My background on mathematics is purely functional. I never understood the idea of a proof. Seems like symbolic tautology. Can someone explain why a proof ain't saying the same thing with 'different words' for me? How can a proof be valid? I know equations work and everything...but isn't it a monstrous waste of time and effort to proove the obvious?

aufbahrung wrote:My background on mathematics is purely functional. I never understood the idea of a proof. Seems like symbolic tautology.

It is. Although I suspect from the way you use those words you might not use them in their mathematical sense.

aufbahrung wrote:Can someone explain why a proof ain't saying the same thing with 'different words' for me?

A proof is a sequence of steps that gets from a set of axioms or assumptions to a conclusion which is not identical with any any of those axioms or assumptions.

Desireable conclusions are often ones which whilst being much narrower than the set of tautological consequences of the axioms of your theory are much more directly applicable. E.g. Cantor's diagonal argument showing that there are different infinite cardinalities for the reals and naturals.

One of the key concepts is that if something is proven then it is true, but if it's not proven (or in some cases even provable) then you don't know if it's true.

aufbahrung wrote:How can a proof be valid?

By the conclusion following from the axioms (or premises) by a sequence of steps all of which follow the rules of the deductive system you're working in.

aufbahrung wrote:I know equations work and everything...but isn't it a monstrous waste of time and effort to proove the obvious?

Mostly you don't prove obvious things. For example Wiles's Theorem (Fermat's last theorem) follows from the axioms of number theory, but it was not obvious for hundreds of years that it was actually a true statement.

Axioms are a arbitary set of conditions. Let's not pretend a proof containing axioms is more than a sticking plaster on the unknowns...

The second sentence does not follow from the first.

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.

A mathematician might say that such and such is the consequence of Euclidean geometry, but not of non-Euclidean geometry (a trivial example being the parallel postulate itself), but this has nothing to do with hiding holes or patching over them. Some groups are abelian and some are non-abelian, both have their own group theory and prove different results. This is a feature, not a bug. In that case real world objects of interest to scientists are successfully modelled by both structures.

Of course from the (purist) mathematician's point of view applicability is neither here nor there.

From your tone I get the feeling you're trying to pick fault with something that you don't understand, rather than trying to understand it. FWIW I'm not sure you'll get terribly far with that approach.

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.

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?

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.

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_universe

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°.

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.

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?

Things that you don't understand are not the same as magic, sleight of hand or voodoo.

Typically rules of inference are chosen with particular well-motivated qualities or systemic properties in mind, and things which are governed by being the consequences of rules of inference are what we call mathematics.

Your question amounts to asking why we don't call "something other than mathematics" "mathematics", which is honestly a rather boring question.

PS: Infinitesimals and limits are totally different things, neither applies to most mathematical results. Limits are very important to calculus, which is very important in physics and engineering applications.

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_universe

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°.

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.

Ah, right. That's what I thought you meant. Just needed to confirm... for my, uh... friend. Thanks!

You're very shifty about this friend.

Thommo wrote:
You can find references to this geometry in articles like this one on the shape of the universe

OK, so we are going to identify 'the real world' with 'the universe'. I know some folks who are going to balk at the very thought.

Cito di Pense wrote:

Thommo wrote:
You can find references to this geometry in articles like this one on the shape of the universe

OK, so we are going to identify 'the real world' with 'the universe'. I know some folks who are going to balk at the very thought.

Those are people that live in the ivory tower, not the real world.