The Argument Of Non-Verification: An Argument against Descartes

How does Descartes know that he is a rational being or, in other words, he may know that he is thinking but how does he know that he is thinking rationally?
How does he know he is not mad?
One way that I assure myself that I am thinking rationally or understand an idea properly is to verify it with other people.
For example, if I do a difficult mathematics sum I might ask someone, "Do you get that answer too?" as a way of checking if I've done the sum correctly.
Or if I have an idea, I will talk it over with someone and ask, "Does that idea make sense to you?", to get a second opinion.
But Descartes, in his Meditations, comes to the conclusion that he is not certain wether anyone else besides himself exists and so, if he is not thinking logically he cannot use the minds of others to try and verify his own thinking.
A  mad person for example, may have little reason to think they are mad if they have no indication that anyone else thinks differently to them.
You might say, "Descartes can prove that he is thinking logically by doing mental mathematics, for example, starting with the statement 2+2 is four and subjecting it to various tests to prove that it is true".
But have you never had a dream in which something appeared both true and logical but then awoken only to think, "What appeared true and logical to me in my dream was crazy".
The same would be the case if you were doing mathematical calculations in a dream; you would write 2+2 = 5 and think it quite true and logical only to wake up later and think, "It should have been 2+2 = 4".
Thus Descartes, because he couldn't rely upon feedback from other people, couldn't use them to verify his own powers of reasoning and if he couldn't verify his own powers of reasoning he couldn't be sure whether his argument was a completely rational one.