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Note that this doesn't apply to proofs by construction, as in Euclidean geometry. Those are visual in nature, but the rules must be rigorously followed, if the

Note that this doesn't apply to proofs by construction, as in Euclidean geometry. Those are visual in nature, but the rules must be rigorously followed, if they are the resulting proofs are reliable. The steps of the proof may also be written out textually, but that's merely a translation, the construction itself is a proof. It's how we can tell the difference between a diagram which looks like a trisected angle, and a construction of a trisected angle, which has been proved impossible in the general case.

zamadatix 3y ago
It kind of doesn't and kinda does at the same time. You can follow these visual mappings of rules to get something that is visually very convincing but subtly not actually a correct result (e.g. something very closely approximating a construction of a trisected angle). If you want to be sure the construction works you've got to have already mathematically proven how the construction method used is valid, applies in the way you used it, and really did come out with the right answer instead of something very close to the right answer which really isn't any less pure math than just doing it symbolically altogether. Something that reasonably makes sense and seems to come out to the right answer is often a great way to shortcut to finding such a validation though, it's just also sometimes a disappointment when it turns out to be arbitrarily close instead of exact.
- samatman 3y ago
  Correctly following the rules of construction results in a valid proof. That proof is in the steps taken, not the final diagram, because there's no way to tell by looking at a diagram if the rules were in fact followed. Part of the point I was making is that proof by construction isn't really a "visual proof" at all, it's a proof system where the steps in the proof are added to a diagram, according to exact rules which must be followed at every step.
  - deleted 3y ago
    [deleted]
  - lupire 3y ago
    How do you know the proof is valid for the theorem you are trying to proof? You have to check the logic of the construction, to know things like which measurements are exactly equal. So the visual part of the construction doesn't suffice. It's the same issue as the infinite chocolate triangles glitch.

zamadatix 3y ago

It kind of doesn't and kinda does at the same time. You can follow these visual mappings of rules to get something that is visually very convincing but subtly not actually a correct result (e.g. something very closely approximating a construction of a trisected angle). If you want to be sure the construction works you've got to have already mathematically proven how the construction method used is valid, applies in the way you used it, and really did come out with the right answer instead of something very close to the right answer which really isn't any less pure math than just doing it symbolically altogether. Something that reasonably makes sense and seems to come out to the right answer is often a great way to shortcut to finding such a validation though, it's just also sometimes a disappointment when it turns out to be arbitrarily close instead of exact.

samatman 3y ago
Correctly following the rules of construction results in a valid proof. That proof is in the steps taken, not the final diagram, because there's no way to tell by looking at a diagram if the rules were in fact followed. Part of the point I was making is that proof by construction isn't really a "visual proof" at all, it's a proof system where the steps in the proof are added to a diagram, according to exact rules which must be followed at every step.
- deleted 3y ago
  [deleted]
- lupire 3y ago
  How do you know the proof is valid for the theorem you are trying to proof? You have to check the logic of the construction, to know things like which measurements are exactly equal. So the visual part of the construction doesn't suffice. It's the same issue as the infinite chocolate triangles glitch.