Post B6NbJZTbXb83fx4dHM by djsumdog@djsumdog.com
 (DIR) More posts by djsumdog@djsumdog.com
 (DIR) Post #B692kQSUxafKuwBfIu by EdBoatConnoisseur@poa.st
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       @karna @hazlin
       
 (DIR) Post #B69seCF26MUYhcEv8S by karna@poa.st
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       @EdBoatConnoisseur @hazlin this technique also "proves" that the diagonal of a unit square has a length of 2 rather than sqrt(2)
       
 (DIR) Post #B69seCW3566RYOCVeq by nachtrabe@fsebugoutzone.org
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       @karna @EdBoatConnoisseur @hazlin ok now how do you express whats wrong with the approach such that you are guarded for much less obvious problems?
       
 (DIR) Post #B69seCgKSsKI4H0igK by karna@poa.st
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       @nachtrabe @hazlin @EdBoatConnoisseur the trite answer is that, in the circle case, all that has been shown is that 4 is an upper bound for pi (you can do something similar with an inscribed square instead of a circumscribed square and get a lower bound that is smaller than pi too though). the problem here is that the "repeat to infinity" step sweeps some details under the rug, namely that functions that can calculate perimeters (or more generally arc-length functions) need stronger conditions than what are shown here for convergence to make sense.somewhat more formally, you have a sequence of curves C_n where C_0 is the square, C_1 is the square with the first set of notches, etc. for all n, the perimeter of these curves is 4. take the limit as n goes to infinity and these curves converge to a circle curve in a C^0 sense (the "limit curve" is the circle, but the derivatives of the C_ns do not converge to the derivative of a circle curve). the function that calculates the perimeter of a closed curve is not continuous unless you restrict your space of curves to C^1 functions (where all sequences of curves converge to some limit curve and the derivatives also converge to the derivative of the limit curve - see an analysis textbook for a proof of this). it should be obvious why the derivatives of the C_n curves dont converge to a circle curve's derivative (intuitively (and loosely), the former is always zig-zagging, the latter is smoove as butter).putting all that together:lim_{n->\inf} P(C_n) =/= P(lim_{n->\inf} C_n)we cannot "pull the limit inside the perimeter function (P)" since the perimeter function is not continuous over general curves.
       
 (DIR) Post #B6NbJZTbXb83fx4dHM by djsumdog@djsumdog.com
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       before modern mathematical notation, a lot of proofs were written out in plain language or done using shapes. I do wonder if one day someone will discover a critical edge case or nuance with a commonly accepted axiom that radically alters our understand of mathematics.