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It's a critique apparent made by Nietzsche. The "I" or entity doing the thinking is supposed, in part because the norms of language. If an action is being done, it's assumed that something is doing that action.

In the quote's case, there being doubt proves that there is thinking, but it does not prove that anything is doing that thinking. Doubt occurs, therfore thinking occurs, therfore thinking occurs. Nothing in that logic proves that there is a self, or really anything doing the thinking. The verb is assumed to come from a noun, but that's not a proof. The most famous and fundamental principle can only say that thinking occurs, not that you exist.

This reality is actually indicated by scientific understandings of cognitive development. Initially, babies don't draw a distinction between themselves and other people. They don't understand object permanence, or that other people don't know the baby's thoughts because they have unique perspectives. The separation between self and not self is not innate, but constructed as the child matures.

There's also the fact that we can be wrong about what our own identity is, as I personally experienced. I thought I was a man, but that identity did not match up with who I could tolerate being. I literally had to rewrite my sense of I. Even though I had an identity for my mind, it was inaccurate to how my mind worked.

There are also meditative practices that allow you to temporarily suspend the sense of having a perspective in your own mind. By training yourself to observe yourself as an observer, you can cause it to disappear for a period of time. That leaves the thoughts without a point of view, until it reforms and you disperse yourself again. We think in terms of "I" because it's helpful, but "I" is just a useful tool for survival.