Mathematics - seing the big picture, how it all comes together.

[haki]

For everything I do I want to see the big picture. How the pieces of puzzle come together. For math I have trouble picturing it. From my view Physics are well structured. If I were to picture Physics as a tree, I would imagine it having 6 big branches.
1. Classical Mechnics(kinematics,work,energy,rotation,etc)
2. Relativity(movement when approaching c, time dialation, the twins paradox,etc)
3. Thermodynamics(heat engines, the kinetical theory of gases, etc)
4. Electromagnetism(electrostatics, electrical current, induction, etc)
5. Optics(reflection, refraction of light, optical instruments, etc.)
6. Quantum Mechnics(the emission spectre, hydrogen model, etc.)

the branches all have sub-branches and are ful of leaves also there are connections etc.This I can imagine. It helps me see how it all comes together when I study physics.

But as Math goes my "tree" is not a tree is just a collection of different pieces of puzzle. There is Calculus, Geometry, Linear Algebra, Number Theroy, Algebra without being Linear, Statistics, Probability, Logic, Discrete Structures, all this stuff. I am wondering if you were to imagine math as a tree or a map of a country or whatever how would it look like?

 

[JasonRox]

For everything I do I want to see the big picture. How the pieces of puzzle come together. For math I have trouble picturing it. From my view Physics are well structured. If I were to picture Physics as a tree, I would imagine it having 6 big branches.
1. Classical Mechnics(kinematics,work,energy,rotation,etc)
2. Relativity(movement when approaching c, time dialation, the twins paradox,etc)
3. Thermodynamics(heat engines, the kinetical theory of gases, etc)
4. Electromagnetism(electrostatics, electrical current, induction, etc)
5. Optics(reflection, refraction of light, optical instruments, etc.)
6. Quantum Mechnics(the emission spectre, hydrogen model, etc.)

the branches all have sub-branches and are ful of leaves also there are connections etc.This I can imagine. It helps me see how it all comes together when I study physics.

But as Math goes my "tree" is not a tree is just a collection of different pieces of puzzle. There is Calculus, Geometry, Linear Algebra, Number Theroy, Algebra without being Linear, Statistics, Probability, Logic, Discrete Structures, all this stuff. I am wondering if you were to imagine math as a tree or a map of a country or whatever how would it look like?

I don't think I'm the best candidate to answer the question, but mathematics will certainly have a tree.

I'm guess it starts with Set Theory then moves on to things like Fields, Groups, Rings, and so on.

Calculus, and Linear Algebra use the concept of fields. When you do Calculus, you are using the properties of the Field of Real Numbers.

Anyways, I'll let someone else answer the question, but I hope this gives you an idea on how fundamental it may start.

 

[matt grime]

linear algebra is just a subset of algebra, and not a very tricky part, analysis leaks into algebra via measure theory, and algebra moves back into analysis by topology and hence algebraic topology, but algebra and topology can lean another way and go into geometry via algebraic geometry which leads us back to analysis. (Unifying all those ideas is the Atiyah-Singer Index theorem, or Riemann-Roch, or Riemann-Hurwitz theorems). Measure theory also makes probability and statistics interact with analysis. algebra also branches off on its own apparently only to come back to geometry (and vice versa) when sheaves come up, sheaves also interest topologists and physicists.

topology led to category theory, which leads into alll of these areas all over again.

one thing I've not mentioned yet is combinatorics, and number theory, now i come to think of it. well, i can fit those into this picture too, and they'd be spread out amongst the many things laready mentioned.

but perhaps that is misleading, and represents my interests in mathematics. someone else doing this could, would, start with number theory and analysis, say, and weave algebra into afterwards.there is no simple tree, perhaps you want to think of them as clouds of elementary particles mingling and interacting. there are for instance interactions between category theory, combinatorics and computer science. and then there are completely different ideas that collide in combinatorics, analysis and game theory.

 

[haki]

Thank you very much for you view on Math. Specially for pointing out the unification theorems. Since I am bit of math novice it will take some time to "see" the connections. For now I have a bit naive picture: Math at Fundations is the Logic and Set Theory -> from the Set Theory you can define some basic number sets to define Quantity-> when you have quantity you can make Structure that would be the case in Arithmetics, Algebra, Number and Group Theory-> from Structures you can move to Space such is the case in Geometry, Trigonometry, Topology. Now we have quantity, structure and space. We can then study Changes with Calculus, Dynamical Systems and Chaos Theory. There are some spatial cases if we restrict number set to only integers(discrete values) we can then have a special form of math called Discrete Math. Like I said my view is in its infancy.

 

[quasar987]

And if you want a trunk for your tree, you can also use this quote of Kronecker:

God created the integers, all else is the work of man.

But is it true that the natural numbers can be defined unambiguously from the ideas of set theory? If so, I guess Kronecker's trunk is obsolete.

Another unification theorem, I believe is the Taniyama-Shimura conjecture (now a theorem), which was a key to Wiles proof of Fermat's last theorem. It says here on mathworld that it is "a very general and important theorem connecting topology and number theory".

 

[matt grime]

The Peano axioms define the natural numbers on a firm mathmatical footing. There are 'true' theorems that in number theory that are not derivable merely from the axiomatic description of them as cardinals of finite sets. I don't see why any of that makes you think Kronecker's statement is obsolete.

 

[quasar987]

The Peano axioms define the natural numbers on a firm mathmatical footing. There are 'true' theorems that in number theory that are not derivable merely from the axiomatic description of them as cardinals of finite sets. I don't see why any of that makes you think Kronecker's statement is obsolete.

Simply because if the natural numbers can be spawned from set theory, set theory "precedes" the existence of the natural numbers, and hence it is the trunk and makes the natural numbers a simble branch on the tree.

 

[matt grime]

But it is a deliberately artificial way to define them. We could do everything entriely in categories and not have to have set theory as the trunk of anything at all.

 

[neurocomp2003]

start with the concept of 1(&0, always start with 1 because from 1 you can remove one...but if there is no concept of one you can't add 1 to 0 =], yes i know if there's no concep tof 0how do you get0...but what would you call removing one/adding one{predecessor/successor})
Then from 1/0 you go to counting/labelling
counting you go to
[A] NumberSystems & discrete
Real...
[C] Size-->basic Geometry

[A] Number Systems&Discrete lead too the basic fields
NOTE not all mutually exclusive, and set theory(with fields) will probably the first node.
->Number THeory
->Set Theory->Algebra(linear alg/fields)->all other algebras including the integrated or cross-disciplined (see advanced areas), eg clifford algebreas,
->Set Theory->Graph Theory
->Set Theory->Combinatorics
->Set Theory->language THeory
->Set Theory->Logic/Truths
->Set THeory->Algorithms->
->Language/Logic/Graph-> Computability THeory & Complexity Theory

[1] All the above lead to COmputer Science
->Set/Computer science lead to Datastructures
->language THeory/Logic lead ot boolean & Compiler THeory
->Number Theory Leads to Cryptography
->Graph Theory Leads to Networking
->Optimization is somewhere in the mix.
REAL (the basics areas)
->Calculus(limits,functions,series/sequencesderivatives,integration,plotting,and the trig/natural functions)
[1]->Calc->Real Analysis->Functional Analysis +(see matgrimes post and below)

[2]->calc->Multivariable Calculus and below
[3]->Calc->Complex Analysis
[A+B]->Calc+lin.alg->Vector Calculus(understanding Flow)->DEs,ODEs,PDEs,Dynsys,Bifurcation Theory

Linalg/Functoinal Analysis/ODE/PDE leads to Numerical Methods

[1] Vector Calculus & DEs-> study of math as applied to the sciences
thus math.phys, math.bio, math.chem,math.psyc etc.
Vector Calculus -> Calculus of Variations

[ADVANCED AREA]
Lastly if you did a "tree"(web),the bottom branches(outer ring) will consist of all hte cross-disciplined studies...as mentioned in mattgrimes, like analytic Geometry, analytical number theory,

Can't rememebr where Tensors fit in.
And I dont' like stats but its the measure of percentage so will probably be a branch of REAL & maybe Set as you needs properties to make a probability
Stats(language)->Markov Models, Stochastics,Brownian, Operations Research -> COUNTING DEAD PEOPLE

 

[matt grime]

->Number THeory
->Set Theory->Algebra(linear alg/fields)->all other algebras see the complex area

you don't include a complex area.

 

[neurocomp2003]

wups knew i forgot somethign...and i knew i forgot a basic subject(complex analysis)...changed the "complex area" to "advanced area"

mattgrimes btw how did you like my breakdown?