4 ms·

Very nice explanation! In particular; the idea of breaking down a polynomial as a sum of terms, mapping each term to a graphical view (i.e. analytic to coordin

Very nice explanation! In particular; the idea of breaking down a polynomial as a sum of terms, mapping each term to a graphical view (i.e. analytic to coordinate geometry) and then realizing such a sum of terms can be a complex curve (i.e. a complex graphical view) which can be an approximation of almost any function. The teaching of mathematics has become so abstract that students are not taught how to map it to geometry which is THE way to build intuition. You can understand a lot by just imagining 2D/3D mappings before generalizing to n-dimensional vector spaces. Every student/beginner should study I.M.Gelfand's Functions and Graphs so that they can train themselves to imagine the graphs corresponding to a string of symbols.

RossBencina 1y ago
> Every student/beginner should study I.M.Gelfand's Functions and Graphs so that they can train themselves to imagine the graphs corresponding to a string of symbols. Also I.M. Gelfand and A. Shen's "Algebra" for the strings of symbols.
- rramadass 1y ago
  Also his The Method of Coordinates and Trigonometry and Geometry to round out his set of five books aimed at high school students.
godelski 1y ago
> You can understand a lot by just imagining 2D/3D mappings before generalizing to n-dimensional vector spaces. While I'm a big fan of visualization I must also stress that it can equally be misleading. It's to make mistakes because you can only visualize anything above 3D like a blind man can describe an elephant[0]. It can be useful but it is also common to get caught up in one aspect while ignoring the rest. The trouble is you're not ignoring because you're rejecting what's there, you're ignoring because you can't see it and it is really hard to see it. I also think there's this misnomer of being good at geometry or good at algebra. I lied about math being a language. It is a family of languages. It's good to be able to speak several of these languages, just like it is beneficial to be able to program in multiple languages. The best language is only the best one in a certain context, but technically any of them will work. That's why I suggest multiple perspectives. I'll even say that programming is one of those math languages ;) [0] https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2Fthinkingispower.com%2Fwp-content%2Fuploads%2F2021%2F01%2F56860237-DEC4-4B47-9BE3-AB715EB31982.png&f=1&nofb=1&ipt=7e88ddd0de31307dfbc6e445d4caac144bf7faaeeb34a670e3d8d5c12f6324d7 https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...
- rramadass 1y ago
  The point is that once you learn to visualize your analytic algebraic equations in 2D/3D through the lens of coordinate geometrical views (there are multiple perspectives here since you can have many coordinate systems) your intuition and understanding of the mappings has now been developed well enough that you can comfortably deal with higher dimensions in the abstract algebraic manner. Imagination and Visualization are fundamental to learning Mathematics and Problem Solving. Any technique (both geometric and/or other graphical ones) which helps one to gain comprehension must be taught and shown how to map to algebraic notations. While mathematics has developed by continual abstraction and generalization the teaching of it should be from the concrete simple to abstract general.
  - godelski 1y ago
    I mean I mostly agree but the part that I'm warning against is "you can comfortably deal with higher dimensions [visualizations]". You should always feel a bit uncomfortable in high dimensions least you be lulled into a false sense of security. In fact, this is frequently a pain point for people as they advance. They over leveraged the utility of visualization and hit a wall abstracting those same concepts to domains where you can't do any visualization. Just look at how many people claim to be able to visualize the 4th dimension and point at a tesseract. I'm not disagreeing imagination and visualization aren't critical tools. They are. But read my comment again like a mathematician. If you misread you'll overfit the wrong concept.
    - rramadass 1y ago
      You are just placing too much emphasis on caveats and limitations rather than focusing on the usefulness of the technique itself. Also techniques like Dimensionality Reduction exist to help tame higher dimensions. None of your cautions are really that big of an impediment in practice and i would not want students to take away the wrong message from these comments. Mathematicians/Practitioners know when to switch from simple geometry to algebra and that is taught as a matter of course. As an example, the farin and hansford book Practical Linear Algebra: A Geometry Toolbox that i mention at https://news.ycombinator.com/item?id=45060305 https://news.ycombinator.com/item?id=45060305 precisely takes this approach starting with visualizing vectors in 2D/3D geometry and then moving on to higher-dimensions. Its 1st edition was actually named The Geometry Toolbox: For Graphics and Modeling.

RossBencina 1y ago

> Every student/beginner should study I.M.Gelfand's Functions and Graphs so that they can train themselves to imagine the graphs corresponding to a string of symbols. Also I.M. Gelfand and A. Shen's "Algebra" for the strings of symbols.

rramadass 1y ago
Also his The Method of Coordinates and Trigonometry and Geometry to round out his set of five books aimed at high school students.

godelski 1y ago

> You can understand a lot by just imagining 2D/3D mappings before generalizing to n-dimensional vector spaces. While I'm a big fan of visualization I must also stress that it can equally be misleading. It's to make mistakes because you can only visualize anything above 3D like a blind man can describe an elephant[0]. It can be useful but it is also common to get caught up in one aspect while ignoring the rest. The trouble is you're not ignoring because you're rejecting what's there, you're ignoring because you can't see it and it is really hard to see it. I also think there's this misnomer of being good at geometry or good at algebra. I lied about math being a language. It is a family of languages. It's good to be able to speak several of these languages, just like it is beneficial to be able to program in multiple languages. The best language is only the best one in a certain context, but technically any of them will work. That's why I suggest multiple perspectives. I'll even say that programming is one of those math languages ;) [0] https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2Fthinkingispower.com%2Fwp-content%2Fuploads%2F2021%2F01%2F56860237-DEC4-4B47-9BE3-AB715EB31982.png&f=1&nofb=1&ipt=7e88ddd0de31307dfbc6e445d4caac144bf7faaeeb34a670e3d8d5c12f6324d7 https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...

rramadass 1y ago
The point is that once you learn to visualize your analytic algebraic equations in 2D/3D through the lens of coordinate geometrical views (there are multiple perspectives here since you can have many coordinate systems) your intuition and understanding of the mappings has now been developed well enough that you can comfortably deal with higher dimensions in the abstract algebraic manner. Imagination and Visualization are fundamental to learning Mathematics and Problem Solving. Any technique (both geometric and/or other graphical ones) which helps one to gain comprehension must be taught and shown how to map to algebraic notations. While mathematics has developed by continual abstraction and generalization the teaching of it should be from the concrete simple to abstract general.
- godelski 1y ago
  I mean I mostly agree but the part that I'm warning against is "you can comfortably deal with higher dimensions [visualizations]". You should always feel a bit uncomfortable in high dimensions least you be lulled into a false sense of security. In fact, this is frequently a pain point for people as they advance. They over leveraged the utility of visualization and hit a wall abstracting those same concepts to domains where you can't do any visualization. Just look at how many people claim to be able to visualize the 4th dimension and point at a tesseract. I'm not disagreeing imagination and visualization aren't critical tools. They are. But read my comment again like a mathematician. If you misread you'll overfit the wrong concept.
  - rramadass 1y ago
    You are just placing too much emphasis on caveats and limitations rather than focusing on the usefulness of the technique itself. Also techniques like Dimensionality Reduction exist to help tame higher dimensions. None of your cautions are really that big of an impediment in practice and i would not want students to take away the wrong message from these comments. Mathematicians/Practitioners know when to switch from simple geometry to algebra and that is taught as a matter of course. As an example, the farin and hansford book Practical Linear Algebra: A Geometry Toolbox that i mention at https://news.ycombinator.com/item?id=45060305 https://news.ycombinator.com/item?id=45060305 precisely takes this approach starting with visualizing vectors in 2D/3D geometry and then moving on to higher-dimensions. Its 1st edition was actually named The Geometry Toolbox: For Graphics and Modeling.