Mayor aims of Mathematics

[Guille]

The actual title of this thread should be: The mayor aims of mathematics for the creation of the theory of everything. But as it is clearly too long, I choosed the present one. Ok, the question I want to make to aeveryone is:

What are the mayor goals that mathematics shoudl acheive, or on which mathematicians should centre, to acheive the TOE?

Please don't tell me: the development of mathematical physics. THis is obvious. It's clear that the TOE's mathematics shoudl mostly be mathematics for physics, such as lagrangian and hamiltonioan mechanics, thermodynamics, electrodynamics... What I'm looking for is the decision of brnahce so fmathematicsin which to centre the study and development, to impulse the ability to create a theory of everything.

In my opinion, the main aim of mathematics shoul dbe to unify and find relationships between the parts of itself that are important and haven't been related yet. Of course, developing new mathematics branches coul dbe helpfull, but we shoudl first devleop the present branches, and then go into new branches. The tow main areas of mathematics that haven't been unifyed and have if at all very little relationships are: PROBABILITY AND GEOMETRY. Yes, I know that they are apparently very far subjects of maths, but I have ocme along many problems that lead to the necesary unfication of these into something that could be called: Probabilistic Geometry or Geometric Probability. It doesn't matter hwo it's done, the fac tof managing something like that, would be the most important mathematical development since the invention of calculus by Newton or Leibniz (not and, or: it's either of them, one had to copy the other, coudln't be the same development, same time). I have tried to find relationships between the two, but the ahve really vry little,. so probably most of what connects the two areas would be new maths all. I now remember that when i asked Antonio if there exists any connection between probability and geometry and he could remember the works of someone I who's name I don't remember, but I remember it was about fractional dimensions, and all that. And this was not very logical in geomtrical forms.

Do you think the unification of probability and geometry will be important for the TOE?

I haven't included any math in this post because I didn't know where to start. But pleace, if you have or know about a principle by which we could start the unification, post the maths here.

[AntonioLao]

Do you think the unification of probability and geometry will be important for the TOE?

This question is too open ended to have a simple answer. Geometry as we know it today is divided into Euclidean and non-Euclidean geometries. Although Euclidean foundation had not changed since Euclid, the non-Euclidean ones have gone through many distinct revolutions. A good example is differential geometry, a combination of calculus and geometry. Since calculus is a study of the infinitesimals, there is a good reason to believe that some kind of infinitesimal geometrical structures can be distinguishable in similar character as the square and the circle. When probability is attached to these structures then some related questions would be under what boundary conditions will the evolution of a state function chooses one structure out the many possible structures.

[Guille]

I have read more pages on the book about chaos that I'm reading. Now I have come along turbulence explained by strange atractors and fractals. Can probability be enhaced to fractal geometry?

[AntonioLao]

Can probability be enhaced to fractal geometry?

As much as computer programmings of fractal geometry, they use pseudo-random number generators to produce those fantastic images of fractals.

The difficulty with chaos theory is its nonlinearity. Bear in mind that nonlinear equations imply infinite solutions. FYI, Einstein's field equations of general relativity are nonlinear. They have infinite solutions, which naturally include a solution for static universe, a solution for expanding universe, maybe a rotating universe, a contracting one, a nonsymmetric universe, an ellipsoid, a spherical, hyperboloid, cylindrical, etc.

[Guille]

But most of the nonlinear partial differential equations, in contrast, have no possible solution. How can this be?

Is nonlinearity so chaotic, random and uperiodic?

[AntonioLao]

But most of the nonlinear partial differential equations, in contrast, have no possible solution. How can this be?

The key point is that there are no constants (scalars: real or complex) in nonlinear math. In linear physics there are constants like the speed of light of special relativity, Planck's constant in QM; in nonlinear general relativity, there is no explicit constants although the universal constant of gravity, G, is implied. Dirac used to think that by the principle of extremely large numbers, G is not a constant. If numbers are all very large, it is very difficult to measure which one among them is the largest.

Is nonlinearity so chaotic, random and uperiodic?

another word to describe nonlinearity is its unpredictability.

[Guille]

Actually, your response to the first question (that there are no constants) impplies your response to the second question (that it's un predictable).

But if this is true, why does it say in my book something about thisstudent of the Massachusetts Institute of Technology called Mitchell Feigenbaum realised this constant, 4.66920.... whcih was a nonlinear pattern, if there are supposed to be no constants?

[AntonioLao]

Feigenbaum's number is an attractor. It is like a mathematical limit. All processes are just evolutionary tendencies toward the attractor but never actually reached it. It's like the speed of light. We can't reached this speed unless we have infinite energy to waste.

[Guille]

Feigenbaum's number is an attractor. It is like a mathematical limit. All processes are just evolutionary tendencies toward the attractor but never actually reached it. It's like the speed of light. We can't reached this speed unless we have infinite energy to waste.

This is the physical proof of my philosphical theory about unachievable perfection or any of it's properties.