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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.)

Galileo writes:

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?

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MmmHmm
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    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
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    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
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    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!" – MmmHmm 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 01:40
  • Related: Is mathematics a language? on Philosophy, Is mathematics a truly universal language? on Worldbuilding – Ooker Jun 17 '18 at 11:48
  • It's Ironic that phon-, as in the first component summarized in the quote by Searle, "phonology", derives "From Proto-Indo-European bʰoh₂neh₂, from bʰeh₂- (“to speak”), from which comes φημί (phēmí, “to say, speak”)". So the definition is recursive. Eventually it's circular, but I'm not saying that were a bad thing. A rose is a rose is rose and all that. I mean, at least it's not inconsistent. Not to mention -logy "teaching, wisdom, ..?", logic, logos ("tongue, speach"); cp. also glosse (and gloss). – vectory Mar 14 '19 at 17:58

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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.

Alenanno
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  • Re 'you certainly cannot "communicate" with it' - whoa dude... of course you can communicate with it without natural language. One could make a case that math is closer to a platonic ideal of language, and natural language is a 'poor' attempt. – Mitch Jan 19 '17 at 14:00
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    @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
  • 'Math' doesn't have a good vocabulary for the concepts of objects and concepts in the experiential world. But surely that's just a matter of creating such a notation which math is good at. – Mitch Jan 19 '17 at 14:41
  • Mathematics has no vocabulary at all; what's called its "vocabulary" is simply a bunch of language that's been repurposed (using language's internal modification facilities) to suit the abstract milieu of topics discussed. Everything in math is a part of language; mathematics is certainly not a language itself, any more than auto mechanics is. – jlawler Jan 19 '17 at 16:22
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    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
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    @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. – MmmHmm Jan 19 '17 at 21:05
  • @Mr.Kennedy the genetic code of rice is informational relative to an interpreter/interpretation method, which is all the machinery in a cell, the most important of which of ribosomes which converts DNA into proteins. – Mitch Jan 19 '17 at 21:16
  • @Mitch precisely, and chemistry as informational machinery is analogy, i.e. relative to an observer. – MmmHmm Jan 19 '17 at 21:27
  • @user6726 Your first question is fine, but the others are weird. How is particle physics a language in that sense? Does anyone use it to speak to other people? And the same goes for the genetic code. Also, keep in mind that that sentence was a simplification of what a language does and to show that maths lacks even just that one (ignoring all others), but it's certainly not the only feature that characterizes a language. – Alenanno Jan 19 '17 at 22:20
  • @Alenanno, you have to start with some kind of definition of what a "language" is. ASL is not spoken to anyone, so is it not a language? Nobody "speaks" math, and it can't be considered a language using the ordinary meaning of language. If you're willing to seriously consider math a language, then I don''t know what you think "language" refers to. Disclaimers like "could be considered" are wrong-headed: the answer is, simply, "No". – user6726 Jan 19 '17 at 22:51
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    @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 00:16
  • You are knocking down a strawman. The comments digress over the definition of language, it does so without establishing a definition of math, as if that was unimportant if only enough requirements for languagehood are found to exclude the question for math as off-topic. Your statement "assigned arbitrary meanings to numbers [would] be using a natural language" is very misleading if, as I read it, it implies giving a name to a symbol (which would be redundant) or giving an extrinsic definition of a number (i.e. by example). Neither one is not needed for the visual representation I + I = II – vectory Mar 09 '19 at 02:28
  • ASL is certainly a language, as the expansion of the acronym American Sign Language implies. What a silly comment. Anyhow, this answer is incongruent. Either mathematical language is a subset of English and therefore a language or English is a subset of mathematical language, or both are orthogonal and joined at the. Maths is structural. Some things numbers are so foundational, that they hardly qualify as "maths". The real question, that your answer fails about, remains: What would be a narrow definition of language? – vectory Mar 09 '19 at 02:36
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    @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 '19 at 22:44
  • @Draconis, For a mathematician, pizza=pi*z*z*A, pi=Euler's constant, z=radius, A=area [picture a sketch of the geometry instead of the equations]. That could be extended. However, baking the thing is, after all, not a matter of language or maths. A recipe on the other hand is the introductory example of an algorithm. You can infer that I must like it, just from how much I talked about it. ,,, PS: I'm sorry if this is distracting. A matter of fact treatment on the meta level about, the philosophy of maths and language doesn't fit in 500 characters. – vectory Mar 14 '19 at 18:59
  • @vectory I don't see what that has to do with anything. Sure, you can give each letter a numeric value, then claim that the English word "pizza" is mathematical. But that isn't expressing the concept of pizza in mathematics, it's just grafting a nonsensical meaning onto a common English word. – Draconis Mar 14 '19 at 19:04
  • This was an old joke to show that maths has culture, one very important aspect of a living language. – vectory Mar 14 '19 at 19:05
  • A more important notion would be that a natural language has to be a primary language. A secondary is merely grafted on top of a primary language. The pertinent question for me is still, whether natural language depends on mathematical abilities equivalent to those expressed on the meta level in maths. However, I don't think it's an important distinction. – vectory Mar 14 '19 at 19:22
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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".

Natalie Clarius
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    Math is axiomatic and science empirical, no? Also, philosophy is not a science, it is love of wisdom (i.e. respect for obtaining knowledge) – MmmHmm Jan 20 '17 at 00:49
  • @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. – Natalie Clarius Jan 20 '17 at 07:29
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    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? – MmmHmm Jan 20 '17 at 08:29
  • @Mr.Kennedy It is a science to my understanding in that it seeks knowledge about a real aspect of our world with methods accepted by the scientific community. For a more detailled discussion of what the subject of mathematical study is, you should consult a mathematician. My point was simply that mathematics ≠ language of mathematics. – Natalie Clarius Jan 20 '17 at 10:02
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    If your account of "if" is correct, then I'm a monkey's uncle. – Greg Lee Jan 20 '17 at 12:38
  • @Greg Lee I judge you statement false both in the mathematical and in the ordinary sense of "if". – Natalie Clarius Jan 20 '17 at 14:35
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    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
  • @Greg Lee What about my account of "if" is it that you consider wrong? I was intentionally leaving the assumed meaning rather vague; I am not trying to give an acceptable definition, just pointing out the obvious contrast to the mathematical "if". – Natalie Clarius Jan 22 '17 at 21:55
  • I didn't quite say your account of "if" was wrong. I said there was an exception to it, noticed by the logician Reichenbach. It is an exception because it simply means that the antecedent is false and does not concern any causal relationship between antecedent and consequent, just like the material implication of logic. It's just a curiosity -- I thought you'd be interested. – Greg Lee Jan 23 '17 at 00:38
  • @Greg Lee Oh, okay. So your point was that "if" can be used in this way to show only the falseness of the antecedent, without implying any relationship to the consequent? – Natalie Clarius Jan 23 '17 at 15:01
  • Yes, that's the point. – Greg Lee Jan 23 '17 at 17:28
  • @Greg Lee Ah, I see. Well, that's pragmatics and pragmatics is pretty much unpredictable ;) – Natalie Clarius Jan 23 '17 at 18:26
  • @Mr.Kennedy You're correct. Math is not simply a science like biology, etc. That is a grave epistemological error, not hair splitting. Well worth noting. – sfmiller940 Jan 04 '20 at 00:00
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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}}?

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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"

jaam
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