## “Language is not mathematics”

It has become something of a platitude to claim that “language is not mathematics”. It shows up in several variants—a quick search in Google gives results for all combinations of the variants using “is not” / “isn’t” and “mathematics” / “maths” / “math”. The origin of this phrase seems to be Otto Jespersen’s The Philosophy of Grammar (1924). Jespersen’s actual argument is that “language is not mathematics, and … a linguistic negative cannot be compared with the sign − (minus) in mathematics; hence any reference to the mathematical rule about two minus’s is inconclusive” (p. 331).

The mathematical rule to which he is referring, of course, is an arithmetic result that holds when multiplying real numbers: for all $a, b\in\mathbb{R}$, if $a, b < 0$, then the product $ab > 0$. This result relies on the definitions of addition and multiplication, the existence of additive inverses, and the order properties of the real numbers that are compatible with the operations on these numbers. Any analogy between linguistic negatives and a result that relates to the product of additive inverses in an algebraic structure would appear to be hopelessly wrong. In this case, Jespersen’s comment amounts to a claim that the semantics of natural languages cannot be adequately modeled as an ordered field—not that anyone would ever claim such a thing in the first place.

The more important point of Jespersen’s observations is that linguistic negatives do not correlate with logical negation. In propositional logic, the meaning of the statements $\neg\neg A$ and $A$ are identical for all propositions $A$. Cumulative (double, triple, quadruple) linguistic negatives behave differently than logical negation in propositional logic. Consequently, language is not (propositional) logic.

There is another respect in which language differs from both mathematics and logic, which is more fundamental than what is suggested by the “is not” assertions. This difference has to do with what language, logic, and mathematics are rather that with what they aren’t. Let’s start by considering the primary sense of each word as defined in the OED:

language The system of spoken or written communication used by a particular country, people, community, etc., typically consisting of words used within a regular grammatical and syntactic structure. Also fig.

logic The branch of philosophy that treats of the forms of thinking in general, and more especially of inference and of scientific method. (Prof. J. Cook Wilson.) Also, since the work of Gottlob Frege (1848-1925), a formal system using symbolic techniques and mathematical methods to establish truth-values in the physical sciences, in language, and in philosophical argument.

mathematics Originally: (a collective term for) geometry, arithmetic, and certain physical sciences involving geometrical reasoning, such as astronomy and optics; spec. the disciplines of the quadrivium collectively. In later use: the science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis; mathematical operations or calculations. Colloq. abbreviated maths, (N. Amer.) math.

Mathematics and logic share the property of being, in effect, the study of a specific subject. Mathematics is the study of “space, number, quantity, and arrangement”; such study has people—mathematicians—who undertake this effort. Logic is the study of “the forms of thinking”; such study has people—logicians—who undertake this effort. We also say, especially as students, that we are “studying mathematics” or “studying logic”; typically, this means that we are actively learning the techniques used and results previously obtained within that science.

Language, however, is not the study of anything; we don’t have ‘languageians’ who undertake the effort of language in the way that mathematicians and logicians undertake their respective efforts. Language is, rather, a “system of spoken or written communication”. It correlates not with mathematics and logic, but rather with space, number, quantity, arrangement, and the forms of thought (and, in Frege’s sense, with formal systems to establish truth values). The study of language is what we call linguistics, and the people who undertake this study are called linguists.

What this suggests is that we have the following parallels:

Field of study Practitioners Objects studied
linguistics linguists language
logic logicians forms of thought
mathematics mathematicians space, number, quantity, and arrangement

The question of whether language “is” mathematics or logic is easily answered at this point. A more interesting question is to what extent language, forms of thought, and arrangement overlap.

### 9 Responses to “Language is not mathematics”

1. […] and linguistics (part 1) Last Friday, I suggested that one of the main reasons that language isn’t mathematics is that mathematics, […]

2. Moreno says:

Greeting from a fellow linguist!

I wonder about your views on Greenberg and his mathematical postulation of language. I am personally interested in the mathematics/linguistics interface and would not consider it completely nonsensical.

If we just take into account three basic linguistic sets, (1) phonological, (2) morphological and (3) syntactic sets. How do you find applying, for instance, some form of an integration function to, e.g., a set of sounds that may produce (algorithm-accordingly?) a morpheme, a word. These are just some of my pondering ideas …

All the best!

3. Hi Moreno,

I haven’t had an occasion to look specifically at Greenberg’s work, so I can’t really comment on how he approached the subject. At first glance, the term “integration function” sounds like it’s related to the mechanisms used in (minimalist) generative grammar to get from the lexicon to the sentence. If so, I’ll actually have quite a bit to say about these types of things in some of my upcoming posts, once I have some background material out of the way.

If I recall correctly, Halle and Marantz’s proposals about distributed morphology suggest some ways in which the morphological and syntactic structures can be integrated within a generative framework, which extends the range of linguistic phenomena that are treated computationally within that framework.

Any idea how closely Greenberg’s approach is to the various flavors of generative linguistics, or to Lambek’s syntactic calculus?

4. Moreno says:

I am currently reading some of Lambek’s works. I should say, though, that I haven’t really heard of him before, so thanks for the information. So far, what I’ve seen, both Greenberg’s & Lambek’s approaches may seem similar.

5. Moreno—

I still haven’t looked at Greenberg’s work (it doesn’t happen to be in the scope of my thesis that I’m trying to finish up), so I don’t know what its intellectual lineage is. Have you encountered Bar-Hillel’s work yet? I’ve recently skimmed through his 1953 article “A Quasi-Arithmetical Notation for Syntactic Description”, which seems to be similar to Lambek’s approach.

6. Moreno says:

The title sounds prominent! Thanks for that! Hope you don’t mind my asking what your thesis is about? Good luck with it, though …

In generative grammar, there’s an assumption that the acquired language interacts with the performance and conceptual-intentional systems, and that these allow us to actually use the language (produce it, consume it, and interpret it as something “meaningful”). I’m looking at the consequences of this assumption, especially in a biolinguistic context; I’m especially interested in whether there are some infeasible theories (in the sense that the theory leads to a language that can’t be used to produce / consume observable language under any assumptions about the conceptual-intentional and performance systems).

The series of posts I just did on mathematics and linguistics essentially sketch a justification of my research methodology.

8. Moreno says:

This is really interesting! Does, then your competence/performance “interface theory” scrutinise the very partition of Generative grammar into competence and performace? It reminds me — plausibly erroneously — of one of the objecting theories to the one of Chomsky (and others, of course), by Chambers (2003:34), where he says that “it is certainly not true — and Chomsky might even agree with this in view of the way categorical linguistics developed since 1965 when he first stated the idealisation — that variation theory must incorporate or in any other way take account of the specific postulates of *categorical grammar*, [and notions like] the Katz-Postal principle (Chomsky 1965: 132), the specified-subject condition (Chomsky 1973), the root clause filter (Chomsky and Lasnik 1977: 486) […]”

Your thesis does not seem, though, to be of sociolinguistic nature, and I very excitedly see it as “psycholinguistic variationist theory.” I am really happy to hear from a doctoral student like yourself as I already have some ideas about my PhD.

You also mention biolinguistics, and I would very much like to invite to check out my University’s summer conference on biolinguistics, which is rumored to gather a plethora of interesting people. The link is: http://www.york.ac.uk/conferences/bale2008/index.html
The title of the conference, as you will see, seems to coincide quite well with some of your keywords.

9. I’m not a doctoral student… at least, not yet. The thesis I’m working on is for a master’s program. :-)

It’s not so much my theory as it is minimalist theory, following the basic framework laid out in Chomsky’s The Minimalist Program (1995). The current literature doesn’t focus so much on competence and performance as it does on the notions of cognitive systems and the interfaces between them, although the division between competence and performance seems to be implicit still in the distinction between the computational system (roughly analogous to competence, but probably not exactly so) and the conceptual-intentional and sensorimotor systems (roughly analogous to performance, but probably not exactly so).

Generative grammar has undergone some fairly significant changes since the 1960s–1970s work cited in that quote. Again looking at Chomsky’s publications, and picking just a couple of works, there were major shifts in at least Lectures on Government and Binding (1981/1993), Knowledge of Language (1986), The Minimalist Program (1995), and Derivation by Phase (2001). If you want a relatively gentle introduction to minimalist grammar, you might want to take a look at Radford’s (2004) text; its what we used in the syntax course where I first learned about minimalist theory.

I’m not convinced that one could classify modern generative theory (minimalism in particular) as “psycholinguistic variationist theory”, since sociological factors really don’t play any role in the theory; they’re relegated to the conceptual-intentional system, and generative theory usually focuses just on the computational system and the interface conditions.