The Crystal Palace (Or "The Palace of Crystal" in Constance Garnett's translation)
So…what exactly is the Crystal Palace? In 1851, The Great Exhibition was held in London. It was basically like a World's Fair, where everyone got together to show how cool and modern things were those days, and especially how much cooler and more modern England was than France. Architect Joseph Paxton said something like, "Hey, let's build a building made out of glass and iron, but mostly glass." The Crystal Palace became a symbol of modernity and technology. When Nikolai Chernyshevsky wrote his radical socialist novel What Is to Be Done?, he used the Crystal Palace as a metaphor. If we all become socialist, he said, we can turn society into a Crystal Palace. And then we can do anything we want!
And that's where Dostoevsky comes in. The Underground Man first brings up the notion of the Crystal Palace in Part I, Chapter Seven when he's talking about how awful things will be if we ever figure out all the rules of nature (because then man won't have free will anymore). It is at this moment, he says, that the Crystal Palace will finally be built. Then he digresses for a bit. When he comes back to it, he's got a few reasons why we shouldn't go along with Chernyshevsky.
Reason #1: Hen houses are not mansions
The Underground Man talks for a bit about the dangers of establishing an ideal: once we do, we'll never settle for anything less. This is part of the danger with wanting or expecting a Crystal Palace of utopian perfection; we'll spend our whole lives being dissatisfied if we never get it.
Reason #2: Suffering is stylish right now
The Underground Man knows that "in the Crystal Palace, it [suffering] is unthinkable." As he's told us time and time again, suffering is man's best friend. We enjoy it – inflicting pain is our way of proving that we have free will and aren't beholden to the rational laws of nature. Living in a Crystal Palace would mean giving up suffering, which would mean abandoning free will.
Reason #3: You can't make fun of the Crystal Palace.
More than anything, the Underground Man rejects the idea that he can't "stick [his] tongue out at" the Crystal Palace. This is highly problematic for a man who makes it his life's work to mock everything.
2+2=4 and the Stone Wall
These two ideas come up a lot. And they really piss off the Underground Man. His argument goes like this: 1) free will means man gets to do whatever he wants. 2) The laws of nature say that 2+2 always has to equal 4. 3) He wants 2+2 to equal 5. 4) The laws of nature are a stone wall standing in the way of his free will.
The metaphor continues when the Underground Man compares the way a man of action would act as opposed to a man of consciousness. The man of action sees the wall and goes, "Oh, it's a wall," and leaves it at that. But the Underground Man is "not going to be reconciled to it simply because it is a stone wall." Rather, he's going to smash his head against it, even knowing the futility of his actions.
So how is this a symbol? Well, when the Underground Man talks about the rule of 2+2=4, he isn't just talking about 2+2=4. He's also talking about…3+3=6? Well, yes, but even more than that, he's talking about all rules of rationale and logic: the rule that if you're hungry, you'll eat; if a dog is threatened, he'll show his teeth; if someone hits you, you will feel pain. He finds all of these rules to be restrictive in the same way that this very specific example – the 2+2=4 example – is limiting to his free will. The stone wall, then, is a great way to represent the barrier that these rules construct.
We know that the Underground Man has been underground for twenty years, but he tells stories about his interactions with the world sixteen years ago. So when he talks about being "underground," he's speaking metaphorically.
So think about what this indicates. Underground = isolated. The Underground Man is solitary. When he witnesses other people speaking, he describes it as "listening through a crack under the floor." When he speaks to others, he imagines himself coming "out into the light of day." When he oscillates between wanting friends and preferring his solitude, it is a decision between the real world and his imagined world underground.
Why underground? That is, why doesn't the Underground Man imagine himself indoors, or in a closet, or behind a tree? What is it about being underground that is such a powerful image? Well, taken literally, if the Underground Man is under the ground, he's beneath us. What we're getting at is the connotation of inferiority – fascinating, since all the guy seems to do is tell us he's "more intelligent" than any of the men walking around above him. This conflict – the fragile ego afraid of being "beneath" and convincing itself it's "above" – is central to Notes from the Underground.
"The Sublime and Beautiful" (or "the lofty and beautiful" in some translations)
This phrase comes from Immanuel Kant's 1764 essay, Observations on the Feeling of the Beautiful and Sublime. In this work, Kant (an enlightenment philosopher) explores these two emotions we feel when looking at things like mountains or roses or breathtaking artwork. Dostoevsky appropriates the term and uses it to describe, somewhat satirically, the Underground Man's "appreciation" for all that is fine in the world. The Underground Man tells us of his "attacks" of the sublime and beautiful, moments when he is "conscious of goodness" in the world. Unfortunately, those moments end up driving him into mire and filth. The sublime and beautiful, then, come to represent one half of the Underground Man's crazy flip-flopping between a life underground and an aesthete's awe of the best the world has to offer.