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The Convergence of Linguistics and Mathematics

  1. Jul 20, 2011 #1
    I have a question that I've been pondering recently. As far as I can tell, it's original to the boards or at least hasn't been discussed in a long time so I think it's fair to start a new topic.

    This concerns initial bases for thought. It would seem that both language and mathematics are the two main foundations upon which we build communication in a functional human society. Now, of course one can say mathematics is a language; or there are even ongoing debates here about mathematics being universally relevant in places where language may not be. But disregarding all of that, my question is this: does it seem that as structural mathematics becomes more abstract, in some ways culminating in category theory, the relation between formal linguistics (such as functional grammar) starts resembling mathematics? For example, a large notion that begins to permeate each subject is the distinction/relationship between sets and functions. Each subject (formal linguistics and mathematics) deal with these, albeit in their own way. Are these all offshoots of the same cognitive thought process? If so, does it only seem to be this way since they are both based on an assumed logic?

    (I recognize that linguistics isn't as rigorously defined as mathetmatics, however it is still a logically-based system, though its aims are different. Primarily mathematical efforts construct and linguistic efforts deconstruct. However, each can go both ways.)

    Hopefully this makes sense. I can clarify anything that needs it. Thank you for your time and opinions! I hope everyone gains from this conversation.
     
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  3. Jul 20, 2011 #2

    disregardthat

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    The fact that language necessarily must consist of ostensive definitions will make it fundamentally different from mathematics. Mathematics cannot have ostensive definitions, they don't make any sense mathematically (though we may be tricked into believing they do make sense, by appeal to mental imagery or suchlike).

    You can't either boil language down to grammar. Using language correctly is knowing when it makes sense to say/hear/read/write this-and-that in such-and-such a situation. Grammar is a structure in which the actually words is written and sound like, but will have nothing to do with the actual meaning. We may mathematically model the structure of grammar, but it will not touch upon the actual meaning of what it can express.
     
  4. Jul 21, 2011 #3

    apeiron

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    As disregard(ing)that :smile: says, there is a basic divide between syntax and semantics. But because you are talking about the formal aspect of languages, then I think you can make the comparison.

    Syntax does then boil down to words and rules, which is perhaps a dichotomy like structures and morphisms.

    But also, all human languages have a particular grammatical form. A sentence encodes a causal statement. It is composed of a subject, verb and object and so tells a little reductionist story of who did what to whom.

    So this is rather Newtonian. It tracks a triadic tale of cause and effect, a pusher, the push and the pushee. It is a way of modelling reality that is designed to give control over events by analysing effective cause.

    Maths is more timeless and spaceless and makes statements about mappings, I would suggest. It is not tracking a trail of effective actions (that break symmetries) but instead describing over-arching symmetry states. You have a structure or object, and then the complete set of all its possible morphisms or mappings. It is more a description of formal cause.

    So the syntax of both grammar and maths seems similar in needing a definite division into words and rules, objects and relations. But in grammar, the concern is with the broken symmetry of some actual path of action, with maths it is with the symmetry of all possible allowable actions.

    You can then make general statements in grammar, or particular statements in maths, but it looks like they are coming from opposed poles of logic - the inductive and the deductive.
     
  5. Jul 21, 2011 #4
    per=/
    of(in context)=*
    is=(=)
    difference=-

    thats as far as I would extend language and mathematics.

    Language is a shorthand method for communication.

    Look at the sentence above... It has a subject (language) that cannot be mathematically (at least simply) described, instead it relies on our brain's ability to link to other related physical objects upon the stimuli of "language".

    Math on the other hand is a system of reason.
     
  6. Jul 24, 2011 #5
     
  7. Jul 26, 2011 #6
    Your overall point about symmetries is intriguing and not something I've thought of before in that manner (although, of course I'm familiar with mathematics as a study of symmetry).

    Is that something you came up with or is there a source you based that off of?

    One thing I guess I'm not clear about is that in grammar (language not mathematics) certain words DO SEEM to retain symmetry or have an ontological equivalent in mathematics. For example, any present tense conjugation of the verb 'to be' form the equality operator. I don't know... I guess I have a hard time thinking about this because any mathematical symbol can be stated in a sentence. We use math because it's more efficient, but in reality we could just create definitions with words. The only other difference is that we allow multiple definitions and colors of meaning for singular words in language.

    Does my plight make sense? Lol. I'm honestly trying! You guys have all made great points.
     
  8. Jul 26, 2011 #7
    I definitely agree, I think, on most of what you say. Just for clarification, what do you mean by the highlighted sentence above?
     
  9. Jul 26, 2011 #8

    apeiron

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    Symmetry and symmetry breaking are the most basic philosophical concepts, even if that is not widely realised. The whole of Greek philosophy was about dichotomies like flux~stasis, chance~necessity, substance~form, etc. All the fundamental metaphysical notions came in complementary pairs.

    But I've not seen it applied to this particular issue in the way I suggest.

    I said grammar itself is a broken symmetry in that it restricts utterances to a linear tale of who did what to whom. And 'to be' would be an example of a word, a verb, not a grammar.

    But you are quite right that it is a special kind of verb in being so general. It has a higher symmetry than most other verbs. And so also plays a special role in grammar. It does map an object to its properties for example (the car is green). Which is a statement symmetric with regard to how the car became this way (painted green, seen through green glasses, whatever).

    The symmetry being broken here is the standard one of particular~general.

    So the argument I was making was not that maths and language are completely different, but that they were coming from opposite sides of the particular~general spectrum. You can still find ways to talk generalities in language and talk specifics in maths.

    And both of them are built on the dichotomy of word~rule - local variables and global actions.
     
  10. Jul 27, 2011 #9

    chiro

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    With regards to things like "human" languages like spoken and written languages, these are atomic (discrete). The number of characters or phonemes or what have comes from a countable set.

    But if you are talking about "sentences" that are non-atomic (like for example the raw signal for an image or sound), then that representation is continuous.

    Linking language with mathematics is actually very natural. Let us discuss the written word.

    As you probably know, language has a probabilistic structure. The best way to describe it, is to say that it has markovian properties: or in other words, that the probability of the next word, character and so on depends on the recent prior data that came before it. It is a pretty simplified view but it does do a good job of describing many characteristics of language.

    Also with language you should be aware of how "definitions" in language relate to "sets" in mathematics.

    Think of the universal set of containing anything that has the ability to be described within the limits of the language. Note that this is usually different for each language. A point of this is found in some tribal languages where a proper number system doesn't exist. For example in one tribe anything above 4 was considered as "many". Clearly it is impossible to compare the descriptive ability of something like japanese, english or arabic with respect to numbers in the same way for the tribal language.

    So given this you have a definition. The definition of the language has a a boundary of some sort. The definition together with its complement define the limits of the language.

    But here is the catch: defining the boundary explicitly like you would define the boundary of a set in mathematics with respect to its universe is a little "wishy washy" when it comes to natural definitions. Making the definition of the boundary clear is an evolving process.

    Also another thing with language: language = structure, and structure = probability. I mentioned the idea of probability (in particular markovian probability). This is important because language is completely a context dependent thing.

    A general purpose language like say english or spanish has a lot of redundancy, but as such also has the ability to be extended much more than other languages with less redundancy (think of all the post and prefix terms, all the modifications of words and so on).

    Also due to the markovian nature of language, we are able to continually reduce the state-space quite naturally because prior information dictates the most likely description and when this continues it makes identification of some descriptive element in a language very effective. Think of a tree diagram where branches correspond to probability.

    Custom languages are formed to describe and analyze a particular subset of phenomena. For example I don't use the same language to describe both Romeo and Juliet and the Mona Lisa. I use each language for a specific purpose.

    Magically though mathematics seems to have this wonderful property that it can both be so broad and yet so specific and precise at the same time. As you know numbers can be a placeholder for literally anything, but at the same time we can write very explicit restrictions on definitions formed with numbers whether they relate to behavior of systems with respect to other systems, or notions of probability and so on. I don't know if this kind of development is done on purpose as part of some intelligent design, but I do wonder what we would be like without the language of mathematics.
     
  11. Jul 27, 2011 #10
    I see. I think I get what you're saying now. However, did you mean "specifics in language and generalities in maths"? It would seem language is mostly general and in special instances can be specific, and vice versa in maths.
     
  12. Jul 27, 2011 #11
    Interesting perspective. Yes, I was quite aware about the Markovian properties of languages, particularly in sentence structure, but I guess I never connected that with some of the things you said.

    On a more general question: would it be fair to say that the language is the essence of thought and mathematics is a specialized subsection that falls underneath this? This is more the question I was getting at, actually.

    Thank you for your reply! :)
     
  13. Jul 27, 2011 #12

    apeiron

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    This is the point of a dichotomy. You cannot have the one without the other. So if you create the possibility to be more crisply specific, you are also creating the "antithesis" of the more crisply general.

    So you are trying to frame this as either/or - well, language, is it specific or general? But in being one thing, it enables the other.

    A sentence, to be grammatical, has to have a subject/verb/object structure. That is pretty specific, right? But that structure can then be used to encode just about any idea. So that is pretty general.

    Animals don't have structured communication so are very limited in their abilities to either make specific statements about the world or form rational generalisations.

    Language separated syntax from semantics and so put humans in a whole new intellectual space of possibility.

    Maths arose out of that eventually as a further development on syntactical operations. Semantics was reduced to a minimum. Maths deals in abstractions or generalities, leaving the specifics, the variables, to acts of measurement - we plug the numbers into the abstract forms to actually "get specific" with some mathematical model.
     
  14. Jul 27, 2011 #13
    That's basically how I was looking at it, but didn't know how to communicate it. Language creates specificities out of generalities, and mathematics creates generalities out of specificities. But sometimes they can each go the other way as well. Yes, I see that one enables the other.

    So... maybe even more to the point... we could perhaps say that language and mathematics are both generated by two separate abstractions, respectively, word and number. In my mind I'm trying to figure out how different (or not) these two concepts are.
     
  15. Jul 27, 2011 #14

    apeiron

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    If you wanted to get hard science about it, you could look up the entropy content of natural language. For English, the ratio is about 8 bits of message to 1 bit of information for the general features of speech. For a string of numbers, it might be near 1 to 1? So that much more specific?

    But that is focusing on the syntax rather than semantics. And so on syllables as your units. When you are really meaning "words" as anchors for semantics, or actual mental concepts.

    So there are quite a few things you need to separate to answer the question you want to ask. Semantics and syntax are very different, if complementary. And semantics - the symbol grounding problem - is still a poorly handled foundational issue in science. It certainly is not measured by current information theoretic approaches!

    A good way to think about semantics is perhaps a set theoretic approach. Sets are a way to specify general meanings.

    So take a sentence like the cat sat on the mat. You have a general subject (cat) that becomes more highly specified via the structure provided by a grammatical statement.

    The cat is now doing something somewhere. So we have {cat {sat {mat}}}. The cat could be doing anything. But it is sitting (a symmetry broken). It could be sitting anywhere, but it is sat on that mat (a further symmetry broken).

    Constraints are used to particularise. Before I said anything, you were in a high state of entropy (uncertainty, vagueness). Then I said cat and that information constrains what you imagine to be the case, or my message, to some crisper set of possibilities. Then further constraints (sat, mat) narrow your mental picture still further. A lot of entropy has been disposed.

    So every word in natural language is general, though most are still quite specific. Cat is more specified than animal for example. But every word is capable of being constrained to more specific meanings by further syntactically organised constraints. By being shoved into sentence structures. Or even just qualified by syllables like -ed or -s that add information about tense and number.

    And words can also be generalised beyond their usual standalone meaning too. So cat can become cat-like, catty, etc.

    Language is flexible and pragmatic in that way. The level of generality/specificity is tuned by everyday needs. Individual words can be sent in either direction by further syntactical operations.

    With maths, it all becomes much more black and white. Semantics is squeezed out of the picture almost entirely. So I don't think you can equate words and numbers in a simple, direct way.

    Numbers are general in that 1 is "one kind of anything". So to use number to talk about the world, you have to add constraints. You need to add information that ties it down to apples or cats or whatever. But the proud boast of maths is of course that it can do without the real world. A number is just a syntactic placeholder, a pure Platonic form without substantial content.

    In fact, a number just minimises semantics, it does not eliminate it. But you can make mathematicians very angry by telling them this. So best avoided. :smile:
     
  16. Jul 27, 2011 #15
    LOL. Technically, I'm a mathematician, and it doesn't bother me. But I definitely know what you mean. God forbid we shake the foundations of an entire system, although that's for another discussion.

    Your explanations make sense. I was leaning towards the direction you stated about numbers and words being separate in that sense.

    I had a strange thought just now.

    *(Deftly leaving any concrete realm that we were in and jumping to crazy hypotheses)*

    So if 'numbers' are just an intermediary form of abstraction that happens to categorize things in a useful way for us humans to use, as you pointed out, we can get much less abstract (more specific) than numbers, such as "cat", or much more abstract (less specific), such as a mathematical ring. But they are all a part of the same spectrum.

    Perhaps the reason numbers themselves seem so universal is because they match the size of our human system, basically our bodies, on this intermediary plane. I'm referencing here the idea that the average physical human size is approximately halfway between the smallest known objects/systems and the largest known objects/systems. So perhaps the concept of number is halfway between the most specific thing and the most abstract thing?

    Does this make sense in any way to you?
     
  17. Jul 27, 2011 #16

    apeiron

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    Again, if you follow the logic of dichotomies, it is a good thing to make things "far apart" because there is then a larger space of possibilities between them.

    So cat (the word) and cat (our averaged mental/perceptual experiences of felines) are not that far apart. The word cat is not that general because it relates to a narrow-ish range of concrete experience.

    But the number 1 is a very long way from any particular concrete instance of "oneness". It is as general as possible. And that gives it extra power, extra universality - though it must then be tied into very specific formal structures to make it do anything.

    The word cat already does quite a lot when one human speaks to another. But saying "one" really does little to narrow down another's mental state. The other person would say "one what - that could be one anything".

    So numbers are universal because we have universalised our experience in this particular way. Which could lead to an interesting philosophical question about how basic the notion of counting actually is.

    Is it in fact a pure formal notion? God invented the integers, the rest is the workings of man, etc?

    Or does number itself smuggle in semantics. Do we find number meaningful due to some aspect of our subjective construction of the world?

    Clearly I favour the second story. Even number is a "helpful fiction" we have invented and which has some deeply buried assumptions which, indeed, could reflect human scale - our particular scale of reality experiencing.

    In fact, humans find it very natural to think in terms of one, two, perhaps three, then after that "many". Even animals think of number in this way.

    So counting has a scale that is natural up to three or four or five. The idea of nothing is a bit trickier. The idea of 20 or 10,000 is getting very abstract. And infinity becomes way outside of possible experience and we can fall into very faulty thinking about such a "number" because we are way beyond personal experience - the semantic grounding for the symbols we are using.

    This is only part of it. We are also used to linear progression. But when it comes to describing the scale of the universe, then counting in orders of magnitude could actually be more "real" as we are describing geometric or expanding processes.

    If you think about time counted off as orders of magnitude since the big bang, you will get a very different (semantically) notion of the speed of events. The age before the appearance of matter, for example, is suddenly "large" enough to have its own rich history.

    So yes, our idea of number is indeed infected with semantics. It is the view from a particular scale of experience. And I think this is huge in fact. Our current maths (and physics) treats scale as non-fundamental. So fractals and other kinds of maths that exhibit scale symmetry(!) are treated as quirky extras, stuff to be tacked on to the regular stuff.

    And the same with non-linear dynamics - which is about change. Again, the regular view is that reality is static existence rather than dynamic process. So that is a semantics that infects our mathematical generalisations. But we have now discovered the universe is a dynamic and scaled thing.

    Our counting system may well indeed not match that reality particularly well. It is not yet sufficiently general.

    I'm not saying this is a crisis as regular numbers can be used to create more complex models that cope with scale and dynamism. But you can see that regular numbers may not be the most natural units for doing this.

    Again, don't mention this to any mathematicians. They will be really annoyed. It is part of their self-image that maths is beyond semantics. :wink:
     
  18. Jul 28, 2011 #17
    Haha... for sure.

    This makes me want contact from another intelligent life form even more now! Questions like this would be extremely interesting to have another perspective on.
     
  19. Jul 28, 2011 #18
    "the output is a function of the input plus a constant" isn't that much more complicated :/

    True, if you are describing a simple linear function, or a direct proportion, or something with three or less variables, but try describing a car mathematically, or even a molecule of water, I used "car" three symbols, and "water molecule"... can you write an equation to describe it with that many symbols... I don't think so. Of course simple effects are simpler with mathematics, but even "gravity" is simpler than "F=G(m1m2/d^2)", words are used to describe things in a general non specific yet convenient way...
     
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