Is thier a topic of mathematics , that you can learn the most from?

[Samardar]

I've heard people around me say , they have learned more mathematics then they have in grad school just by reading this topic.

Is this true , I know mathematics has a huge diverse background and specializing in many topics , well I don't know if it's even possible.

So I'm asking the question , is their a topic of mathematics that is very multi-specialized , where it encompasses most fields of mathematics , that you learn from?

Would it be too advanced considering that their is a sort of ladder analogy in mathematics , would you be able to learn it from a top-down sort of way? Would the person have to combine most or all fields? Is it like a fundamental theorem of mathematics? How specialized would it be?

 

[chiro]

I've heard people around me say , they have learned more mathematics then they have in grad school just by reading this topic.

Is this true , I know mathematics has a huge diverse background and specializing in many topics , well I don't know if it's even possible.

So I'm asking the question , is their a topic of mathematics that is very multi-specialized , where it encompasses most fields of mathematics , that you learn from?

Would it be too advanced considering that their is a sort of ladder analogy in mathematics , would you be able to learn it from a top-down sort of way? Would the person have to combine most or all fields? Is it like a fundamental theorem of mathematics? How specialized would it be?

Hello Samardar and welcome to the forums.

I haven't come across a book per se that fits your description, but when you do enough math you start to get some core ideas for what math is about.

In terms of mathematics, three important ideas for every kind of math include assumptions, transformations and representations.

Assumptions include any initial data and constraints that apply to your problem of interest. It could be pure or applied mathematics (or even statistics). Usually pure mathematics have constraints that are more relaxed (i.e. broader) and this can make it harder.

The transformations are simply converting one thing to another. It is another way of way of "decomposing" things. You might do this with a simple algebraic trick, you might do it via some integral transform, or you might do it in a way that doesn't preserve the object itself (like doing an integral transform and keeping only the dominant terms). Learning transforms and their uses is absolutely critical, but what is more critical is understanding why different transforms are used.

The representation is just the explicit definition of the object you are dealing with. In order to analyze or even think about solving a problem, you need to explicitly define both the problem and the "stuff" that the problem refers to. You would be surprised that it's rather recent for mathematicians to define something like continuity in a fairly rigorous way.

So yeah in the back of your mind think about representation, transformation (or decomposition), and assumptions as key staples in any mathematical field. Put these into context of the area you are working on, and it should make a lot more sense.

 

[homeomorphic]

Dynamical systems, maybe?

The people I know in that area say they use everything, but everyone sort of uses everything to an extent.

Another might be category theory, but in a very different way. Category theory sort of applies to everything, but it doesn't have much to say because it's too abstract. But it does provide a convenient language and some unifying concepts.

I don't think I would really recommend going for either of those to a newcomer. Dynamics uses tons of advanced stuff, and category theory doesn't have many prerequisites, but the problem is, it doesn't really make sense until you have lots of examples under your belt and things to apply it to, so it's not really worth learning right off the bat, I think.

Maybe something like elementary complex analysis would be better for a newcomer. It draws on a lot of different things, but it's not super-advanced.

There's no fundamental theorem of mathematics. There are just certain areas that are kind of at the crossroads of a large number of other areas.

 

[Samardar]

assumptions, transformations and representations - the first and last part are critical to what I'm looking for.

Interesting research thanks!

---

Is it just dynamical systems , what about their related fields?

Ya , Category theory which considers mathematical objects to be the same but it doesn't rely on content; meaning it's too complex , thus calling it abstract nonsense that breaks away from the route of concrete problems.

Your right , but their have been several attempts by great mathematicians in finding a unifying theory in mathematics they failed , instead their have been attempts at uniting theories , examples include the langlands program , fundamental theorem of galois theory , taniyama-shimura conjecture and other's like the monstrous moonshine.

 

[mathwonk]

for thousands of years people have recommended euclid's elements. today i would suggest combining it with hartshorne's companion book - Geometry: euclid and beyond.

 

[kings7]

Category theory and/or a related offshoot to it will probably move from being 'abstract nonsense' to something akin unifying theories in topology and algebraic geometry later this century. It may eventually even have something to say about physics.


Remember... even linear algebra was considered 'abstract nonsense' for along time before there were uses for it in applied scientific fields.

 

[homeomorphic]

Dynamical systems was just one example.

The term "abstract nonsense" is not always meant in a negative way.

Category theory is actually very useful for certain things, particularly the kinds of things I am doing, myself. It can be done without it, but categories make it much nicer.

Maybe it's more of a tool than can be used in various situations than a unifying framework, although I think it does have unification value, too.

Abstraction doesn't necessarily take you away from concrete problems. It may, on the contrary, provide a more elegant way of solving them.

 

[Gerder10]

My candidate is point set topology. It is based on set theory, which is the starting point for most mathematical theories. When metric spaces are studied in point set topology most of the stuff encountered in elementary analysis course become crystal clear. The rather contrived epsilon delta definition of continuity is best expressed thus: a function f is continuous if its inverse carries open sets into open sets. Newbie Btw!