4 ms·

Sure. The overall proof of Thales is then about as complicated as the “standard” one with two isosceles triangles.

by akho 2y ago

jacobolus 2y ago
Breaking proofs of trickier theorems up into a few parts and hiding the details of each part behind another theorem whose proof is self-contained and easy to follow (even better if it's "obviously true") is the basic idea of mathematics. Trying to make the top-level proof very clearly demonstrate, without a lot of bookkeeping or head scratching, that the theorem must be true is a wonderful goal. In my opinion it's a lot more obvious that a parallelogram with 2 equal diagonals must be a rectangle than that an inscribed angle intersecting a circle at ends of a diameter must be a right angle.
- akho 2y ago
  Whether this proof is complete depends on what was known before. Was the lemma introduced by a teacher? was it not actually proven, and the student missed the hole in their proof? these are different teachable moments. I think the two facts, by themselves, are roughly equivalent in difficulty. Moving from Thales to equal diagonals parallelogram requires drawing an extra circle; moving from parallelogram facts to Thales requires either the construction in the article, or continuing the missing radius to get the same rectangle.
  - jacobolus 2y ago
    I don't understand why you are invested in nitpicking the work of an anonymous 7th grader from decades ago. Clearly, filling in all of the details of any formal proof in Greek style is going to require a careful knowledge of the axioms in use and some list of previously proven theorems which are allowed to be used without re-proof. Depending on which theorems are at hand already, one or another proof might be shorter or longer or more or less obvious. When trying to construct a whole mathematical treatise, the best order for the theorems so that each proof depends only on previously proven theorems is a tricky choice involving some trade-offs between pedagogical goals and concision of individual proofs. But all of that is missing the point.

Sure. The overall proof of Thales is then about as complicated as the “standard” one with two isosceles triangles.

by akho 2y ago

Sure. The overall proof of Thales is then about as complicated as the “standard” one with two isosceles triangles.

Breaking proofs of trickier theorems up into a few parts and hiding the details of each part behind another theorem whose proof is self-contained and easy to follow (even better if it's "obviously true") is the basic idea of mathematics. Trying to make the top-level proof very clearly demonstrate, without a lot of bookkeeping or head scratching, that the theorem must be true is a wonderful goal. In my opinion it's a lot more obvious that a parallelogram with 2 equal diagonals must be a rectangle than that an inscribed angle intersecting a circle at ends of a diameter must be a right angle.

akho 2y ago
Whether this proof is complete depends on what was known before. Was the lemma introduced by a teacher? was it not actually proven, and the student missed the hole in their proof? these are different teachable moments. I think the two facts, by themselves, are roughly equivalent in difficulty. Moving from Thales to equal diagonals parallelogram requires drawing an extra circle; moving from parallelogram facts to Thales requires either the construction in the article, or continuing the missing radius to get the same rectangle.
- jacobolus 2y ago
  I don't understand why you are invested in nitpicking the work of an anonymous 7th grader from decades ago. Clearly, filling in all of the details of any formal proof in Greek style is going to require a careful knowledge of the axioms in use and some list of previously proven theorems which are allowed to be used without re-proof. Depending on which theorems are at hand already, one or another proof might be shorter or longer or more or less obvious. When trying to construct a whole mathematical treatise, the best order for the theorems so that each proof depends only on previously proven theorems is a tricky choice involving some trade-offs between pedagogical goals and concision of individual proofs. But all of that is missing the point.