# Is math a language?

Is math a language? Is "language" a comprehensive description of mathematics? Maybe this is just a trivial description... or possibly there something about mathematics that is missed by describing it so... Certainly math is far more restricted than ordinary languages, but is math itself a proper language?

As a basis for what a constitutes a language, I refer to John Searle's article, "What Is Language?" Also, my thinking about math is informed by Wittgenstein's "Foundations Of Mathematics" and Russell's "Introduction To Mathematical Philosophy".

Obviously mathematics are not a language in the sense of an ordinary or commonly spoken and written language (whether colloquial, vernacular or something more formal). Inasmuch as language is a means of communication, however, it seems like "a language" is an accurate description of the abstract and formal means of dialogue and expressing ideas through intentionalistic utterance.

Searle writes:

The standard textbook accounts of language say that specific languages such as French or German consist of three components: a phonological component that determines how words and sentences are pronounced, a syntactical component that determines the arrangement of words and morphemes in sentences, and a semantic component that assigns a meaning or interpretation to words and sentences. More sophisticated accounts add that there must also be a pragmatic component that is not a component of specific languages; rather, it sets certain constraints on the use of language and is not internal to specific languages in the way that the syntax of French is internal to French and the syntax of German is internal to German. ... The relation of syntax to semantics is however crucial. Syntax organizes semantics according to three principles: discreteness, compositionality and generativity. Discreteness is that feature by which syntactical elements retain their identity under the various syntactical operations. ... Compositionality is both a syntactic and a semantic property. Syntactically, a complex element such as a sentence is built up out of simple elements, words and morphemes, according to the formation rules of the language. Semantically, the meaning of the whole sentence is determined by the meanings of the simple elements together with the syntactical structure of the sentence. ... Generativity, as I am using the term, implies that the syntactical operations of the language allow the speakers to generate an indefinite number of new sentences. There is, strictly speaking, no upper limit to the number of sentences in any natural human language.

Though he notes that this is an incomplete picture of what constitutes a language, I think mathematics meet these criteria of phonology, syntax, semantics, as well discreteness, compositionality and generativity. Also, the deontological considerations he advances when considering the question "what is language?" I think are also met by mathematics (when you say "2+2=4" and mean it, you are committed to the utterance.)

Philosophy is written in this grand book, which stands continually open before our eyes (I say the 'Universe'), but can not be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to humanly understand a word; without these one is wandering in a dark labyrinth.

...I don't speak Italian, but in this translation "mathematical" is adjectival, possibly suggesting that mathematics are not necessarily a language, but sufficiently an aspect of ("what can be done with"?) language. Other translations say "the language of mathematics" and perhaps this is merely poetic license upon the translators part. For what it is worth, Galileo's "Egli è scritto in lingua matematica" from The Assayer is translated by Google Translate as the adjectival form. Not that interpreting Galileo is the final word on answering the question, but as a possible counter-example demonstrating that mathematics is instead a formal aspect or sub-set of the human capacity for language.

Considering the answer here to the question "what is a language?" perhaps the mathematics lack "biplanarità"? (For what does three refer to except for the taxonomy of counting and measuring?)

Is math a language?

• Mathematics is simply a subject which provides a context for certain linguistic affordances and prohibitions. It does not provide for them itself. Without language there could be no mathematics. How would mathematicians validate a proof? Every symbol they use is a part of language (and its attendant technology, literacy), and without the capabilities that language provides they would be unable to describe anything, let alone prove it. – jlawler Jan 19 '17 at 16:18
• This might have better traction on Philosophy.SE. Math is certainly not a 'natural language' nor a 'natural' language. It is definitely a communication device and has a lot in common with natural languages. Define language first then see. – Mitch Jan 19 '17 at 17:07
• Mathematics is a theory of natural language -- interesting, but incomplete. – Greg Lee Jan 19 '17 at 19:42
• @Mitch the question has been asked there and, to my thinking, not answered satisfactorily, e.g. nonsense abounds, such as "mathematics is the language which encodes everything which is existing in material plane!" – Mr. Kennedy Jan 19 '17 at 21:10
• @Mr.Kennedy, That's decipherable -- which encodes everything which is existing in material plane. An intensional language which includes reference to other possible worlds has been suggested for natural language. Restricting reference to just one real world seems plausible for math. – Greg Lee Jan 20 '17 at 1:40

The thing is that a language, when you get to the core of it, is a system of communications. It is used a means of communicating to talk to others about the world and so on. Math can be considered a language in the sense that it's a system with well-defined rules and that can convey some meaning.

However the range of concepts it can treat is very limited and you certainly cannot "communicate" with it, unless you assigned arbitrary meanings to numbers but then you'd be using a natural language with it. You could say A=1, B=2, and so on, but it wouldn't be just math anymore, it'd be "insert natural language" + math. However English, as any other natural language, can be used by itself satisfactorily.

Even if you were to use the language of mathematics, as in adopting mathematical notation, you can only talk about math-related subjects or anything that can be discussed mathematically, and even then you'd still be using a natural language around it.

So my answer is: It could be considered a language, using the broad definition, but not in the same way natural languages are considered languages.

• @Mitch I meant in the sense of "Yesterday I went grocery shopping, but the stores were all closed!" you cannot communicate this with pure math (as far as I know at least), and the only way would be together with another language. I'd like to see an example of the contrary though, were it possible. – Alenanno Jan 19 '17 at 14:03
• Do you have any evidence that language is a system of communication? Aren't there well-defined rules to particle physics? Isn't the genetic code of rice a language? After all, it communicates information. – user6726 Jan 19 '17 at 17:16
• @user6726 No, genetic code is not a language at all. The genetic code of rice is only informational relative to an observer. See Searle's review here. – Mr. Kennedy Jan 19 '17 at 21:05
• @user6726 My use of "speak" was not in the strict sense. I meant, do you use particle physics to communicate with another human being? Also, natural languages are not the only systems we call languages and "language" by itself is a very broad term. C is a language, yet it obviously is not a natural language the same way English is. Math is also not a language in that sense but it can be considered as such the same way programming languages are "languages". Finally, if you want to see what other factors make a language as such more in depth, check my answer at the other link in the OP. – Alenanno Jan 20 '17 at 0:16
• @vectory A subset of a language isn't necessarily a language. For example, the word "pineapple" (and only that word, nothing else) is a subset of English, but I wouldn't call it a language on its own. But I'd personally define a language, in the linguistic sense, as a system that can be used by humans to communicate arbitrary concepts. Number theory, for example, can't communicate the concept of "I like pizza", therefore number theory is not a language. – Draconis Mar 13 at 22:44

Mathematics itself, just like biology, philosophy, linguistics, ... is not a language or a communication system at all, but simply a science. You may argue about the precise scope of mathematics, but roughly you might describe it as "the study of such things as numbers, structures, and logical reasoning".

Obviously, in order for humans to talk about the subject of study, they require some kind of language or communication system.
You may ascribe literal conventions as used in mathematical notation some properties of language in that it is a system of rules on how to build expressions to convey a specific meaning, this encompassing both specialised symbols and a mathematical dialect of natural language.
For example, there is a clear difference between ordinary English "if" (which is normally associated with some kind of causal other otherwise complex semantic relationship) and a mathematical "if" (which operates solely on truth values)1. There is also a certain amount of highly conventionalized expressions which are used in mathematical proofs and descriptions that have a very specific meaning in the scope of mathematics which differs from what it would mean if used in ordinary everyday English. In that sense, you could indeed talk about some kind of "mathematical English" as a language in the sense of a set of expressions with a fixed meaning and usage that are a different variant of language than their equivalents in the "ordinary English".
And of course, symbols or notational conventions like f(x) = x², ∀, π, ... are strings that convey a specific information that doesn't have an equivalent in natural language at all.

But the use of these tools (specialized symbols and a mathematical variant of English) in order to talk about the subject of study doesn't make the subject of study itself a language.
Mathematics is as much of a language as biology is - pretty much not at all: It is a science, and the science itself is not equivalent to the linguistic and symbolic tools that you make use of in order to talk about that science.

1 In a situation where the sun is not shining and no pig is green, the sentence If the sun shines, then some pigs are green is a true statement in the mathematical sense. However, any speaker of English you'd ask about their intuition would tell you that this sentence is pretty much nonsense and probably even wrong. This is because a mathematical "if" actually has a different meaning than an ordinary English "if".

• Math is axiomatic and science empirical, no? Also, philosophy is not a science, it is love of wisdom (i.e. respect for obtaining knowledge) – Mr. Kennedy Jan 20 '17 at 0:49
• @Mr. Kennedy Split hairs about the precise definition of science if you want. That still doesn't make your system of axioms the same entity as the tools that are used to talk about that system. – lemontree Jan 20 '17 at 7:29
• not splitting hairs, trying to understand what you mean by mathematics is "simply a science" - that it is a subject of research, study or investigation? – Mr. Kennedy Jan 20 '17 at 8:29
• If your account of "if" is correct, then I'm a monkey's uncle. – Greg Lee Jan 20 '17 at 12:38
• It was an example to display an exception to your account of English "if" when you say it involves "some kind of causal [or] other otherwise complex semantic relationship". The correctness of your account obviously does not cause me to be a monkey's uncle. (I got the example from Hans Reichenbach.) – Greg Lee Jan 20 '17 at 18:20

of course mathematics is not a language. but we do have a "language" of mathematics, that allows us to express mathematical ideas. Or so it would seem. Is there really such a thing? Not really. It looks like a formal language, but it isn't, it's just a tightly constrained variety of natural language. Mathematicians do not write their papers in a formal language, generally. Some of them even oppose doing that.

we cannot (so far) even define "language", although we can formally define very impoverished formal structures that we like to call languages.

regarding the classic triumvirate of phonology, syntax, and semantics: its obvious that the language of mathematics does not have its own phonology, for example. what is the phonology of the symbol used in differential equations? {\frac {\partial z}{\partial x}}?

• – Mr. Kennedy Jan 19 '17 at 22:12
• good point about phonology. – Mr. Kennedy Jan 19 '17 at 22:15
• Not so good, actually. A phonology is not the same thing as a pronunciation, – Greg Lee Jan 19 '17 at 22:23
• @Greg Lee: ok. what is your point? i don't recall claiming that a phonology is the same as a pronunciation. – mobileink Jan 19 '17 at 22:34
• You asked about the phonology of a symbol? I guessed that you meant pronunciation. Was I wrong? – Greg Lee Jan 20 '17 at 1:28

The question "What is a language?" in fact asks "What is a natural language?". The accepted answer starts "A language is a complex system of communication..." which, perhaps coincidentally, is a passable definition for any language whatsoever: A language is a (comparatively) complex system of communication. Clearly, maths isn't a natural language but, even more clearly, it is a language by the latter definition -- in fact, the most complex language we know. A possible "downside" to the definition is that some animal communication systems (e.g. honeybee's dance) are also comparatively complex (and thus, possibly, languages)

Quoting lemontree:

Mathematics itself, just like biology, philosophy, linguistics, ... is not a language or a communication system at all, but simply a science. /---/ "the study of such things as numbers, structures, and logical reasoning".

Maths isn't just another science -- in fact, it's routinely contrasted w/ sciences. Even if we forget about Galilei, there's a lot of highly qualified talk about maths as the language of science and the language of mathematics, which are two main ways of construing maths as language

In principle, I don't have a problem w/ honeybee's dance being a language, but it lacks symbols, so we could strengthen our definition w/ "A language is a (comparatively) complex system of symbolic communication"