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Would you say that it is possible to give a unified, general definition of the different structures of mathematics and draw a clear distinction between them? I have been repeatedly trying to come up with such a distinction myself, but they either all fail to cover the entire discipline, or they end up covering something that is also dealt with in other disciplines. The best one I have myself is

  • Analysis is the study of the general idea of limits (which is perhaps a bit unprecise), of the real numbers (and those number structures that are based on them), and --- perhaps a bit controversially --- the study of the applications of the Axiom of Choice.
  • Algebra is the general study of operations on sets.
  • Topology is the general study of the global structures of spaces and maps between spaces.

I only covered three major branches, and it already starts to go wrong, particularly when you want to draw a distinction between algebra and topology. Both deal with abstract structures in sets in one way or another. Perhaps someone has a better distinction?

Another part of what makes this task difficult could perhaps be that the disciplines are more divided with regards to how they operate than what exactly they deal with. Any thoughts of this?

Gaussler
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    Check the first pages of the Princeton Companion to Mathematics, where the authors try to classify each discipline and give a short overview of what they're about. – Hakim Jun 28 '14 at 12:00
  • Everything is both similar and different from everything else. The general ideas of the above three categories solidify around their respective centers, but they cross over quite a bit. – Mitch Jun 28 '14 at 12:25
  • I usually say something glib like "algebra is about equations, but, analysis is about inequalities". I have no place for topology in this slogan. Honestly though, any sentence or even page cannot hope to truly encompass what these things are. – James S. Cook Jun 28 '14 at 13:22
  • Mathematical Fields. – Lucian Jun 28 '14 at 17:26

2 Answers2

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I think those first pages of the Princeton Companion to Mathematics are what you're looking for:

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Hakim
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Here is a scheme I came up with for my own amusement. I post it here, but I am sure it will annoy some and confuse others and since I have no taste for contoversy Ill remove it soon.

Basically the idea is that there are four main structures, $\mathbb{N}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$ and most disciplines have a strong tie to one of these. (I consider $\mathbb{Z}$ and $\mathbb{N}$ to be so similar and not to create a new category, Just to anticipate that question.)

Further there is a second axis according to methods, Analytic, Intermediate, Algebraic and Combinatoric. This axis also has a historic component, "Intermediate" is to corespond to the "Nineteenth century".

Some entries and well defined others include a large body of things, and more elements can be added. Note that it is not a table of dependence.

\begin{matrix} &\rm \mathbb{N}&\rm \mathbb{Q}&\rm \mathbb{R} &\rm \mathbb{C}\\ \rm Analytic &\rm \rm Elementary&\rm Geometry&\rm Real&\rm Complex \\ &\rm Number Theory&\rm &\rm Analysis&\rm Analysis\\ &\rm &\rm &\rm &\rm \\ \rm Intermediate&\rm Quadratic &\rm Group/Field &\rm Differential &\rm Complex \\ &\rm Forms &\rm Theory&\rm Geometry&\rm Geometry\\ &\rm &\rm &\rm &\rm \\ \rm Algebraic&\rm Algebraic &\rm Ring &\rm Algebraic &\rm Algebraic \\ &\rm Number Theory&\rm Theory&\rm Topology&\rm Geometry\\ &\rm &\rm &\rm &\rm \\ \rm Combinatorial&\rm Logic/Recursion &\rm Graph&\rm Set &\rm Model\\ &\rm Theory&\rm Theory&\rm Theory&\rm Theory\\ \end{matrix}

Hakim
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Rene Schipperus
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  • Thanks for the edit how did you do it so fast ? – Rene Schipperus Jun 28 '14 at 12:42
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    Neither annoyed nor confused, rather curious to see if anybody takes this classification seriously. Exercise: enumerate the mathematical domains it obliterates. – Did Jun 28 '14 at 12:43
  • @ReneSchipperus I've put the TeX code in notepad++ and used the tool find and replace so that it replaces & with &\rm automatically. – Hakim Jun 28 '14 at 12:44
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    Its not intended to obliterate diciplines but rather to give them a place to be classified, so for example under Real Analysis there is a long list of things. – Rene Schipperus Jun 28 '14 at 12:45
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    Very interesting answer indeed! There are some parts I would like you to comment on (because I'm curious): For instance, why is Model Theory grouped under $\Bbb C$? – Gaussler Jun 28 '14 at 16:22
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    Modern model theory has much to do with algebraic geometry, $\mathbb{C}$ is algebraically closed and ACF is one of the basic examples in Model theory, model theory also says things about the other structures, eg real closed fields, but these are unstable structures. $\mathbb{C}$ is the only one of the four basic structure that is not naturally ordered. Those are some of my reasons. Set theory is definitely under $\mathbb{R}$ there is a lot of set theory about the reals. – Rene Schipperus Jun 28 '14 at 19:36
  • Interesting view indeed! And why is geometry placed under $\Bbb Q$? – Gaussler Jun 30 '14 at 09:52
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    @Gaussler Because you can't construct some irrational numbers only based on geometry. For instance $\sqrt[3]{2}$ is not constructible. You can learn more here. – Hakim Jun 30 '14 at 22:42
  • Ah, you're talking about classical geometry. To me, "geometry" has become closely associated with differential geometry, which it shoudn't be. ;-) – Gaussler Jul 01 '14 at 06:21