Descartes famously says "I think therefore I am." Does it follow that anything I can think of exists?
This is an interesting question, there are several ways of looking at it. Here are two:
Alexis Meinong looked at it from the point of view of intentionality. Intentionality is the fact that any thought we have has to be about something. We don't just think, we think about people, objects, events, conversations etc. Because our thoughts always have to have some sort of "target", every thing we think about has to have some form of existence, even if it isn't material existence in the real world.
So although you might have only one sister in the real world, per Meinong, you also have a very large amount of siblings that exist virtually, as long as you think of them. This very crowded universe where all sorts of hypothetical siblings, imaginary girlfriends and dragons and ewoks exist was called Meinong's Jungle by other philosophers.
This was a beautiful thought while it lasted, but it was first put in doubt by Frege and Russell with their logical atomism (see the famous "the present king of France" example).
More recently Meinong's Jungle was completely wiped out by modern information science and neuroscience. Of course all of these imaginary entities have a real existence: They exits as neural patterns in your mind or bytes stored in a computer memory and there is nothing really special or metaphysical about them.
The second more interesting approach is more closely related to your Descartes example. But first you have to be very careful with what Descartes was doing exactly when he came up with it.
What Descartes really meant isn't "I can think of myself, that's why I know I exist", he wasn't implying that conceiving of something necessarily means that it exists somewhere. He meant something much more subtle. His reasoning is the following:
- I can imagine that my body is an illusion. I can never be certain that I have a body.
- Because I can think, I cannot imagine that my mind is an illusion. The very fact that I think means that I am certain I have a mind.
- My body and my mind have different properties, since it is possible that one is an illusion, while it is impossible for the other.
- By Leibniz's law, if two objects have different properties, than those two objects are different.
- Therefore my mind is different from my body, and has a separate existence from it.
Descartes claims that he has thus definitely proved the existence of a mind or a soul that is separate from the body. I won't go into the details of whether his proof stands or not.
The key point for your question is the following:
- The fact that we can imagine an object doesn't mean that it has any real existence. But the fact that we can imagine the difference between two objects does mean that logically they are different (they are not identical). A modern modal logic version of this conceivability argument was later proposed by Kripke and others. This "conceivability trick" is used in all sorts of places to prove various metaphysical facts, although its validity is under debate. See this question.
Here some further references: http://plato.stanford.edu/entries/possible-objects/ http://plato.stanford.edu/entries/nonexistent-objects/
No. Descartes didn't say: "I think about me therefore I am". He said "I think therefore I am". That is, the thing you're asserting the existing of is the thing that is thinking. His rationale is that if something is thinking, it must exist. The rationale is not that anything he can think of exists.
This basic idea is called the "Principle of Plenitude", and it is considered valid in realms like classical mathematics. Anything you can describe in a mathematically exact form, is assumed to be available for you to use mathematically.
But even then, blatant inconsistencies force you to choose some imaginary objects over others. The clearest traditional example is that you cannot have a single set of everything, and the ability to know what is in and outside of every set in the universe with perfect precision. Choosing both nets you Russel's Paradox: the set of all sets that do not contain themselves as elements must both include and exclude itself.
So you cannot have your cake and eat it, too, if you want to be able to navigate your imaginary universe in a way that preserves the ability to do things like use language, and map your imaginings onto the outside world occasionally.
Mathematics itself is therefore not uniquely determined, you can choose some concepts over others. But, aside from that latitude, which is not very broad, every 'plenary' imaginary universe that is internally stable is equivalent to some model of Mathematics as a whole, and every less complete one is equivalent to some mathematical model.
The idea of a concept (notion) which implies its own existence is due to Descartes, but it is not the Cogito, ergo sum, but it's his ontological proof of the existence of god: "There is a notion which implies its own existence, and that concept is god."
This became, incidentally, the pivotal point in Hegel's Science of Logic, the transition of the subjective notion into objective reality by necessity. See §1528, §1529 at the very end of the section Subjectivity and §1530, §1531 at the very beginning of the following section Objectivity, where it says:
Of the Notion, now, we have shown that it determines itself into objectivity. [...] this latter transition is identical in character with what formerly appeared in metaphysics as the inference from the notion, namely, the notion of God, to his existence, or as the so-called ontological proof of the existence of God [...] Descartes' sublimest thought, that God is that whose notion includes within itself its being
I recommend to distinguish the following points:
1) Descartes‘ „I think therefore I am“: I can always deceive myself concerning the content of my perception. Possibly I am dreaming and my perceptions relate to a dream world. But the fact, that I reflect my perceptions and that I think about the possibility of deception, proves that I exist – according to Descartes.
2) Independently from Descartes and confirmed by general experience: We cannot create objects in the real world just by thinking about them. Except from some fairy tales wishing does not create the content of my wish.
3) Anselm of Canterbury thought that certain ideas necessarily imply the existence of the content of the idea. His example from the ontological proof for the existence of God:
- a greatest being, having such attributes that nothing greater could exist (id quo nihil maius cogitari possit)
If such being would not exist, it were not the greatest being, because existence increases magnitude.
Today only a minority accepts his reasoning as a valid proof. Already his contemporary Gaunilo pointed out the weakness of the argument. Kant rejected Anselm’s argument by stating: Existence is not a property which increases the scope of a concept.
Hence my answer to your question: Nothing can be derived about an object from the fact that I can think about the object. The domain of thoughts and the real world are two ontological different domains.