Is a 3D Equation System the Future of Advanced Mathematics?

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Discussion Overview

The discussion revolves around the potential benefits and feasibility of representing mathematical equations in three dimensions, as opposed to the traditional two-dimensional representations. Participants explore the implications of such a system for advanced mathematics, visualization, and computational tools.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant suggests that a 3D system could enhance the representation of complex mathematical concepts that are currently limited to 2D surfaces.
  • Another participant questions the practical benefits of visualizing equations in 3D, particularly how one could perceive information beyond the surface of a 3D shape.
  • A response proposes that a computer program could allow users to navigate through a 3D structure, offering insights that are not available in a flat representation.
  • One participant argues that arithmetic expressions are fundamentally one-dimensional, and expresses skepticism about the utility of a multi-dimensional notation system due to its complexity.
  • Another participant acknowledges the idea may not be useful but believes it is worth considering.
  • One participant asserts that while 3D representations exist, the practicality of 2D notation on paper remains a significant advantage, leading to a preference for reducing complex math to 2D.
  • Another viewpoint suggests that functions with multiple arguments could be represented in 3D, but questions the overall usefulness of such representations when they ultimately need to be presented in 2D.

Areas of Agreement / Disagreement

Participants express a mix of skepticism and curiosity regarding the usefulness of a 3D mathematical representation system. There is no consensus on whether such a system would be advantageous or practical.

Contextual Notes

Participants highlight limitations related to the complexity of using a multi-dimensional notation system and the inherent challenges of visualizing information beyond the surface of 3D representations.

Jonnyb42
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I realize that we describe many dimensional and abstract things with only shapes that fit on 2 dimensions. All the operators, objects, sets, Integrals, derivatives, higher math I don't know yet, is all described by equations that could be written down on a 2-dimensional surface, such as paper. I believe we could get help in describing more advanced things if we used 3 dimensions.

Equations have been written on paper and other surfaces for a long time, and with our advances, (and many more advances to come) in computers, I believe we should try to devise a system of representing equations in 3 dimensions. Such a system would only be accessible on a computer or some other device.

If it is not advantageous whatsoever, then I still think we should do it because it would be very interesting.

(PS. If there already is such a system in development, PLEASE show me.)
 
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How does being able to visualize an equation in 3 dimensions do us any good in the way you're talking? If I were to make an equation "cube"... how would I see anything that isn't simply on the surface?
 
You would be able to see more than what's on the surface because a computer program that implements such a system would allow you scan through the "cube" (by the way it doesn't necessarily have to be a cube, but let's say for example,) so you can see a three dimensional structure.

If you are talking about literally a cube, with 6 sides, that is not what I am talking about. I take you saying a cube to mean some structure in the general shape of a cube, there isn't technically the literal surface of a cube.
 
Actually, arithmetic expressions are rather one-dimensional. An example of a truly two-dimensional arithmetic would be that of string diagrams.


I find it unlikely that any physical notational system with more than two dimensions would be useful, due to the sheer difficulty of actually using it.
 
It is probably not useful, but I wanted to throw the idea out there.
 
It is certainly useful and it is used, but being able to write stuff simply on paper is so useful that people are more interested in being able to reduce the math to the 2d case than in figuring stuff out about the 3d case.
 
Sure, whenever you have any function or operator that has multiple arguments, you can represent all the arguments and the result as coming out of it in different directions in 3 dimensions. I don't think it would be all that useful though since it has to be presented in 2D to the user anyway. You might as well use a conventional representation and just move around the terms in a computer the usual way rather than making it 3D and moving them around that way.
 

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