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Hilary Putnam, in "Reason, Truth And History", attempts to argue against Cartesian (or hyperbolic) doubt, by proving that a 'brain in a vat' cannot actually think that it is a brain in a vat.

Part of the argument (if I understood it correctly, which is unlikely) hinges on the fact that in order to be a valid 'thought', an idea must be related to something in reality. Because the brain has a thought about itself, it must exist in reality.

Writing it in this way, the argument seems so obviously incorrect that I'm sure I'm missing something. How does Putnam define 'thought' in a way that makes it necessary for a thinking thing to exist as it perceives itself?

I found a summary of the argument here, in "Philosophers Explore The Matrix" (not that I'm recommending the book or the movie which is referenced in the title) but it appears similarly flawed: if I merely have an illusion of a hand, isn't my 'thought' or speech regarding my hand a valid reference to my illusory hand? The word 'hand' still is referencing something, even if it has no ontological existence.

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Putnam holds that the meaning of terms for physical objects such as "rock," "lamp," "tiger," and yes, "brain," or "vat," just is the object to which they refer. Furthermore, the connection between the term and the object is made through ostension; by pointing to an object and naming it, a speaker causes the name to gain a reference. Hence we cannot use those terms in a coherent way without referring to the objects in the world they are causally connected to through the act of naming.

Now if it turned out that what we were to call a "vat" and what we were to call a "brain" were just collections of pixels, the sentence "I am a brain in a vat" would be equivalent to "I am the collection of pixels we call a 'brain' in the collection of pixels we call a 'vat.'" Needless to say, for Cartesian skepticism to have any weight, this equivalence must not hold (the skeptic needs reference to extend beyond our immediate surroundings), so Putnam's argument is a powerful one. In short, you are right to say that words such as "hand" must refer to something (at least when they are used properly), but it is important to remember that they reference the objects surrounding us, whether they are made from atoms or pixels.

Now, is everything that passes through your consciousness a thought? I doubt many people would say that everything they are aware of is a thought; I would say that I am conscious of thoughts, objects, sensations, etc. That is not to say that arguments have never been made supporting the idea that everything is in some sense a thought (Spinoza, Berkeley), only that it is very counterintuitive and often unsuccessfully defended. Furthermore, Putnam's argument does not require that everything is a thought, in fact, it implicitly assumes that there exist things that are not thoughts...

I'm not sure how Putnam would define "thought," but I would say it would be similar to how he defines "concept." For Putnam a speaker's concept c of an object x is a set of descriptions about x that the speaker associates with x. Therefore, if you take "thought" to be synonymous with "concept" then Putnam's argument assumes a distinction between thoughts and objects.

For further reading I suggest Kripke's Naming and Necessity and Putnam's Meaning and Reference.

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