One can roughly divide mathematical education into three stages:
I'm of the opinion that computational mathematics is at the first stage. We write code that "sort-of works" and lacks any attempt at formality, even failing to provide literature references. Moving to stage 2 will be a long and tedious task. Axiom has the connections to the proof machinery and is being decorated to provide some early attempts at proofs using ACL2 and COQ. This effort is an interesting combination of mathematical proof and computational proof since both fields underlie the implementation. Moving computational mathematics up Tao's tower is going to be a long, slow, painul effort but, if memory serves me correctly, so was graduate school. Tim _______________________________________________ Axiom-developer mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-developer |
I don't have a copy of the Classical Mechanics book so I can't comment. re: When do you write code versus when do you use CAS systems like Axiom?I write code when I need to understand something. For example, I'm writing I also write code when I'm working at the basic level of the world. For example, I'm pushing Gustafson's ENUM representation of numbers so I can use it as an Axiom domain. That is so primitive it needs raw code. And maybe an FPGA so there is VHDL to learn. leap from GA to code and Axiom is the right middle ground. struggle is whether Axiom can't do it or I'm just too uneducated to know. phone every year. You don't even know how to replace the battery anymore. I don't want to create "this year's shiny new thing" only to have it die. But if we explain it so it can be maintained, modified, and extended without spending years doing research, which is why highways survive and evolve, then Axiom can live. If we can make it possible to teach computational mathematics "from the ground up" we win. Students use what students learn. Explain your code. Reference the literature. Make it possible to teach it. Think long term. (stepping off the soapbox) Tim On Sat, Dec 3, 2016 at 3:57 PM, Lawrence Bottorff <[hidden email]> wrote:
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See Feynman's Knowing versus Understanding: https://www.youtube.com/watch?v=NM-zWTU7X-k Calculation and "the right answer" are never enough. See also: Wholeness and the Implicate Order by Bohm On Sat, Dec 3, 2016 at 3:57 PM, Lawrence Bottorff <[hidden email]> wrote:
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Tim,
You recently mentioned Clifford Algebra a couple of times so I thought I would mention an idea that I had on the subject. The idea is still too vague and hand-wavy to turn into code but I would be interested to know if anyone thinks its viable? The idea is to implement a 3-layer architecture in the CAS. The layers being: 1 - Physics 2 - Geometry 3 - Algebra Each of these layers would have the ability to slot in different options, for instance: Physics - classical, relativity or quantum. Geometry - Euclidean, projective, conformal or Minkowski space. Algebra - Clifford or matrix/tensor. The general idea being that, when working at the higher level there is some independence from the layers below. For instance, if you are working on a physics problem and you have a issue like: * It does not scale up. * It has an awkward singularity. * Need for different type of transform not supported by algebra. Then you can slot on a different algebra or geometry to see if that fixes the problem without changing the physics code. Unfortunately I can see some difficulties with this idea: 1) How to refer to literal values? Even when working at the physics level we may still need to refer to concrete values for fixed points, planes, transforms and so on. Is there a way to specify these literal values in a way that does not use actual Clifford or matrix values? Even if the answer is no then I think the model could give some independence between the layers. 2) The model may need 2 or more geometry layers. For instance, we may be working in 3 dimensions but we want to combine translate and rotate into a single transform, or we want to get rid of singularities, so we add extra dimensions but we still want to translate back to original coordinates so we need multiple coordinate systems. As I say, this is just a vague idea but I just thought I would mention it. Martin _______________________________________________ Axiom-developer mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-developer |
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