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Is there a branch of linguistics using Calculus as a mathematical tool? I mean, can we use differential or integrate in linguistic study?

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Calculus helps in continuous phenomenon , discrete calculus is used in statistical natural language modelling for text documents which involves summations and discrete probabilistic distributions. –  Ali Feb 26 '13 at 14:23
@Ali Thanks for the reply,I am just wondering if we could introduce calculus into linguistic,that would be amazing. –  Ave Maleficum Feb 26 '13 at 15:09
there is lambda calculus for linguistics too –  Ali Feb 26 '13 at 16:49
en.wikipedia.org/wiki/Lambda_calculus –  Ali Feb 26 '13 at 17:04
@AveMaleficum Hello and welcome to Linguistics. :) Since you're new here, check our FAQ and About pages. :D Your question here is good by the way. –  Alenanno Feb 27 '13 at 10:39

3 Answers 3

up vote 6 down vote accepted

Calculus can be a useful tool in quantitative linguistics. One simple example would be the deduction of the theoretical equation for the Piotrowski law modeling language change by Altmann et al.

The increase of new forms p' is proportional to the product of the proportions p of new and 1-p old forms: p' ∝ p(1 - p). Introducing the factor of proportionality b then yields the differential equation p' = bp(1-p), which can be solved for p as a function of time: p = 1/(1 + a exp(-bt)). (Skipping a few details here, like Piotrowski's earlier fitting of a tangens hyperbolicus curve, which is compatible with the above, since 1/(1 + a exp(-bt)) = 1/2 tanh w(t-t₁) + 1/2, where 1/2 tanh w(t-t₁) was one of Piotrowski's original proposals. Also, this can be (and was) generalized for partial change, and there are other viable approaches besides the interactionistic model.)

If you are interested in quantitative linguistics, I'd suggest the works of Prof. Gabriel Altmann and Prof. Reinhard Köhler.

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Source: Best, Karl-Heinz (Hrsg.) ; Kohlhase, Jörg (Hrsg.): Exakte Sprachwandelforschung. Göttingen : Edition Herodot, 1983. –  danlei Feb 26 '13 at 15:31

Lambda calculus is often used in formal semantics to express the meaning of a sentence compositionally as a series of successive function applications. On Barbara Partee's website there's a nice lesson on how lambda calculus (or, as some say, the lambda calculus) can be used to model English semantics. Lesson 7 introduces lambda calculus, but it might be worth looking at earlier lessons to see why it is useful.

For sure, lambda calculus is not "calculus" in the typical sense of differential calculus that we learn in our high school math classes, but it is a kind of calculus in that it involves an operator (lambda) that operates on a function and returns another function, binding a variable in the process. In that sense the lambda operator gives you something like the "derivative" of the original function.

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I'm curious as to why this answer received a downvote (it had received an upvote before that, making its score 0 as of the writing of this comment). If someone deems something in the answer inaccurate or unclear, it would be helpful to know specifically what that is so that I can improve the answer. Thanks! –  musicallinguist Feb 27 '13 at 14:58
Probably because "calculus" by itself usually refers to differential calculus, at least as far as I know. Here, have an upvote. (Also, note the second sentence in the question.) –  danlei Feb 27 '13 at 15:42
Thanks, @danlei. I've edited the answer accordingly. –  musicallinguist Feb 27 '13 at 17:48

Definitely. This is a little obscure, probably to obscure to be of any real use, but there is a chapter from the book Aspects of Automatic Text Analysis, The Mathematics of Semantic Spaces (PDF), that models semantics as a metric space. It's general calculus on metric spaces, and this is probably the limit of mathematical abstraction in applied linguistics that I've seen. Unless you've been introduced to some very high level math, this material won't make any sense.

Other then that, I know acoustic phonetics has derivatives, integrals, limits, series, etc. For example, one way to describe a sound wave quantitatively is take it's Fourier transform which involves the evaluation of an integral. Then there's statistical analysis of corpora, where calculus isn't directly involved, but the formulas won't make any sense unless your familiar with the concepts of basic calculus. Things like weighting, infinite sums, neighbourhoods, continuity, finding volumes and lengths, optimization etc. are implicit in a lot of statistical measurements. Including vector calculus. A lot of linguistic models employ continuous sets, not just discrete sets, and require a metric or a topology to define the set as a mathematical space.

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