# Mathematics Branches and Foundation

Hi I hope every one is fine , I am an Electrical Engineer. I asked before about real and complex analysis because I am interested in Signal Processing also I am interested in coding theory and information theory. I felt I am in need of more mathematics so I am searching the internet continuously to get the idea how could I find my path to my interest but I was encountered by two classification scheme , I don't know the difference between them ;

1) Wikipedia :

• quantity

• structure

• space

• change

2)Princeton companion :

• Algebra
• number Theory
• Geometry
• Algebraic geometry
• Analysis Logic
• logic
• Combinatorics

so what is the difference between the two classifying schemes, Also what accounts as a foundation and if I want to read about philosophy of Mathematics and Foundation and their evolution pls suggest a good book

thanks

• At least as a place-holder: for me, although I understand the appeal of "classification schemes", I think they fundamentally fail both as descriptions and as prescriptions... or even as genuine classification. Yet, as we know, people are fond of such things. Not-ironically, many people interested in mathematics also have a predilection for classification schemes... :) One sort of objection to "classification" is that classifying issues is not necessarily a resolution to any of them. But may give one a sense of accomplishment? So don't worry about "classification"... but, then, what? ... :) Dec 2 '16 at 23:02
• I'll tell you :) ... when you are self educating yourself you are faced at least at first by a too complicated subject that have many branches each of which are linked together ... in mathematics the same part have multiple complexities or in other words you need to pass the same part many times ..after reading in another branch ... the key from my prespective to resolve the complexity is two things 1) history and evolution 2) classification ... both are linked together..example : set theory you have many versions 1) cantorian (naive) 2) axiomatic ... each is sufficient for a particular task . Dec 2 '16 at 23:12
• so you need to have such a map to tell you what you need .. you don't have a tutor who will help you .. for example after long search i realized that all the algebra we are doing to day is based on "van der waerden" classical text in Algebra it was translated from german .. so if you want to learn algebra know the prerequisites of that book then read it . Dec 2 '16 at 23:17
• I am interested in doing such a project about "math map" if any one is interested , tell me we can work together and share information and material... at the end we can share the final map in the forum here ..to benefit all people..:D Dec 2 '16 at 23:19

(1) The American Mathematical Society (AMS) maintains a Mathematics Subject Classification (MSC) for the mathematics literature. Here is a graph of the topics built by Andrius Kulikauskas:

Image by AndriusKulikauskas.
And here is a more readable detail (from roughly the center of the chart):

(2) And here is another chart, more idiosyncratic, emphasizing physics, from this paper:

Tegmark, Max. "Is 'the theory of everything' merely the ultimate ensemble theory?" Annals of Physics 270.1 (1998): 1-51. (PDF download.)

Image by Max Tegmark.

The OP is looking at the Wikipedia article on Mathematics in general. In the first line it defines mathematics as:

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change.

You should be aware that this specific terminology is not used in mathematics practice itself as defined terms (at least, none that I've ever seen). Rather, this article is written at the outset from a hypothetical perspective of a reader who's never heard of any mathematics at all, and in that respect, I'd say it's a halfway clever way of encapsulating what the discipline is about.

Further down in the article there are links provided to more fully developed articles on a variety of topics, and those links associate these (undefined) terms to more normal mathematical topic terminology, to wit:

• Quantity → Arithmetic
• Structure → Algebra
• Space → Geometry
• Change → Calculus

Each of the second items in the list above is a full article on Wikipedia, a common course in schools, and one of many books you could find on that subject. Of course, the opening Wikipedia definition uses the qualifier "such as", so this list is not represented as being complete. The article also has links to other topics, such as: Foundations and philosophy (logic, set theory, category theory, computation), applied mathematics, statistics, etc.