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That's actually untrue. 3 Blue 1 Brown has an excellent example of how you can use visuals to aid intuition in higher dimensional thinking. It's just not the na

That's actually untrue. 3 Blue 1 Brown has an excellent example of how you can use visuals to aid intuition in higher dimensional thinking. It's just not the naive representation.

mjburgess 9y ago
Yes, I think this advice isn't quite right. A pictorial formalism is possible for most if not all mathematical concepts as much as a symbolic. It just won't be naive and may not have been systematically developed and communicated / taught anywhere. The issue here is mostly just an awareness that, naively, the pictures change fast than the formula. (r^1, r^2, r^3... look more a like than lines, circles and spheres).
- dsacco 9y ago
  I agree with what you and the parent are saying here, and I suppose I didn’t well enough communicate my meaning. To clarify, I am focusing on the naive representations, which I find people rely on too much. It is easy to learn lower dimensions naively and feel like they make sense, but you won’t bridge that understanding to higher dimensions as seamlessly as going from a point to a line to a circle to a sphere, etc. And for the pictorial representations which do exist, implicit in understanding them is the symbolic information. Conversely, you can understand common naive representations without an implicit understanding of the formalism. For example, we all know what a circle is before we know about irrational numbers, but it’s silly to try and understand a 5-cell or a tesseract without a broader understanding of k-dimensional space. It’s precisely that naive intuition that doesn’t generalize well. In other words, my point more simply was caution: use pictures, but don’t replace symbolic formalism with them, just use them augment your understanding once you’ve at least sort of got it.
dsacco 9y ago
Yeah, good point. I like 3Blue1Brown’s videos but in fairness I would say that video is quite a bit closer to the formal side of things than the visual side. I was talking about naive representations - if you build up to the compound visualization 3B1B walks through you’ve covered a lot of the rigor implicitly.

That's actually untrue. 3 Blue 1 Brown has an excellent example of how you can use visuals to aid intuition in higher dimensional thinking. It's just not the naive representation.

mjburgess 9y ago

Yes, I think this advice isn't quite right. A pictorial formalism is possible for most if not all mathematical concepts as much as a symbolic. It just won't be naive and may not have been systematically developed and communicated / taught anywhere. The issue here is mostly just an awareness that, naively, the pictures change fast than the formula. (r^1, r^2, r^3... look more a like than lines, circles and spheres).

dsacco 9y ago
I agree with what you and the parent are saying here, and I suppose I didn’t well enough communicate my meaning. To clarify, I am focusing on the naive representations, which I find people rely on too much. It is easy to learn lower dimensions naively and feel like they make sense, but you won’t bridge that understanding to higher dimensions as seamlessly as going from a point to a line to a circle to a sphere, etc. And for the pictorial representations which do exist, implicit in understanding them is the symbolic information. Conversely, you can understand common naive representations without an implicit understanding of the formalism. For example, we all know what a circle is before we know about irrational numbers, but it’s silly to try and understand a 5-cell or a tesseract without a broader understanding of k-dimensional space. It’s precisely that naive intuition that doesn’t generalize well. In other words, my point more simply was caution: use pictures, but don’t replace symbolic formalism with them, just use them augment your understanding once you’ve at least sort of got it.

dsacco 9y ago

Yeah, good point. I like 3Blue1Brown’s videos but in fairness I would say that video is quite a bit closer to the formal side of things than the visual side. I was talking about naive representations - if you build up to the compound visualization 3B1B walks through you’ve covered a lot of the rigor implicitly.