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Can formal language theory be correctly characterized as a branch of discrete mathematics?

If so, what is the correct antonym for "discrete" here?

Reason I'm asking is that for rhetorical purposes in a draft paper, I need to contrast language-theoretic objects with objects such as quadrics and manifolds arising in "non-discrete" mathematics.

In this regard, see these questions:

Suppose it can be shown that any right(left)-regular finite-state grammar implicitly generates a "hypar"" surface

Is there any surface approximation algorithm which uses "hypar" patches only?

Hyperboloids of one sheet, hyperbolic paraboloids, and Hilbert's famous "three skew lines"

Calum Gilhooley
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David Halitsky
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    See Discrete math: "the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic– do not vary smoothly in this way, but have distinct, separated values." – Mauro ALLEGRANZA Dec 01 '17 at 14:54
  • @MauroAllegranza - Thanks for taking the time to respond - much appreciated. But would you simply use the word "non-discrete" in contrast to "discrete"? If not, is there a single word you COULD use? I don't think it would be proper to say "continuous mathematics" . . . – David Halitsky Dec 01 '17 at 15:00
  • Correct; discrete math is a recent "label" for topics that have some "family resemblances" but it is not a single theory. Arithmetic is clearly "discrete" but was already there well before the new label. – Mauro ALLEGRANZA Dec 01 '17 at 15:11
  • @MauroALLEGRANZA - ok thanks again - unless someone has a better idea, I'll use the term "non-discrete", perhaps with an elaboration in a parenthesized remark or a footnote . . .) – David Halitsky Dec 01 '17 at 15:14

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To answer your question, let me quote a part of Wikipedia entry for Discrete Mathematics concerning number theory.

Number theory is concerned with the properties of numbers in general, particularly integers. (...) Other discrete aspects of number theory include geometry of numbers. In analytic number theory, techniques from continuous mathematics are also used. Topics that go beyond discrete objects include transcendental numbers, diophantine approximation, p-adic analysis and function fields.

I would say that the situation is similar for formal languages. Formal languages do belong to Discrete Mathematics, but some tools go beyond discrete objects.

Indeed, formal language theory makes use of topology, especially profinite topologies. See for instance the papers Duality and equational theory of regular languages and A Topological Approach to Recognition. By the way, this approach shares some analogy with the $p$-adic metric on the integers. For instance, the $p$-adic metric can be extended to words: see the paper (in French) Topologie $p$-adique sur les mots or Thue sequence and p-adic topology of the free monoid.

J.-E. Pin
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  • thank you so much for taking the time to respond, and for providing such an illuminating response regarding how some tools of formal language theory do go beyond Discrete Mathematics. Since you do believe that formal language theory can benefit from non-discrete tools, you MAY be interested in my recent question here: https://math.stackexchange.com/questions/2537513/suppose-it-can-be-shown-that-any-rightleft-regular-finite-state-grammar-implic and in this site https://www.thesyntacticretina.com/ – David Halitsky Dec 03 '17 at 13:23
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While I agree that one could not use the adjective continuous as an antonym of the adjective discrete in this context (although the two terms are at least roughly antonymic in other contexts, even within mathematics), I think one could nevertheless use the noun phrase mathematical analysis (although this would admittedly be in an uncomfortably extended sense) as an approximate antonym of the noun phrase discrete mathematics. (But whether a phrase can legitimately be called an 'antonym' of another phrase is another question!) I also agree that formal language theory belongs to the latter.

Calum Gilhooley
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  • thanks so much for taking the time to respond - I am glad that you've OK'd "mathematical analysis", albeit with a caveat. I can certainly tailor my prose to express the contrast using two noun phrases instead of two adjective on the noun "mathematics", and I actually think that might work well. – David Halitsky Dec 01 '17 at 18:49