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To be honest, I prefer the unattractive and inelegant proof given alongside that one in Lockhart's essay.

lupire 2y ago
Lockhart missed the point a bit. His student's "proof" is an illustration, not a proof. There's no way to know if its circular or simply unfounded, since it is purely an appeal to intuition. That's only a part of mathematics. The correct thing to do, mathematically, is to validate the intuition by formatting it as a proper proof, based on non-circular axioms and theorems. In the paper itself he admits that his student's work was incoherent and needed him to rewrite it. He's right that 2 column geometry proofs are ugly, and could be presented better. This has been known for centuries. https://www.c82.net/euclid/en/book3/#prop31 https://www.c82.net/euclid/en/book3/#prop31 For children and for starting out, the pictures are great. But for mathematics, pictures are extremely limiting. 2D and 3D are notgod models for N-Dimemsions. ("Spiky balls" , for example). Mathematics is far, far more powerful than human eyes. The amazing thing about geometry is that the whole thing works without pictures! A blind person can be a great geometer, because geometry is axiomatizable. Meanwhile, Euclid's Elements, while an incredible achievement in its day, is not well-founded, relying on unstated axioms.
- JadeNB 2y ago
  > Lockhart missed the point a bit. I think it's fairer to say that his point may not have been what you expected it to be. As a teacher myself, trust me, if a student comes to you and says "I came up with a proof!" and you say "no, you see, what you have is an intuitive explanation that can possibly be turned into a proof," then all that will happen is that that student will not be interested any more in exploring, or at least will not be interested any more in sharing their explorations with you. At that point, you have both lost. Lockhart is well aware of the standard of mathematical proof, and knows that, all else aside, this theorem is not in want of proof. His focus is on the fact that, if we want mathematics to remain a live profession, then we must improve our pedagogy, and help to train students who enjoy and want to pursue mathematics—even if it means occasionally accepting less than maximally rigorous mathematics from a seventh grader.
- zyklu5 2y ago
  While I respect your point about rigour I should say one has to be careful not least because the way the subject is presented in textbooks are most often backwards -- axioms are really an end not the starting point. Here's another view: Euclidean geometry is euclidean because the underlying transformation group (the group of those transformations which preserve what we want to preserve -- in this case the metric) is the euclidean group (the semi-direct product of the orthogonal group and translations). This is the symmetry that encodes our intuition -- the same intuition the kid is using to prove the above result. If we were to change the underlying space to the real projective space instead of R^2, and instead of choosing to preserve the metric we choose incidence and cross-ratio, we'd get a different group (GL(3,R)) and different geometry, viz. projective geometry. This is an ancient dialectic that runs within mathematics -- embodied in modern math by Hilbert on one side (the formalist) and Poincare on the other.

Lockhart missed the point a bit. His student's "proof" is an illustration, not a proof. There's no way to know if its circular or simply unfounded, since it is purely an appeal to intuition. That's only a part of mathematics. The correct thing to do, mathematically, is to validate the intuition by formatting it as a proper proof, based on non-circular axioms and theorems. In the paper itself he admits that his student's work was incoherent and needed him to rewrite it. He's right that 2 column geometry proofs are ugly, and could be presented better. This has been known for centuries. https://www.c82.net/euclid/en/book3/#prop31 https://www.c82.net/euclid/en/book3/#prop31 For children and for starting out, the pictures are great. But for mathematics, pictures are extremely limiting. 2D and 3D are notgod models for N-Dimemsions. ("Spiky balls" , for example). Mathematics is far, far more powerful than human eyes. The amazing thing about geometry is that the whole thing works without pictures! A blind person can be a great geometer, because geometry is axiomatizable. Meanwhile, Euclid's Elements, while an incredible achievement in its day, is not well-founded, relying on unstated axioms.

JadeNB 2y ago
> Lockhart missed the point a bit. I think it's fairer to say that his point may not have been what you expected it to be. As a teacher myself, trust me, if a student comes to you and says "I came up with a proof!" and you say "no, you see, what you have is an intuitive explanation that can possibly be turned into a proof," then all that will happen is that that student will not be interested any more in exploring, or at least will not be interested any more in sharing their explorations with you. At that point, you have both lost. Lockhart is well aware of the standard of mathematical proof, and knows that, all else aside, this theorem is not in want of proof. His focus is on the fact that, if we want mathematics to remain a live profession, then we must improve our pedagogy, and help to train students who enjoy and want to pursue mathematics—even if it means occasionally accepting less than maximally rigorous mathematics from a seventh grader.
zyklu5 2y ago
While I respect your point about rigour I should say one has to be careful not least because the way the subject is presented in textbooks are most often backwards -- axioms are really an end not the starting point. Here's another view: Euclidean geometry is euclidean because the underlying transformation group (the group of those transformations which preserve what we want to preserve -- in this case the metric) is the euclidean group (the semi-direct product of the orthogonal group and translations). This is the symmetry that encodes our intuition -- the same intuition the kid is using to prove the above result. If we were to change the underlying space to the real projective space instead of R^2, and instead of choosing to preserve the metric we choose incidence and cross-ratio, we'd get a different group (GL(3,R)) and different geometry, viz. projective geometry. This is an ancient dialectic that runs within mathematics -- embodied in modern math by Hilbert on one side (the formalist) and Poincare on the other.