Peter Ivanovich provides a framework for depicting the character and circumstances of Ivan Ilyich. They are of the same kind in every way: classmates, court officials, bridge lovers. They belong to the same social class. The only difference between them is that Ivan is now dead, while Peter is alive. By presenting us Peter Ivanovich, in all his mediocrity, Tolstoy wants to tell us that this is what Ivan had been.
Peter is quite confused by the news of Ivan's death. He can't really mourn him, as they were never true friends (even though we are informed that Peter was one of two Ivan's nearest acquaintances). Peter is only interested in the opportunity for his brother in law to take Ivan's position.
"'Now I must apply for my brother-inlaw’s transfer from Kaluga,' thought Peter Ivanovich. 'My wife will be very glad, and then she won’t be able to say that I never do anything for her relations.'
'I thought he would never leave his bed again,' said Peter Ivanovich aloud. 'It’s very sad.'"
So, he considers the benefits he could get from Ivan's death, while pretending to mourn him.
