It turns out his schedule is very similar to yours though, and you start to see him more frequently. You nod, say 'hello', and eventually he is a familiar face and you start making small talk. 'Look at this weather!' 'So glad spring is here.'
Now that you know his face, you see him more places. At the park, even the grocery store occasionally. You get a feel for his life and routine, and the small talk becomes a little more familiar. You ask after families, and maybe talk about trips you're taking.
And it's happened. This person has become an acquaintance, almost a friend, but you still don't know his name.
Now you're stuck. You've been chit-chatting on the elliptical machines for months, you know about his children's chicken pox, you told him about your grandmother in Jersey, how can you ask his name now??
It gets worse, if you discover that somehow he's learned your name. You hear him them greet you from across the parking lot, and your gut clenches in embarrassment. You're the only fool who doesn't know!
So you start small. When you are both pulling mail from the community mailboxes, you surreptitiously glance at his envelopes. "Occupant" -- dammit! Or maybe his mail is always wrapped in a newspaper, no clues at all.
Maybe his apartment buzzer will show a name? So you tail him home one day, to determine which building is his. He enters, and you look at buzzers, but the labels are all haphazard and meaningless.
Next you try the classic Introduction Trick. You walk into the swimming pool with a buddy of yours, knowing that he'll be there too. Casually you walk up you say "Oh hey, have you two met?" Then you hope against hope that they'll introduce themselves to one another.
But they don't, instead they just stand there with polite smiles on their faces and stare at you, waiting for you to make the introductions. So you're forced to fake a severe nosebleed and rush away with your hands over your face.
The situation growing desperate, you start going through his trash. There must be something in here to tell you who this person is! But he is one of those over-paranoid types who shred all their personal documents. And not just in strips, but in those little confetti squares, so you still can't identify him. What is this world coming to, that people can't even trust simple trash-pickers any longer?
Left with no options, obviously next you start stealing his mail. Unfortunately all the damn mail is addressed to his wife. Maybe now you have a family name? But maybe she kept her own last name -- you have no way of telling.
And you're still seeing this guy every morning in the gym, and his friendly hellos are eating away at your insides and filling you with rage. You hate him! Who is this guy, and how in the hell did you get into this ridiculous situation?!?
At this point you are left with two options: burn down his apartment or move away. Either is justifiable, but I'd recommend the second if at all possible....