Linguistics Is More Mathematical Than You Think

Most people, if you stop them on the street and ask what a linguist does, will say something about grammar. Maybe languages. Possibly — if they've read a pop-science book recently — something about how children learn to talk. What they almost certainly won't say is: Fourier transforms.

Dr. Taylor Jones, a linguist with a PhD from the University of Pennsylvania who runs the YouTube channel languagejones, has heard every version of the grammar answer. In a recent video, he walks through something that surprised even him when he entered the field: a substantial portion of modern linguistics is, without apology, mathematics. Not as a peripheral tool, not as a quirk of one subfield, but as the load-bearing infrastructure of empirical language research.

The argument he makes is genuinely interesting — not just because the math inventory is surprising, but because of the tension he identifies at the center of it. Math is necessary. Math is not sufficient. And an over-reliance on either the equations or the intuitions, without the other, produces research that looks rigorous and isn't.

The Math Is Already Everywhere

Jones starts with the most familiar-sounding subfield: syntax. Sentence structure, he explains, is formally modeled as trees — or more precisely, as graphs. Graph theory and formal language theory provide the vocabulary for determining what counts as a grammatically well-formed structure. The Chomsky hierarchy (yes, that Chomsky) and automata theory determine whether a given grammar can even generate the sentences a language allows — and what computational resources would be required to parse it. This is the same mathematical terrain that draws the lines between what computers can and can't efficiently compute.

Phonology borrows from computer science through optimality theory, which treats pronunciation choices as a constraint satisfaction problem. Historical linguistics — which sounds about as far from a spreadsheet as you can get — now imports phylogenetic methods from evolutionary biology, using the same algorithms that reconstruct species trees from DNA to rebuild language family trees from cognate data. Computational linguistics and NLP are largely statistics and linear algebra rebranded, first as machine learning, now as artificial intelligence.

The pattern Jones identifies is that different subfields borrow different branches of math depending on the structure they're trying to capture. Not because linguists decided math was cool, but because the questions kept demanding it.

What Sociophonetics Actually Requires

Jones's own work sits in sociophonetics — the study of how speech sounds vary across speakers, regions, and time, and what that variation reveals about social structure. He uses this as his primary case study, and it's the most illuminating part of the argument.

Consider something as basic as identifying a vowel. You might assume a trained linguist could just... listen. Jones describes a case in which a researcher conducting dialect interviews for a large oral history project compromised large amounts of data because he had a phonological merger that his interview subjects didn't share — and couldn't hear the distinction he was supposed to be documenting. The ear has limits. The ear has biases.

The solution is the Fourier transform: a mathematical operation that takes a sound wave — pressure varying over time — and decomposes it into its component frequencies, showing which frequencies are present and how strongly. This is how vowel formants get measured with any precision. And the math behind it involves complex numbers, which, as Jones puts it, "is not something that I expected to need before I could measure a vowel formant."

The catch — and this is where it gets interesting — is that the Fourier transform doesn't know what a vowel is. It will decompose any pressure wave you feed it: speech, a creaking door, line noise from a faulty microphone cable. It has no mechanism for distinguishing spectral peaks that correspond to actual vocal tract resonances from artifacts of the recording environment or the analysis window you chose. That requires phonetic knowledge. Without it, you can produce a spectrogram that looks completely legitimate and is measuring nothing useful.

One layer up, the same dynamic plays out with statistics. Mixed effects regression models are the standard tool for the inferential questions in sociolinguistics: Is this apparent vowel shift across age groups a real change in progress, or an artifact of sample size? How much of the variation is explained by social class, gender, or speech style once you control for the others? These models handle the clustering problem — the fact that multiple tokens from the same speaker aren't independent observations the way classical regression would assume.

But as Jones points out: "A mixed effects regression model will dutifully tell you that some predictor is statistically significant. It will not tell you that you've coded a sociolinguistic variable in a way that conflates two historically distinct mergers, that your speech style variable is actually capturing recording session order, or that the social category that you've used as a predictor is itself a contested or anachronistic grouping for the community that you're studying."

Better statistics cannot recover from that. The judgment calls — about how to operationalize a variable, about which social categories are meaningful for which communities, about the history of a speech form — require knowing the language, the community, and the relevant research tradition. The equation doesn't contain that. The equation can't supply it.

The Mathematician Can't Just Step In

This leads to the most provocative section of Jones's argument. If linguistics requires all this math, why not just bring in a mathematician? He doesn't dismiss the idea entirely — collaboration with people from STEM backgrounds can be genuinely valuable — but he's clear about the limitation.

The math is domain-agnostic. A Fourier transform processes whatever signal you give it. A regression model fits whatever data you put in. The knowledge about what those signals mean — which spectral features correspond to which phonological categories, which social variables carry which kinds of prestige in which communities — lives outside the equations entirely.

Jones extends this to the dynamical systems modeling he does when tracking how linguistic changes spread through populations, treating competing variants the way a population biologist treats competing alleles. Eigenvalues will tell you whether a variant is heading toward dominance, stable coexistence, or something more complicated. They will not tell you whether the selection pressure you've modeled as "prestige" actually represents overt prestige in one community and covert solidarity in another — and conflating those gives you a model that fits the historical curve while being wrong about the actual mechanism.

The inverse failure is equally real. Jones notes that physicists and economists have produced highly publicized linguistics papers that are, in his assessment, essentially worthless — competent math applied without adequate knowledge of what language is or how it works.

The Tool Problem Is Getting Worse

There's a third actor in this dynamic, beyond the linguist-who-only-knows-math and the mathematician-who-doesn't-know-linguistics: the linguist who knows neither particularly well and outsources the thinking to software.

Jones is pointed about this. Some researchers have relied on legacy statistical packages — he names Varbrul specifically — that operate as black boxes, accepting data and returning results without requiring the user to understand the assumptions being made or whether those assumptions hold. The concern isn't hypothetical. Jones describes a pattern in which researchers opposed to developing coding skills entirely showed up to conferences with data coded by hand by teams of undergraduates, then analyzed with outdated software.

In the age of AI coding assistants, he sees this dynamic accelerating rather than resolving. The ability to generate functional code on demand doesn't substitute for knowing what the code should be doing or why — what he calls understanding "the scientific method from first principles": statistical power, operationalization, collinearity, the relationship between inference and sampling. Those things matter more going forward, not less.

What Remains Open

Jones's core argument is essentially a claim about what empirical linguistics actually is: not a humanities field that occasionally borrows a statistical test, but a scientific enterprise in which mathematical tools are constitutive of the research questions themselves. The acoustic signal is a frequency decomposition problem. The question of whether a vowel shift is real is a statistical inference problem. How a change spreads through a population is a dynamical systems problem.

That's a defensible position — and the evidence he marshals for it is concrete and specific. But the flip side of his argument raises questions he gestures at without fully resolving. If the field requires both deep mathematical fluency and deep linguistic knowledge, how realistic is it to expect individual researchers to hold both? His own trajectory — spending graduate school learning Python, R, and signal processing when he'd planned to study languages — represents one answer: you acquire what the questions demand. Whether that's a sustainable model for training researchers, or whether it quietly narrows who can succeed in the field, is a different question.

What's undeniable is that the next time someone tells you they study linguistics and you picture a library full of old grammars, you're looking at maybe half the room.

— Nadia Marchetti, Unexplained Phenomena Correspondent, BuzzRAG