mathematical proof

[HydrogenBond]

I would like to ask a question to the forum. If one can prove something mathematically does that imply it is true?

[Tormod]

The definition of "true" will depend on the context.

[HydrogenBond]

Much of theorectical physics is based on mathematical calculations. If one can mathematically prove a theory within physics does that imply that the theory is based on natural truth? Or can one also prove imaginary theories with mathematics?

[rockytriton]

things can be proven mathematically, but that doesn't mean that people will believe it until they see it demonstrated.

[HydrogenBond]

I posed this question because I figured out an example where mathematical proof can be used to support a theory that is out of touch with reality, but which can hold up to experimental proof. Here it is. Say I was to propose that gravity was due to the repulsion of matter by space. As ridiculous as this theory is, it would not be too difficult for someone with excellent math skills to take a reciprical of all the equations that define gravity, i.e., presto mathematical proof for an illusion. Because it is massaged from proven math, this theory would correlate with experimental data just as well as the correct theory, yet it would be an illusionary theory based on mathematical proof.

If math can prove illusionary theory that can correlate experimental data than how does one know which theories are true and which are illusions? Maybe illusion is the wrong word. Maybe correlation is a better word. Let me rephrase the question; how do we tell the difference between a theory that is a correlation and a theory that expresses natural validity. For example, the many many dimensions of string theory, is this real or just an artifact of a correlation based on mathematical proof?

[UncleAl]

A mathematical proof is internally self-consistent. "Truth," certainly in an absolute metaphysical sense, is irrelevant. Mathematics is not science; there is no empirical constraint or real world falsifiability.

The whole of Euclidean (plane) geometry arises from a mere five postulates. There are no mistakes within Euclid. For all that, Euclid does not correspond to observable reality. Example: In Euclidean geometry the sum of the three internal angles of any triangle is exactly 180 degrees - neither more nor less. It is a trivial proof in high school geometry. Take a globe of the Earth. Let us make a triangle on the surface of the Earth:

1) We take a segment of the Equator. That is the base of the triangle.

2) At either end of the Equator segment, we have a line of longitude as the other two sides. All lines of longitude intersect the Equator at exactly 90 degrees. We already have 180 degrees.

3) We follow the lines of longitude north unil they intersect at the North Pole. That is the third angle and adds more degrees beyond the 180 we already have.

(Note that the Equator and all lines of longitude are geodesic paths on the sphere - true "straight" lines.) A spherical triangle's three interior angles always sum to more than 180 degrees. They can sum to as much as 540 degrees. 540 is bigger than 180. How do you define "truth"? There are in fact eight primary geometries possible in three dimensions,

WP Thurston, "Three-dimensional geometry and topology," Vol. 1. Princeton Mathematical Press, Princeton, NJ, 1997.

GP Scott, "The geometries of 3-manifolds," Bull. Lond. Math. Soc. 15(5) 401-487 (1983)

Two more things:

1) Euler's equation, e^([i(pi)] = -1, unites algebra and analytic geometry. If you know what you are doing, math and reality are indistinguishable. Create a good model. Economics is not a good model.

2)

Say I was to propose that gravity was due to the repulsion of matter by space. As ridiculous as this theory is, it would not be too difficult for someone with excellent math skills to take a reciprical of all the equations that define gravity, i.e., presto mathematical proof for an illusion.

There's facile grandiloquent bullshit in the world. Don't step into it, the foregoing swill or economics.

[Erasmus00]

Originally Posted by HydrogenBond

. Let me rephrase the question; how do we tell the difference between a theory that is a correlation and a theory that expresses natural validity. For example, the many many dimensions of string theory, is this real or just an artifact of a correlation based on mathematical proof?

Generally it works like this: I have some experimental data, and I say "alright, here is a theory that explains that data." And I make a theory. Now, I take my new theory and I say "alright, if my theory is true, it predicts this" and I predict some NEW behavior, preferably a bunch of new behavior. Then I give my predictions to an experimentalist and I say "measure this." He does, either supporting or destroying my theory.

With string theory, we cannot make the measurements required to test many of its predictions. However, with string theory another effect occurs. String theory makes many different aspects of our current standard model fall out of one theory. It "unifies" as is often said. However, we cannot "prove" string theory untill we can perform experiments.
-Will

[Qfwfq]

Originally Posted by rockytriton

things can be proven mathematically, but that doesn't mean that people will believe it until they see it demonstrated.

The only "thing" that can be proven mathematically is mathematics.

Originally Posted by HydrogenBond

Because it is massaged from proven math, this theory would correlate with experimental data just as well as the correct theory, yet it would be an illusionary theory based on mathematical proof.

If two theories can't be distinguished experimentally, they are equivalent and there's no sense in saying which is wrong and which is right.

Originally Posted by Uncle Al

For all that, Euclid does not correspond to observable reality.

Why do you keep claiming that Euclid's geometry is wrong? It isn't a theory, it's a formal system. The surface of the Earth has nothing to do with with ironing tree trunks. A triangle on a flat surface will have a sum of 180 degrees.

[Qfwfq]

Originally Posted by UncleAl

The whole of Euclidean (plane) geometry arises from a mere five postulates.

I've always heard that it has ten axioms, five of which are called postulates.