I Math Categorization: Number Theory, Calculus & More

[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?

 

[jedishrfu]

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

 

[Ehden]

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

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.

 

[jedishrfu]

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

 

[micromass]

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.

 

[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.

 

[Demystifier]

I always categorize math in three compartments: analysis, geometry and algebra.

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?

I WOULD say probability theory is a part of analysis though.

Even for discrete probability?

Moreover, I would not say statistics is a part of math to begin with.

Then what is it? A part of engineering?

This said with all due respect to statisticians, but the goals of statistics and math are very different.

What are those goals?

 

[pwsnafu]

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?

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

What are logic and set theory then? Metamathematics?

Probably it's own thing.

Even for discrete probability?

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

Then what is it? A part of engineering?

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.

 

[micromass]

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?

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.

Even for discrete probability?

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.

Then what is it? A part of engineering?

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.

 

[StatGuy2000]

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.

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.

 

[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?

 

[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

 

[Ssnow]

I always categorize math in three compartments: analysis, geometry and algebra.

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

 

[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/dp/0521592712/?tag=pfamazon01-20
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.

 

[Demystifier]

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?

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.

 

[lavinia]

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.

- 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.

 

[Demystifier]

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

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.

 

[S.G. Janssens]

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.

 

[Demystifier]

What problem are we trying to solve?

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

 

[Demystifier]

For instance, if you consider the homology theory of groups, then algebraic topology is being used to better understand group theory.

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.

 

[lavinia]

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.

A simple example is the study of group extensions. Another example is Euler characteristics of groups. The study of group cohomology is an active area of research.

- I have tried to read this paper. I do not see how it is trivial.

http://www.ams.org/journals/tran/1991-328-01/S0002-9947-1991-1031239-1/S0002-9947-1991-1031239-1.pdf

 

[FactChecker]

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.

Combining these two statements makes classical electricity a branch of number theory. I couldn't disagree more. Just because complex analysis uses numbers does not make it a branch of number theory. Applying that logic transitively would make everything a "branch" of set theory. Actually complex analysis and number theory go off in very different directions.

 

[lavinia]

Combining these two statements makes classical electricity a branch of number theory. I couldn't disagree more. Just because complex analysis uses numbers does not make it a branch of number theory. Applying that logic transitively would make everything a "branch" of set theory. Actually complex analysis and number theory go off in very different directions.

Difficult problems in number theory are solved using complex analysis not to mention algebraic geometry.
For instance, the study of elliptic curves has lead to profound results in number theory.

But I agree that these hierarchies of what is a subject of what don't work very well.

 

[Demystifier]

For instance, the study of elliptic curves has lead to profound results in number theory.

It certainly did, but the converse is not true. Study of elliptic curves belongs to algebraic geometry, and study of elliptic functions belongs to analysis. If your goal is to understand elliptic curves or elliptic functions, a sophisticated knowledge about number theory will not help you. On the other hand, a sophisticated knowledge about elliptic curves and elliptic functions will help you to understand number theory. The relation is unidirectional.

 

[lavinia]

It certainly did, but the converse is not true. Study of elliptic curves belongs to algebraic geometry, and study of elliptic functions belongs to analysis. If your goal is to understand elliptic curves or elliptic functions, a sophisticated knowledge about number theory will not help you. On the other hand, a sophisticated knowledge about elliptic curves and elliptic functions will help you to understand number theory. The relation is unidirectional.

- Right. So by your criterion the study of elliptic curves is a subfield of number theory. There are other applications of complex analysis to number theory. Hmm.- If you want to continue this nonsense then why haven't you responded to my post about group cohomology? Did you read the paper I posted? It is still too hard for me. I could use some help.

-

 

[Demystifier]

- Right. So by your criterion the study of elliptic curves is a subfield of number theory.

I don't know how did you arrive at that conclusion. A book entitled "Elliptic Curves" belongs to algebraic geometry. A book entitled "Elliptic Curves in Number Theory" belongs to number theory.
(Occasionally, however, the book titles may be misleading. For instance, the book "Sets for Mathematics" by Lawvere and Rosebrugh belongs to category theory. Or who would guess that "A Course of Pure Mathematics" by Hardy is a course in analysis?)

- If you want to continue this nonsense then why haven't you responded to my post about group cohomology? Did you read the paper I posted? It is still too hard for me. I could use some help.

I'm not an expert, but as far as I can see, group cohomology is a tool for studying groups, not a tool for studying topology. The logic is this: First one identifies a problem in topology, then one develops an algebraic tool (cohomology theory) to deal with this topological problem, and finally one finds that this algebraic tool can also be used to study some aspects of algebra itself. In my book shelf, a book on cohomology theory lies at the topology section, while a book on group cohomology (if I had one) would lie at the algebra section. I still don't see a problem for categorization, whenever I see the goal of certain study.

Intuitively, a group can be viewed as a topological space, so a part of understanding of the group is understanding the topology of the group. By algebraic topology some topological properties can be described in terms of groups, which means that some properties of groups can be described in terms of other groups. This should be no more surprising than the fact that some properties of numbers can be described in terms of other numbers, e.g. 6 is divisible by 2 and 3. I can use a geometric tool to see this by drawing a 2x3 rectangular, but it does not make geometry a branch of number theory.

 

[Demystifier]

For those who still don't get it, let me further trivialize the matter. Suppose that I write
6=2x3
Is it about prime numbers or about non-prime numbers? It depends on the context. If it is written together with
6=1+5
6=2+4
6=3+3
then one can see that it is about 6, so 6=2x3 is not about prime numbers. Or if it is written together with
5=2+3
1=3-2
-1=2-3
then one can see that it is about 2 and 3, so 6=2x3 is not about non-prime numbers.

The moral is, a sub-branch of mathematics can only be categorized by seeing a big picture, not by identifying its parts. Otherwise, all modern text on mathematics could be categorized as LaTeX applications.

 

[zinq]

Statistics is unquestionably a field of applied math. Its main tool is a field of pure math, namely probability theory, which is itself a subset of measure theory.

 

[Svein]

Statistics is unquestionably a field of applied math. Its main tool is a field of pure math, namely probability theory, which is itself a subset of measure theory.

And it is even applied very effectively in practice. Quality control is all about designing test criteria that gives as much information as possible with as little work as possible - and giving an estimate of how reliable the test results are.

 

[Demystifier]

Statistics is unquestionably a field of applied math. Its main tool is a field of pure math, namely probability theory, which is itself a subset of measure theory.

If most mathematicians agree that probability theory is a branch of measure theory, does it mean that most mathematicians are not Bayesians?
https://en.wikipedia.org/wiki/Bayesian_probability

Or perhaps that they don't even care whether they are Bayesians or not?

I opened a new thread on such issues:
https://www.physicsforums.com/threads/is-probability-theory-a-branch-of-measure-theory.890795/

 

[micromass]

If most mathematicians agree that probability theory is a branch of measure theory, does it mean that most mathematicians are not Bayesians?
https://en.wikipedia.org/wiki/Bayesian_probability

Or perhaps that they don't even care whether they are Bayesians or not?

I opened a new thread on such issues:
https://www.physicsforums.com/threads/is-probability-theory-a-branch-of-measure-theory.890795/

Why do you think Bayesian statistics cannot be done with measure theory?

 

[Demystifier]

Why do you think Bayesian statistics cannot be done with measure theory?

I was talking about Bayesian interpretation of probability.

 

[micromass]

I was talking about Bayesian interpretation of probability.

And why do you think the Bayesian interpretation is inconsistent with measure theory and the frequentist one is?

 

[Demystifier]

And why do you think the Bayesian interpretation is inconsistent with measure theory and the frequentist one is?

Frequencies are objective, so they can be measured directly. Bayesian interpretation has something to do with the state of knowledge, and knowledge is somewhat subjective so cannot be directly measured. I know this explanation is a bit vague, but I am here to polish my understanding during the discussion.

 

[micromass]

Frequencies are objective, so they can be measured directly. Bayesian interpretation has something to do with the state of knowledge, and knowledge is somewhat subjective so cannot be directly measured. I know this explanation is a bit vague, but I am here to polish my understanding during the discussion.

First of all, frequencies are not objective. They rely on a pretty shady limiting argument. Nobody can actually toss a coin infinitely many times. Furthermore, there is no way to define a probability using frequencies and verify the axioms. It's religion, you assume it holds. While I do use frequentist methods and consider them valid, I reject the frequentist interpretation completely.

Contrary, the Bayesian point of view can be made rigorous without any shady business. Furthermore, the Kolmogorov axioms can actually be proven in the Bayesian approach if you rigorize the Bayesian interpretation in a reasonable way. That said, there are very convincing reasons why some of the Kolmogorov axioms might have to be weakened (usually sigma additivity), but while they yield a very beautiful (bayesian) interpretation, it doesn't give us a very useful and comprehensive statistical methods.

 

[Demystifier]

First of all, frequencies are not objective. They rely on a pretty shady limiting argument. Nobody can actually toss a coin infinitely many times. Furthermore, there is no way to define a probability using frequencies and verify the axioms. It's religion, you assume it holds. While I do use frequentist methods and consider them valid, I reject the frequentist interpretation completely.

Contrary, the Bayesian point of view can be made rigorous without any shady business. Furthermore, the Kolmogorov axioms can actually be proven in the Bayesian approach if you rigorize the Bayesian interpretation in a reasonable way. That said, there are very convincing reasons why some of the Kolmogorov axioms might have to be weakened (usually sigma additivity), but while they yield a very beautiful (bayesian) interpretation, it doesn't give us a very useful and comprehensive statistical methods.

So, what do you think about the Jaynes approach to probability I referred to in post #14? Your arguments above look very similar to his arguments, but he does not seem to think of probability as a branch of analysis.

 

[martinbn]

If your goal is to understand elliptic curves or elliptic functions, a sophisticated knowledge about number theory will not help you.

I'd say that depends on what you mean by elliptic curves and what you want to understand about them. If you are studying elliptic curves over number fields then number theory could be helpful.

p.s. The title of the thread made me think it was about viewing any part of mathematics as part of category theory.

 

[Demystifier]

I'd say that depends on what you mean by elliptic curves and what you want to understand about them. If you are studying elliptic curves over number fields then number theory could be helpful.

I meant over field of real numbers, and real numbers, as far as I can understand, are not very interesting from the point of view of number theory.

 

[lavinia]

I'm not an expert, but as far as I can see, group cohomology is a tool for studying groups, not a tool for studying topology. The logic is this: First one identifies a problem in topology, then one develops an algebraic tool (cohomology theory) to deal with this topological problem, and finally one finds that this algebraic tool can also be used to study some aspects of algebra itself. In my book shelf, a book on cohomology theory lies at the topology section, while a book on group cohomology (if I had one) would lie at the algebra section. I still don't see a problem for categorization, whenever I see the goal of certain study.

You said that group cohomology could only yield "trivial" results about groups. Explain to me what is trivial in this paper.

Secondly, group cohomology although certainly linked to topology - as is a lot of other math e.g differential geometry, dynamical systems, Riemann surfaces and so on - it is a field of algebra.

 

[Demystifier]

You said that group cohomology could only yield "trivial" results about groups.

That's not exactly what I said. Try to read again what I said.

 

[lavinia]

That's not exactly what I said. Try to read again what I said.

This is what you said.
"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."

 

[Demystifier]

This is what you said.
"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."

Yes. There I say something about (co)homology and its relation with group theory, but nothing about group cohomology. Even whole textbooks on cohomology usually say nothing about group cohomology.

Nevertheless, I do say something about group cohomology, but in another post, post #26.

 

[zinq]

"Or perhaps that they don't even care whether they are Bayesians or not?"

Not everyone is either a Bayesian or non-Bayesian. Lots of people see the methods of Bayesian statistics as one of several methods to use, including frequentist and maximum likelihood, to get good results, with no method superseding the others, since each seems suited to certain circumstances.

Such people would be unlikely to care about estimating the probabilities of their being Bayesian or not being Bayesian, given their state of self-knowledge.

 

[lavinia]

Yes. There I say something about (co)homology and its relation with group theory, but nothing about group cohomology. Even whole textbooks on cohomology usually say nothing about group cohomology.

Nevertheless, I do say something about group cohomology, but in another post, post #26.

And there are whole textbook on the cohomology of groups.

https://www.amazon.com/dp/0387906886/?tag=pfamazon01-20

Cohomology theory of groups says a lot about group theory.

 

[Demystifier]

And there are whole textbook on the cohomology of groups.

https://www.amazon.com/dp/0387906886/?tag=pfamazon01-20

Cohomology theory of groups says a lot about group theory.

And this is consistent with what I said in #26.

 

[lavinia]

To me the more mathematics develops, the more various fields combine. For instance, Differential geometry is inextricably linked to differential topology and combinatorial topology of manifolds. to the modern theory of partial differential equations and differential operators.The study of cohomology on manifolds leads to new geometric and combinatorial invariants. Complex analysis which once was restricted to the study of meromorphic functions in the plane is now integral to geometry.
The calculus of variations which can be viewed in isolation asa method of analysis yields both topological and geometric properties of smooth manifolds.

Studying the mapping properties of complex functions leads to point set topology, new topological spaces (Riemann surfaces), conformal structures on Riemann surfaces and ultimately to algebraic geometry which in turn folds back into differential geometry, cohomology theory, and algebra.

The interconnections between fields is an overlapping that blurs their boundaries.

 

[zinq]

I completely agree with the wise post #46. There is little value in debating whether an area of math "does" or "does not" belong to another related field. Rather, fields of math are like overlapping probability distributions on fuzzy sets. In fact, it is difficult to find two fields of math that don't have some degree of overlap. Some propositions in topology hold if and only if the Continuum Hypothesis in set theory is taken to be true. Probability distributions can be assigned to random simplicial complexes and then various properties of them — such as whether they are manifolds — can be assigned probabilities. The study of vector fields in differential topology depends fundamentally on the existence and uniqueness of solutions to ordinary differential equations in analysis. Number theory is obviously a (very important) part of algebra and algebra obviously has important applications to topology. Topology is an essential portion of geometry. Regardless of whether some people think it is silly terminology, algebraic number theory is an important generalization of number theory to subfields of the field of complex numbers having finite dimension over the rationals. Geometry can be applied to algebraic number theory to determine which rings of algebraic integers have the unique factorization property. The theory of analytic functions — which I would call mainly part of analysis — leads to a beautiful proof that all simply connected open sets of the plane are homeomorphic to each other — and even conformally equivalent, with one exception.

I would probably say that the main fields of math that stand out as *mostly* distinct from one another are algebra, analysis, geometry, probabiliity, combinatorics, and foundations — but with major overlaps nonetheless.

Rather than a bunch of disconnected subsets, mathematics most resembles a bunch of blurry amoebae having an orgy.

 

[Demystifier]

I would probably say that the main fields of math that stand out as *mostly* distinct from one another are algebra, analysis, geometry, probabiliity, combinatorics, and foundations — but with major overlaps nonetheless.

Where would number theory fit in this scheme? Mostly part of foundations? Mostly part of algebra? Strong overlap between foundations and algebra? Something else?

 

[lavinia]

Where would number theory fit in this scheme? Mostly part of foundations? Mostly part of algebra? Strong overlap between foundations and algebra? Something else?

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

 

[Demystifier]

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

Yes, but I had basic number theory in mind which does not rest on analytic techniques. Moreover, even when you need analysis to prove a theorem in number theory (like the last Fermat's one), you don't need analysis to state the theorem.

If one science uses tools from another science does not mean that two sciences cannot be distinguished. For instance, medicine and nuclear physics are clearly different sciences, despite the fact that there is a branch of medicine called nuclear medicine. As I stressed several times (and you failed to understand) it's the goals, not the tools, that distinguishes different branches.