Is Language infinite?

Question

1. Can a Natural Language (like English for instance) describe anything?
2. Are our thoughts limited by our language?
3. If the number of words in a Natural Language is a finite set, can this set describe an infinite number of concepts, notions or ideas?
4. Is there a core or reference set of vocabulary that can describe any concept?

Editing:

What drove me to ask this question is my adoration to mathematics and its nature. Although any mathematical concept can in principle be formalized i.e. it can be expressed in terms of the axioms of set theory (ZFC), rarely the mathematicians do that. Instead they use natural language (English in particular) to describe and communicate mathematical concepts. So the following two questions came to my mind:

• Is natural language limited?
• Does natural language put constraints on the amount of mathematical discoveries that humans can make?

Editing:

Steven Pinker has a nice presentation that touches upon this issue (particularly at the minute 20):

https://www.youtube.com/watch?v=Q-B_ONJIEcE

Answer 1

The headline question: Is language infinite? should perhaps invite more scrutiny than it's generally given these days. It was posited by Chomsky in the context of a particular view of language: "A small set of rules operating on a large but finite set of words, generates an infinite number of sentences." The problem with this definition is that it assumes a very bounded and discrete view of all elements: rules, lexical items, sentences. Language is not organized in sentences and the difference between words and rules is likely an artifact of dictionary making and grammar writing. But even if we did assume that, there is no guarantee, that the actual number of possible sentences is infinite rather than just unimaginably and practically inexhaustibly large (as was argued by Pullum and Scholtz in 2010). As a matter of practical fact, while the set of possible expressions in a given language may or may not be infinite, the actual set of all expressions ever uttered (even if we ignore language death) is going to be finite (because we cannot ignore the heat death of the universe). Now, the set of expressions actually produced by humans (or even machines) is going to be too large to practically enumerate by current (and possibly any) technology but it's going to be finite.

So it really does not matter whether language is infinite at all (other than to keep certain formal theories internally consistent). What matters is that any language is going to allow a set of expressions that is sufficiently large for any purpose a human language can be put towards.

However, this does not mean that the answers to your other questions are straightforward.

1. Can any language describe anything? Well, this depends on what you mean by describe. There are two ways you can approach it:

   1. Exhaustive definition that will communicate between any two people all the relevant features of a phenomenon in such a way that cannot be confused with anything else?
   2. Talk about it.

   The answer to the first point is no, language cannot describe everything in a perfectly communicative manner. In fact, I'd argue that language cannot describe most things. However, the answer to the second way of putting the question is yes. Language can talk about everything. It uses deixis to point and analogy to compare. These ways are never definitive but they are good enough for language to carry the explicit cultural and interpersonal knowledge it does.

   This issue is often tied to infinity of expressions. But that's a non-sequitur. We could have an infinity of expressions in a language that does not match a single item in an infinite set of phenomena.

2. Are our thoughts limited by our language? The problem with this question is that we don't really know what thoughts are (or really what language is). So the answer is yes, our language limits what we tend to think, but no language does not preclude the possibility of any thought (in the way we think of thoughts).

3. Can a finite set of words describe an infinite set of ideas. This makes several unwarranted assumptions. First, words don't describe ideas, they are used to create expressions (by rules if you subscribe to the words/rules theory of language or unification if you don't). Second, expressions do not really directly map onto a discrete set of ideas/notions/concepts. They co-create them. So in some way the question makes no sense. But even if you did subscribe to the simplistic notion of 'String of words'='Idea', you could generate enough strings in a language to match enough ideas to last humans the rest of the existence of Earth regardless of whether a language is finite or not.

4. Is there a core or reference set of vocabulary that can describe any concept? Again, remember all the caveats about the limits of 'description'. However, even with a simplistic view of what 'describe' means, the answer is that in practice there is not. However, different theoreticians have tried to come up with different sets of semantic primitives that can be used to describe anything. Sometimes this only applies to a subset of reality (logic, mathematics) but not necessarily. Anna Weirzbicka constructed a Natural semantic metalanguage that only consists of about 70 words found across all languages. She claims that these can be used redescribe any human concept. However, these descriptions are not very practical for every day use, even if you did subscribe to the very notion of semantic primes.

Note: There has also been some debate as to how large languages actually are, with some linguists claiming that language is actually not just infinite but uncountably infinite (see here for details of the discussion).

Answer 2

1. Yes: "The sky is blue." I just described something.
2. Only in much more limited ways than might seem to be intuitive. For an introduction to this area you could start with the Wikipedia article on Linguistic relativity, the famous "Sapir-Whorf hypothesis".
3. No natural language satisfies your precondition of having a finite set of words (until they become dead languages). But even a hypothetical language with a very small vocabulary would have no limit on combination of words.
4. There are surely many of these. Many dictionaries develop them to help prevent circular definitions among other things. The "Simple English" edition of Wikipedia is based on such a premise.

Answer 3

"Wilhelm von Humboldt's phrase that language is 'the infinite use of finite means' has been cited by Noam Chomsky, who in turn has frequently referred to the concept of infinite language, a speaker's competence, which in Chomsky's terminology is their ability to produce a potentially infinite number of correct sentences." http://usingenglish.com/speaking-out/language-infinite.html

If by sense that language can grow to evolve to describe and analyze any idea, language is countable to a point.

"While formal language theory usually concerns itself with formal languages that are described by some syntactiical rules, the actual definition of the concept "formal language" is only as above: a (possibly infinite) set of finite-length strings composed from a given alphabet, no more nor less." https://wikipedia.org/wiki/Formal_language

A question measuring language may have many answers, towards examining what's countable to language.

Answer 4

I would jump up one level and along the way disagree with your statement regarding the capabilities of mathematics. There is no system that can describe anything. Reference Godel, Escher, Bach and other like works. or more poetically "The Terrain is not the map".

Answer 5

This question targets mental processes, cognition, far more than language. Can one formulate for oneself some (mathematical) concept.

There are many non-linguistic examples, where symbolic manipulation is hampered or accelerated by the fitting view.

Mathematical differentiation started with Newton and Leibnitz, two competitive notations, one being far inferior.

Soroban is an Asian abacus, that is mentally much more operative, because it basically is turned 90 degrees, so one forms the digits of numbers. The European abacus got forgotten and in disrespect relatively fast, but the soroban has much more mental merits.

With pure language to describe things, imagination/projection, plays a significant role. As Wittgenstein expressed, in language we play, use shifting meanings, contextual meaning. To express the behaviour of quarks in physics, they came up with "colour", three quarks forming a indivisable unity, like Red, Green, Blue. Of course "colour" is inadequate, but the interesting point is, that one sees a wish to project more than there really is, and literally is a very colourful image.

You pose a notion that language can be purely formally used to describe phenomna like formal building blocks. And question to what degree this is possible. I think such formal essays are often the results of first searching the right tool to model the world.

As a last example, physics is based on equations (E = mc²). Science philosphy may put this to debate. In linguistics we know other systems than equations like rewrite rules. Wolfram Mathematica's author tried discrete recursive engines to model physics.

In conclusion: ideas are much more powerful than words. Afterwards those ideas are formalized in words - after the "aha" moment of insight. And vice versa the prose in words needs an intellectual project of insight.

Especially: it is easy to err in a unsatisfactory notation. Study the soroban or latin numbers.

Answer 6

Besides the answers above you might as well take a look at the https://libraryofbabel.info/

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The Library of Babel is a place for scholars to do research, for artists and writers to seek inspiration, for anyone with curiosity or a sense of humor to reflect on the weirdness of existence - in short, it’s just like any other library. If completed, it would contain every possible combination of 1,312,000 characters, including lower case letters, space, comma, and period. Thus, it would contain every book that ever has been written, and every book that ever could be - including every play, every song, every scientific paper, every legal decision, every constitution, every piece of scripture, and so on. At present it contains all possible pages of 3200 characters, about 104677 books.

Since I imagine the question will present itself in some visitors’ minds (a certain amount of distrust of the virtual is inevitable) I’ll head off any doubts: any text you find in any location of the library will be in the same place in perpetuity. We do not simply generate and store books as they are requested - in fact, the storage demands would make that impossible. Every possible permutation of letters is accessible at this very moment in one of the library's books, only awaiting its discovery. We encourage those who find strange concatenations among the variations of letters to write about their discoveries in the forum, so future generations may benefit from their research.

Answer 7

1. Yes, anything that is describable, up to a point, as we can only describe on the basis of shared understanding that comes only from shared experience. I.e., we can always describe the objective (the wavelength of color red) but not the subjective (redness to a blind person). Regarding the indescribable, I should mention "colorless green ideas" whose state of slumber is an important linguistics research topic). Can you describe more precisely what you mean by "describe"?

2. Given that our thoughts are able to extend languages, which get richer as time passes, one would think that language does not limit our thoughts. But since thinking is sensitive to training, linguistic habits do influence our thinking.

3. since anything that can be described, alluded to, defined, computed, lied about can be thus mishandled with only two symbols usually noted "0" and "1", I would guess the answer is yes. We can describe an infinity of things, but only a denumerable infinity. Contrary to popular belief, it is likely that the reals do not exist ... most of them.

   This said, while the number of word currently in use is undoubtedly a finite set, as is the number of words that have been in use. The number of words that can be is undoubtedly infinite. The number of words that will be depends on the future of the universe, but will most likely be finite at any time.

4. Yes, the sign language together with a few grunts. That is all our ancestors had, and they developed the rest from that. As long as shared experience is available.

The point is, there is more to language than the language itself. Is the working of our brain part of language or not? Is language separable from experience? How much of language is limited by the physical concept of representation? How much of language is dictated by intrinsic concepts of communication (logic)?

P.S.
To emphasize the importance of shared experience in language and communication development, I would like to point out the Pioneer plaque and the Voyager golden record. The plaque was intended to communicate with unknown being, and the designers tried to communicate their message by referring to physical and atronomical experience that any advanced civilisation should normally have (as far as we can predict), or referring to the only non-natural experience we can share: the spacecraft itself. The record contains images and sounds, which is more direct experience sharing than actual communication language.