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Originally Posted by Robh
I've had a long, hard think about that one. Up to a point, I think we are in some agreement.
Interestingly, as you mentioned, Pythagoras' Theorem (like a lot of the early maths) was formulated from practical observations.
It stands as a provable theorem from defined axioms in the plane (flat geometry). However, the Earth is close to a sphere and Pythagoras' Theorem is decreasingly accurate for points at greater distances. At greater distances, spherical geometry must be applied. On a larger Universe scale, Pythagoras' Theorem only applies where the Universe is relatively flat. So the theorem is not a logical result of the Universe itself- it just happens to be true where its is locally flat.
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The observable Universe is flat.
The geometry of the Universe is governed by the amount of mass and energy present.
A geometrical measurement therefore only confirms the underlying geometry, it doesn't determine it.
Pythagoras' theorem (or spherical geometry for the Earth's surface) only reflects the nature of the underlying geometry.
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Now consider this hypothetical. On planet X in some far away galaxy, a right-angle has no significance. 60 degrees is the angle of importance. All dwellings are hexagonal pyramids. The great mathematician Alpha has developed a theorem for triangles with one angle 60 degrees- c^2=a^2+b^2-ab with c being the side opposite the angle of 60 degrees. Like Pythagoras' Theorem the expression was first hinted at from early measurements.
My question is- was Pythagoras' Theorem or Alpha's Theorem there to be discovered or was it just an invention of the planet's conscious beings. Sure, they are both provable from defined axioms of plane geometry, but is not the theorem a logical result of the path of development taken by each planet civilisation.
However, having said this, either culture could prove the other's theorem from certain basic axioms. So the question I'm asking myself is what is it that is universal in the maths? Is it the logic itself? Theorems can be constructed but are not necessarily there to be found. The mathematical path taken will determine whether a conjecture is made or a theorem is constructed. Conscious beings from different Universes, with entirely different mathematical theorems and constructs, should be able to navigate each other's mathematics from some logical base.
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Let's take the analogy further. Suppose the inhabitants of Planet X have been building these structures for centuries before Alpha came on the scene. Let's suppose they have built the same shape but at different scales. The formula c^2=a^2+b^2-ab applies for all scales and represents a limitation or mathematical constraint. For example the inhabitants can't independently alter dimension c without changing the overall shape of the structure.
When Alpha comes along he discovers the limitation. He may have used logic, lateral thinking etc to derive the formula but it is still a discovery. If he invented the theorem he has also invented the limitation, in which case there was no limitation prior to the invention which of course is a logical contradiction.
There is no right or wrong in these type of discussions always a perpetual grey, it's why this debate has gone on for centuries.
Regards
Steven