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 , if , then the product . 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 and are identical for all propositions . 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|
|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.
Copyright © 2008 Michael L. McCliment.