A novel can tell you everything you want to know about what it's trying to say, but plays are by definition incomplete. They are instructions for performance, like musical scores, and they need players to become music.

Working on Hamlet, Rory Kinnear and I repeatedly found that Shakespeare simply left stuff out – stuff that would have made the play last as long as War and Peace if he'd put it in. What, for instance, are we supposed to think has really gone on between Hamlet and Ophelia before the play starts? That things have gone on is plain from the pile of letters she returns to him. "I did love you once," he says, though he never says why he's stopped loving her; and I have seen this done so sardonically that it's impossible to believe. And a couple of lines later, he says "I loved you not." Which doesn't make it any easier to know whether he did, though it's the kind of contradiction lovers go in for. In any event, it feels like there's a missing scene near the start of the play that shows you how they are with each other before things start to go wrong.

But the genius of the play, as opposed to, say, War and Peace, is that it implies multitudes as much as it contains them. Shakespeare was an actor, and he leaves a lot of the work to his actors. The text forces any Hamlet to ask questions, which he answers through the way he delivers it. What did he feel for Ophelia? What does he feel now? What does he want from her? What – within himself and in Denmark – makes it impossible for him to trust himself or trust the world around him? A play's meaning is conferred on it by the act of playing it. In the way he said "I did love you once," Kinnear told you in five words what might have taken Tolstoy five chapters.

Simon Russell Beale is fond of describing acting as three-dimensional literary criticism. And in my personal experience, the most mind-expanding insights into Shakespeare have come from actors in the rehearsal room, usually without the long introductory preamble with which directors generally preface even the most banal of suggestions. As a tribe, we can barely ask an actor to move to the left without making a speech about it – but actors just get on with it. One day in rehearsal, without warning, David Calder – who played Polonius in Hamlet – approached the end of his speech of advice to Laertes and flinched. He seemed to dry. And then, under the heavy weight of what felt like deep personal shame, he said: "This above all: to thine own self be true,/ And it must follow, as the night the day,/ Thou canst not then be false to any man." From the heart, like many fathers, Polonius wants his son not to make his own mistakes. But, mired in a corrupt court, he is incapable of dealing truthfully with others, and of being true to himself. And Calder's Polonius knew it.

It would be equally plausible to present the Polonius of tradition, a man devoid of self-knowledge, puffed up with self-regard. But I was electrified by Calder's illumination of three lines worn thin by their relentless repetition, out of context, often by habitual liars. I knew immediately that the Calder Polonius had helped Claudius assassinate the old king, and was tortured by his own treachery. I started to think that the old king was probably a disaster for Denmark, that – like Richard II – he had to go. This was the real Shakespeare: an actor who provides for other actors a myriad of ways of telling his stories and of being his characters. His intuitive openness to interpretation is mistaken for complexity. His relish for ambiguity is taken as a challenge to those who would pin him down. But they are functions of his calling: he writes plays.

This is not to deny that his texts are often dauntingly complex. I cannot be alone in finding that almost invariably in performance there are passages that fly straight over my head. In fact, I'll admit that I hardly ever go to a performance of one of Shakespeare's plays without experiencing blind panic during the first five minutes. I sit there thinking: I'm the director of the National Theatre, and I have no idea what these people are talking about.

It's worth saying, therefore, that a huge amount of our work in rehearsal goes into achieving the maximum amount of clarity. What is pompously called verse-speaking seems to me properly to involve the kind of acting that reveals the boundless felicities of the text not by drawing attention to them, but through a commitment to clarity and simplicity. The actors I value most are those who speak Shakespeare as if it is their first language. They are aware of the rhetorical and rhythmic substructure of the text, but have no wish to reveal it for its own sake.

But for all the craft and talent of the best actors, it's hard to deny that there are occasions when most of the audience don't understand what's being said. Sometimes, it's deliberate: Leontes in The Winter's Tale stops making sense to the audience when he stops making sense to himself. More often, it's because the passage of 400 years has taken a toll on the immediacy of the language.

This has led me to think that to be true to Shakespeare you sometimes have to confront the incomprehensible stuff head-on, by cutting it – or even by rewriting it. How many precious seconds in a performance of Othello would it take for most of the audience to understand Iago's plan to cause unrest amongst the Cypriots, whose qualification shall come into no true taste again but by the displanting of Cassio? More seconds than they have available to work out that he means the Cypriots will not be pacified again but by the displanting of Cassio, because everyone in the audience wants to be ready for the next line, and the next. So why can't Iago say what he means?

I don't want to overdo this: 99% of the original text does just fine. But it's all of it provisional in the sense that it's waiting to be completed by its actors.

It has often been noted that Iago's "motiveless malignancy" in fact comes, in his soliloquies, with a superfluity of motives, as if he himself has difficulty locating the source of his depravity. What Shakespeare has done, of course, is to pay his fellow actor the compliment of trusting him to complete Iago for himself. He provides the actor with a solid enough starting point: Iago is consumed by the promotion of Cassio. But thereafter, the play works overtime not to lock Iago down, and seems to invite the actor to allow himself to be surprised by what happens to Iago: a man driven by envy and hatred, who isn't fully in control of what happens next (as none of us is), to whom the action of the play occurs spontaneously – as life happens to all of us.

The desire of literary critics over four centuries to solve Iago as if he were a puzzle seems to me to be missing the point. The solution is the actor. The playwright writes from the premise that the dots can't be joined on the page, and writes with the confidence of an actor who knows that, if they are any good, his colleagues will do the rest of the job for him. Shakespeare knew what he was doing, and what he knew was that he had no idea who Iago or Cleopatra – or even Snug the Joiner – were going to turn out to be.