I'm doing a math project that requires the research of an application of differential calculus or statistics, and as someone who is very interested in linguistics, I was wondering if there is any area of linguistics that implements these mathematical branches.
Statistics are very useful in linguistics, but the relevant techniques go far beyond AP level. Calculus is not directly useful, but calculus (especially integral) is essential background for higher-level courses in statistics.
I doubt whether any single person could recite a comprehensive list of applications of statistics to linguistics, but I can name some, based on my personal experience and interests.
Hypothesis testing: Many so-called “laws” in historical linguistics are inferred from paltry numbers of examples. They would better be called “correlations on stilts”. To test hypotheses rigorously, one must understand the concept of statistical significance, i.e., the probability that a weak correlation or other observed phenomenon is the result of a random fluke or sampling error. (This is obligatory in experimental fields that can gather abundant data, but one must make allowances for the fact that historical linguists are exploring a fossil record. They don’t make more dead languages while you sleep.)
Bayesian inference is a powerful tool for assessing alternative hypotheses, but it is often too hard to apply because it requires mathematically precise specifications of the hypotheses, numerical estimates of their a priori likelihoods, and conditional probabilities of outcomes.
Rates of vocabulary replacement: Swadesh pioneered this topic by examining a limited list of words common to the older IE languages, but his conclusions are controversial. His list featured common concepts such as family members and body parts, but what if words for more abstract concepts had higher replacement rates?
The information content of language is an intriguing but slippery topic. Some languages use more syllables than others to say exactly the same thing (Spanish vs. French). At another level, some speakers say little of interest at great length. Shannon’s measure of information content (“entropy”) is most relevant to data compression, but other measures may be more relevant for other purposes.
The productivity of root templates is a pet interest of mine. The triliteral root template of Arabic and other Semitic languages is capable of generating more than enough phonemically distinct roots to meet Arabic’s semantic needs, but other language families have more restrictive root templates. One must specify the root shape precisely, measure the diversity of phonemes allowed in each “slot” of the template, and apply known constraints. Shannon’s measure of diversity may not be the best choice because the key issue now is the likelihood of “collisions” between quasi-homophonous roots. If the number of phonemically distinct roots so generated seems inadequate, one must ask: Is the math wrong? Is the template wrong? How did/do languages differentiate quasi-homophonous roots?
Etymological dictionaries such as Pokorny’s lexicon of PIE roots are often flawed by the inclusion of spurious cognates. (Pokorny’s critics think he applied low standards of phonetic and semantic matching, or in other words, that he erred on the side of including unreliable material rather than that of excluding possibly valid material.) Statistics may illuminate the tradeoff or help in assessing unreliable entries.
For sure, statistics and probability theory has been extensively used in linguistics, you can see for example the papers published in the Journal of Quantitative Linguistics. On the other hand, I do not explicitly remember any significant piece of work in linguistics, where differential calculus has a key rôle.