Math Categorization: Number Theory, Calculus & More

[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.

 

[zinq]

If it's the goals, not the tools, that distinguish different branches, then why separate non-analytic number theory from analytic number theory? It's all number theory.

 

[Demystifier]

If it's the goals, not the tools, that distinguish different branches, then why separate non-analytic number theory from analytic number theory? It's all number theory.

To be consistent with what I said before, I would say that analytic number theory and algebraic number theory are not two branches but two approaches within the same branch. (Alternatively, one could also say that they are two sub-branches, and stipulate that a sub-branch is not a branch.)

But one should not forget that the goal of categorization of math is not to establish a truth. The goal is merely practical, e.g. when you want to organize a lot of math books and papers into directories and sub-directories at your computer. There is no perfect categorization, but some categorization scheme simply needs to be chosen for practical reasons. The scheme I have described works very well for me. Someone else can choose a different scheme, and if it works well for him/her, that's fine too.

 

[lavinia]

To be consistent with what I said before, I would say that analytic number theory and algebraic number theory are not two branches but two approaches within the same branch. (Alternatively, one could also say that they are two sub-branches, and stipulate that a sub-branch is not a branch.)

But one should not forget that the goal of categorization of math is not to establish a truth. The goal is merely practical, e.g. when you want to organize a lot of math books and papers into directories and sub-directories at your computer. There is no perfect categorization, but some categorization scheme simply needs to be chosen for practical reasons. The scheme I have described works very well for me. Someone else can choose a different scheme, and if it works well for him/her, that's fine too.

There are many practical reasons for classifying things and they can be completely personal.

But let's take your example of "basic number theory", propositions about numbers that can be stated without reference to analysis or other fields.

For instance the Fundamental Theorem of Algebra.

There have been many proofs of the FTA all of which have used something other than basic number theory. The ones I have seen use Complex Analysis, General Topology, or Differential Topology. The Complex Analysis proofs and the Differential Topology proof demonstrate that every complex polynomial determines a surjective map from the Riemann sphere onto itself. One can therefore view the FTA as a statement about mappings of the Riemann sphere. The more algebraic proofs still use topology since they rely on the intermediate Value Theorem. There is also one of Gauss's proofs that the two surfaces determined by the real and complex parts of a polynomial must have non-empty intersection. I am not sure how this proof works.

One might ask whether there are"purely algebraic" proofs and it seems that maybe there are. The following synopsis describes an analysis free proof but requires using the hyperreal numbers. That also seems far away from "basic number theory."
A Purely Algebraic Proof of the Fundamental Theorem of Algebra
Piotr Błaszczyk
(Submitted on 21 Apr 2015)
Proofs of the fundamental theorem of algebra can be divided up into three groups according to the techniques involved: proofs that rely on real or complex analysis, algebraic proofs, and topological proofs. Algebraic proofs make use of the fact that odd-degree real polynomials have real roots. This assumption, however, requires analytic methods, namely, the intermediate value theorem for real continuous functions. In this paper, we develop the idea of algebraic proof further towards a purely algebraic proof of the intermediate value theorem for real polynomials. In our proof, we neither use the notion of continuous function nor refer to any theorem of real and complex analysis. Instead, we apply techniques of modern algebra: we extend the field of real numbers to the non-Archimedean field of hyperreals via an ultraproduct construction and explore some relationships between the subring of limited hyperreals, its maximal ideal of infinitesimals, and real numbers.

Subjects: History and Overview (math.HO)
MSC classes: 08A40, 26E35
Cite as: arXiv:1504.05609 [math.HO]
(or arXiv:1504.05609v1 [math.HO] for this version)