The cogito comes in different forms. "I think, therefore I am." "This proposition, I am, I exist, is necessarily true whenever it is put forward by me or conceived in my mind."
This part of it, at least, is not rocket science. Note the I. Cogito, ego cogito, both "I think." The idea of a thought without someone (thing) having it is not part of Descartes' thinking. Indeed, it cannot be. If it were possible for there to be thought without it being thought, then Descartes could be fooled as to his being. So he concludes ergo sum, therefore I am. I am a being who thinks.
The difficulty here is that an active self has been introduced into thought, through the back door without having either an ivitation or any credentials to gain entry to the 'party'.
What descartes has proven is that it is possible for there to be thought, and thought is inescapable through thought itself. But he has not shown the thinking... that he or we are in control of our thoughts. He admits this point in Meditation 5 when he asserts that
"There is certainly further in me a certain passive faculty of perception, that is, of receiving and
recognising the ideas of sensible things, but this would be useless to me
[and I could in no way avail myself of it], if there were not either in me
or in some other thing another active faculty capable of forming and
producing these ideas. But this active faculty cannot exist in me
[inasmuch as I am a thing that thinks] seeing that it does not presuppose
thought, and also that those ideas are often produced in me without my
contributing in any way to the same, and often even against my will; it
is thus necessarily the case that the faculty resides in some substance
different from me in which all the reality which is objectively in the
ideas that are produced by this faculty is formally or eminently
contained, as I remarked before. And this substance is either a body,
that is, a corporeal nature in which there is contained formally [and
really] all that which is objectively [and by representation] in those
ideas, or it is God Himself, or some other creature more noble than
body in which that same is contained eminently."
Neitzsche in aphorism 17 (BG&E) Reiterates Descartes own criticism of himself, with the empirically correct observation that 'a thought comes when it wills' and not when this 'I' thing wills it. Therefore if thought simply comes when
it wills, and is not generated by the entirely presumptive 'I', we must conclude that thought is independent of the 'I' and return to the fundamental principle that thought exists apriori. Although this is an unpleasant self negation, there is much evidence empirical and otherwise to point to its truth, and it seems to me that Philosophy all too often appears to fear the implications rather than explore them fully. And this 'fear' has led to much published and convicted nonsense.