There is something surprising about the Cogito: it is not merely trivially true like, “I think, therefore I have states of mind” or “I think, therefore there is a non-empty set of my thoughts”. What is it that is surprising? Not the claim that I have thoughts, and not the claim that I exist, but the claim that my existence follows from my having thoughts. I take it for granted that I think and that I exist, but I don’t take it for granted that these two facts are logically connected—the Cogito informs me of that connection. That is the point of the Cogito. It is the “therefore” in it that commands interest. It is surprising that the existence of a self should be implicit in acts of thinking—that such acts should require a thing to perform them. For could there not be acts without an actor? Compare the Corporea: “I breathe, therefore I have a body”. I know that I breathe and I know that I have a body, but it is surprising to be told that the former demonstrates the latter—that having a body is logically implicit in acts of breathing. That is, it is surprising that we can infer the existence of a thing that breathes from the mere fact of the existence of breaths. For could there not be events without objects? Descartes invites us to accept that we are thinking things (“substances”) based on the premise that we perform thinking acts—as someone might try to derive bodily things from bodily acts. But the inference in both cases is notoriously questionable: all we can derive logically from acts and events are acts and events, not things that are the subject of such acts and events. We might be committed to a metaphysical theory that recognizes only acts and events, with no objects in the picture, or we might believe in the metaphysical possibility of free-floating acts and events; in neither case will we see a logical connection between the existence of acts or events, on the one hand, and the existence of objects that perform or undergo these acts or events, on the other. We certainly won’t think that we can prove to a skeptic that there are things that think or breathe merely from the fact that there is thinking or breathing (hence Lichtenberg’s objection). Such a proof requires a substantive metaphysical assumption and is certainly not self-evident. So the interesting part of the Cogito is also the most questionable part. Such is philosophy.
Now consider Hume’s anti-Cogito. Hume invites us to agree that we do not exist, or at least that we have no rational reason to suppose that we do (we might believe it instinctively). The argument for this is that we cannot find the self by looking within: we find only perceptions, thoughts, and feelings, not a self that has them. I know that I have thoughts because I can be conscious of my thoughts introspectively, but I cannot know that I exist by being conscious of myself introspectively—for I do not encounter myself in this way (nor in any other way).  I have thoughts and yet I do not exist (or there is no reason to believe that I do). We might even put Hume’s argument in the form of a deduction: “I think, therefore I do not exist”.  For it is the known existence of my thoughts that reveals by contrast that my self is unknowable: I am not the kind of object of knowledge that my thoughts are—they can be encountered and I cannot. Moreover, for Hume I am rendered redundant once thoughts are introduced: if I have a robust mental life, why do I need to be a thing that has this mental life—isn’t the mental life enough? According to Hume, the self contrasts with thoughts epistemologically, not being knowable as they are; and it is surplus to requirements. Thoughts are therefore not a reason to believe in the self, but a reason to doubt its existence. We can never know the self to exist in the way we know thoughts to exist—by being presented with the self. And there is no other way we could know the self to exist. Thoughts crowd out the self instead of ushering it in.
Hume’s argument is also surprising. It is certainly surprising to be told that I do not exist, based on the premise that I can’t be encountered in introspection; but this is actually a bad argument—and I don’t think Hume is guilty of it (though many of his readers have been). For it does not follow from the fact that I cannot encounter Xthat X does not exist—that would be to infer an ontological conclusion from an epistemological premise. But Hume’s argument is surprising even without that familiar non sequitur, because it is surprising that the self does not possess the kind of accessibility that thoughts possess: we assume that thoughts can be encountered, and we assume likewise that the self can be, but Hume persuades us that the latter assumption is false. It is surprising to discover that something whose existence we take for granted (rightly so, in my view) is not knowable in the way we naively supposed. The surprise here is analogous to the surprise delivered by Hume’s argument about causation: we assume causation to exist (rightly so, in my view) but are surprised to discover that it cannot be encountered in the perception of causally related objects (because of the imperceptibility of necessary connection). The self similarly eludes encounter—and that is surprising. What Hume’s argument tells us (or purports to) is that we have no reason to believe in the self that is based on the known existence of thoughts, still less on direct experience of the self. Hence it is an anti-Cogito: “I think, but I cannot infer from this that I exist”.
Both Descartes’ Cogito and Hume’s anti-Cogito rank as paradigms of philosophy: they find (or claim to find) something surprising in even the most familiar and intimately known facts. This tendency to find the surprising in the commonplace is characteristic of philosophy. You can learn a lot about philosophy just by considering these two arguments. 
 Descartes’ argument can be read as conceding Hume’s main point (setting aside chronology), since Descartes never claims that we have direct conscious awareness of the self—hence it must be inferred from things of which we do have such awareness, viz. thoughts. From Hume’s point of view, however, no such argument could ever work, since selves and thoughts are “distinct existences”, so that one never logically implies the other.
 I am here calling “Hume’s argument” an argument that many have derived from Hume—to the effect that there is no such thing as the self. In fact, I don’t think Hume argued this way, concluding only that we have no knowledgeof the self. But I am trying to avoid exegetical questions and focus in on a well-known line of thought.
 The main purpose of this essay is pedagogical: setting Descartes’ argument and Hume’s argument side-by-side, contrasting and comparing. I think this would be a good way to begin an introductory philosophy course: two compelling lines of thought that lead in opposite directions.