Timeline for How should you respond to a student who asks whether a very nice physical example constitutes a proof? [closed]
Current License: CC BY-SA 3.0
32 events
| when toggle format | what | by | license | comment | |
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| Aug 21, 2018 at 14:22 | history | edited | Kimball |
removed proofs tag
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| Jan 11, 2014 at 15:19 | comment | added | Benjamin Dickman | Just came across this (closed) question. Perhaps you would find this MO post to be of interest: mathoverflow.net/questions/104714/… | |
| Sep 10, 2013 at 13:01 | review | Reopen votes | |||
| Sep 10, 2013 at 17:17 | |||||
| Sep 9, 2013 at 17:00 | comment | added | Todd Trimble | I was just asked (as a reviewer) whether this should be reopened. My answer is 'no'. But if I were teaching such a high school course, then (to repeat what has already been said) "seeing is not always believing"; otherwise we should believe magicians are really capable of producing magic. On the other hand, "a proof is any completely convincing argument" (Errett Bishop), and has nothing to hide. A good proof will moreover be illuminating; in the present case, if you assume the parallel postulate and the sum of three angles being 180 degrees, the Pythagorean theorem is a beautiful consequence. | |
| Sep 9, 2013 at 12:58 | review | Reopen votes | |||
| Sep 9, 2013 at 17:00 | |||||
| Aug 9, 2013 at 18:19 | comment | added | Emerton | ... verification even in that case. But, so as not to be purely negative, one can also remark that the fact that real-world measurements give such good accordance with the theorems of Euclidean geometry shows that the geometry of the world (at least on a small scale near the surface of the earth) is very close to Euclidean, something that is not automatic (and not exactly true). [Looking at the answers below, I guess my comment is similar in spirit to Paul Garrett's answer.] Regards, | |
| Aug 9, 2013 at 18:18 | comment | added | Emerton | Dear Amir, Here is my reaction, which might also be what I would say to the student: first of all, we could just measure the sides of a right-angled triangle, square them and compare the sum of two to the third; if we did so carefully, we would get good agreement. The demonstration with water is basically just a fancier way of doing the same thing. Your student probably understands that checking one example is not the same as proving a general theorem, and could also probably understand that even in one example, measurements and so on are imprecise, so this is really only an approximate ... | |
| Aug 8, 2013 at 13:56 | comment | added | Mariano Suárez-Álvarez | You can also build physical models of this: en.wikipedia.org/wiki/Missing_square_puzzle and they do not prove anything :-) (One can use larger and larger Fibonacci triangles to do this, and the error can be made as small as most sensible measuring instruments) | |
| Aug 8, 2013 at 13:52 | history | edited | Amir Asghari | CC BY-SA 3.0 |
Shorten the previous updates ito one single update.
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| Aug 8, 2013 at 13:03 | review | Reopen votes | |||
| Aug 8, 2013 at 14:44 | |||||
| Aug 8, 2013 at 4:58 | comment | added | Aaron Meyerowitz | I'd maintain that (very?) arguably this is a (not that great) demonstration that $a^2+b^2=c^2$ for one certain particular triangle. It shares some relation to a drawing on graph paper. It is a motivation for a formal proof. Some science museums have whispering galleries demonstrating a reflective property of ellipses. | |
| Aug 7, 2013 at 13:57 | comment | added | Todd Trimble | Well, it's a fun device, but aside from the obvious objections given before, it also doesn't give any insight into why the theorem is true. Isn't that really what a proof is good for? | |
| Aug 7, 2013 at 13:49 | comment | added | Amir Asghari | @JoelReyesNoche I am sure MO users are faced with such down-to-earth situations in their everyday teaching life. I am happy that there are several occasions that MO has accepted that fact in the past. Let's hope for the future. | |
| Aug 7, 2013 at 13:42 | comment | added | JRN | It is a pity that MO does not seem to be a good place to ask mathematics education-related questions. While there are other mathematics education-related question-and-answer websites, the ones I know of do not have an active and representative community of users. | |
| Aug 7, 2013 at 13:42 | comment | added | Amir Asghari | @HenryCohn Indeed, as I mentioned before, he is a highly competent mathematics student. I asked him why he asked such a question. Here is the story. He was teaching a high school math course when his students came up with the question. His answer was "as far as I know, in mathematics a proof is constructed based on axioms". I feel, his answer works for him (us) individually, but it is not a constructive answer for his (our) students as such. That is why I asked the question. | |
| Aug 7, 2013 at 13:40 | comment | added | JRN | There are two things in your image that are different from what is commonly accepted as a mathematical proof. First, it involves a physical (concrete) system, instead of an idealized (abstract) one. Thus, it cannot show an equality, only an approximation. That is, it is not precise. Second, it involves a specific instance of the problem, not a generalization. It shows that the statement is true for that triangle, but not necessarily for all triangles. | |
| Aug 7, 2013 at 13:34 | history | closed |
user6976 HenrikRüping Andrey Rekalo David White Steven Landsburg |
Not suitable for this site | |
| Aug 7, 2013 at 13:30 | comment | added | Henry Cohn | It's unclear to me what your student is even asking. To me, it sounds like the student has seen this presented online as a proof and is skeptical, but it's hard to be sure. | |
| Aug 7, 2013 at 13:20 | history | edited | Amir Asghari | CC BY-SA 3.0 |
Modify the description to go better with the change of the title. Remove one of the tags.
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| Aug 7, 2013 at 13:15 | answer | added | paul garrett | timeline score: 10 | |
| Aug 7, 2013 at 13:04 | comment | added | Amir Asghari | @TheoJohnson-Freyd It is indeed a very constructive modification. Thanks. Consequently I need to modify my description. I'll do it now. | |
| Aug 7, 2013 at 13:00 | comment | added | Theo Johnson-Freyd | I modified the title to make it closer to the question asked. | |
| Aug 7, 2013 at 12:59 | history | edited | Theo Johnson-Freyd | CC BY-SA 3.0 |
edited title
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| Aug 7, 2013 at 12:45 | history | edited | Amir Asghari | CC BY-SA 3.0 |
Clarify the question
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| Aug 7, 2013 at 12:22 | answer | added | Dietrich Burde | timeline score: 3 | |
| Aug 7, 2013 at 11:56 | review | Close votes | |||
| Aug 7, 2013 at 13:36 | |||||
| Aug 7, 2013 at 10:47 | comment | added | Uwe Stroinski | No proof. A magician could use containers with different depths to 'prove' fake theorems. | |
| Aug 7, 2013 at 10:30 | comment | added | A.B. | I think the first question would be: A proof of what? Clearly not a proof of pythagoras, perhaps a proof that the volume of the 2 small containers is equal to the volume of the 3rd. Even then I'm not sure, although that's how I'd test it. | |
| Aug 7, 2013 at 10:24 | comment | added | Paul Reynolds | I want one of those. | |
| Aug 7, 2013 at 9:44 | comment | added | Colin McLarty | I'm with Asaf. A very pretty device. But you know the angles, lengths and volumes are not exact. And there are a lot of other shapes of right triangle. A proof has to cover all that. | |
| Aug 7, 2013 at 9:41 | comment | added | Asaf Karagila♦ | No. It's not an actual proof. It's an excellent example, though. | |
| Aug 7, 2013 at 9:36 | history | asked | Amir Asghari | CC BY-SA 3.0 |