http://dl.dropbox.com/u/315/random_pics/derivative_wheel.png The red circle is a "wheel" which happily rolls back and forth along our f(x) graph. The green line is the derivative of f(x). It's intuitively obvious what that green line is -- if you had to fit a flat piece of paper between the wheel and where it touches the graph, then you'd get that line. That's the derivative, also called the "tangent line" or "slope of the graph" at that point. (The wheel will always touch the graph only once, because we can visualize it as small as we need it to be --- infinitely small, even.) It's common knowledge that a curve's derivative is zero at its maximum / minimum. (In other words, "when the wheel reaches the top or bottom of a 'hill', then its derivative is zero, aka the tangent line is perfectly horizontal".) So right away, we immediately understand where the zeroes of the f'(x) graph must be located. But the real power is, we also have a rough idea of how the f'(x) graph must look, by "watching the wheel's tangent line as we roll it along the graph", in our mind's eye. So that's a neat trick, but why does it matter? Well, here's one example: It's the most natural and obvious thing in the world that the derivative of cos(x) is -sin(x). http://dl.dropbox.com/u/315/random_pics/derivative_wheel_cos.png http://dl.dropbox.com/u/315/random_pics/derivative_wheel_cos... We start at x=0; we see that the wheel is "resting on the very top of the slope", meaning the tangent line is flat, so we know right off the bat that the derivative of cos(0) is 0. From there, the wheel "rolls downhill"; therefore the derivative "goes negative"; and it goes back to zero when the wheel reaches the bottom of the hill (which we see is at x=pi). The wheel then begins climbing uphill, so its derivative "goes positive"; and it goes back to zero when the wheel reaches the top of the slope (at x=2pi). So we see that the graph of the derivative of cos(x) has zeroes at 0, pi, and 2pi; and we also intuitively understand that it's negative from 0 to pi; and we see that it's positive from pi to 2pi. Now all that's left is to think about the graph of -sin(x). It's exactly what we just described: zero at x=0, x=pi, and x=2pi; negative from 0 to pi; and positive from pi to 2pi. So that's why "it's obvious to us that the derivative of cos(x) is -sin(x)", QED. So if you've had to worry about rote memorizing those sorts of formulas, I'd urge you to give these visual techniques a shot. You'll never again need to memorize seemingly-arbitrary equations, which is pretty sweet IMO. Sorry for rambling on; I just get so excited about presenting math "in a visual way". It's a lot of fun, and likely wouldn't hinder the ambitious mathematician at all --- just the opposite, in fact: Feynman, for example, had an arsenal of similar techniques, which probably contributed to his intuitive understanding of the physics behind the formulas and his ability to "see past" the raw equations.

kalid 15y ago
Awesome example about the derivative! I think the best part of geometric thinking is running ideas on your "native hardware". Our brains are massively parallel, visual-processing monsters... and we try to think about ideas linearly, like we're a turing machine. No -- use that GPU! :). When I think visually, I can immediately grok the idea (it's running in hardware) and my software layer (my conscious thought process) can work on stitching together deeper ideas. Shameless plug, but I've written about visualizing Euler's Identity (http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ http://betterexplained.com/articles/intuitive-understanding-...): * Radians are distance from the mover's point of view * e is continuous growth * i is a rotation (so e^i*anything is imaginary growth, i.e. growth in a perpendicular direction) Combining ideas like this is just so enjoyable! It brings out the natural beauty in learning.

Dn_Ab 15y ago

I would say it was a matter of interest led by cultural biases, not lack of imagination. It is impossible to look at the ideas of Archimedes and not be dumbfounded by his supreme creativity. Archimedes was able to think of infinitesimals and come up with some of the basics of integral calculus but seemed to think of his methods as an unwholesome but useful tool so he never published them. Who knows where things would have gone if he had. http://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems http://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theore...