[AntonioLao]

what are the key characteristics, if any, for the separation of each branch of math?

1. probability (highest level) including discrete and continuous random functions.
2. analyses including real, functional, vector, tensor, fourier, complex, etc.
3. calculus includes differential and integral partial and ordinary equations.
4. analytic geometry includes conic sections linear and nonlinear transformations
5. trigonometry includes plane and spherical.
6. algebra
7. geometry
8. arithmetics (lowest level)

[Guille]

Have 1 (probability) and 3 (calculus) been found any connection? What is it?

[AntonioLao]

Originally Posted by GUILLE Have 1 (probability) and 3 (calculus) been found any connection? What is it?

The theory of probability uses all the math under it since it is considered as the highest point of math. In continuous random functions, the integral(s) of the probability density function over the random variable(s) is always equal to unity. One of these continuous functions is the well known Gaussian function or normal function. It is used in IQ assessment and hypothesis testings, etc.

[Guille]

How can it be used to test hypotheses?

Can a branch of mathematics be made that is over probability?

[AntonioLao]

How can it be used to test hypotheses?

First, you hypothesize what is the expected value based on your probability density function then perform an experiment. If the value derived from the experiment falls within some standard deviation then it proves that your hypothesis is correct.

All branches of math cover over probability since it only works with numbers between 0 and 1.

[Guille]

Oh, yes,

But I meant if there is possibility for a math of higher level than probability to be discovered? If so, how would it be?

[AntonioLao]

Originally Posted by GUILLE if there is possibility for a math of higher level than probability

Personally, I think this would be the mathematics of absolute certainty. However, if everything is certain then why do we need math?

[Guille]

Originally Posted by AntonioLao Personally, I think this would be the mathematics of absolute certainty. However, if everything is certain then why do we need math?

But why should it be absolute certanty at all?

You say that any mathematics level contains all the previous mathematics levels, so, if this is true, then the mathematics of higher level of probability must contain probability, which impplies non absolute certainty, thus, the new math cannot be absolute certainty.

[AntonioLao]

I can give you two replies
1. absolute certainty means that the cumulative probability function is exactly 1. However, if the random variable is time, probability 1 means the end of one's life.
2. All math is invented in order to find a solution to a problem. Absolute certainty means that there is one and only one solution to the problem. When there is one solution, mathematicians call this type of problem linear. However, there are an infinite number of linear problems with different slopes. Solving them can be done by solutions of simultaneous equations. If lines are parallel then no simultaneous solution exists. But for nonlinear problems there are infinite number of solutions to a problem. One can then always select a solution that is applicable to a particular situation.

[Guille]

Are there developments done on the relationship between calculus and trigonometry?

That unification would definatelly be good, because it would make a connection between difference, angles, and legths. Pythagoras' theorem could then be studied by calculus. And probably we could find another theorem just as fundamental but more usefull than theoretical euclidean geometry, usefull to physics, in particular.

Also, where are langragians and hamiltonians in the list of math branches?