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aufbahrung wrote:My background on mathematics is purely functional. I never understood the idea of a proof. Seems like symbolic tautology.
aufbahrung wrote:Can someone explain why a proof ain't saying the same thing with 'different words' for me?
aufbahrung wrote:How can a proof be valid?
aufbahrung wrote:I know equations work and everything...but isn't it a monstrous waste of time and effort to proove the obvious?
Thommo wrote:The fact that Euclidean geometry contains axioms that aren't necessarily reflective of the real world doesn't stop you landing a shuttle on the moon using it.
LucidFlight wrote:Thommo wrote:The fact that Euclidean geometry contains axioms that aren't necessarily reflective of the real world doesn't stop you landing a shuttle on the moon using it.
What might be an example (or examples) of axioms in Euclidean geometry that aren't necessarily reflective of the real world? Asking for a friend.
The curvature is a quantity describing how the geometry of a space differs locally from the one of the flat space. The curvature of any locally isotropic space (and hence of a locally isotropic universe) falls into one of the three following cases:
Zero curvature (flat); a drawn triangle's angles add up to 180° and the Pythagorean theorem holds; such 3-dimensional space is locally modeled by Euclidean space E3.
Positive curvature; a drawn triangle's angles add up to more than 180°; such 3-dimensional space is locally modeled by a region of a 3-sphere S3.
Negative curvature; a drawn triangle's angles add up to less than 180°; such 3-dimensional space is locally modeled by a region of a hyperbolic space H3.
Curved geometries are in the domain of Non-Euclidean geometry. An example of a positively curved space would be the surface of a sphere such as the Earth. A triangle drawn from the equator to a pole will have at least two angles equal 90°, which makes the sum of the 3 angles greater than 180°. An example of a negatively curved surface would be the shape of a saddle or mountain pass. A triangle drawn on a saddle surface will have the sum of the angles adding up to less than 180°.
aufbahrung wrote:What if there was another way of doing mathematics entirely? Rather than 'magic spells that work' which is all it amounts to at present something radically original and containing no axioms, no infintestimal limits or any of that slight of hand voodoo?
Thommo wrote:LucidFlight wrote:Thommo wrote:The fact that Euclidean geometry contains axioms that aren't necessarily reflective of the real world doesn't stop you landing a shuttle on the moon using it.
What might be an example (or examples) of axioms in Euclidean geometry that aren't necessarily reflective of the real world? Asking for a friend.
Well, Euclidean geometry would be a model for a so-called "flat" spacetime. So conditions here at the human scale on Earth are pretty well modelled by it, but famously Einstein observed that gravity bends space, so on larger scales the universe is curved. This curvature allows phenomena like gravitational lensing - lines which seem parallel in one region of space might converge elsewhere because the space itself is curved by the mass of objects.[Reveal] Spoiler:You can find references to this geometry in articles like this one on the shape of the universe:
https://en.wikipedia.org/wiki/Shape_of_ ... e_universeThe curvature is a quantity describing how the geometry of a space differs locally from the one of the flat space. The curvature of any locally isotropic space (and hence of a locally isotropic universe) falls into one of the three following cases:
Zero curvature (flat); a drawn triangle's angles add up to 180° and the Pythagorean theorem holds; such 3-dimensional space is locally modeled by Euclidean space E3.
Positive curvature; a drawn triangle's angles add up to more than 180°; such 3-dimensional space is locally modeled by a region of a 3-sphere S3.
Negative curvature; a drawn triangle's angles add up to less than 180°; such 3-dimensional space is locally modeled by a region of a hyperbolic space H3.
Curved geometries are in the domain of Non-Euclidean geometry. An example of a positively curved space would be the surface of a sphere such as the Earth. A triangle drawn from the equator to a pole will have at least two angles equal 90°, which makes the sum of the 3 angles greater than 180°. An example of a negatively curved surface would be the shape of a saddle or mountain pass. A triangle drawn on a saddle surface will have the sum of the angles adding up to less than 180°.
If you have a globe in your house (or a strong visual imagination) you can see this "triangle" on Earth with three right angles, going from two points on the equator to one of the poles. Pretty cool if you ask me.
Thommo wrote:
You can find references to this geometry in articles like this one on the shape of the universe
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