# I Categorizing Math

1. Oct 19, 2016

### Ehden

I'm currently taking calc 2, and I plan to take courses in linear algebra, statistics, diff eqs/partial and complex analysis. I was wondering, do these courses fall under a certain umbrella in math?

I heard that math can be separated into "two" fields that involve number theory, combinatorics and etc and the other involves calculus, statistics and etc. Is there any truth to this, and if so, what exactly differentiates these two fields?

2. Oct 19, 2016

### Staff: Mentor

Sometimes folks will divide math into pure and applied with number theory, combinatorics... in the pure camp and Calculus, Differential Equations, Linear Algebra... in the applied camp. One reason for the division is that applied math is crucial for scientists and engineers whereas pure math is the route of many mathematicians.

There was a PF thread on this a few years ago that may elucidate the diferences even more:

https://www.physicsforums.com/threa...tics-vs-statistics-degree-differences.610946/

and these wikipedia articles on pure / applied maths:

https://en.wikipedia.org/wiki/Pure_mathematics

https://en.wikipedia.org/wiki/Applied_mathematics

3. Oct 19, 2016

### Ehden

Thanks for the response! I didn't know that the entirety of calculus can be categorized into applied. Thought pure math has a lot to do with theories and proofs, which seems to be present in calculus. As applied is in the field of finding results with such theories and proofs. Anyways, thanks for the thread suggestion, will look into it.

Also where would you categorize complex analysis? Sounds like a pure math course, however, the description for the course says it has applications in engineering.

4. Oct 19, 2016

### Staff: Mentor

Categorizing things is never so absolute, I'm sure there are portions of Calculus and other math topics that are pure in nature.

Pure and applied means whether or not it can be used in a practical sense to solve some real world problem.

Prof GH Hardy an esteemed mathematician from England and a pacifist, once said he was proud of his contributions to pure math and that he was sure they would never be used to wage war. However, some of his work eventually did come into use in advanced cryptography.

https://en.wikipedia.org/wiki/G._H._Hardy

5. Oct 19, 2016

### micromass

Staff Emeritus
It is not a good idea to classify math in pure and applied, since all of math is applied math in some sense. Calculus, complex analysis, group theory all have very useful and important applications.
Rather, I think you should classify mathematicians in pure and applied. A pure mathematicians cares about math results only, while an applied mathematician wants to apply it to a specific problem. Both pure and applied mathematician might deal with complex analysis and the same theory, but with very different focus.

I always categorize math in three compartments: analysis, geometry and algebra. And this is a very rough categorization, so not everything will fit well, furthermore there are some very important overlaps like algebraic geometry which uses both geometry and algebra.
Anyway, if you follow this very rough (maybe too rough) categorization then calculus belongs to analysis. Complex analysis belongs to analysis. Diff eq belongs to analysis. And linear algebra belongs to algebra (although there are very significant links with geometry). The three-way categorization already breaks down for statistics. The methods used in statistics are analytic and linear algebraic in nature. But I would not say statistics is a part of analysis (I WOULD say probability theory is a part of analysis though). Moreover, I would not say statistics is a part of math to begin with. This said with all due respect to statisticians, but the goals of statistics and math are very different.

6. Oct 19, 2016

### lavinia

Some mathematics is designed to model practical problems e.g. statical control models for industrial processes. Other math is designed to model theoretical but still empirical problems for instance mathematical formulation of the laws of Electricity and Magnetism by Maxwell. Other math is the study of mathematical objects in and of themselves for instance the field of Differential Topology. Strangely even the most abstract areas of math often end up having empirical applications. Here is a quote from a physicist.

"The beauty and profundity of the geometry of fibre bundles were to a large extent brought forth by the (early) work of Chern. I must admit, however, that the appreciation of this beauty came to physicists only in recent years."

— CN Yang, 1979

Chern was a theoretical mathematician and the study of fiber bundles came out of purely mathematical research. I think that Chern's first work on this was in the 1930's and 40's.

It seems that almost every area of mathematics has both pure and applied sides to it.

Last edited: Oct 19, 2016
7. Oct 21, 2016

### Demystifier

What is number theory then? A part of algebra? (If so, then "algebraic number theory" sounds a bit silly.)

Or combinatorics? Also a part of algebra?

What are logic and set theory then? Metamathematics?

Even for discrete probability?

Then what is it? A part of engineering?

What are those goals?

8. Oct 21, 2016

### pwsnafu

I would. The difference between analysis and algebra is whether or not you are working with finite operations or infinite.

Probably it's own thing.

Absolutely. It's still measure theory. If you want applications, you'll also want martingales.

At my university, we used to have a Department of Mathematics and a Department of Statistics, in separate buildings. It was still part of the Faculty of Science though.

9. Oct 21, 2016

### micromass

Staff Emeritus
I don't pretend my three way classification of algebra, analysis and geometry is very comprehensive. You've already found some that don't fit well, I can find many others. But I still think it's a good first order approximation. You might want to put in a special label "Foundations" to capture logic and set theory. There are other modifications which can be made.

Absolutely. Take a look at Feller's first volume probability. It deals only with discrete probability, but you'll see a lot of analysis in there.

No, I'd say it's a separate field. There are links, and a certain part of stats (mathematical statistics) is part of mathematics. But overall, I think it's a separate field. Just like physics is a separate field from math even though there's a huge amount of links and interactions.

10. Oct 21, 2016

### StatGuy2000

As a statistician, I would agree with you that probability theory is a part of analysis, particularly if you delve deeply into the foundations of probability theory, which can be defined as a form of measure (and thus measure theory -- an intrinsic part of analysis -- applies). Indeed, there is an entire research field involving stochastic differential equations and dynamical systems that can be classified as part of analysis.

I should also add that many statisticians (including myself) would agree with you that statistics is not a branch of mathematics. Instead, I would much rather think of statistics as a mathematical science, in which the methods and theories underlying the different branches of mathematics are utilized/applied to specific scientific questions. I would also classify economics, theoretical physics, and computer science as mathematical sciences.

Now one may ask, isn't "mathematical science" the same as applied mathematics. In my mind, the answer is no -- in this, I agree with you as well that I don't want to classify math as pure or applied, but instead focus on mathematicians focusing on pure or applied problems, since the fundamental goal among mathematicians is in the development of the mathematical methodologies itself.

11. Oct 21, 2016

### lavinia

One might think of probability theory as Analysis together with the idea of independence.

Nowadays the old distinction into analysis, algebra, and geometry seems less useful. For instance how would you categorize Algebraic Topology? Or Algebraic Geometry? Or algebraic versus analytic number theory?

How does one classify the theory of heat flow on manifolds to derive topological invariants?

Are differential extensions of homology theories a subfield of analysis, or algebra?

What field is the study of Ricci flow on Riemannian manifolds? Geometry? Analysis?

12. Oct 21, 2016

### FactChecker

There are a lot of ways to classify mathematical subjects. In the most simple classification of applied / theoretical, I would put your list of subjects in the applied category. And it is a good list for that. For more theoretical, abstract subjects, I would add Abstract Algebra and Topology

13. Oct 22, 2016

### Ssnow

I agree with prof. Micromass, as first approximation is a good classification. There are branches as probability considered as part of Analysis and other fields as algebraic geometry or analytic number theory that are between these categories. In my opinion these hybrids contribute to make " smooth" the micromass categorization.

Ssnow

14. Oct 24, 2016

### Demystifier

I don't like the view of probability as a branch of analysis. I am more with Jaynes who views it as a scientific method
https://www.amazon.com/Probability-Theory-Science-T-Jaynes/dp/0521592712
making probability more related to statistics than to analysis.

Let me explain. Informally, probability can be described as a measure of certainty. The "measure" part belongs to analysis, but here "measure" is only the method, while the true goal is to say something about certainty. Concentrating only on the analytic aspect, you can never grasp the main idea of probability as a topic about certainty and uncertainty. Just like algebraic geometry is not a branch of algebra (but of geometry), analytic number theory is not a branch of analysis (but of number theory), and differential topology is not a branch of analysis (but of topology), in the same sense probability (as a measure of certainty) is not a branch of analysis.

Last edited by a moderator: May 8, 2017
15. Oct 24, 2016

### Demystifier

One should always distinguish the method from the goal. The goal of algebraic topology is to better understand topology, while algebra is only the method. In that sense, algebraic topology is a branch of topology (not of algebra). In the same sense algebraic geometry is a branch of geometry, algebraic and analytic number theory are branches of number theory, the study of heat flow on manifolds to derive topological invariants is a branch of differential topology which is a branch of topology, etc. When you know the goal of certain study, there should be no doubts how to classify it.

Let me also add that many mathematicians and physicists don't understand the difference between mathematical physics and physical mathematics. But there should be no confusion. Mathematical physics is a study of physical systems (e.g. fluid mechanics, general relativity, or quantum mechanics) by applying high standards of mathematical rigor. Physical mathematics is a study of mathematical tools (e.g. differential equations, algebraic topology, or group theory) at a practical not-too-rigorous level suitable for physicists.

Last edited: Oct 24, 2016
16. Oct 24, 2016

### lavinia

- I see your point but again these lines are not clear cut and your distinctions are too vague e.g " to better understand topology" is essentially meaningless, at least to me.

For instance, if you consider the homology theory of groups, then algebraic topology is being used to better understand group theory. Yet homology of groups can be also interpreted as also homology of certain special topological spaces.

Algebraic geometry may also be thought of as a branch of algebra since one is trying to understand the zeros of polynomials.

Complex analysis is used to study number theory so complex analysis is a branch of number theory.

The Physics of static electric fields is used to understand conformal mappings, so classical Electricity and Magnetism is a branch of complex analysis.

- The study of certain partial differential equations on manifolds helps understand a large variety of geometric problems. So partial differential equations is a branch of geometry. But geometry is used to understand solutions to a large variety of partial differential equations. So geometry is a subfield of partial differential equations

But some partial differential equations e.g. the heat equation help to understand the Pontryagin classes of manifolds and these are defined as topological invariants of real vector bundles. So partial differential equations is actually a subfield of topology.

Examples like this are endless.

-BTW:You are admitting topology as the fourth category along with algebra,analysis, and geometry.

17. Oct 24, 2016

### Demystifier

I certainly do. My full categorization of math would be something like

the king:
- foundations (logic, set theory, category theory, philosophy)

the queen:
- number theory

4 princes:
- algebra
- analysis
- topology
- geometry

the commoners:
- discrete math (combinatorics, graph theory, etc.)
- computation
- probability
- statistics
- mathematical modeling (dynamical systems, game theory, etc.)

I know it's far from perfect, but it works quite good for me.

Last edited: Oct 24, 2016
18. Oct 24, 2016

### Krylov

What problem are we trying to solve? There is no canonical and timeless way to do this, it depends far too much on personal preferences and experiences. Even the AMS subject classification (useful for journaling purposes) is updated every decade or so.

19. Oct 24, 2016

### Demystifier

Well, I have a lot of math e-books, so I have to categorize them somehow into directories and sub-directories.

20. Oct 24, 2016

### Demystifier

I disagree. In a study of homology theory it is assumed that one already knows some basics of group theory. Indeed, if you already know the basics of group theory, the homology theory will probably not teach you anything new about group theory. Homology theory is a kind of "trivial" from a group-theory point of view, while the non-trivial aspect of homology theory is in its topological content.

Similar comments apply to all the other examples you mentioned.