[AntonioLao]

Originally Posted by GUILLE

What does the fact that lightning doesn't follow straight line (it follows geodesic ones) have to do with what happens in the brain? Can we explain thought throughout this?

geodesic line is related to the intrinsic curvature of a particular spacetime. For the Euclidean spacetime, the geodesic is a straight line since the intrinsic curvature is zero. Since electric field and magnetic field are spherically symmetric in a curvilinear spherical coordinates of spacetime, the intrinsic curvature varies from point to point, the straightest path is not necessarily always the path of less resistance for the electric current.

[baudrunner]

I like to imagine the Universe as balls within balls as it expands outwards in a colosssal wave. That makes sense of the non-curvature of a straight line in a euclidian Universe which exists in a Universe where where space-time is curved. Keep going forward and you'll wind up where you started. It makes perfect sense.

[mkirkpatrick]

life is eternal-that is an absolute-certantly,and potentuality of course.there is a neverending supply of certantly.

[hanzoganz]

Originally Posted by <<<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?

The geometry you are talking about is Differential and Algebraic Geometry. Dynamical systems is a branch of math that studies hamiltonian systems.

[hanzoganz]

Originally Posted by 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)

I believe this is a good clasification if what you are trying to do is implement the high school curricula. For advanced math i would classify as discrete and continous math. That gives enough. For a university level i would say: Topology, Geometry, Algebra, Differential Equiations, Probability, Education, Computers and Physics All that you mentioned before is needed for all this.

[AntonioLao]

Originally Posted by hanzoganz

All that you mentioned before is needed for all this.

Thanks. Before learning how to integrate first is to learn how to add. Before learning how to differentiate 2nd is to learn how to subtract. Before learning how to do math 3rd is to learn how to count.

[hanzoganz]

And I`m not completelly sure that counting is needed to do math. Let`s take the first geometry, the euclidian. That school invented a way of doing math, goemetry in this case, without the need to count or measure. So math is much more than counting.

Now, if we want to take a look at modern math, the theory of wavelets or the one of cellular automata, gives a lot of information just by looking at the patterns...

cheers

[AntonioLao]

the process of counting implies the use of cardinal numbers (quantification) while that of pattern recognition implies the use of ordinal numbers (qualitification). So math is all about numbers. But as you said complex numbers can do both.

[hanzoganz]

But not everything in math in counting. There are uncountable topological spaces that can not be defined by numbers. In its most primitive forms, math doesn`t deal with numbers or shapes but with abstract spaces.

[baudrunner]

Speak English, you guys!

Euclidian and cartesion geometries involve lines and angles, trigonometry, and vectors such as plotting velocity versus time, and plane mathematics, and the construction of three dimensional polygons using plane surfaces. Differential Calculus is concerned with the rate of change of a moving object, implicating expansion of a three dimensional object or the expansion of a plane area over time etc. Integral Calculus is concerned with the reduction of a range of the changes of the rates of a moving object or a variable over time with varying rates of change over the time frame into a mean plotted in two dimensions for example, but remember it's all just math and geometry and drafting. Turn on the Simpsons and when it's over just go to bed. That's what I do.