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I'd rather be wary of explaining mathematical concepts with pictures. Pictures are known to be highly misleading and force the reader to imagine a thing in a ce

by Grue3 9y ago

I'd rather be wary of explaining mathematical concepts with pictures. Pictures are known to be highly misleading and force the reader to imagine a thing in a certain way. For example, most pictures of triangles have all 3 angles being acute. This leads people into forgetting the case when one of the angles is obtuse.

Falkens_Maze 9y ago
Completely agree. It's what Prof. Susskind states in one of his lectures on QM explaining quantum fields. Eventually pictures and mental abstractions break down, and it's better to not visualize. At some point, one's limited biological brain will be unable to fully perceive or visualize some mathematical object. That's where intuition comes in.
- jules 9y ago
  You can still reason by analogy with something you can visualise. You can't visualise 3+1 = 4 spacetime dimensions, but you can pretend that space has 2 dimensions so that you can visualise it as 2+1 dimensions. Blind algebra rarely works. I think that many have an optimistic view of blind algebra because proofs are often stated as algebra without geometric intuition, even when the author of the theorem almost certainly used geometric intuition to come up with it. This can give a false impression. I've seen people struggle tremendously with simple proofs in Hilbert spaces, when the corresponding proof in 2 dimensions is easy by drawing a picture and translating that into algebra that works just as well in the more general setting. For example, let S be a closed subspace of Hilbert space H and x in H. Prove |x-y| is minimised over y in S iff x-y is orthogonal to S.
  - AstralStorm 9y ago
    Analogies break down as easily as they are helpful. On average, they help mathematical intuition less than expected. (Check Bose-Einstein thought experiments and the mathematics they spawned for an example.) QM went into blind math side way past any analogies. Yet it is currently matching observations which is as good as it gets with true.
    - jules 9y ago
      QM did not go into blind math mode. The way physicists usually do QM is by glossing over a lot of the subtleties of Hilbert spaces and act as if the results from finite dimensional linear algebra work. Concepts physicists have been using for a long time, such as the Dirac delta function and path integrals, only got formalised much later. Many revolutions in QM, such as Feynman diagrams, are very much based on intuition. You can transfer all of that back into formal math, as Dyson did, but Dyson could not have done so without Feynman having invented them first. Especially in mathematics it is fashionable to write proofs in a style that completely erases the reasoning that lead to the proof. I think that's a pity. So much unwritten knowledge gets lost that way when the originators die. Differential forms, for example, have a beautiful geometric interpretation, but the way they're taught nowadays obscures that completely and makes it seem like they're a formal algebraic tool only. In fact, we're now several generations later, so even some of the instructors may be unaware of that, because their own instructors failed to transfer the original geometric intuition!
mathgenius 9y ago
"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine." — Sir Michael Atiyah, 2002
yummy 9y ago
On the contrary, graphs is a good example, when pictures make perfect sense and any other way is rather meaningless for human brain. But yes, algebra specifically and pictures don't gel.
wglb 9y ago
Ok, so how would you explain venn diagrams without pictures? How would you describe Mandelbrot sets without pictures? How would you explain hyperbolic parabaloids without pictures? How would you explain conic sections without pictures? How would you explain a trapezoid without pictures? How would you describe a sine wave without pictures? Eversion of a sphere? How would you illustrate the calculation of the slope of a curve when talking about derivatives without pictures? How about a cycloid? How would you show the representation of a signal in both time domain and frequency domain?
- Grue3 9y ago
  >How would you describe Mandelbrot sets without pictures? It's literally easier to describe Mandelbrot set without the picture than with it. The set of complex numbers c where recursive equation "z_n = z_(n-1)^2 + c, z_0 = 0" does not diverge. In fact, the picture of a Mandelbrot set is actively misleading! You think you can see the set, but it's a fractal! It's infinitely complex in a way that cannot be shown in a picture. >Eversion of a sphere? Pretty sure that one was figured out way before anyone was able to visualize it. Same with Banach-Tarski paradox. These things resist visual intuition in the first place, so it's much safer to rely on equations when dealing with problems like these.
  - wglb 9y ago
    If you read of the early work that Mandelbrot did on this equation, you will recall that despite having those symbols in front of him, he had to plot it out, crudely at first on a line printer, before he fully understood the implications. Thus pictures are not only tutorial but a fundamental necessity for understanding mathematics.
  - Simon_says 9y ago
    You picked possibly the worst example for what you're trying to say. Nobody knew it was a fractal until it was plotted. Proving it's a fractal was done with a written, rigorous proof, but the intuition for where to look came from the pictures.
i2om3r 9y ago
You can explain things with more than a single picture. You can also play with pictures in your mind, and you can encourage students to do so by giving them more than one picture. Visualizing corner cases is often very useful. For triangles, you can change angles and side lengths and see which facts still hold and which don't and how certain values/function results develop. Moreover, it very much depends on who your target group is. My brain, for example, works entirely inductively (in the beginning). I won't be able to develop an intuition of something if I don't start with examples. Pictures are often good examples. During my undergrad studies, my linear algebra prof was as critical about pictures as you and other commenters here. I hated it. I was never able to get an intuition about the more abstract topics until I saw concrete examples including pictures in later lectures and projects. Moreover, not everyone is going to be a theoretical mathematician or quantum physicist. I suspect that by not showing pictures, you usually lose more students along the way than pictures would ruin students that need a fully abstract understanding (later). It would be interesting to see some data on this, but I guess its going to be difficult to collect.

by Grue3 9y ago

Falkens_Maze 9y ago

Completely agree. It's what Prof. Susskind states in one of his lectures on QM explaining quantum fields. Eventually pictures and mental abstractions break down, and it's better to not visualize. At some point, one's limited biological brain will be unable to fully perceive or visualize some mathematical object. That's where intuition comes in.

jules 9y ago
You can still reason by analogy with something you can visualise. You can't visualise 3+1 = 4 spacetime dimensions, but you can pretend that space has 2 dimensions so that you can visualise it as 2+1 dimensions. Blind algebra rarely works. I think that many have an optimistic view of blind algebra because proofs are often stated as algebra without geometric intuition, even when the author of the theorem almost certainly used geometric intuition to come up with it. This can give a false impression. I've seen people struggle tremendously with simple proofs in Hilbert spaces, when the corresponding proof in 2 dimensions is easy by drawing a picture and translating that into algebra that works just as well in the more general setting. For example, let S be a closed subspace of Hilbert space H and x in H. Prove |x-y| is minimised over y in S iff x-y is orthogonal to S.
- AstralStorm 9y ago
  Analogies break down as easily as they are helpful. On average, they help mathematical intuition less than expected. (Check Bose-Einstein thought experiments and the mathematics they spawned for an example.) QM went into blind math side way past any analogies. Yet it is currently matching observations which is as good as it gets with true.
  - jules 9y ago
    QM did not go into blind math mode. The way physicists usually do QM is by glossing over a lot of the subtleties of Hilbert spaces and act as if the results from finite dimensional linear algebra work. Concepts physicists have been using for a long time, such as the Dirac delta function and path integrals, only got formalised much later. Many revolutions in QM, such as Feynman diagrams, are very much based on intuition. You can transfer all of that back into formal math, as Dyson did, but Dyson could not have done so without Feynman having invented them first. Especially in mathematics it is fashionable to write proofs in a style that completely erases the reasoning that lead to the proof. I think that's a pity. So much unwritten knowledge gets lost that way when the originators die. Differential forms, for example, have a beautiful geometric interpretation, but the way they're taught nowadays obscures that completely and makes it seem like they're a formal algebraic tool only. In fact, we're now several generations later, so even some of the instructors may be unaware of that, because their own instructors failed to transfer the original geometric intuition!

mathgenius 9y ago

"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine." — Sir Michael Atiyah, 2002

yummy 9y ago

On the contrary, graphs is a good example, when pictures make perfect sense and any other way is rather meaningless for human brain. But yes, algebra specifically and pictures don't gel.

wglb 9y ago

Ok, so how would you explain venn diagrams without pictures? How would you describe Mandelbrot sets without pictures? How would you explain hyperbolic parabaloids without pictures? How would you explain conic sections without pictures? How would you explain a trapezoid without pictures? How would you describe a sine wave without pictures? Eversion of a sphere? How would you illustrate the calculation of the slope of a curve when talking about derivatives without pictures? How about a cycloid? How would you show the representation of a signal in both time domain and frequency domain?

Grue3 9y ago
>How would you describe Mandelbrot sets without pictures? It's literally easier to describe Mandelbrot set without the picture than with it. The set of complex numbers c where recursive equation "z_n = z_(n-1)^2 + c, z_0 = 0" does not diverge. In fact, the picture of a Mandelbrot set is actively misleading! You think you can see the set, but it's a fractal! It's infinitely complex in a way that cannot be shown in a picture. >Eversion of a sphere? Pretty sure that one was figured out way before anyone was able to visualize it. Same with Banach-Tarski paradox. These things resist visual intuition in the first place, so it's much safer to rely on equations when dealing with problems like these.
- wglb 9y ago
  If you read of the early work that Mandelbrot did on this equation, you will recall that despite having those symbols in front of him, he had to plot it out, crudely at first on a line printer, before he fully understood the implications. Thus pictures are not only tutorial but a fundamental necessity for understanding mathematics.
- Simon_says 9y ago
  You picked possibly the worst example for what you're trying to say. Nobody knew it was a fractal until it was plotted. Proving it's a fractal was done with a written, rigorous proof, but the intuition for where to look came from the pictures.

i2om3r 9y ago

You can explain things with more than a single picture. You can also play with pictures in your mind, and you can encourage students to do so by giving them more than one picture. Visualizing corner cases is often very useful. For triangles, you can change angles and side lengths and see which facts still hold and which don't and how certain values/function results develop. Moreover, it very much depends on who your target group is. My brain, for example, works entirely inductively (in the beginning). I won't be able to develop an intuition of something if I don't start with examples. Pictures are often good examples. During my undergrad studies, my linear algebra prof was as critical about pictures as you and other commenters here. I hated it. I was never able to get an intuition about the more abstract topics until I saw concrete examples including pictures in later lectures and projects. Moreover, not everyone is going to be a theoretical mathematician or quantum physicist. I suspect that by not showing pictures, you usually lose more students along the way than pictures would ruin students that need a fully abstract understanding (later). It would be interesting to see some data on this, but I guess its going to be difficult to collect.