Quote:
Originally Posted by sjastro
Rob,
Consider geometrical theorems/conjectures. The catalyst for geometrical concepts is through measurement. In a subtle way a geometrical theorem is analogous to a scientific theory except that individual measuresurements themselves do not prove the theorem.
As an example consider the Pythagorean theorem for right angle triangles c^2=a^2+b^2.
Did Pythagoras invent this geometric property for right angle triangles? Clearly no.
Mathematicians before Pythagoras knew of the relationship on the basis of measurement but where not able to prove it for all "a" and "b". Before Pythagoras the geometrical property was based on conjecture.
The geometrical property has always been around, measurement has simply confirmed it's existence.
The element of invention is through proof. There are at least 8 ways to prove the Pythagorean theorem. Each method is a product of logical processes at work but the ultimate objective of each proof is to confirm the existence of the mathematical property.
This example can be extended to mathematics in general.
In essence the theorems are "already there", how we proceed to prove them is where invention comes into the picture.
Regards
Steven
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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.
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.
As a digression, if we were to communicate with an alien civilisation, maths would be a logical starting point but what would be in that set to get the ball rolling? It is unlikely they would count in base ten, but is base 2 (numbers made up from zeroes and ones) a logical starting point?
Regards, Rob.