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People often mention that there is an isomorphic nature between language and the world in the Tractatus' conception of language. As far as I can see it, according to Wittgenstein (it's been a few months since I actually read through it, so apologies if the terminology is wrong), the objects in the world correspond to the elementary names of the language. Thus, a sentence (a complex of elementary names) is a picture for a possible state of affairs. I guess the isomorphism is mentioned because in the ideal version of language Wittgenstein is talking about, there is a bijective mapping between the basic objects and the names for these objects. I imagine the relation that is preserved through the isomorphism is roughly: The relationship between the objects in the world is mirrored by the formal relationship between the names in a proposition.

So maybe in a simple natural language version of Wittgenstein, a statement like "the ball is 10m from the door", transformed through the isomorphism, corresponds to the state of affairs where this is the case.

I'm assuming that Wittgenstein must be talking about a radically new form of language here, because as it stands in natural language is more of a homomorphism than an isomorphism. That is to say, a statement like "The ball is 10cm from the door" corresponds to many different states of affairs (the ball could be 10cm north, or 10cm south, etc). So there is not a one-to-one mapping, as multiple states of affairs all get mapped to the same sentence. Given the common mention of isomorphism in relation to the picture theory, does that mean that, in Wittgenstein's view, when he is staking about propositions, he is imagining a type of language where literally every different state of affairs has its own isomorphic description, so that there would be an infinite number of propositions which would collectively make up the natural language idea of the ball being 10cm from the door.

If this is what Wittgenstein is talking about, then I guess it's an isomorphism, not a homomorphism, although in reality people's descriptive sentences could be better viewed as involving a homomorphism from reality to language. I guess Wittgenstein is talking about some language where even the simplest notion (the ball being 10cm away) would be an extraordinarily long proposition formally, at least as complicated formally as the world is.

Also, I am more of a fan of late-Wittgenstein, so I'm aware of how he came to the view that the work was fundamentally flawed, but I'm trying to get an idea of how it was supposedly going to work.

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  • Wittgenstein, in the Tractatus, does not use the word "isomorphism". – Mauro ALLEGRANZA May 18 '17 at 19:13
  • Having said that, the "applicabile" sense of isomorphism is the non-mathematical one: similarity of form. – Mauro ALLEGRANZA May 18 '17 at 19:40
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You're right, the tractarian language is a logically perfect language, a language which we don't actually know yet, but which must, so Wittgenstein believed at the time, underly every natural or scientific language. This perfect language is, at the level of elementary propositions, isomorphic to the world. This follows from a clear order of dependence: Facts are primary. Thoughts are "pictures" of facts. Propositions "express" thoughts, and thereby are also pictures of facts. The picturing/depicting relation is bijective.

4.014 A gramophone record, the musical idea, the written notes, and the sound-waves, all stand to one another in the same internal relation of depicting that holds between language and the world.

The Tractatus does not offer a theory of natural languages. What it says, in a commenting way, is that the structure of a natural language can, and does, totally contort and obscure the underlying perfect, logical language. So that a natural language is not expected to correspond to the world in any apriori specifiable way. So, no homomorphisms, or any other morphisms, are expected to hold.

4.022 Man possesses the ability to construct languages capable of expressing every sense, without having any idea how each word has meaning or what its meaning is — just as people speak without knowing how the individual sounds are produced. Everyday language is a part of the human organism and is no less complicated than it. It is not humanly possible to gather immediately from it what the logic of language is. Language disguises thought. So much so, that from the outward form of the clothing it is impossible to infer the form of the thought beneath it, because the outward form of the clothing is not designed to reveal the form of the body, but for entirely different purposes. The tacit conventions on which the understanding of everyday language depends are enormously complicated.

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