In Act 3, Hamlet can see the ghost but Gertrude cannot. Is this evidence that he is truly mad or that he is pretending to be mad?
Gertrude herself actually concludes that "alas, he's mad" but Hamlet later says "it is not madness that I have utter'd." In short, the Queen is completely unaware of Hamlet seeing the ghost in Lines 131-135 ("To whom do you speak this?/ Do you see nothing there?").
So if Hamlet is mad, can we not doubt the existence of the ghost?
We have to remember that everybody in the audience sees the Ghost when it appears to Hamlet the first time and again when it appears to him in his mother's chamber. If Shakespeare wanted us to think Hamlet was mad, then he would have had Hamlet speaking to the empty air. The Ghost is real. We not only see him but we can hear everything he says to his son. If Hamlet is mad, then we are all mad. The whole play is based upon the fact that Hamlet has gotten information from the Ghost which is of the most vital importance. Note that after his first encounter with the Ghost he takes great pains to make sure that the other men who have seen it and who know that he has communicated with it will not tell anybody about the encounter or even hint that they know anything about it. Hamlet takes elaborate precautions to make them swear they will never reveal what they know about the Ghost. The point is to keep Claudius in complete ignorance of the Ghost's existence. Polonius has to be killed, because otherwise he would know about the Ghost and report it to Claudius immediately. Gertrude doesn't know about the Ghost because she can't see it. I don't see how anybody can doubt the existence of the Ghost when they have actually seen and heard it themselves. It may only be a ghost, but it is a real ghost.
It is true that Gertrude questions Hamlet’s sanity in this scene. Hamlet can see and hear the ghost, but his mother cannot. But look at what the ghost says to Hamlet:
Do not forget: this visitation
Is but to whet thy almost blunted purpose.
But, look, amazement on thy mother sits:
O, step between her and her fighting soul:
Conceit in weakest bodies strongest works:
Speak to her, Hamlet.
At first, the ghost tells him to remember his purpose—to avenge his death. If Hamlet were mad and seeing visions of ghosts that did not really exist, you would expect the ghost to say this sort of thing. But then the ghost tells him to attend to his mother, who is watching him with “amazement.” It would be a little surprising for Hamlet to conjure up an imaginary ghost that told him to think about someone other than himself. Hamlet’s attitude toward his mother has been accusatory, but the ghost seems to pity her.
Also, we must keep in mind the beginning of the play. In Act 1, Scene 1, Bernando and Marcellus are on watch outside the castle. When Horatio arrives, their conversation reveals the fact that Bernardo and Marcellus have seen a ghost and that Horatio is skeptical. Marcellus says:
Horatio says 'tis but our fantasy,
And will not let belief take hold of him
Touching this dreaded sight, twice seen of us:
Therefore I have entreated him along
With us to watch the minutes of this night;
That if again this apparition come,
He may approve our eyes and speak to it.
Moments later the ghost arrives. Horatio sees it and says:
...it harrows me with fear and wonder.
So, as we can see, Hamlet is not the only person who sees the ghost. Shakespeare does not explain why some see the ghost while others do not.
We should also note that Hamlet does doubt the purpose of the ghost on several occasions. He is not completely sure if it is the ghost of his father or if it is some evil apparition trying to influence him to kill Claudius.
It appears that we have to accept Shakespeare’s depiction of the supernatural at face value.
Lets look at it this way... it is reasonable for the ghost not to appear in front of Gertrude. Why? It is because it only took Gertrude so long to marry his brother after his death it could be speculated that something was going on between them for he died. So why would the ghost appear to her when it will only agitate the ghost?
Then there is ghost trying to protect Gertrude so she can live in her sin and I say protect her because Hamlet wanted to get revenge on her. Now if you think this ghost was not really and rather a figment of Hamlet's imagination he would have been encouraging Hamlet to take revenge on his mother? Exactly! So this whole Hamlet is "mad" is just Hamlets plan to make him so seem dumber than he is so his strike is more surprising.
No. Gertrude doesn't see the Ghost because she no longer remembers her deceased husband as he lived. Visions of the Ghost are accompanied by remembrances of King Hamlet. Those that see the Ghost identify him/it by what they remember of the King. From 1.1
Bernardo: In the same figure, like the king that's dead.
Marcellus: Is it not like the King?
Horatio: As thou art to thyself.
Such was the very armour he had on
When he th' ambitious Norway combated.
So frown'd he once, when, in an angry parle,
He smote the sledded Polacks on the ice.
And then at the end of 1.2
Hamlet: His beard was grizzled--no?
Horatio:It was, as I have seen it in his life,
A sable silver'd.
Then in 3.4 in Gertrude's closet.
Hamlet: Why, look you there! look, how it steals away!
My father, in his habit as he lived!
Remembrance is one of the themes of the play. Now contrast that with the theme of bestial oblivion. Hamlet says in his first soliloquy "a beast that wants discourse of reason would have mourned longer." That first soliloquy is Hamlet reflecting on what his mother fails to remember. This is why Gertrude can't see the Ghost.
Hamlet is pretending to be mad so as to prompt claudius to give himself away. He is also quite smart in setting up the whole play scenario, not something that a madman would have the patience to do.