Infinite Signs & Löwenheim-Skolem Theorem

Posted on February 24, 2009
Filed Under Philosophy | 2 Comments

Here’s a good summary ofthe Löwenheim-Skolem Theorem. I’ve been discussing infinities relative to Peirce, Derrida, Plotinus, Marion and Badiou of late and I found this relevant. I really ought put together a post on some of what I’ve been discussion. Putnam applies this theorem in some ways I find interesting (if not necessarily ones I agree with) Putnam thinks this makes a mess of traditional realism. Consider two similar entities. (Say horses and unicorns) We can axiomize them by listing all their attributes. Then we discover two new creatures that share the same attributes but have some new ones. If our axioms that initially did distinguish horses from unicorns (and thus the real from the fictitious) then these new creatures pose a problem. (I’m following Floyd Merrell here) The claim then is that if our axioms can fail in one case (as above) then they can fail in an infinite number. That is we can always think of new axioms in addition to our defined axioms.

In this reading the Löwenheim-Skolem Theorem entails that there will always be an infinite number of interpretations for any set of axioms. Now at this stage you’ll probably be thinking of some of Quine’s writings. Putnam though uses this to argue that no classification can be unambiguously constituted. Since for Putnam traditional realism (what he calls metaphysical realism) depends upon this (a perfect 1:1 relationship between thought and object) realism fails.

Semiotically this is the idea that for any set of signs there can always be at least two new signs that would differentiate the series. This is key for Derrida’s thought which is ultimately about semiotics. Différance is the idea that one never reaches the end of semiosis. Thus we can never have stability since the new sign in experience can always change our expectations. There is a fundamental undecidability yet we must make a decision.

One way of thinking about this is to say that meanings are not in any model or theory. (The representations) They are also not in the axioms. (The properties we might encounter in any particular phenomenal experience) Meaning is essentially future oriented. This isn’t anti-realism. (Quite the opposite, it entails that what is present to consciousness can’t be sufficient) So we end up with a realism that isn’t traditional realism and also isn’t traditional idealism.

Related posts:

  1. Realism and Naive Realism
  2. Heidegger and Realism
  3. Heidegger’s Realism
  4. Realism
  5. Two Realisms
  6. Cosmology and Realism

Comments

2 Responses to “Infinite Signs & Löwenheim-Skolem Theorem”

Its is a good post for readers

Yeah, I was reading Dewey last night and he makes a similar argument. Back then he was being attacked by the New Realists on one side and various Idealists on the other. He argued that both kinds of argument end up reducing to more or less the same thing.

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