In a word, no. In more than a word, it depends, of course, upon how one defines "knowledge." Knowledge is determined or produced through specific sets of procedures and practices such as logical inference, corroborated observation, inspection, controlled experiment, etc.; and even the best of these procedures, at least in their application, admit the possibility of doubt. Now, it's true that some philosophers (e.g. Descartes) have cast knowledge as requiring absollute certainty or the complete absence or elimination of doubt. (I'm assuming here that by impossibility of doubt you mean the absenceor elimination of doubt.) But, as far as I can see, it makes more sense and people are better off acknowledging human finnitude and abandoning this requirement. Interestingly, you didn't ask whether "truth" or "certainty" require the impossibility of doubt. Consider how those questions might lead to different answers.

You raise an interesting issue. At the outset, I'm afraid, I must say that much depends upon what in this sentence is meant by "knowable." On the face of it, however, the statement "X is unknowable" is paradoxical, even incoherent. To use the name or term, "X." meaningfully seems possible only if something is known about X. Still, it seems to make sense to say things like, "The velocity and position of an electron are unknowable" (by the Heisenberg uncertainty principle); "The temperature of every meter foot of atmosphere on the planet currently closest to Alpha Centauri is unknowable"; "The Power Ball number for the next lottery is unknowable today." "The last thought of Abraham Lincoln is unknowable." "The name of every human being is unknowable." Of course, none of these statements imply that nothing at all is known about the topics they address. We can know, for example, that the last thought that crossed through Abraham Lincoln's mind was a thought, that the Power Ball number will be...

This dimension of Socrates' thought has been, of course, highly influential with skeptics. Indeed, it was in part on the basis of this sort of gesture in Plato's works that the Academic skeptics regarded themselves as inheritors of Platonic philosophy. Later the idea became known as "learned ignorance," for example in Nicholas of Cusa's work by the same name. It's an interesting thing to examine the different ways philosophers have tried to cope with the constellation of ideas involved with coming to understand one's ignorance, as well as other dimensions of human finitude. Hellenistic and Greco-Roman skeptics explored the ways in which doubt my characterize humanity's relationship to knowlegdge and whether skeptical arguments advance any positive wisdom or simply tear things down. Montaigne formulated the now-classic, "What do I know?" Erasmus called himself a "foolospher." Hume explored concepts of "natural," "common," ordinary, and non-dogmatic forms of belief while still acknowledging skeptical...