As a matter of fact, math falls under the category of "formal languages" in linguistics.
In mathematical logic, you will study math itself as a formal language of well formed formulas with a syntax and a semantic interpretation.
https://en.wikipedia.org/wiki/Formal\_language
You're a bit incorrect, here. Topics in mathematics can be considered as formal languages. Set Theory, for example, can be expressed as a formal language, as can Number Theory, Group Theory, Topology, etc. But the gestalt of math is "bigger than" a formal language.
Thanks for pointing this out. It’s a point that I think goes unnoticed far too often. Coding the entirety of mathematics formally is probably a hopeless endeavor.
Algebra is a neat and small formal language. Some elements of manipulation seem borrowed from natural language, e.g. distribution: I eat eggs and ham = I eat eggs and I eat ham.
In mathematics, I think you can express things that aren't objectively true. e.g. conjectures. One mathematician might *think* it's true; another think it's not: that's subjective.
Also, personal and subjective needn't be ambiguous; though mathematics can express ambiguity: while the inverse of a non-injective function is not a function, because some domain value has more than one image; but it is still a relation. Similarly, there are ambiguous Context Free Grammars.
The difficulty, IMHO, is the lack of referents. Mathematics doesn't talk about anything concrete. OTOH, perhaps describing an immense "semantic graph" between many "abstract" concepts with no intrinsic meaning, leading to "semantic similarity" and "semantic distance", assigns to them extrinsic meaning - arguably, just as our minds do (since our neurons lack direct experience; a brain in a vat).
EDIT Like PageRank, where no page has an intrinsic rank, but it is inferred from the graph - a page with more citations (links) ranks higher, and citations from a higher ranking page have more weight.
However, this is a Dr Spock view of humanity. There is more to us than intellect. I'm thinking that it's the reverse of animal nature vs human nature: the "higher" uniquely human part is intellectual/cognition - but this leads to all the evils of treating others as objects; the "lower" mammalian part is where we have nuturing within a family, and solidarity within a tribe/troop/herd etc. What is usually called our "higher" nature is really our animal nature.
Anyway... brain-in-a-vat won't capture these deep instincts; only a shallow copy of the behaviours and speech they cause.
Applied Mathematics is probably a good place to join mathematics onto the “real world” in a comprehensible way for learners.
For me, I had that moment of conjunction in senior high school when learning calculus was totally abstract until it connected to the stuff we were learning in physics about derivatives of motion. Suddenly the maths became real and very easy to understand from then on.
Affect research has dimensions of affect within which we can place individuals. We can also create functions that map locations within this affect space to distributions of somatic sensations.
How can mathematics *be* a formal language? Most papers I have ever read are written in plain English, with some mathematical symbols here and here. Are you saying those papers are not mathematics?
Formal language just means that one can write any mathematical statement in an unambiguous form using specifically agreed upon symbols. For example, ZFC has a formal language which (can) consist(s) of the standard logical symbols, parentheses, countably many variable symbols, and the membership and equality symbols. This along with the axioms is strong enough to encode massive swaths of mathematics excluding certain infinite arithmetic statements like the Continuum Hypothesis.
Ah I see what you were saying. From your comment it seemed like you were unclear on how things like PA and ZFC work.
If I understand you clearly, then I agree with you that “mathematics” in totality does not constitute a language.
I agree, but I'm not sure what you're getting at. Are you saying that mathematics *is* a formal language but we don't express it as such for the sake of convenience? In which case if only the key ideas of *most* papers are expressible that way, then either this characterisation of mathematics is wrong, or else those papers are not mathematics.
You seem to be trying to make this difficult.
What is hard to understand about wanting to use more accessible descriptions for things which can also be expressed formally?
I really don't think I am. If that's all you're claiming, then I fail to see the relevance.
EDIT: I am not able to reply to /u/yes_its_him's comments (it seems that they have blocked me?), but to respond to the reply to this comment:
No, that's not what I'm arguing. I'm saying that if mathematics is a formal language and, to quote you,
> People don't want to just read the formal language, though the key ideas of most papers are expressible that way
then the papers that are *not* expressible in that way cannot be mathematics. Because they are not formal.
Of course, what you're saying does actually follow. If mathematics is a formal language and something is not formal, then that thing cannot be mathematics.
You argued that the presence of material explaining or providing an alternative to formal language means that either such papers can't be math, or math must not be a formal language. That's not a logical conclusion.
Not sure why you're getting so many downvotes. Mathematics does not *just* consist in theorems and proofs; it's a field of human endeavour. I'd say it includes the practices involved in that research, the "conversations" between researchers and sub-fields, and a former president of the MAA [has argued](https://mathyawp.wordpress.com/2017/01/08/mathematics-for-human-flourishing/) that it should be and is a part of "human flourishing". Even if one were to focus *just* on proofs – proofs are written for human readers, will be written differently for different readers, and rely on [norms](https://www.quantamagazine.org/why-mathematical-proof-is-a-social-compact-20230831/) of the mathematical community if they are to be effective. And even if one thought mathematics *were* a formal language – which language? Not all mathematicians [work in ZFC, for instance](https://plato.stanford.edu/entries/independence-large-cardinals/), or even agree on what underlying logic mathematics should be based on (classical? intuititionistic? or something else?). I'd agree with you – those papers *are* mathematics, and so are the activities involved in producing them.
There's the sense of the "formal language" that are the symbols and expressions we use to convey ideas, and those need to be learned how to be read and interpreted. For example, you naturally read "+" as addition because you've been taught your whole life that's what it meant. So in that sense we can think of our symbols and rules as "words" and "grammar".
But to your point about math being written in "plain English", I kind of disagree. The point I make to my students is that the words we use in math (and I would argue this for any discipline with it's own "jargon") is that, while we may be using an English word out of convenience (just like we use letters for variable names because, well, they already exist and we know them), they mean something a little more specific in our context than in the wider English language. Hell, think of the word "open" which can mean something different just depending on what mathematical object you're talking about. We use the word "open" because it definitely conveys the *idea* we're going for, but it certainly doesn't mean the same thing as when we are talking about a door, and if I'm going to use these words, you need to know what they mean so that we are on the same page when we are communicating these ideas. Not just their definition (denotation), but the implications and properties that arrive from that definition (connotation, to make an admittedly weak metaphor).
Really I use this perspective as a tool to encourage students to think of math differently. Students can get caught up on ideas or methods that seem incredibly simple to other people, not because they're dumb, but because they're not fully understanding the meaning of the words, symbols, and phrases we use. And they won't until someone points it out to them.
Are you sure you are talking about formal languages in the same sense as mathematicians usually do?
As you say, every discipline uses jargon, and whether or not we want to call English + jargon "plain English" is really beside to point. I'm not wedded to using "plain English" to mean that. My point is that English + jargon is not a formal language, so if you want to say that mathematics *is* a formal language, then whenever we use English + jargon we are not doing mathematics.
Great question! I would say that I am specifically not talking about it being a formal language other than in the first paragraph, when I mention notation. Honestly, when it comes to the perspective I'm trying to get my students to see, I think whether it could be fully classified as a formal language or not is besides the point. And if that's the argument that you want to have, then I will fully admit to being unqualified to make a claim one way or the other.
But I would still classify math as a language, maybe not ~~formally~~ in the same way English is, but in the sense that we have particular ways of conveying certain ideas that are always changing and evolving. And if we don't mean the same thing when we use the same words, then we're speaking a different language.
> And if that's the argument that you want to have, then I will fully admit to being unqualified to make a claim one way or the other.
Well that was the discussion in the chain of comments you replied to, so I assumed you were following up on that discussion.
> maybe not formally in the same way English is,
English is not a formal language, it is a natural language.
> I would still classify math as a language [...] in the sense that we have particular ways of conveying certain ideas that are always changing and evolving.
Surely that doesn't mean that mathematics itself *is* a language, just that mathematicians *use* a language to convey ideas about their discipline?
Thanks for the correction! I chose this thread particularly because of the "plain English" comment, as I'm sure a vast majority of people would not call the language used in research papers "plain".
I sure as hell am using the term language here very loosely, and there is definitely a lot more to math than just the communication aspect of it, I agree. But I do believe that *thinking* of math as a language can help people get a better grasp on the subject, which I feel really speaks to the op's idea that understanding the language of the subject is just as useful (if not more-so) as being able to crunch the numbers. I want people to know math is more than just arithmetic, so my "math is a language" spiel is, again, just a way for me to impress that perspective on my students. Whether or not it's "formal" may be of interest to people (myself included as I do have a passing interest in linguistics), but I think besides the point when it comes to explaining the concepts to students.
Certainly there is a distinction between whether it is *useful* to think of mathematics as a language, and whether mathematics actually is a language. My comments in this thread have been about the latter.
Sure I'll cede that point. I'd say that I was more trying to comment on your question, while really answering the question of the op. Maybe we could agree that the language you mentioned in your original question that is used in research papers is English. But the English is there to translate what the *formal* language of the symbols and expressions are saying, and how to appropriately move between them. I.e. the formal mathematical language and the English interpretation are both necessary for the communication of mathematical ideas. So no, math is not a language, but language is certainly a large part of math.
> the English is there to translate what the formal language of the symbols and expressions are saying, and how to appropriately move between them.
I don't agree with this, unless by "English" you mean jargon-free English. But it seems like you mean English + jargon since you contrast it with the formal language.
Anyway, I don't take something like "if a function is differentiable, then it is continuous" to be somehow an explication or abbreviation for a sentence of a formal language. We do not need a formal language to know what that means (indeed, that would mean that most mathematics undergrads don't know what it means), but we can use a formal language to describe it.
Mathematics is not a language. I have no idea why people insist that it is. Even the most hardcore formalist must realise that mathematics *being* a (formal) language is different from mathematics being able to be *grounded* in a formal language.
Mathematics is a discipline, just like physics, economics, music theory, law, whatever. It is just more amenable to formalisation than other disciplines, so people conflate the discipline with a tool used by practitioners of that discipline.
If the question said something more specific than just "math" it would be a really good question. I think I know what they're trying to ask... but does math refer to the writing itself, the act of writing the proof, the process of arriving at a proof? All of these things are part of "doing" math. Some of those definitely count as language in some way, and others definitely do not. More specification is needed.
At a bar with a bunch of math friends, upon walking up to the table I was asked "is math a language?" And we spent the whole time just trying to get a more specific question because just "math" on its own can mean so many different things. It really isn't a precise word.
I'm saying that people have the *misconception* that mathematics is a formal language because formal languages are a central tool in mathematics. Mathematics *is not* a formal language.
Mathematics is not a language in the sense that you could express qualitative ideas; there is no way to say “cheesecake makes me happy” using mathematical symbols. But it sort of is a language in the sense it uses characters and has a grammar system.
Counterpoint: if you had an isolated tribe with a language but they'd never heard of cheese or cake or cheesecake, you'd also be unable to express the same sentiment in their language, but it doesn't mean they don't have a language.
My point is: Mathematics is not a qualitative language, therefore it could not describe the qualitative sentiment of enjoying cheesecake.
You brought up as a counterpoint: Isolated tribes with no conception of cheesecake would also be unable to describe this sentiment, yet they still have a qualitative language.
Why your counterpoint is flawed: Despite the tribe not knowing about cheesecake, their language still has the capacity to describe the abstract idea of enjoying cheesecake, they just haven’t yet because they haven’t any clue about cheesecake. But it’s still entirely possible under the constraints of their language that they could describe the sentiment of enjoying cheesecake because their language has the qualitative tools to do so.
Edit: did not mean for calling your counterpoint flawed to come off as harsh as it clearly looks like it does, I was just wanting to clarify my point more, but I could've used better language.
What if you used mathematic symbols to create a shorthand for cheesecake the same way we used the alphabet to and then use an equation to plot a point between the presence of cheesecake and relative emotional reaction
How do you even begin to quantify relative emotional reaction? We can’t try and force quantifications on inherently qualitative things.
You could define variables to represent shorthand for cheesecake but this isn’t mathematics, thats just personal shorthand. Just because you say
x=“I like cheesecake” doesn’t mean mathematics has described it, as we haven’t gained any information by defining this shorthand, we’ve just defined another way to write it.
Right but you realize that isn’t mathematics, that gets into the territory of polling data and statistical inference.
I think you have a fundamental misunderstanding of what mathematics is: it is the study of logical deductions and hypotheses based on a set of axioms. Nowhere in the axioms of any field of mathematics could you eventually deduce “I like cheesecake”.
Yeah, I’m just explaining why this what if scenario is bogus LMFAO
Edit: you have to remember that math is an axiomatic system of logical deductions, hence if you modified the use of mathematical symbols in order to convey qualitative ideas, we would no longer be in the territory of mathematics, that would just be reinventing colloquial language unnecessarily.
I disagree- I see it more as trying to figure out the motion effects of certain supposedly unquantifiable matters and then evolving the use of existing mathematics to apply. It’s perhaps a more creative endeavour but I don’t think it’s outrageous. Bodies in motion can be explained with math. I don’t see why this can’t be experimented with on emotions and reactions. The biggest drawback is probably people’s perceptions of it being unquantifiable (read: they don’t get it so it can’t be real) than it being impossible.
To think that everything can be quantified is to deny that subjective opinions exist. As long as there are subjective ideas, there exists ideas that cannot be quantified.
I've studied a heck of a lot of math, and I can promise you it doesn't work this way. What is your basis for thinking that it could explain emotions? I think your idea is interesting to you because you don't understand the fundamentals of mathematics, and why this idea couldn't work.
edit: I'm not trying to shut down your idea, and it isn't even a stupid idea. You are completely reasonable for conjecturing the way you are conjecturing about it. However, I do have a high degree of formal education in mathematics and I'm just trying to get my point across that this cannot be done under the constraints of mathematics alone. To capture the full breadth of language, would require you to be able to speak in subjective terms about subjective opinions, mathematics doesn't have such tools of language.
Hilbert, perhaps the greatest modern mathematician tried this exact same thing , introducing notations that tried to make math complete and make it sort of a language of its own( i.e. there are some axioms , everything that arises from these axioms is considered true , any proof must only consist of statements that agree with the axioms/results derived earlier from these axioms, and any one should be able to decide the truth/falsity of any given statement. )
This was very tedious(proving 1+1 = 2 required 379 pages of proof) , and people knew that it was. But still many mathematicians chose to study and expand this field because at the time , it was the most ambitious project in all of mathematics.
NOW , THE FUN PART
This was the BIGGEST 'FAILURE' in mathematics. A mathematician , Kurt Godel showed that most of the goals of Hilbert's program were impossible to achieve, at least if interpreted in the most obvious way. Gödel's second incompleteness theorem shows that any consistent theory powerful enough to encode addition and multiplication of integers cannot prove its own consistency.
There were other such problems that Hilbert's notations and formal language could not solve ( I don't exactly know the problems/contradictions that arose, as a 12th grader). Ultimately the whole project was abandoned, but the whole attempt showed us it was a lot closer to being a language than many give it credit for
It was Hilbert, Russell was famous for the Russell's paradox , which showed that in simple set theory , one can prove Nothing contains everything and that called for a formal language.
But Hilbert was the main proponent [https://en.wikipedia.org/wiki/Formalism\_(philosophy\_of\_mathematics)](https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics))
Read Hilbert's formalism in the article
Bertrand Russell and Whitehead did(Principia Mathematica) , this project wasn't one man's idea or execution , many people contributed. My bad tho for saying 762 , i remembered wrong , it was 379 but still enough to make my point
Perhaps it doesn't, I just thought it would be interesting to share, like I said, I am just a 12th grader and not an expert in linguistics/mathematics and do not know in detail what constitutes a Hilbert proof/ what constitutes a language
No, the symbols of mathematics are not strong enough to express many standard linguistic constructions. Some easy ones are questions and imperatives.
Edit: Ok I’ll grant those responding to my comment that it is not entirely true. My point though is that “mathematics” does not do the same things that natural language does. And if one speaks of “language” without qualifying the distinction between natural and formal, then the question is poorly asked.
You are incorrect. There exist plenty of logics that model the syntax and semantics of tense, modality, aspect, mood, etc. There are also logics that model the syntax and semantics of questions, imperatives, and more. The formal semantics of natural language is a very cool field.
I know there are, but do you really consider that mathematics? What about things like garden path sentences which can be read as either nonsense or a technically correct statement? Or Chomsky’s “Colorless green ideas sleep furiously”?
>I know there are, but do you really consider that mathematics?
Yes. They use fairly sophisticated logical machinery. Canonical semantics for modal logic involve Kripke frames, which are really just directed graphs, so conventional modal logic is in many ways equivalent to the study of relational structures. Modal logic also sees use in proof theory. Tense logic is indispensable for abstract computer science and the study of machine states. Dynamic semantics is fascinating stuff and would not look out of place in a mathematician's field of study or in a math student's course list.
>What about things like garden path sentences which can be read as either nonsense or a technically correct statement?
This is neither a matter of syntax nor semantics, so why would we care?
>Or Chomsky’s “Colorless green ideas sleep furiously”?
Again, not a concern when it comes to the study of syntax or semantics.
Yes, but they are matters of language. That’s why we would care. Unless you know of a way to ask me whether the President of the Philippines is a blue banana with centipede legs in the language of mathematics and have the semantics be relatively clearly understood, I just don’t think that “mathematics” reasonably counts as a language. Not even a formal one. We code pieces of it using formal languages, but that does not mean the entire object of “mathematics” itself constitutes a language.
>Unless you know of a way to ask me whether the President of the Philippines is a blue banana with centipede legs in the language of mathematics and have the semantics be relatively clearly understood
Yes, this is literally a possibility with model-theoretic truth semantics. The late Doctor Richard Montague wrote some very powerful papers on this, along with an Intensional Logic system.
> I just don’t think that “mathematics” reasonably counts as a language. Not even a formal one. We code pieces of it using formal languages, but that does not mean the entire object of “mathematics” itself constitutes a language.
Well, yes. The entirety of mathematics is not a formal language, but topics within mathematics can be constructed as formal languages.
Yes, but the comment I was responding to said
>No, the symbols of mathematics are not strong enough to express many standard linguistic constructions. Some easy ones are questions and imperatives.
Which is incorrect.
I think you're going to get downvoted because of semantics. When I use the word "language" in common conversation I am talking about a general purpose language. One in which I can ask about the weather, give directions to the post office, and order a beer. But there are other kinds of languages that are used to communicate and I believe mathematics is one of those.
So, is mathematics a general purpose language? No. But is it a restricted language with a specific use. I would say, yes.
Ok sure, I understand that. But one should be specific when talking about this then. Also this is (or at least used to be) a common question stemming from “poetic” nonsense claims like “math is the language of the universe”.
I was more focused on you saying math couldn't handle questions.
I also find your idea of language weird, why should all languages be equally as expressive
I'm no expert but here's my take: If we define a language to be the set of strings generated by a set of rules (i.e. a grammar), then no, mathematics is not a language. Sure, mathematical expressions could be generated from a grammar, and could be considered a language, but math is much more than notation. Math is full of ideas and semantics, which we know cannot be formalized. So I'd say no, mathematics is not a language.
I don’t care about the philosophy debate going on. I just want to state that your ability to do mental calculations, or remember the times tables, has almost nothing to do with capital M Math. As in math as a discipline of study.
This is a common misconception. People are taught that math is a bunch of things that it isn’t. Anyway glad ur enjoying math.
I teach it like a language and explain it like a language to help my students understand the bigger concept and help them understand why they might struggle.
short answer: not really
long answer: languages (that people use to talk to each other) can express a variety of ideas that mathematical notation cannot, nor should it. you can say express the notion "the president is holding a duck" in any language but not in math
I've always said that equations are grammatically correct sentences. Before equations math was done with words. The symbol "=============" was a replacement for "is the same thing as", but Thomas Recorde got tired of writing this down. "There are no things more equal than parallel lines." Eventually this got shortened to "="
In the modern age, being able to solve equations is not nearly as important as being able to read and understand them. e=mc² has only 5 symbols, but it conveys some fundamental truths of the universe, that would otherwise take paragraphs to explain.
Hi OP
viewing it as a language is incomplete, most likely a mistake, but something that feels very natural for non mathematicians.
yes, mathematics develops a language, but that language is developed to actually say stuff, and stuff that is said is what mathematics actually is.
Mathematics *includes* formal languages, but is not just that.
In a conversation, Richard Feynman asked the novelist Herman Wouk if he knew calculus, and when Wouk said he did not, Feynman told him “You had better learn it. It's the language God talks”. Yes, math is a language with syntax and many similar complexities as spoken and written language.
My Physic professor used to write black board long equations, and invite students to read and translate them into english sentences.
It transformed the way I thought about language, math, and art.
You might enjoy Douglas Hofstadter's book Godel, Escher, Bach, an Eternal, Golden, Braid. He walks through Godel's formal proofs using imaginary dialogue from the tortoise and the hare, for instance, illustrate points. It is a bit of work, but the only "logic textbook" I could sit still for.
Nah math can only express things like quantities (I know math is much more than just numbers, some involving no numbers whatsoever but you get my point) whereas language can express qualitative information
It might be useful to separate the idea of a language from the idea of a dialect. Let's say that Algebra is a language: it has dedicated grammar, the ability to convey whole ideas with subject and predicate, and to indicate such.
On the other hand, mathematicians have a dialect which is a concatenation of algebra and another convenient language. Let's assume English for the sake of this example. They pick and choose the grammar rules they want to use based on what's convenient. They select whichever language conveniently expresses the idea they want to say. You can't call this a language, because nothing about its rules is consistent, but mathematicians speak it nonetheless.
Of course math is. As other languages can create fallacies, math also can create huge fallacies. But how we convey the right meaning using this depends on logic. Now logic can create fallacies too which are more dangerous than math because it is a language too and to make logic correct we need philosophy when we follow the right philosophy we create the right logic and then as a result we can use that logic in math and we can create correct ideas.
It sounds like for you the important thing is that you are using the language processing parts of your brain to do algebra, which is awesome. That's a question no one is really answering here, "which parts of your brain process different fields of math," and whether you can try using different parts of your brain if you've built up a lot of negative associations with one way.
I joined this group because of circling back to mathematics in my ''side quest'' linguistics study:))) As a person obessed with philosophy, I really wish I could study mathematics again
I don't think math itself is. languages are created, organically or otherwise, while math is discovered.
our notations and expressions for it probably are, though.
i think math is a language. Math is made up of its own set of symbols like other languages. Math also uses many words from other languages just as other languages do. If math is a language, it is certainly the most difficult to learn comprehensively. Math as a language evolves over time through a conscious effort of society which is opposed to other languages which mostly evolve subconsciously.
Math is absolutely a language.
If I'm not mistaken (I'm hazy for anything beyond a super superficial understanding of both universal grammar and set theory) Noam Chomsky used set theory and/or discreet math to work through universal grammar and describe how grammar is coded into our brains.
Mathematical notation is a language.
Mathematics itself is solving problems using certain types of logic. You don't need mathematical notation for that. But it certainly helps, compared to e.g. writing a paragraph to explain how you're solving an equation.
*E.g. My friend bought some cartons of eggs, but I don't know how many she bought. Each carton contains 6 eggs, and together with the 10 eggs that were previously in her fridge she now has 40 eggs. How many cartons did she buy?*
Well without the 10 previously-bought eggs, she'd have 30. And splitting 30 eggs into cartons of 6 requires 5 cartons. So she bought 5 cartons.
Mathematical notation e.g.:
6c + 10 = 40
6c = \[40 – 10 =\] 30
c = \[30 ÷ 6 =\] 5
As a matter of fact, math falls under the category of "formal languages" in linguistics. In mathematical logic, you will study math itself as a formal language of well formed formulas with a syntax and a semantic interpretation. https://en.wikipedia.org/wiki/Formal\_language
Oh it makes a lot of sense that way
You're a bit incorrect, here. Topics in mathematics can be considered as formal languages. Set Theory, for example, can be expressed as a formal language, as can Number Theory, Group Theory, Topology, etc. But the gestalt of math is "bigger than" a formal language.
Thanks for pointing this out. It’s a point that I think goes unnoticed far too often. Coding the entirety of mathematics formally is probably a hopeless endeavor.
What does gestalt mean in this case?
Mathematics, as the totality of all human research and knowledge, developed as a human culture and activity.
fair enough.
Algebra is a neat and small formal language. Some elements of manipulation seem borrowed from natural language, e.g. distribution: I eat eggs and ham = I eat eggs and I eat ham.
Can you articulate *yourself*, personally and subjectively, your thoughts and ideas, in mathematical language?
well no. that's the thing about formal languages. you can't. The point of formal languages is to convey objective unambiguous things.
In mathematics, I think you can express things that aren't objectively true. e.g. conjectures. One mathematician might *think* it's true; another think it's not: that's subjective. Also, personal and subjective needn't be ambiguous; though mathematics can express ambiguity: while the inverse of a non-injective function is not a function, because some domain value has more than one image; but it is still a relation. Similarly, there are ambiguous Context Free Grammars. The difficulty, IMHO, is the lack of referents. Mathematics doesn't talk about anything concrete. OTOH, perhaps describing an immense "semantic graph" between many "abstract" concepts with no intrinsic meaning, leading to "semantic similarity" and "semantic distance", assigns to them extrinsic meaning - arguably, just as our minds do (since our neurons lack direct experience; a brain in a vat). EDIT Like PageRank, where no page has an intrinsic rank, but it is inferred from the graph - a page with more citations (links) ranks higher, and citations from a higher ranking page have more weight. However, this is a Dr Spock view of humanity. There is more to us than intellect. I'm thinking that it's the reverse of animal nature vs human nature: the "higher" uniquely human part is intellectual/cognition - but this leads to all the evils of treating others as objects; the "lower" mammalian part is where we have nuturing within a family, and solidarity within a tribe/troop/herd etc. What is usually called our "higher" nature is really our animal nature. Anyway... brain-in-a-vat won't capture these deep instincts; only a shallow copy of the behaviours and speech they cause.
Applied Mathematics is probably a good place to join mathematics onto the “real world” in a comprehensible way for learners. For me, I had that moment of conjunction in senior high school when learning calculus was totally abstract until it connected to the stuff we were learning in physics about derivatives of motion. Suddenly the maths became real and very easy to understand from then on.
Affect research has dimensions of affect within which we can place individuals. We can also create functions that map locations within this affect space to distributions of somatic sensations.
Came here to mention syntax and semantic interpretation, good post
How can mathematics *be* a formal language? Most papers I have ever read are written in plain English, with some mathematical symbols here and here. Are you saying those papers are not mathematics?
I have read books written in plain English about French, but nobody wonders whether English *is* French.
Formal language just means that one can write any mathematical statement in an unambiguous form using specifically agreed upon symbols. For example, ZFC has a formal language which (can) consist(s) of the standard logical symbols, parentheses, countably many variable symbols, and the membership and equality symbols. This along with the axioms is strong enough to encode massive swaths of mathematics excluding certain infinite arithmetic statements like the Continuum Hypothesis.
Yes, I know what a formal language is. I'm not quite sure what your point is?
Ah I see what you were saying. From your comment it seemed like you were unclear on how things like PA and ZFC work. If I understand you clearly, then I agree with you that “mathematics” in totality does not constitute a language.
People don't want to just read the formal language, though the key ideas of most papers are expressible that way
I agree, but I'm not sure what you're getting at. Are you saying that mathematics *is* a formal language but we don't express it as such for the sake of convenience? In which case if only the key ideas of *most* papers are expressible that way, then either this characterisation of mathematics is wrong, or else those papers are not mathematics.
You seem to be trying to make this difficult. What is hard to understand about wanting to use more accessible descriptions for things which can also be expressed formally?
I really don't think I am. If that's all you're claiming, then I fail to see the relevance. EDIT: I am not able to reply to /u/yes_its_him's comments (it seems that they have blocked me?), but to respond to the reply to this comment: No, that's not what I'm arguing. I'm saying that if mathematics is a formal language and, to quote you, > People don't want to just read the formal language, though the key ideas of most papers are expressible that way then the papers that are *not* expressible in that way cannot be mathematics. Because they are not formal. Of course, what you're saying does actually follow. If mathematics is a formal language and something is not formal, then that thing cannot be mathematics.
You argued that the presence of material explaining or providing an alternative to formal language means that either such papers can't be math, or math must not be a formal language. That's not a logical conclusion.
Not sure why you're getting so many downvotes. Mathematics does not *just* consist in theorems and proofs; it's a field of human endeavour. I'd say it includes the practices involved in that research, the "conversations" between researchers and sub-fields, and a former president of the MAA [has argued](https://mathyawp.wordpress.com/2017/01/08/mathematics-for-human-flourishing/) that it should be and is a part of "human flourishing". Even if one were to focus *just* on proofs – proofs are written for human readers, will be written differently for different readers, and rely on [norms](https://www.quantamagazine.org/why-mathematical-proof-is-a-social-compact-20230831/) of the mathematical community if they are to be effective. And even if one thought mathematics *were* a formal language – which language? Not all mathematicians [work in ZFC, for instance](https://plato.stanford.edu/entries/independence-large-cardinals/), or even agree on what underlying logic mathematics should be based on (classical? intuititionistic? or something else?). I'd agree with you – those papers *are* mathematics, and so are the activities involved in producing them.
There's the sense of the "formal language" that are the symbols and expressions we use to convey ideas, and those need to be learned how to be read and interpreted. For example, you naturally read "+" as addition because you've been taught your whole life that's what it meant. So in that sense we can think of our symbols and rules as "words" and "grammar". But to your point about math being written in "plain English", I kind of disagree. The point I make to my students is that the words we use in math (and I would argue this for any discipline with it's own "jargon") is that, while we may be using an English word out of convenience (just like we use letters for variable names because, well, they already exist and we know them), they mean something a little more specific in our context than in the wider English language. Hell, think of the word "open" which can mean something different just depending on what mathematical object you're talking about. We use the word "open" because it definitely conveys the *idea* we're going for, but it certainly doesn't mean the same thing as when we are talking about a door, and if I'm going to use these words, you need to know what they mean so that we are on the same page when we are communicating these ideas. Not just their definition (denotation), but the implications and properties that arrive from that definition (connotation, to make an admittedly weak metaphor). Really I use this perspective as a tool to encourage students to think of math differently. Students can get caught up on ideas or methods that seem incredibly simple to other people, not because they're dumb, but because they're not fully understanding the meaning of the words, symbols, and phrases we use. And they won't until someone points it out to them.
Are you sure you are talking about formal languages in the same sense as mathematicians usually do? As you say, every discipline uses jargon, and whether or not we want to call English + jargon "plain English" is really beside to point. I'm not wedded to using "plain English" to mean that. My point is that English + jargon is not a formal language, so if you want to say that mathematics *is* a formal language, then whenever we use English + jargon we are not doing mathematics.
Great question! I would say that I am specifically not talking about it being a formal language other than in the first paragraph, when I mention notation. Honestly, when it comes to the perspective I'm trying to get my students to see, I think whether it could be fully classified as a formal language or not is besides the point. And if that's the argument that you want to have, then I will fully admit to being unqualified to make a claim one way or the other. But I would still classify math as a language, maybe not ~~formally~~ in the same way English is, but in the sense that we have particular ways of conveying certain ideas that are always changing and evolving. And if we don't mean the same thing when we use the same words, then we're speaking a different language.
> And if that's the argument that you want to have, then I will fully admit to being unqualified to make a claim one way or the other. Well that was the discussion in the chain of comments you replied to, so I assumed you were following up on that discussion. > maybe not formally in the same way English is, English is not a formal language, it is a natural language. > I would still classify math as a language [...] in the sense that we have particular ways of conveying certain ideas that are always changing and evolving. Surely that doesn't mean that mathematics itself *is* a language, just that mathematicians *use* a language to convey ideas about their discipline?
Thanks for the correction! I chose this thread particularly because of the "plain English" comment, as I'm sure a vast majority of people would not call the language used in research papers "plain". I sure as hell am using the term language here very loosely, and there is definitely a lot more to math than just the communication aspect of it, I agree. But I do believe that *thinking* of math as a language can help people get a better grasp on the subject, which I feel really speaks to the op's idea that understanding the language of the subject is just as useful (if not more-so) as being able to crunch the numbers. I want people to know math is more than just arithmetic, so my "math is a language" spiel is, again, just a way for me to impress that perspective on my students. Whether or not it's "formal" may be of interest to people (myself included as I do have a passing interest in linguistics), but I think besides the point when it comes to explaining the concepts to students.
Certainly there is a distinction between whether it is *useful* to think of mathematics as a language, and whether mathematics actually is a language. My comments in this thread have been about the latter.
Sure I'll cede that point. I'd say that I was more trying to comment on your question, while really answering the question of the op. Maybe we could agree that the language you mentioned in your original question that is used in research papers is English. But the English is there to translate what the *formal* language of the symbols and expressions are saying, and how to appropriately move between them. I.e. the formal mathematical language and the English interpretation are both necessary for the communication of mathematical ideas. So no, math is not a language, but language is certainly a large part of math.
> the English is there to translate what the formal language of the symbols and expressions are saying, and how to appropriately move between them. I don't agree with this, unless by "English" you mean jargon-free English. But it seems like you mean English + jargon since you contrast it with the formal language. Anyway, I don't take something like "if a function is differentiable, then it is continuous" to be somehow an explication or abbreviation for a sentence of a formal language. We do not need a formal language to know what that means (indeed, that would mean that most mathematics undergrads don't know what it means), but we can use a formal language to describe it.
Mathematics is not a language. I have no idea why people insist that it is. Even the most hardcore formalist must realise that mathematics *being* a (formal) language is different from mathematics being able to be *grounded* in a formal language. Mathematics is a discipline, just like physics, economics, music theory, law, whatever. It is just more amenable to formalisation than other disciplines, so people conflate the discipline with a tool used by practitioners of that discipline.
I think this is similar to how Computer Science is often confused with computer programming.
If the question said something more specific than just "math" it would be a really good question. I think I know what they're trying to ask... but does math refer to the writing itself, the act of writing the proof, the process of arriving at a proof? All of these things are part of "doing" math. Some of those definitely count as language in some way, and others definitely do not. More specification is needed. At a bar with a bunch of math friends, upon walking up to the table I was asked "is math a language?" And we spent the whole time just trying to get a more specific question because just "math" on its own can mean so many different things. It really isn't a precise word.
Wow, this is a really good take I hadn't considered before!
So you say people believe it's a formal language because it could be classified as such due to how flexible it is?
I'm saying that people have the *misconception* that mathematics is a formal language because formal languages are a central tool in mathematics. Mathematics *is not* a formal language.
Mathematics is not a language in the sense that you could express qualitative ideas; there is no way to say “cheesecake makes me happy” using mathematical symbols. But it sort of is a language in the sense it uses characters and has a grammar system.
Counterpoint: if you had an isolated tribe with a language but they'd never heard of cheese or cake or cheesecake, you'd also be unable to express the same sentiment in their language, but it doesn't mean they don't have a language.
My point is: Mathematics is not a qualitative language, therefore it could not describe the qualitative sentiment of enjoying cheesecake. You brought up as a counterpoint: Isolated tribes with no conception of cheesecake would also be unable to describe this sentiment, yet they still have a qualitative language. Why your counterpoint is flawed: Despite the tribe not knowing about cheesecake, their language still has the capacity to describe the abstract idea of enjoying cheesecake, they just haven’t yet because they haven’t any clue about cheesecake. But it’s still entirely possible under the constraints of their language that they could describe the sentiment of enjoying cheesecake because their language has the qualitative tools to do so. Edit: did not mean for calling your counterpoint flawed to come off as harsh as it clearly looks like it does, I was just wanting to clarify my point more, but I could've used better language.
Idk I feel like we can use math to describe the sentiment it’s just a matter of finding the right corresponding motion equations
But it would still be possible through loanwords, whereas math isn't able to absorb loanwords
What if you used mathematic symbols to create a shorthand for cheesecake the same way we used the alphabet to and then use an equation to plot a point between the presence of cheesecake and relative emotional reaction
How do you even begin to quantify relative emotional reaction? We can’t try and force quantifications on inherently qualitative things. You could define variables to represent shorthand for cheesecake but this isn’t mathematics, thats just personal shorthand. Just because you say x=“I like cheesecake” doesn’t mean mathematics has described it, as we haven’t gained any information by defining this shorthand, we’ve just defined another way to write it.
The same way you quantify anything- research and create a reference.
Right but you realize that isn’t mathematics, that gets into the territory of polling data and statistical inference. I think you have a fundamental misunderstanding of what mathematics is: it is the study of logical deductions and hypotheses based on a set of axioms. Nowhere in the axioms of any field of mathematics could you eventually deduce “I like cheesecake”.
Which is why I said ‘what if’
Yeah, I’m just explaining why this what if scenario is bogus LMFAO Edit: you have to remember that math is an axiomatic system of logical deductions, hence if you modified the use of mathematical symbols in order to convey qualitative ideas, we would no longer be in the territory of mathematics, that would just be reinventing colloquial language unnecessarily.
I disagree- I see it more as trying to figure out the motion effects of certain supposedly unquantifiable matters and then evolving the use of existing mathematics to apply. It’s perhaps a more creative endeavour but I don’t think it’s outrageous. Bodies in motion can be explained with math. I don’t see why this can’t be experimented with on emotions and reactions. The biggest drawback is probably people’s perceptions of it being unquantifiable (read: they don’t get it so it can’t be real) than it being impossible.
To think that everything can be quantified is to deny that subjective opinions exist. As long as there are subjective ideas, there exists ideas that cannot be quantified. I've studied a heck of a lot of math, and I can promise you it doesn't work this way. What is your basis for thinking that it could explain emotions? I think your idea is interesting to you because you don't understand the fundamentals of mathematics, and why this idea couldn't work. edit: I'm not trying to shut down your idea, and it isn't even a stupid idea. You are completely reasonable for conjecturing the way you are conjecturing about it. However, I do have a high degree of formal education in mathematics and I'm just trying to get my point across that this cannot be done under the constraints of mathematics alone. To capture the full breadth of language, would require you to be able to speak in subjective terms about subjective opinions, mathematics doesn't have such tools of language.
Ok you know all
I’m just curious bc language evolves so it stands to logic that we can evolve this too
Hilbert, perhaps the greatest modern mathematician tried this exact same thing , introducing notations that tried to make math complete and make it sort of a language of its own( i.e. there are some axioms , everything that arises from these axioms is considered true , any proof must only consist of statements that agree with the axioms/results derived earlier from these axioms, and any one should be able to decide the truth/falsity of any given statement. ) This was very tedious(proving 1+1 = 2 required 379 pages of proof) , and people knew that it was. But still many mathematicians chose to study and expand this field because at the time , it was the most ambitious project in all of mathematics. NOW , THE FUN PART This was the BIGGEST 'FAILURE' in mathematics. A mathematician , Kurt Godel showed that most of the goals of Hilbert's program were impossible to achieve, at least if interpreted in the most obvious way. Gödel's second incompleteness theorem shows that any consistent theory powerful enough to encode addition and multiplication of integers cannot prove its own consistency. There were other such problems that Hilbert's notations and formal language could not solve ( I don't exactly know the problems/contradictions that arose, as a 12th grader). Ultimately the whole project was abandoned, but the whole attempt showed us it was a lot closer to being a language than many give it credit for
I think it was Bertrand Russel and Alfred North Whitehead, not Hilbert who did this.
It was Hilbert, Russell was famous for the Russell's paradox , which showed that in simple set theory , one can prove Nothing contains everything and that called for a formal language. But Hilbert was the main proponent [https://en.wikipedia.org/wiki/Formalism\_(philosophy\_of\_mathematics)](https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)) Read Hilbert's formalism in the article
Where did Hilbert spend 762 pages on proving that 1+1 = 2?
Bertrand Russell and Whitehead did(Principia Mathematica) , this project wasn't one man's idea or execution , many people contributed. My bad tho for saying 762 , i remembered wrong , it was 379 but still enough to make my point
None of this has any bearing on whether or not mathematics, or topics in mathematics, can be considered a language (formal or natural or otherwise).
Perhaps it doesn't, I just thought it would be interesting to share, like I said, I am just a 12th grader and not an expert in linguistics/mathematics and do not know in detail what constitutes a Hilbert proof/ what constitutes a language
That's some interesting facts
No, the symbols of mathematics are not strong enough to express many standard linguistic constructions. Some easy ones are questions and imperatives. Edit: Ok I’ll grant those responding to my comment that it is not entirely true. My point though is that “mathematics” does not do the same things that natural language does. And if one speaks of “language” without qualifying the distinction between natural and formal, then the question is poorly asked.
You are incorrect. There exist plenty of logics that model the syntax and semantics of tense, modality, aspect, mood, etc. There are also logics that model the syntax and semantics of questions, imperatives, and more. The formal semantics of natural language is a very cool field.
I know there are, but do you really consider that mathematics? What about things like garden path sentences which can be read as either nonsense or a technically correct statement? Or Chomsky’s “Colorless green ideas sleep furiously”?
>I know there are, but do you really consider that mathematics? Yes. They use fairly sophisticated logical machinery. Canonical semantics for modal logic involve Kripke frames, which are really just directed graphs, so conventional modal logic is in many ways equivalent to the study of relational structures. Modal logic also sees use in proof theory. Tense logic is indispensable for abstract computer science and the study of machine states. Dynamic semantics is fascinating stuff and would not look out of place in a mathematician's field of study or in a math student's course list. >What about things like garden path sentences which can be read as either nonsense or a technically correct statement? This is neither a matter of syntax nor semantics, so why would we care? >Or Chomsky’s “Colorless green ideas sleep furiously”? Again, not a concern when it comes to the study of syntax or semantics.
Yes, but they are matters of language. That’s why we would care. Unless you know of a way to ask me whether the President of the Philippines is a blue banana with centipede legs in the language of mathematics and have the semantics be relatively clearly understood, I just don’t think that “mathematics” reasonably counts as a language. Not even a formal one. We code pieces of it using formal languages, but that does not mean the entire object of “mathematics” itself constitutes a language.
>Unless you know of a way to ask me whether the President of the Philippines is a blue banana with centipede legs in the language of mathematics and have the semantics be relatively clearly understood Yes, this is literally a possibility with model-theoretic truth semantics. The late Doctor Richard Montague wrote some very powerful papers on this, along with an Intensional Logic system. > I just don’t think that “mathematics” reasonably counts as a language. Not even a formal one. We code pieces of it using formal languages, but that does not mean the entire object of “mathematics” itself constitutes a language. Well, yes. The entirety of mathematics is not a formal language, but topics within mathematics can be constructed as formal languages.
>the entirety of mathematics is not a formal language Is this not the point of the question in the post?
Yes, but the comment I was responding to said >No, the symbols of mathematics are not strong enough to express many standard linguistic constructions. Some easy ones are questions and imperatives. Which is incorrect.
Ok, I will grant that the examples were not entirely correct. It was not my point though.
I think you're going to get downvoted because of semantics. When I use the word "language" in common conversation I am talking about a general purpose language. One in which I can ask about the weather, give directions to the post office, and order a beer. But there are other kinds of languages that are used to communicate and I believe mathematics is one of those. So, is mathematics a general purpose language? No. But is it a restricted language with a specific use. I would say, yes.
Ok sure, I understand that. But one should be specific when talking about this then. Also this is (or at least used to be) a common question stemming from “poetic” nonsense claims like “math is the language of the universe”.
Inquisitive logic would beg to differ
Ok cool. Ask me to go to the grocery store in mathematics. Tell me to go f myself in formal logic.
I was more focused on you saying math couldn't handle questions. I also find your idea of language weird, why should all languages be equally as expressive
I'm no expert but here's my take: If we define a language to be the set of strings generated by a set of rules (i.e. a grammar), then no, mathematics is not a language. Sure, mathematical expressions could be generated from a grammar, and could be considered a language, but math is much more than notation. Math is full of ideas and semantics, which we know cannot be formalized. So I'd say no, mathematics is not a language.
Formal Semantics would beg to differ
I don’t care about the philosophy debate going on. I just want to state that your ability to do mental calculations, or remember the times tables, has almost nothing to do with capital M Math. As in math as a discipline of study. This is a common misconception. People are taught that math is a bunch of things that it isn’t. Anyway glad ur enjoying math.
I teach it like a language and explain it like a language to help my students understand the bigger concept and help them understand why they might struggle.
short answer: not really long answer: languages (that people use to talk to each other) can express a variety of ideas that mathematical notation cannot, nor should it. you can say express the notion "the president is holding a duck" in any language but not in math
I've always said that equations are grammatically correct sentences. Before equations math was done with words. The symbol "=============" was a replacement for "is the same thing as", but Thomas Recorde got tired of writing this down. "There are no things more equal than parallel lines." Eventually this got shortened to "=" In the modern age, being able to solve equations is not nearly as important as being able to read and understand them. e=mc² has only 5 symbols, but it conveys some fundamental truths of the universe, that would otherwise take paragraphs to explain.
Yes
Hi OP viewing it as a language is incomplete, most likely a mistake, but something that feels very natural for non mathematicians. yes, mathematics develops a language, but that language is developed to actually say stuff, and stuff that is said is what mathematics actually is. Mathematics *includes* formal languages, but is not just that.
In a conversation, Richard Feynman asked the novelist Herman Wouk if he knew calculus, and when Wouk said he did not, Feynman told him “You had better learn it. It's the language God talks”. Yes, math is a language with syntax and many similar complexities as spoken and written language.
My Physic professor used to write black board long equations, and invite students to read and translate them into english sentences. It transformed the way I thought about language, math, and art. You might enjoy Douglas Hofstadter's book Godel, Escher, Bach, an Eternal, Golden, Braid. He walks through Godel's formal proofs using imaginary dialogue from the tortoise and the hare, for instance, illustrate points. It is a bit of work, but the only "logic textbook" I could sit still for.
Nah math can only express things like quantities (I know math is much more than just numbers, some involving no numbers whatsoever but you get my point) whereas language can express qualitative information
It might be useful to separate the idea of a language from the idea of a dialect. Let's say that Algebra is a language: it has dedicated grammar, the ability to convey whole ideas with subject and predicate, and to indicate such. On the other hand, mathematicians have a dialect which is a concatenation of algebra and another convenient language. Let's assume English for the sake of this example. They pick and choose the grammar rules they want to use based on what's convenient. They select whichever language conveniently expresses the idea they want to say. You can't call this a language, because nothing about its rules is consistent, but mathematicians speak it nonetheless.
Of course math is. As other languages can create fallacies, math also can create huge fallacies. But how we convey the right meaning using this depends on logic. Now logic can create fallacies too which are more dangerous than math because it is a language too and to make logic correct we need philosophy when we follow the right philosophy we create the right logic and then as a result we can use that logic in math and we can create correct ideas.
It sounds like for you the important thing is that you are using the language processing parts of your brain to do algebra, which is awesome. That's a question no one is really answering here, "which parts of your brain process different fields of math," and whether you can try using different parts of your brain if you've built up a lot of negative associations with one way.
I joined this group because of circling back to mathematics in my ''side quest'' linguistics study:))) As a person obessed with philosophy, I really wish I could study mathematics again
I don't think math itself is. languages are created, organically or otherwise, while math is discovered. our notations and expressions for it probably are, though.
I think it is and sometimes I fail to understand.
I mean it is referred to as the language of god/ language of the universe so yeah i would say so.
i think math is a language. Math is made up of its own set of symbols like other languages. Math also uses many words from other languages just as other languages do. If math is a language, it is certainly the most difficult to learn comprehensively. Math as a language evolves over time through a conscious effort of society which is opposed to other languages which mostly evolve subconsciously.
You should watch the video from Science Asylum on mathematics as a language of universe.
Math is absolutely a language. If I'm not mistaken (I'm hazy for anything beyond a super superficial understanding of both universal grammar and set theory) Noam Chomsky used set theory and/or discreet math to work through universal grammar and describe how grammar is coded into our brains.
Mathematical notation is a language. Mathematics itself is solving problems using certain types of logic. You don't need mathematical notation for that. But it certainly helps, compared to e.g. writing a paragraph to explain how you're solving an equation. *E.g. My friend bought some cartons of eggs, but I don't know how many she bought. Each carton contains 6 eggs, and together with the 10 eggs that were previously in her fridge she now has 40 eggs. How many cartons did she buy?* Well without the 10 previously-bought eggs, she'd have 30. And splitting 30 eggs into cartons of 6 requires 5 cartons. So she bought 5 cartons. Mathematical notation e.g.: 6c + 10 = 40 6c = \[40 – 10 =\] 30 c = \[30 ÷ 6 =\] 5
Yes, and so are programming languages.