The Underground Man goes through a ton of arguments in Notes, each one building on the last. It can get confusing. We're going to give you the quick and dirty here in what we hope to be a delightful 60-seconds of reading. But if you're good with the basic tenets and want to jump right to the analysis, feel free. Key terms are italicized, and definitions in parentheses.
Next comes the subject of romanticism. According to the Underground Man, Russian romantics are better than the French and German for their grounding in reality and traditional values. We also learn that the Underground Man sometimes fakes living because he doesn't know how else to be a person; he has "phases" of wanting friends, but these are short-lived. We also get an attack on Chernyshevsky and the Crystal Palace. Rational egoism (the theory that man will always act according to his best interests) is wrong because it ignores free will. To prove free will, man will intentionally cause destruction and harm. Oh, lastly, the Underground Man has no readers.
Let's start with the Underground Man's early claim that he has never been able to act. His reasoning is that there is no justification for any action – so he remains inert (taking no action). He's so paralyzed by having to choose what to do that his decision is to not choose. OK, but this isn't really an option. If he can't justify going outside, he ends up staying in – he's still deciding to stay in, though. If he's sitting around and brooding all the time, why doesn't he try to justify said brooding? He claims all a man like he can do is babble – what's the primary cause (motive) for babbling?
Then you've got the claim that he's hyper-conscious, acutely aware. Check out the passage when he walks the Nevsky, debating how best to get revenge on the officer, only to pass by the officer before he's realized what happened. Does this sound like someone who is acutely conscious of his surroundings?
And what about his disdain for rational egoism? He rejects the idea that man will act according to his own self-interest, but in many ways he is the epitome of an egoist. "The world may go to pot," he says, "so long as I always get my tea." On top of that, he condemns the "frippery" of French and German romanticism, but he can be quite the romantic himself. He lives in a world of his own idealistic making – look at the "Lake Como" passage in Part II, Chapter One. His fantasies are all something out of a cheesy novel. Come to think of it, most of his life is out of a cheesy novel. The whole idea of the older man redeeming the young and corrupted prostitute is a major theme in Russian literature. His obsession with revenge is right out of The Count of Monte Cristo.
But the most interesting contradiction has to do with freedom. The Underground Man is Mr. Free Will. He's so free, in fact, that he's going to bash his head against the wall of reason just because he won't resign himself to the fact that 2+2=4. All his suffering, all his self-inflicted pain is supposed to prove his freedom. Except in his masochism, he's constricted himself to an underground prison. Prison = freedom? We don't think so. Next, look at the way that the Underground Man explains his actions to us.
He justifies them with a series of logical arguments. He is, in fact, beholden to rationality. In condemning the laws of reason, he uses…reason. And what about living according to books? Isn't this just another system of rules he must follow? "Leave us alone without books and we shall be lost and in confusion at once. We shall not know what to [do]," he says.
And look at the only time the word "fate" is used in the entire novel, at the end of Part II, Chapter V, when he decides "It's fate" for him to go slap Zverkov in the face. If he's not freely determining his own actions, how could he possibly have free will? Lastly, check out the line at the end of the Underground Man's hysterical Liza-induced breakdown: "They won't let me…I can't be good!" That doesn't sound like free will, either.
The big 2+2=5 argument is also rather compelling. Look at it this way: someone tells you that you're fated to eat pancakes tomorrow morning. No doubt about it, you will eat pancakes. Not only that, but you are physically incapable of desiring anything other than pancakes. Are you going to go along with that? Every day for the rest of your life? What if you want cornflakes – who are they to tell you what you want! You'll want whatever you please! We don't know about you, but that certainly makes sense to us.