# Is math (only) a language?

"Math is the langauge in which God has written the universe" ~ Galileo Galilei (no less)

I recall vaguely, dovetailing with Galileo's words supra, reading math is a language. I recognize "2 + 3 = 5" means "the sum of two and three is five" and here it's quite clear that math is a language. In this particular case the need for a new language (math) is evidently efficiency and readability (very much like for programs written high level computer languages like Python or C).

That out of the way, let's shift focus to the stock-in-trade of math viz. theorems. A theorem is most definitely not linguistic in any way. It's better (vide supra) expressed using mathematical symbols, but it has no logical connection to the symbols so used. It is, I guess I want to say, not a property of (mathematical) language. It is a property of a mathematical object, e.g., a² + b² = c² (Pythaogrean Theorem) is a property of right triangles.

Question: Is math (only) a language?

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Commented 10 hours ago

The approach to consider math a language seems interesting, even if it does not capture all of the art of mathematics.

1. Galilei’s statement considers math the language which encodes the laws of nature. I think this view still holds for todays mathematical physics.

2. For those who work in the field of pure - not applied – mathematics, there are other aspects fundamental.

Then mathematics is a discipline to invent games with new rules, a bit like the author Herrmann Hesse described in his book The Glass Bead Game.

During the game new questions about the game, its structure and variants arise. Mathematicians work hard to develop tools to solve these new questions. E.g. questions about the zeros of the Riemann zeta-function or conjectures from the Langlands’ program.

These are some minor thoughts about the character of mathematics. I am curious how other participants of this site will answer your post from different points of view.

I would like to bring this discussion down to earth. I am a retired professor who has taught fluid mechanics to engineering students. Many important fluid mechanics concepts (like vorticity) are next to impossible to describe in natural language. If we were intelligent aquatic creatures we would have a vocabulary that included that concept as something apparent to our senses.

I once set as a question in a midterm exam "describe Helmholz' three vortex theorems" and one student responded in an angry scrawl "I am not here to memorize theorems!!" Since then I have begun the class by reminding the students that they are not fish, and therefore do not have the language to understand what I am going to tell them. I say that first of all I have to teach them a language that will serve the purpose. I add that this language not only expresses important facts, but can also express insight and creativity. Just like a natural language, they can use it to buy potatoes or write a poem.

• Natural language evolves in response to experience and needs. Sometimes we need to discuss concepts for which the vocabulary has not arisen through that route, but happens to have arisen in mathematics. Commented yesterday

Looking at the the artifacts that mathematicians produce - basically, theorems and proofs and algorithms - mathematics is a demonstrative, non-empirical science. You could call that a language; the artifacts are a way for mathematicians to communicate to other mathematicians (and their students) what they discovered; the communication uses a mish-mash of everyday language and highly technical, sometimes highly formalized language. But does calling it a language tell you anything you didn't already know?

Looking at what mathematicians do - math as activity or process - this is trying to solve mathematical problems, problems that all have the peculiar very nice character that you don't have to know any specific facts of physics or history in order to understand them or solve them (if they are solvable). This is essentially just guessing. Hopping around in the dark and hoping to stumble upon a lamp that will light up the space they are hopping around in before they hit a wall again. Until a problem is solved, no one may even have any idea why the problem is so hard or how to attack it. Polya speaks of trying to use "plausible inference" (in contrast to "demonstrative reasoning"):

I don't believe that there is a foolproof method to learn guessing. At any rate, if there is such a method, I don't know it (...) The efficient use of plausible reasoning is a practical skill and it is learned, as any other practical skill, by imitation and practice.

In what follows, I shall often discuss mathematical discoveries, great and small. I cannot tell the true story how the story did happen, because nobody really knows that.

[Induction and Analogy in Mathematics, G. Polya, 1954, vol 1, preface]

As process, as guessing, mathematics may sometimes use language, scribbled notes, signs that are being shifted around, fragments of language, but this really cannot be called a linguistic activity, I think. Language (and logic) can be used after a discovery has been made, after you suddenly "saw" how to solve it, but play no big role in the discovery process, in coming to understand a problem. At least that's my experience when trying to solve math puzzles (I'm only an amateur however; I made the fatal mistake of dropping math after my freshman year and switching to ... you guessed it... philosophy).

• Mathematical symbols like "22" and "+" feel more like shorthand for "twenty two" and "add" than anything substantive by way of being quantiatively enabled. A language that makes it easier to talk about quantity, now that would be a language in which God wrote the universe. Anything more would be a bonus. What about Russell-Whitehead/Frege logicism? Commented 2 days ago
• @Hudjefa - Yes, notation really matters - making further work easier (think of the discovery/invention of '0'). I don't quite know how to think of the relation between notation and understanding, though. Logicism was (partially) an attempt to standardize notations, the script. Commented yesterday

The word math, like nearly any other word in any natural language, can be (and is) used in many ways by many speakers in many contexts. For this particular speaker (me, a mathematician), if I am expressing myself carefully, then the noun math refers to

1. an undertaking,
2. a way of thinking that explores patterns by means of rigorous logical thinking, and even
3. a corpus of results obtained from such exploration.

In such circumstances, there is a body of lexical and notational practices that greatly assists in the unambiguous recording and exchange of mathematical ideas. That body of practices can reasonably be described as a language, but those practices don’t constitute math. Instead, they constitute a medium in which to carry out and to communicate math.

For my money, Galileo’s aphorisms are in that sense somewhat metaphorical.

To someone who took the hard line that math “is only a language,” I would respond, “Then so is, for instance, knitting.” After all it too involves strange things, in its case things like purls and cables and tinking. And what can the uninitiated make of ideas like WIPs, YOs, and DPs, and of such hieroglyphs as K2tog and *K3 S1 wyif P3. Rep from * ? And what’s the significance of the difference between RS and WS? How does one “make gauge,” and when and why should one bother to do so?

Most definitions of mathematics define it as a science or field of study. These definitions indicate that math is more than a language. Mathematical notation is a structured communication designed as a type of shorthand to convey mathematical ideas in a manner that is far more efficient than natural language can provide. So mathematical notation can be considered a natural language supplement used to help convey mathematical ideas.

Chemistry is a science that has similar natural language supplements; Lewis diagrams, chemical formulas, etc designed to convey concepts that are cumbersome to explain using natural languages

• Non liquet (as usual), but from what I can see, you're on the mark as far shorthand or supplement goes. My best shot is, if math is a distinct language, it should have a distinct syntax; this isn't the case. While notation is different from natural languages like English, it is read & understood per natural language syntax. Commented 13 hours ago
• @Hudjefa And how is a complex mathematical equation read per natural language syntax? Philosophers often call math and logic artificial languages in contrast to natural ones. Commented 13 hours ago
• You will never open a Math or Physics textbook and find only equations. Math alone can't provide context which is something that language can do. Commented 13 hours ago
• I don't need any natural language to see 3 pebbles and 3 pebbles makes 6 pebbles on the lake shore. Commented 13 hours ago
• Yes, @JKusin, certainly it is a distinct surface syntax (as linguists would call it). But it nonetheless should read like, say (technically complicated) English sentences. As an example, any mathematical expression of the form "A = B" should be understood as a sentence amounting to the English-language sentence, "A equals B." Thus, A can be understood as the grammatical subject of the sentence, and "=" as the verb. Commented 9 hours ago

Is math (only) a language?

Yes and no, depending which definition of language you are talking about, so let's discuss.

According to modern linguistics after contributions by Noam Chomsky borrowing from Saussure, language can be distinguished in terms of linguistic competence and performance. When most people use the term language, they are referring to performance which is the concrete act and product of language. If someone utters a sentence in Germany to buy a Trabbi, they are living up to an expectation of what a language is, mainly the use of sound to convey meaning to achieve pragmatic ends. But the notion of linguistic competence differs because it emphasizes not so much the process of communication, but the psychological capacity of agent to use language. It is therefore a form of psychological and sociological description. From WP:

In linguistics, linguistic competence is the system of unconscious knowledge that one knows when they know a language. It is distinguished from linguistic performance, which includes all other factors that allow one to use one's language in practice.

Thus, a language is not just the observable behavior of applying a grammar to communication, but is also an ontological framework for understanding how knowledge functions not only within the performance of language proper, but adjacent to language. For instance, one of the primary products of cognition both historically and contemporaneously is the notion of ideas, concepts, and thoughts. Language-as-competence is therefore a way to communicate about human cognition as well as communication.

As an example, consider the difference among the concepts of utterance, proposition, and assertion or judgement. An utterance is phonological and physical in nature; it is the phonemes and the phonics, the concrete implementation of communication. A proposition, however, is the semantic contents, the assembly of morphemes to convey the concept. And an assertion is the assignment of truth conditions to a proposition. Thus, there is a difference between recording the sentence 'The polygon is closed' with a recorder, understanding 'The polygon is closed' when listening, and determining that 'The polygon is closed' is true because the polygon is, indeed, closed in some drawing. Those are important distinctions in mathematical logic, and philosophy in general precisely because they are cognitive notions.

Consider as an example the Chomskyan generative grammar as a description of human cognition applied to the domain of mathematics. One of the most important introductions to the model after transformational grammar was offered in Syntactic Structures was the introduction of the lexicon and complex symbols to represent vocabulary in a language. In essence, it allowed one talk about language in virtue of the semantics. For instance, "The triangle has three sides" can be understood not just in terms of articles, nouns, verbs, and adjectives, but also in terms of quantity, polygons, properties, and line segments. Thus, there is no clear dividing line between mathematical concepts and mathematical language.

And what applies to the definition of a triangle, that there is no strict conceptual dividing line between a syntactic and linguistic interpretation and a semantic and cognitive interpretation applies to the entire endeavor of mathematics. Mathematics in this light, as Brouwer advocated, is a function of our linguistic intuitions (SEP). Part of mathematical language is performance, the explicit and behavioral process of using language faculty to convey ideas within the language community, but part of mathematical language is competence, the implicit and mental process of using language faculty to formulate and introspect mathematical ideas privately.

Therefore, yes, mathematics is a domain-specific language of quantity, quality, relations, operations, directions, and so on in both the the context of performance (submitting a mathematical theory for publication, for instance) and in competence (finding axioms, constructing theorems, corollaries, and lemma, ensuring they fit within the paradigm of the broader theory, finding applications for theories in modeling or other applications, etc.) Thus, the key to answering this question is understanding that language is not just about the syntax of communication; language always encompasses the generation and refereeing of semantics which is often called conceptualization. When seeing language as a linguist sees it, both performance and competence (or langue and parole to use Saussure's terms), math is certainly only a language.

• I wouldn't even know where to begin. 'True answer'??? That's not a thing. 'Language of the universe is mathematics' This a poetic and metaphorical construction. 'Media' that refers to the metaphorical, physical substrate of the message which is generally conceived of the channel of communication, not the language, which is part of a mental ontology. I think Herr Wheler's answer is a good fit for your understanding.
– J D
Commented yesterday

I would say a2 + b2 = c2 is not per se mathematics. Mathematics is about reasoning, not about notation. A lot of people see formula as what mathematics is. There are mathematical proofs without any formulas, and there are formulas that are just nonsense.

P=NP for example also means something in a context where P and NP are explained properly. Because mathematics often tends to be complicated to explain and because details can have extreme impact, they invented a language to denote the concept.

But mathematics has existed already for a very long time, whereas the notation has changed. For examples variables and constants have been introduced mainly by Rene Descartes, but that does not mean people could not reason about variables before the introduction of that notation.

Mathematics essentially deals with making abstraction of problems, and solve this with a set of well-established reasoning techniques for which only a (very) limited set of axioms can be used. A lot of proofs nowadays can be checked with computers, because you can write a proof in a certain language, and denote what inference rule is used between every step, and therefore the algorithm can help find reasoning errors.

• True, Is semantics a property of languages? Commented 5 hours ago