The big idea: After doubting everything, Descartes hunts for a single belief the evil demon couldn't fake.
He finds it in a flash: the demon may deceive me about the world, my body, even maths — but to be deceived at all, I must exist. There has to be a self for the demon to fool.
This is the cogito, from Descartes' Latin cogito, ergo sum: 'I think, therefore I am.' It's the first thing to survive the doubt.
Hold onto this: The cogito isn't a lucky guess — it's self-proving. The very act of doubting your own existence proves a thinker is there doing the doubting.
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The move is simple but airtight — try to doubt it and you prove it.
Checkpoint — the cogito: In one line: to doubt is to think, and to think, I must exist — so 'I am' can't be doubted. Hold that — next we ask what this certain 'I' actually is.
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The cogito proves you exist — but only as one very specific kind of thing.
A thinking thing: Notice what the cogito does not prove: not that you have a body (the demon could be faking that), not that the world is real. All it proves is that there's a thinking thing — a res cogitans. So the first certain fact about yourself is that you are, at minimum, a mind. Whether you also have a body comes much later.
Go further — higher-level insight: The cogito is certain only while you're thinking it — it's a first-person, present-tense proof. It doesn't guarantee you existed yesterday or will tomorrow; it guarantees existence now, each time you think. Spotting that narrowness is a top-band point (and it's exactly where critics like Russell pushed).