Is there any evidence in the play that proves Hamlet's sanity?

The evidence in the play that Hamlet is sane is that he makes a plan to pretend to be mad. He tells Horatio and Marcellus that he might do this, and then later he does it. There is no other evidence of insanity in the play.

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After Hamlet has the meeting with his father's ghost in Act I, Scene 5, he swears Horatio and Marcellus to secrecy and tells them not to give any indication that they know anything of unusual import about him, regardless of

How strange or odd soe'er I bear myself— As...

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I perchance hereafter shall think meetTo put an antic disposition on—

This shows that he is already planning to act mad in the future. Why should he want to do that? Claudius is spying on him. Claudius is a very shrewd and dangerous adversary, and he is already suspicious. Hamlet feels it will be hard to act the same towards his uncle now that he knows all about him, and therefore he is thinking of putting on an act of insanity to keep Claudius from detecting what he is really thinking. He does succeed in convincing Claudius, Polonius, Ophelia, Rosencrantz and Guildenstern that he is mad.

In Act III, Scene 4, when Gertrude asks

What shall I do?

Hamlet tells her not to let "the bloat king"

Make you to ravel all this matter out,That I essentially am not in madness,But mad in craft.

Earlier in that scene he tells his mother

My pulse as yours doth temperately keep time(And makes as healthful music. It is not madnessThat I have utt'red. Bring me to the test,And I the matter will reword; which madnessWould gambol from. 

Then in Act IV, Scene 2, he runs ahead of Rosencrantz and Guildenstern, pretending to be playing a game like Hide and Seek, which he calls

Hide fox, and all after.

Hamlet is obviously only pretending to be mad. We have seen that he is perfectly rational in his meeting with his mother. Then in Act IV, Scene 3, he continues to pretend that he thinks he is playing a child's game and that he thought Polonius was playing it too when he was hiding behind the arras. He tells Claudius that Polonius is at supper and goes into a long speech ending with

Your fat king and your lean beggar is but variable service -- two dishes, but to one table. That's the end.

Claudius at this point says

Alas, alas!

He believes his stepson is completely mad. But we know it is all a pretense. If Claudius did not think Hamlet was mad, he would probably be much more suspicious now and perhaps have Hamlet locked up. Hamlet foresaw such a possibility when he swore Horatio and Marcellus to secrecy in Act I, Scene 5. He needs his freedom if he is going to find a way to assassinate Claudius. It is essential that Claudius have no inkling that Hamlet has ever had any contact with his father's ghost. Claudius would be sure to think that Hamlet knows what nobody else in Denmark knows--that he killed his brother to usurp the throne and marry his wife.

So we see Hamlet pretending to be mad, and telling Horatio and Marcellus that he might do that, and also convincing his mother that he is not mad. On the other hand, we see no real evidence that he might be mad and not just pretending. It seems impossible that Hamlet could be mad sometimes and only pretending at other times.

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In Hamlet, has Hamlet lost his sanity?

The problem with trying to determine whether Hamlet is insane or not in this wonderful tragedy lies in the way that he reveals to Horatio and Mercellus that he actually intends to put on an "antic disposition" as he calls it after confronting the Ghost and discovering the "truth" about his father and who killed him. This does suggest that the scenes where Hamlet appears to be mad are mere disimulation. For example, when he appears to Ophelia in her chamber, as reported to Polonius in Act II scene 1, she certainly thinks he is mad based on his appearance:

Lord Hamlet, with his doublet all unbraced,

No hat upon his head, his stockings fouled,

Ungartered, and down-gyved to his ankle,

Pale as his shirt, his knees knocking each other,

And with a look so piteous in purport

As if he had been loosed out of hell

To speak of horrors, he comes before me.

Based on what Hamlet has just revealed in the previous scene, we can view this as a deliberate strategy to use Ophelia, who he clearly suspects will go running to Polonius who, in turn, will tell the King, to spread concern about his mental state. Appearing to be "mad" is therefore something of a stratagem for Hamlet whilst he considers what to do next and how to avenge his father.

However, at the same time, it is possible to look at the text and determine that there are moments when Hamlet is genuinely mad. Certainly, the above quote is poignant when it talks about his expression reflecting unspeakable "horrors." That this occurs just after meeting the Ghost could indicate that confronting the Ghost and finding out the truth about his father has mentally unhinged Hamlet. There is never enough evidence to clearly state that Hamlet has either become insane or he is just pretending.

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Does Hamlet show any real signs of madness?

I believe Hamlet was extremely anxious over his father's death. He did feel compelled to avenge his father's death. He is driven by his rage and anguish over losing his father. That alone is enough to make one mad. Nevertheless, I believe Hamlet held it together well. Under the circumstances, any normal person would feel anxiety and mental and emotional stress. His father has been murdered. That is enough to push one over the edge, but to add to that, his father has been murdered by his uncle, and his uncle has married his mother. Under these horrific circumstances, anyone would suffer mental anxiety.

I believe Hamlet held it together well. He cautiously reenacted his father's murderous scene, searching for his uncle's guilt. He wanted to be sure that Claudius had killed his father. He is carefully contemplating what his further actions should be. This type behavior is done under well-controlled thinking processes.

If there was a time Hamlet appeared to be mad, it could have been when he stabbed the man behind the curtain, not knowing for certain whom he was killing. But then again, he was thought it was his Uncle Claudius, and his death would have avenged his father's death. This would not be calculating. If he had killed the right man, this would have been justice.

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