Quote:
Originally Posted by sjastro
Alpha comes along and discovers (or invents) the equation.
If he discovered the equation there are no problems, he has simply derived the equation that shows the relationship between c and a,b.
If he has invented the equation there is a problem. By definition prior to the invention, the equation never existed. If the equation never existed then neither did relationship between c and a,b. If the relationship between c and a,b never existed then how could the builders build to the correct shape?
So therefore if the equation is invented then it was impossible for the builders prior to the invention to build to the correct shape. To suggest it is possible is a logical contradiction.
Steven
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Consider this. As a pure mathematician, I derive theorems for right-angle triangles that allow me to find c (opposite the right-angle) for a plane, spherical and hyperbolic geometries. To derive these, I need have no recourse to the real world i.e I need not physically measure anything.
I transport to planet X on the other side of the Universe and, by luck, find the local Universe is flat and my measurements confirm the results of the plane geometry created on Earth. Alpha, on planet X, says he can beam me into parallel Universe U2. In U2, I find the geometry is spherical and my measurements for right-angle triangles confirms this. Getting back to planet X, Alpha tells me that there seems to be no hyperbolic parallel Universes.
So, on Earth, Pythagoras' Theorem is confirmed from measurement and was, truth be told, originally formulated from measurements. On Alpha, Pythagoras' Theorem is unknown but I can also confirm the theoretical geometry from measurements I make there. Back on Earth, one could say that the relationship described by Pythagoras' Theorem was discovered by measurement. However, the proof of the theorem is not so much a discovery but an exercise in logic.
But did I discover spherical geometry or create it- I can certainly confirm it in U2? But the mathematics for a hyperbolic Universe exists even though it seems such a Universe does not. Is it a creation or a discovery? It is not a physical discovery of the world we live in.
Regards, Rob.