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That is a great demo. Why isn't it a proof? In a sense, isn't a concrete fact of reality better than an abstract proof?

by abtinf 9y ago

nerdponx 9y ago
It doesn't show that it's true in all cases, just that one.
- muro 9y ago
  It could leave the hypotenuse intact and let you change the size of other sides (while enforcing the right angle).
  - dagw 9y ago
    Also, imagine if a^2+b^2 was actually equal to c^1.9998724... Your physical model wouldn't be accurate enough to detect that discrepancy.
immad 9y ago
You could use it as a starting point to a proof by induction maybe, but it currently only proves one case of the theorem.
- tuukkah 9y ago
  Further, it doesn't prove exact mathematical equality. Depending on the scale of the demonstration and the accuracy of the observations, it demonstrates approximate equality up to a certain precision. (You could also demonstrate π=3.14.)
cyphar 9y ago
It only shows that the Pythagorean theorem holds for that one particular set of side lengths. A mathematical proof requires that you show it is true for all possible side lengths. An analogy would be that I take two equal-length sticks and say "given any two sticks they will be the same length". I have an example (the two sticks I'm holding) but this does not amount to a proof of my statement (and the statement is obviously incorrect).
- jacobolus 9y ago
  It’s interesting to wonder if a contraption might be built whereby some slider adjusts the sizes of the two smaller squares, while maintaining the right angle (i.e. keeps the corner on a circle with the hypotenuse as diameter). It would be much more mechanically complicated and harder to build, but pretty awesome.
  - srtjstjsj 9y ago
    That's easy enough, but it changes the total area, which means some kind of drainage is required, which complicates the demonstration of equal-area squares.
    - jacobolus 9y ago
      As long as you don’t change the size of the large square, the total area of the two smaller squares must be the same. That’s just a restatement of the Pythagorean Theorem.
soVeryTired 9y ago
Demonstrations based on physical objects aren't mathematically precise. Who's to say that the wood doesn't flex, or that the fluid doesn't compress? You can see that the areas are equal to within measurement tolerance, but not that they're exactly equal. Here's a classic 'physical' pictorial proof that 31.5 = 32.5: https://jeremykun.files.wordpress.com/2011/07/31equals32.png https://jeremykun.files.wordpress.com/2011/07/31equals32.png
- deleted 9y ago
  [deleted]
adrianN 9y ago
Besides the issue that a physical demonstration only shows that the theorem is true for one particular triangle, reality is sometimes more tricky than one thinks. See for example this https://en.wikipedia.org/wiki/Missing_square_puzzle https://en.wikipedia.org/wiki/Missing_square_puzzle

That is a great demo. Why isn't it a proof? In a sense, isn't a concrete fact of reality better than an abstract proof?

by abtinf 9y ago

That is a great demo. Why isn't it a proof? In a sense, isn't a concrete fact of reality better than an abstract proof?

nerdponx 9y ago

It doesn't show that it's true in all cases, just that one.

muro 9y ago
It could leave the hypotenuse intact and let you change the size of other sides (while enforcing the right angle).
- dagw 9y ago
  Also, imagine if a^2+b^2 was actually equal to c^1.9998724... Your physical model wouldn't be accurate enough to detect that discrepancy.

immad 9y ago

You could use it as a starting point to a proof by induction maybe, but it currently only proves one case of the theorem.

tuukkah 9y ago
Further, it doesn't prove exact mathematical equality. Depending on the scale of the demonstration and the accuracy of the observations, it demonstrates approximate equality up to a certain precision. (You could also demonstrate π=3.14.)

cyphar 9y ago

It only shows that the Pythagorean theorem holds for that one particular set of side lengths. A mathematical proof requires that you show it is true for all possible side lengths. An analogy would be that I take two equal-length sticks and say "given any two sticks they will be the same length". I have an example (the two sticks I'm holding) but this does not amount to a proof of my statement (and the statement is obviously incorrect).

jacobolus 9y ago
It’s interesting to wonder if a contraption might be built whereby some slider adjusts the sizes of the two smaller squares, while maintaining the right angle (i.e. keeps the corner on a circle with the hypotenuse as diameter). It would be much more mechanically complicated and harder to build, but pretty awesome.
- srtjstjsj 9y ago
  That's easy enough, but it changes the total area, which means some kind of drainage is required, which complicates the demonstration of equal-area squares.
  - jacobolus 9y ago
    As long as you don’t change the size of the large square, the total area of the two smaller squares must be the same. That’s just a restatement of the Pythagorean Theorem.

soVeryTired 9y ago

Demonstrations based on physical objects aren't mathematically precise. Who's to say that the wood doesn't flex, or that the fluid doesn't compress? You can see that the areas are equal to within measurement tolerance, but not that they're exactly equal. Here's a classic 'physical' pictorial proof that 31.5 = 32.5: https://jeremykun.files.wordpress.com/2011/07/31equals32.png https://jeremykun.files.wordpress.com/2011/07/31equals32.png

deleted 9y ago
[deleted]

adrianN 9y ago

Besides the issue that a physical demonstration only shows that the theorem is true for one particular triangle, reality is sometimes more tricky than one thinks. See for example this https://en.wikipedia.org/wiki/Missing_square_puzzle https://en.wikipedia.org/wiki/Missing_square_puzzle