Descartes thought he proved the "I" part, and that's the usual interpretation.

Others have disputed that since then.

I think that to the extent that he proved the existence of the self, he also proved the existence of some world: whatever the subject and object of the thought are. (Which could potentially be the same thing, but aren't necessarily).

'I cannot doubt that I'm doubting'? Yeah, big whup ... There are no grounds, René, to doubt everything else in toto, and it's this lack of grounds for doubting of which (my) certitude consists (Witty). Sorry Monsieur - 'radical doubt' is a performative contradiction (i.e. global skepticism is self-refuting). Or, if one prefers, 'my doubting' - with or without grounds - presupposes (my) existence (i.e. ontological nihilism is self-refuting (Spinoza)). In sum: 'The Cogito' doesn't prove anything that needs to be proven.

Or did he not prove the 'I' exist part? — Kranky

According to proof theory, Descartes' views do not constitute "proof" in any fashion:

In any area of mathematics defined by its assumptions or axioms, a proof is an argument establishing a theorem of that area via accepted rules of inference starting from those axioms and from other previously established theorems.[7] — Wikipedia on the term 'proof'

Like many philosophers in western philosophy, Descartes' argument is utmost and fundamentally flawed because:

[1] Descartes fails to duly establish the confines of the system-wide premises within which he is reasoning.
[2] Descartes' conclusion is not a purely syntactic entailment of his (actually even unspecified system-wide) premises.

Proof is never semantic. Proof is always syntactic.

Therefore, I have to make the same remark all over again:

Either you reason within a system, or else, you reason about a system, because in all other cases you are simply spouting system-less bullshit.

Descartes' Cogito Ergo Sum is the product of his approach of methodological skepticism which, in my opinion, is the more important piece. Rene's conclusions about the status of the Cogito are the product of a mind limited to a seventeenth-century perspective. As such, it is hardly fair to criticize him from our standpoint of much deeper understanding of psychophysical processes (Popper would be more of a fair fight). I suggest recreating his approach of metaphysical doubt yourself and see where that leads you.

According to proof theory, Descartes' views do not constitute "proof" in any fashion:

In any area of mathematics defined by its assumptions or axioms, a proof is an argument establishing a theorem of that area via accepted rules of inference starting from those axioms and from other previously established theorems. — alcontali

This applies to math, don't it? Yet you said in "any fashion".

Math is not all fashions. Math is math fashion.

Your dispute is invalid, because you quoted a restrctive definition for math proofs, and you mistakenly and arbitrarily, but at any rate invalidly applied this criteria of proof to apply to all other proofs. This is invalid extension of the restrictions and of the necessities for a valid proof in other areas of human thought but math.

Or did he not prove the 'I' exist part? — Kranky

You are being cranky, Kranky.

Descartes proved the existence of his being via experiencing his own thought. He reasoned that thoughts can only be generated by thinkers. No thinker, no thought. But the thought existed. So he, the thinker, must exist.

His proof applies only to himself, and it can be transplanted, but each thinker can only prove his or her own existence by applying this proof.

If you need more explanation, then I am sorry, there is a threshold of understanding that you don't pass. Not my judgment; you are displaying strong signs of it.

Rene's conclusions about the status of the Cogito are the product of a mind limited to a seventeenth-century perspective — Pantagruel

You wish. His proof is as valid today as it ever has been.

No argument from me. All I am pointing out is that the extension of the concept of the ego-cogito is far more complex now than it was in the 17th century.

This applies to math, don't it? Yet you said in "any fashion". — god must be atheist

Your dispute is invalid, because you quoted a restrctive definition for math proofs, and you mistakenly and arbitrarily, but at any rate invalidly applied this criteria of proof to apply to all other proofs. This is invalid extension of the restrictions and of the necessities for a valid proof in other areas of human thought but math. — god must be atheist

A proof is sufficient evidence or a sufficient argument for the truth of a proposition. — Wikipedia on proof

There are two senses of "proof". In math, a proof is deductive reasoning with no room for error (when performed correctly). In a courtroom or a laboratory, proof is just strong evidence, which could be wrong or misleading. — Is the notion of proof meaningless outside of mathematics?

The use of the term "proof" outside the context of mathematical proof is wrong and misleading, because the mere evidence itself could be wrong or misleading. Such evidence is never sufficient for the truth of a proposition, and therefore, does not satisfy the definition mentioned above for the term "proof".

Consequently, it is a widespread misconception that it would be permissible to use the term "proof" outside mathematics. The proper term outside mathematics is "evidence".

You've heard of our greatest scientific theories: the theory of evolution, the Big Bang theory, the theory of gravity. You've also heard of the concept of a proof, and the claims that certain pieces of evidence prove the validities of these theories. Except that's a complete lie. While they provide very strong evidence for those theories, they aren't proof. In fact, when it comes to science, proving anything is an impossibility. — Scientific Proof Is A Myth

Therefore, I repeat, the only legitimate form of proof is syntactic entailment. Descartes did not prove anything in "cogito ergo sum" because his conclusion does not syntactically entail from his premise.

Or did he not prove the 'I' exist part? — Kranky

To me, it is a logically necessarily proven point. But to me the fact that we can never prove that we have evaluated our proofs correctly, trumps all that are seemingly logical necessities. So, it's just the best of proofs - not an absolute proof.

To me it's an absolute proof, but only to the person who does the thinking.

This is not a scientific proof; it is much bigger than that. It is the only proof in existence that is empirical, yet it has the strength of an a priori truth.

The use of the term "proof" outside the context of mathematical proof is wrong and misleading, because the mere evidence itself could be wrong or misleading. Such evidence is never sufficient for the truth of a proposition, and therefore, does not satisfy the definition mentioned above for the term "proof". — alcontali

What about
"The earth is flat.
Yet I can circumnavigate the earth going in one direction constantly, and arriving at the same spot as from where I started.
Therefore the earth is not flat."

Do you think this is an invalid proof that the Earth is not flat?

Now think about it this way: I think. -- is there a way I could be wrong about it? I don't think? No, if I think, then I think. There are no two ways about it. You can't think and be mistaken about it. It is an absolutely true proposition, to the person who thinks. When you say to me "I think", I'm not convinced. But when I think, I know I am thinking.

Please try to understand that part.

Once you got that part, then consider that if I think, then somebody has to be doing the thinking. Could it be somebody else doing the thinking? I hardly think so. I can't mistake my thinking to be done by someone else.

Therefore the person who thinks is I. I am doing the thinking.

And as long as I am thinking, I can be sure I exist.

I really can't convince you, can I.

But that's okay. Because you haven't at all given any reason for me to not consider this as a proof.

------------------------------

@alcontali, you said something to the effect that in the empirical world positive proof is not possible, because all proofs hinge on evidence that could be wrong.

Please tell me what evidence in the foregoing explanation could potentially be wrong. I reiterate it below.

I claimed the following things as premises:

1. I am thinking. Is this wrong to claim it, for myself to believe it? I accept that for you it's possible to be false, but to me, it's not false.
2. If I think, I can't not exist.

Which of the two, from MY point of view, can potentially be false or misleading?

What about
"The earth is flat.
Yet I can circumnavigate the earth going in one direction constantly, and arriving at the same spot as from where I started.
Therefore the earth is not flat."

Do you think this is an invalid proof that the Earth is not flat? — god must be atheist

The same theory will say that the earth did not always exist. How can something that still needs to come into existence be flat or not flat? Furthermore, according to similar views, the earth will not always exist. At some point in time, it will disappear. Will it be flat or not flat?

So, you have to qualify when the earth is supposed to be flat. The idea about scientific incompleteness is that you will never manage to specify all conditions under which your theory will remain valid. That is the difference with mathematics, where we completely know and control the construction logic of the abstract, Platonic world that the theory is about.

But then again, knowledge of the full construction logic may not be enough to specify exactly one Platonic world. If it implicitly contains a concept of infinite model size, then an unlimited number of Platonic worlds may be created by just one theory (=construction logic).

It is an absolutely true proposition, to the person who thinks. When you say to me "I think", I'm not convinced. But when I think, I know I am thinking. Please try to understand that part. Once you got that part, then consider that if I think, then somebody has to be doing the thinking. Could it be somebody else doing the thinking? I hardly think so. I can't mistake my thinking to be done by someone else. Therefore the person who thinks is I. I am doing the thinking. And as long as I am thinking, I can be sure I exist. I really can't convince you, can I. — god must be atheist

I am not against this particular reasoning. I just do not consider the conclusion to be a form of syntactic entailment, and therefore, I do not consider it to be proof. If you provide proof for a theorem, it must be possible to verify it mechanically. How is that possible for your reasoning? If the paperwork cannot be verified objectively, then it is not even proper justification for knowledge.

I claimed the following things as premises:

1. I am thinking. Is this wrong to claim it, for myself to believe it? I accept that for you it's possible to be false, but to me, it's not false.
2. If I think, I can't not exist.

Which of the two, from MY point of view, can potentially be false or misleading? — god must be atheist

An automated bot could post all of that to this forum.

It would mislead everybody in this site into believing that it is a person. In fact, that is already a multibillion dollar business. Just ask any dating site how they make money. They use lots of automated or cheap-labour bots to make users believe that there is a real person talking to them, while there isn't.

Seriously, you cannot "prove" on this site that you are real.

If there is any money to be made in making other people believe that you are real, you will probably not be real.