View Full Version : Is Math a Language?

08-29-2007, 10:28 AM
I was having an interesting conversation/debate/discussion with a friend today, and I wanted to hear some of your opinions. He started off by saying that language is the only way for people to really understand the world. I said that I think language is often used for expression, and that understanding can happen in the brain, and furthermore that sometimes understanding takes place without language. One example I used was mathematics. I mentioned that in my humble opinion math can be as a language, and if not a language, then at least another way of understanding the world. He disagreed that math can be seen as a language, and furthermore stated that language is fundamental and necessary in understanding math. He made the argument that without language, math would not exist. He even noted that language is older then math and we can't really understand math without language.

Some of the main questions that came up during our debate where: Is math a language? Can humans communicate math to one another without language (for example, can 2 mathmaticians express and explain themselves without using a word and only using math symbols, proofs, etc)? Can we think abstractly without language? Can infants and/or nonhumans understand math at all? Can human understanding take place in the absence of language? Is language the most fundamental thing when it comes to understanding the world?

Feel free to weight in on any these questions, or to bring up different questions and comments related to this topic. Furthermore, I would be interested in any resource which discuss this topic.

P.S. Moderator, I hope you will not delete this post. I realize it is off topic, but it is an interesting topic, and only those who find it interesting will reply.

08-29-2007, 12:56 PM

08-29-2007, 12:57 PM
(for example, can 2 mathmaticians express and explain themselves without using a word and only using math symbols, proofs, etc)?

Yes. Proof by picture.

08-29-2007, 01:01 PM
That's awesome. :-)

08-29-2007, 03:59 PM
The answer to the question hinges on another question, "What is a language?"

From www.jcu.edu.au/studying/services/studyskills/essay/language_literacy.rtf

"Language, roughly, can be defined as communicating with others. Language is more than speech and writing, it is the making and sharing of meaning with ourselves and others (Emmitt and Pollock, 1997, p.19). For that meaning to be shared the language signs and symbols are selected and used according to rules. These rules have been developed and agreed upon by the language users and must be learned by new language users (Emmitt and Pollock, 1997, p.11)."

[Emmitt, M., & Pollock, J. (1997). 2nd ed. Language and learning. South Melbourne: Oxford University Press.]

From that definition, mathematics is clearly a language. However, is it interesting to observe that determining whether mathematics is a language seems to require the existence of written language. Noam Chomsky's book Syntactic Structures can, quite convincingly, put the question to rest.

08-29-2007, 04:31 PM
You might enjoy the book Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being by Lakoff & Nunez.

08-29-2007, 08:24 PM
As someone who has been studying mathematics for the last four years, I would say that mathematics is NOT a language; at least, it is not a language like English, French, etc. For one, the range of concepts that one can communicate purely symbolically is pretty limited. All of the theorems, definitions are stated primarily in English, with a few symbols sprinkled in here and there. For example, the Hahn-Banach theorem states:

Let X be a real vector space and let f be a linear functional on a subspace (not necessarily closed) M of X . Say f ≤ p (on M), where p is a sublinear functional on X . Then there is a sublinear functional F : X→ R that extends f , and F ≤ p on X.

Just look at all the words there! OK, that's not really an argument; there have been attempts to reformulate all of mathematics in terms of quantifiers, but they have all failed (not least because of boredom!). Perhaps it is possible to restate all theorems as hideous nested logical formulae, but that would still leave the problem of how to interpret them, and how to make sense of the axioms of set theory.

Hmm. Back-pedalling a little, my argument here seems to be "If I can't express something in [these particular] symbols, there must be something lacking in this symbolic mode of expression, so this is not a language." I'm not convinced by that. At best, the above paragraph (and the cartoon above) allude to the existence of a "mentalese" in which we think, and I was trying to argue that mathematical symbols are not part of our mentalese. But, of course, our thoughts are not in English either (easily seen by the existence of ambiguous English sentences), yet no one would deny that English is a language.

I guess I should (1) stop babbling and (2) try to define "language" in a stricter way.

Anyway, my experience of studying math makes me think it's more like a restriction of philosophy to certain subject matter (abstract ideas of quantity, shape, transformations, etc). By contrast (and this may not be the contrast you guys had in mind) learning a language by itself does not teach you any new content, only a new means to carry across ideas to other people.

I guess the strongest thing I can say about the comparison between mathematics and human languages is that mathematics is the only appropriate way we have of thinking clearly about these abstract ideas like quantity and shape. Even if mathematics (by whatever definition one comes up with) is a "language," it is unique in that it makes analytical thought (at least about certain subjects) substantially easier, whereas I don't think that any particular language is so uniquely suited to the expression of other types of content (moral discourse, gossip, courtship etc.)

Sorry for the long post... I was enjoying myself.

08-29-2007, 09:19 PM
By contrast (and this may not be the contrast you guys had in mind) learning a language by itself does not teach you any new content, only a new means to carry across ideas to other people.

Like most long conversations we discussed many things. Some of the main ideas that we disagreed with where: He claimed understanding of the world only happens because of language and that language is the most fundamental thing for understanding the world. I said, maybe we already understand the world because the brain allows us to think, understand, perceive, etc. and that we mainly use language to get across these ideas to others. Basically, a main theme was whether language was the most fundamental thing that allows people to understand the world. He thinks so, and I think not. In other words, I think humans have the capacity and ability to understand concepts without language. One example I tried to use was mathematics. I said that people understand mathematical concepts even without language. I believe that even infants have some capacity to understand math (simple counting and arithmatic and the like) long before they are speaking and told in words how to think mathematically. Furthermore, animals probably have some concept of numbers and measurements, such as, this is larger than that, or there are more bones on my left then on my right etc. Furthermore, I believe that if I was born on an island and never came in contact with another human being, I would not develop language, yet I would still have some understanding of the world. For example, I would learn how to hunt, fish, and/or farm. I would learn how to build a dwelling of some kind if my environment required it. I would learn how to use fire. I would learn what animals to be afraid of and not afraid of. I would also have some sort of concept of math, albeit a very limited one. I would probably understand that if a tree had 3 bananas one day, but only two the next that there was one less. Or I would understand that one fish is larger than another, and therefore I should attempt to catch the larger one as long as it was just as easy to catch. Allow I didn't think of this example during our conversation, he seemed to disagree, saying that an understanding of the world can only take place because of language. One example we discussed is love. It seems to me that one does not need language to understand love because love happens in the brain. I asked him, does a baby not understand the love for it's mother before it speeks? He claims that no, the baby cannot in anyway understand love because he has no language. I feel that the understanding is happening in the brain independently of language. Mathematics tended to come up often in our discussion because I was trying to use it as an example of a way in which people can think about the world even on an abstract and complex level, even without traditional language. For example, like I was saying before, I believe infants and nonhumans have some capacity for mathematics on a very basic level. Furthermore, I believe that infants and nonhumans can still have some understanding of the world in which they live, even if they do not have language.

Sorry for the long post... I was enjoying myself.

No problem, feel free to write more as I would enjoy your outlook on this topic.

08-29-2007, 09:33 PM
To the OP: First of all, your friend is in a practical sense right that mathematicians cannot communicate without some other language. Perhaps in principle it could all be written down in logical formulae, but that is unimaginably inefficient. And mathematical proofs are highly dependant on the use of human languages. Try looking at any mathematics journal or textbook: "we construct a function f as follows..." or "The existence of a maximal element obviously follows by the _____ Theorem..." are typical turns of phrase. A proof is never entirely self-contained; it is a message from one person to another to be interpreted in the context of a larger discourse (literature).

I'm not going to try to answer the questions about "understanding the world" because I'm not sure, having not read the philosophical literature.

However, you really do need to clarify what you mean by "understand". I have never understood anything empirical from my studies in mathematics; theorems are strictly necessary truths, and of course they hold no implications for the "real world". (I am a naive realist, although my philosophy-major friends tell me I should be less proud of this than I am...) So if by "understand" you mean something about formulating scientific theories, I would say no (e.g. evolution), although it certainly is a useful tool. As far as a more anthropomorphic take on "understanding" goes (i.e. you're willing to admit the appreciation of artistic works as a kind of "understanding") then the answer is definitely no... although I would say that most people don't appreciate that some parts of mathematics itself have aesthetic value.

Your last four questions concern the philosophy of language and the philosophy of mind/neuropsychology, about which I know too little to say anything coherent. But you will find some useful stuff here:

The Language of Thought Hypothesis (Stanford Encyclopedia of Philosophy) (http://plato.stanford.edu/entries/language-thought/)

and for stuff on the philosophy of mathematics, I can't find one general entry, but an almost entirely non-symbolic discussion of set theory is here:

Set Theory (Stanford Encyclopedia of Philosophy) (http://plato.stanford.edu/entries/set-theory/)

I haven't checked Wikipedia, but presumably there's some good stuff there too.

08-29-2007, 10:03 PM
YoungEconomist, a book you might really enjoy is "How The Mind Works" by Steven Pinker. It's a little more "technical" in that he discusses (albeit in a very entertaining way) some more focused questions (e.g. "How exactly do our brains impute visual information from the reflected light around us?" vs. "How do we understand the world?"), but it's meant for the general public, and you get a sense of how it's possible to turn these general philosophical wonderings into scientific questions. He also has a book called "The Language Instinct" in which he essentially argues that we "grow" language... I haven't read it, but I really want to.

I need to go now, but as far as your conversation with your friend goes, I
think you're right in that language is not necessary to the types of cognition we usually call "understanding". I mentioned this above, but an easy way to dismiss his claim is to point out that ambiguous sentences exist. If we really thought in our spoken language, we would be unable to recognise that there are two meanings compatible with a given string of words.

One implication of your friend's claim is that mute/deaf people cannot "understand" (whatever he means by that) anything, which I'll bet will make him pull back from that position. Another problem that it raises is that of the "origin" of understanding: who was the first human to "understand" by your friend's definition? How did he manage this, given that all of his peers were either language-less, or using language without "understanding" anything at all.

There are at least three ways that you guys used the word "understand" in that conversation:

(1) to mean "believe a true statement," as in

"infants have some capacity to understand math (simple counting and arithmatic and the like)"

(2) to mean "know how to perform tasks that are materially useful," as in

"...I believe that if I was born on an island and never came in contact with another human being, I would not develop language, yet I would still have some understanding of the world. For example, I would learn how to hunt, fish, and/or farm."

(3) to mean "experience some emotion," as in

"does a baby not understand the love for it's mother before it speeks?"

I don't think that any of these uses is without merit, but I also don't think they characterise our intuition about understanding either. For example, experiencing sexual desire is not the same thing as believing that genes are the basic unit of natural selection and that it is consequently adaptive for organisms to have pleasure/reward circuitry that delivers endorphins during sex. The latter is a scientific understanding of sexual desire, whereas (I think!) an asexual alien biochemist who could presumably "get" that would not have "understood what it's like" to feel lust. [Although I think Daniel Dennett might disagree with me...
]Error (http://en.wikipedia.org/wiki/Mary_the_color_scientist)

08-30-2007, 05:38 AM
ponder this:
Blue-green across cultures - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Distinguishing_blue_from_green_in_language)

This might be interesting too: Thoughts On Economics: Positions On The Philosophy Of Math (http://robertvienneau.blogspot.com/2007/08/positions-on-philosophy-of-math.html)

Here's something else, from Rubinstein: Economics and Language (http://arielrubinstein.tau.ac.il/el.html)