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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 5th 2022

The term “discrete mathematics”, as it is actually used in the literature, appears to be synonymous with combinatorics.

Its use appears to be limited mostly to introductory textbooks on combinatorics.

I suggest that this article reflects the actual use of this term, and, in particular, does not claim that (∞,1)-toposes are part of discrete mathematics, as it currently does.

Do we really need a separate article here, or should this be merged into combinatorics, with a section on terminology there?

• CommentRowNumber2.
• CommentAuthorGuest
• CommentTimeJun 5th 2022

There is no one unified definition of what discrete mathematics is supposed to mean.

The University of Chicago School Mathematics says on this website:

Discrete mathematics includes logic and mathematical reasoning, mathematical induction and recursion, combinatorics, the analysis of networks, and systematic sorting methods that are important in computer science, business, economics, and the biological sciences. Students are expected to be competent at proofs involving mathematical induction.

Norman Biggs says on page 89 of his Discrete Mathematics textbook that

Discrete Mathematics is the branch of Mathematics in which we deal with questions involving finite or countably infinite sets.

Wikipedia’s article on discrete mathematics has the following subsections under the section “Topics in discrete mathematics”

• Theoretical computer science
• Information theory
• Logic
• Set theory
• Combinatorics
• Graph theory
• Probability
• Number theory
• Algebraic structures
• Calculus of finite differences, discrete calculus or discrete analysis
• Geometry (i.e. Discrete geometry and Computational geometry)
• Topology (i.e. combinatorial topology, topological graph theory, topological combinatorics, computational topology, discrete topological space, finite topological space)
• Operations research
• Game theory, decision theory, utility theory, social choice theory
• Discretization
• Discrete analogues of continuous mathematics
• Hybrid discrete and continuous mathematics

This is already a very broad set of topics, which cannot be reduced to combinatorics.

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 5th 2022
• (edited Jun 5th 2022)

Re #2: Biggs’s textbook is a textbook in combinatorics, as is clear from its table of contents.

And Wikipedia further confirms this:

In university curricula, “Discrete Mathematics” appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time. The curriculum has thereafter developed in conjunction with efforts by ACM and MAA into a course that is basically intended to develop mathematical maturity in first-year students; therefore, it is nowadays a prerequisite for mathematics majors in some universities as well. Some high-school-level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics is sometimes seen as a preparatory course, not unlike precalculus in this respect.

Chicago’s description is quite similar.

Wikipedia’s list of supposed topics in discrete mathematics is not particularly competent. For example, here is what they write about topology:

Topology (i.e. combinatorial topology, topological graph theory, topological combinatorics, computational topology, discrete topological space, finite topological space)

Combinatorial topology is a rather obsolete name for algebraic topology, and I have yet to see a topologist who refers or thinks about his field as “discrete mathematics”.

I am pretty sure the situation is the same for logicians, set theorists, number theorists, probability theorists, etc.

As far as I can see, “discrete mathematics” is nothing more than a term for a undergraduate course and associated textbooks, comprising primarily combinatorics and similar elementary topics. The article has to reflect this.

There is zero evidence for the wild claims that it somehow includes topology, number theory, set theory, probability theory, given a complete absence of acknowledgment of this “fact” by the respective experts in these fields.

• CommentRowNumber4.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 5th 2022

Perhaps an even better evidence is the journal Discrete Mathematics. MathSciNet makes it clear (by its classification codes) that almost all articles are in combinatorics: of the 15368 articles that it published, 11854 (77%) are explicitly classified as combinatorics, and the remainder is classified to closely related fields: coding theory, convex geometry, posets, etc.

• CommentRowNumber5.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 5th 2022

Rewrote the article to accurately describe the current mathematical use of the term.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJun 5th 2022

I have reverted the change; I agree with Guest #2. At my university we teach a “discrete mathematics” course that includes, among other things, number theory. Certainly a large part of discrete mathematics in practice is combinatorics, but I think that properly understood the term “discrete mathematics” refers to all mathematics that is discrete, as the article originally said.

That said, I do think the article could be improved a lot.

• CommentRowNumber7.
• CommentAuthorGuest
• CommentTimeJun 6th 2022

The main issue here is that different branches of mathematics has different notions of “discrete”. Whoever originally wrote this article is a homotopy type theorist who interpreted “discrete” as being “discrete infinity-groupoids” compared to non-discrete cohesive infinity-groupoids in cohesive modal homotopy type theory. Category theorists who don’t go all the way up to infinity would dispute that, claiming that their notion of “discrete” is 0-truncated. Meanwhile, some constructive field theorists would argue that “discrete” is a synonym for decidable (i.e. discrete field, etc), which in many models of constructive mathematics is only provable for the countable sets.

1. added contents and a better description

Anonymous

• CommentRowNumber9.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 6th 2022
• (edited Jun 6th 2022)

Re #6: Well, the text of the article was now completely replaced again.

The “number theory” included in such courses amounts to elementary properties of Z/nZ, and their presence does not fundamentally alter the validity of what I wrote. It can be mentioned explicitly.

The current version of the article promotes claims that simply have no basis in any published literature (whether research or textbooks), e.g., that discrete mathematics could mean “The study of discrete infinity-groupoids and mathematical structures on discrete infinity-groupoids”.

Neutrally describing the content of an internationally recognized journal in the field seems like a much less controversial stance than presenting (marginal) opinions that are likely to raise eyebrows among pretty much all experts and/or teachers in the field. (Exactly what fraction of all mathematicians performing research in, or teaching discrete mathematics have ever heard of an ∞-groupoid?)

My version, at the very least, rather accurately described the content of various undergraduate textbooks named “Discrete Mathematics”, as well as the content of the research journal with the same name.

2. reverted back to Dimitri Pavlov’s version, but made the undergraduate curriculum the primary focus of the article, and added in a few references

Anonymous

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJun 6th 2022
• (edited Jun 6th 2022)

There is enough room in an nLab page to discuss both the verbatim meaning of a term as well as its de facto use in practice. I suggest we explain both.

Digging through the page history, I see that rev 5 made a sensible point with observing that “discrete” may be read as “non-cohesive”. On the nLab, of all places, this should be worth mentioning, as long as we don’t mislead the reader into thinking that this is a widely appreciated insight.

In this vein I have (renamed the “Definition”-section to “Idea” and then) prefixed the Idea-section by the following paragraphs:

Taken at its verbatim face value, the term discrete mathematics refers to mathematics concerned with mathematical structures which are discrete in the sense of discrete topological spaces, hence which do not involve topology and in particular do not involve analysis (“calculus”).

With the hindsight of the nPOV one could usefully say that discrete mathematics, in this sense, is the topic of (models of) bare homotopy type theory, in contrast to its cohesive refinement to cohesive homotopy type theory.

However, in common parlance the term discrete mathematics is used much more restrictively: $[$$]$

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJun 7th 2022

I have slightly re-arranged and adjusted the wording of the paragraph on undergraduate courses (here), to make it clearer (hopefully). This should be uncontroversion, but please check.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeJun 16th 2022
• (edited Jun 16th 2022)

• Kenneth Rosen, Discrete Mathematics and its Applications, McGraw Hill (2019) $[$ISBN:978-1-259-67651-2, pdf$]$