Friday 7 October 2011

Mathematics as a native language

A couple of days ago, I was in a discussion in which I expressed one of my favourite views. This is that we should not be disturbed by our puzzlement at the meaning of quantum mechanics, or anything else of that nature. Explanations that try to picture the quantum world in the terms that are appropriate to our life in the macroscopic world are bound to be inadequate. The reality is in the mathematics, and the only thing to do is to start speaking mathematics as a native language.

I think my interlocutor agreed. But afterwards, I wondered whether there would be any reason why mathematics could not be a native language, in the way that French and German can be. Obviously, it is learnt later than other languages. The pattern of human development prevents its being a native language in the sense of the first language one learns. But that would not prevent its being a native language in the sense of a language in which one thinks, a language that is transparently meaningful. Likewise, someone who moves to a country with a different language, even as an adult, can go native, and take on his or her new language as a native language.

One possible difficulty would be that languages do not exist in isolation from the world. Just as one needs agency and a sense of agency in order to have a full sense of self, one needs to live in the world and navigate it, as well as living within a linguistic community, in order to know what words mean. Words need to latch on to the things with which one has dealings. This is not true of all words, but it needs to be true of at least some of them. How can this be true of mathematics?

The obvious answer is our starting-point. We live in the world that physics describes. Various mathematical statements can be interpreted to be about that world. So the terms and structures that are used latch on to the world. They could latch on to different things, but so could terms and sentences of a natural language. There is more scope for mathematical terms to latch on to something different than for terms of natural languages to do so. The mathematical terms are more neutral as between forms of life. Moreover, a given mathematical theory could latch on to any one of a set of isomorphic worlds, and the concept of isomorphism here underlines the neutrality as between forms of life. But this does not strike me as debarring us from seeing mathematical talk as latching on to the world in which we live.

The main problem would seem to be that the form in which the mathematical expression of quantum mechanics represents that world is utterly unlike the form in which the world is experienced by us. But even this does not strike me as a decisive objection to going native. As someone learns quantum mechanics, he or she learns how the quantum world gives rise to our everyday experience of medium-sized objects. The new picture of the world is linked to our experience through that understanding. Furthermore, the mathematical terms and methods that are used also have direct application in the world as it is experienced. We can come to understand them in that context, then apply them in the quantum context.

That leaves one last gap to close. Is anything lost when we transfer our understanding of mathematical terms and methods, acquired in the world as experienced, to our conception of the quantum world? I think it is not. At least, there is no loss that would debar us from taking on mathematics as a native language for the purpose of making sense of quantum mechanics.


  1. Would it be just a written language ?
    And if a spoken language too, then how would you pronounce ?
    Like in English ? Or like in French or Italian ?

  2. I do not think the distinction between writing and speaking matters much. We could, after all, communicate using English, French, German or Italian by writing everything down, and never speaking, although children would find it difficult to learn a language that way.

    But your question does make me think of something else. Can a language be a transparently meaningful native language if it is not regularly used in rapid conversation in everyday life? I think that is a psychological point about the capacities of human brains, rather than a philosophical point. So even if the answer is that such use is in practice essential, I do not think that would debar mathematics from being used as a native language.

  3. The pronounciation does matter, I would say.
    "=" is pronounced differently in French and in English, not to mention other languages.
    And there are no letters to attach signs to.
    Which sounds would you attach to "=" ?

    In the so-called "dead language" Latin there are at least letters we can try to pronounce in the way we think right or most probable and the ancient Romans are no longer there to disagree.
    But mathematical signs do not correspond to any sounds.

  4. Mathematics can express logical thinking well.
    But how about emotions ?
    Emotions could be expressed by melodies, written down in notes or pronounced by singing.

  5. If the language is to be spoken, a speaker can pick whatever words will be understood by the people who are listening (English if they are English, French if they are French, and so on). It is like Chinese: the symbols correspond to meanings, not sounds, so a single written language can support several spoken languages.

    It may be worth stating the limits to my suggestion. I only say that the way to understand the physics is to use mathematics as a native language. That is, the mathematical statements need to be transparently meaningful, without any feeling that something is lacking because they cannot be translated into any other language without loss. I am not suggesting that mathematics should be the only native language of people who use it like that. Natural languages can be used for anything to do with emotions. People can be multilingual.

  6. Now I see what you mean by "native language": being fluent in it to a degree that one grasps the meaning immediately and not by translating first into another language.

    But in this sense every child uses mathematics as a native language very soon after he started to learn mathematics. 2 plus 2 makes 4.
    Hardly any child translates it into: two apples and 2 tomatoes makes four items.
    And if the numbers get very high then a visualization is not possible anyway. So you have to grasp the meaning without any translations or visualizations.

    Habe ich Sie richtig verstanden ?

  7. Yes, you are right, we all speak simple arithmetic as a native language, in the sense that I mean. But very few people have the same easy relationship to the Schrödinger equation or to the Einstein field equations.

  8. I see. Yes, it would be great to be fluent in a very abstract and very precise language.
    Maybe the BBC could start a course in such a language. For interested listeners worldwide.

    Would the Aristotalian system of syllogisms meet the requirements of such an abstact and precise language ?
    There are also the languages of the code writers (software writers).
    There is the language of logic.

    The language of mathematics does not seem to be uniform.
    Is it the language of high level algebra or quantum mechanics which you would like to see more widely understood ?

    (I have never studied maths.)

  9. Aristotelian syllogistic logic is not nearly precise enough. It was only when it was replaced by symbolic logic in the nineteenth and twentieth centuries that we could start to study logical systems properly.

    The mathematics I have in mind is whatever is needed for the field of study. For physics, this means quite a wide range of areas of mathematics. And one of the most beautiful features of mathematics is that different areas turn out to be connected in deep ways.

  10. I suggest you create the syllabus, try it on your respondents, then guage the results. You will then be able to test your theory.

    We are waiting ...