Measuring a hypotenuse that calls for an irrational number

[martinl]

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The length of the hypotenuse of a right angle triangle having sides that are each 1 cm long is the square root of 2 since a² + b² = c² according to Pythagorean theorem. But the square root of 2 is an irrational number. So how is it possible both to know yet be able to measure the hypotenuse in this example?

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[pka]

The length of the hypotenuse of a right angle triangle having sides that are each 1 cm long is the square root of 2 since a² + b² = c² according to Pythagorean theorem. But the square root of 2 is an irrational number. So how is it possible both to know yet be able to measure the hypotenuse in this example?

You my dear fellow need a good lesson in the philosophy and history of mathematics. There is a very long story to answer your question which begins in sixth century BCE. Here is a bit of it.

You seem to hold a non-Platonic idea about mathematics: "be able to measure the hypotenuse in this example?"
Here is a quote. “As far as the lawsof mathematics refer to reality, they are not certain; and as far as they are certain,they do not refer to reality.” Albert Einstein in Geometry & Experience,1929.

 

[martinl]

Thank you pka for your reply. The short lesson contained on the posted link to the philosophy and history of mathematics is interesting reading, if not a little obtuse, but unfortunately it still leaves me unable to reconcile the paradox that whereas in the physical world, at least, one can measure 'precisely' the length of the hypotenuse, in the mathematical world one can never precisely know it in an example of the kind I presented. I'd be interested to know whether the same is true of all isosceles triangles. If there's no solution to this apparent paradox then I'll just have to content myself knowing such questions remain in the void of the (presently) unknown, along with paradoxes such as measuring a circle: one can measure the length of a circle (presumably a finite figure) even though a circle has no beginning or end or, viewed differently, has an infinite range of beginnings and ends.

 

[stapel]

...it still leaves me unable to reconcile the paradox that whereas in the physical world, at least, one can measure 'precisely' the length of the hypotenuse, in the mathematical world one can never precisely know it in an example of the kind I presented.

Define "precisely measured in the physical world". Define "can not know precisely in the mathematical world".

Thank you!

 

[pka]

Thank you pka for your reply. The short lesson contained on the posted link to the philosophy and history of mathematics is interesting reading, if not a little obtuse, but unfortunately it still leaves me unable to reconcile the paradox that whereas in the physical world, at least, one can measure 'precisely' the length of the hypotenuse, in the mathematical world one can never precisely know it in an example of the kind I presented.

You have a biblical type faith in the ability to apply mathematics to reality. Please reread that Einstein quote again.
My painter/art-historian wife once showed a paper she was reviewing for possible publication. The author marveled at the places mathematical constants show up. I showed her that the Egyptians used wheels to measure. They put a marker(a big spike) on the wheel. They then rolled out so many lengths. But one length is \(\displaystyle 2\pi r\) the circumference of the wheel. Is there any mystery why \(\displaystyle \large\pi\) would show up? NO! it is simply the result of the easiest way to measure.

One of the great influences on my views is the book The number sense: how the mind creates mathematics by Debaene.
Read it. It will make you a Non-Platonist.