Levin and his brother have an extended discussion about Levin's failure to stay on the District Council and try to improve living conditions (i.e., schools, health care) for those in the district.
Koznyshev points out that Levin's district is in bad shape: they contribute taxes that pay local council salaries, but there are still no public schools, medical care, midwives, or pharmacies ("dispensaries," as he calls them). How can Levin turn his back on the peasants he claims to love?
Levin thinks to himself that he never said he loved the peasants. What's more, these issues don't concern him directly. He's half-listening and thinking more about the ploughing in his fields, which seems to have stopped prematurely.
Koznyshev continues. Either Levin can't see all the good that he could do, or else he's unwilling to give up his time to do it.
Levin's feeling unable to defend himself and offended by his brother's questions. He knows he's about to be pushed into admitting that he doesn't care about "the common cause." Levin replies that the kinds of reforms Koznyshev is talking about seem impossible in a district of three thousand square miles, with its bad weather and seasonal work. What's more, Levin doesn't see the point of either medicine or schools.
Koznyshev asks how there can be any doubt about the usefulness of education? If it's good for Levin himself, why wouldn't it be good for the peasants?
Levin gives his main reason for not caring about "the common cause." Why should he build schools that neither he nor his children will ever use, and that the peasants don't want to attend? Not only do the peasants not want to bother with education, but schooling makes them useless for their actual jobs.
Levin asks Koznyshev to prove to him philosophically that education is good when it comes to the peasantry. When Koznyshev says that he doesn't see what philosophy's got to do with it, Levin gets upset.
Levin furiously argues that the motive of all actions is personal happiness. As a nobleman, he sees nothing in the zemstvo District Councils that contributes to his personal well-being.
The roads are still bad, and won't improve. He has no personal need of a justice of a peace or a doctor. Finally, he thinks that schools are bad for accomplishing his agricultural goals. It's all a giant waste of his time and money, since he has to pay for the privilege to participate in these councils.
Koznyshev thinks he has Levin at this point. The emancipation of the serfs was not motivated by self-interest, and yet we still did it.
(By the way: Russian serfs were functionally slaves of their feudal, aristocratic masters. Strongly influenced by anti-slavery sentiment from Europe and the United States, the reformist Czar Alexander II freed the serfs in 1861, sixteen years before the final publication of Anna Karenina in 1877. See The Encyclopedia Britannica on the subject.)
Levin's getting worked up: he says that freeing the serfs was a matter of throwing off the yoke oppressing noblemen and all good people. But working on the council and arguing about pipe installation in a town he doesn't even live in is a different thing.
Koznyshev asks if Levin were put on trial tomorrow, would he rather be tried by the criminal courts or trial by jury (introduced to Russia in 1864 by the same Alexander II who liberated the serfs. See this article on jury trials in Russia for more information).
Levin says that Koznyshev's arguments are illogical because he's not going to go on trial tomorrow. All this zemstvo stuff is a sham taken from current institutions in Europe. None of it's motivated by real personal investment.
Koznyshev thinks Levin's being a hypocrite, since he too has a philosophy of his own that fits his own desires. He adds that philosophy is about finding precisely that connection between personal and general interests. What's more, the only nations that have a future are the ones that carefully look after their institutions.
Levin now feels totally left out of the argument: he can't argue with Koznyshev's greater intellectual and philosophical skill, but he still feels that his brother just doesn't get what Levin has been trying to say.
Instead of restarting the argument, though, Levin just retreats into thinking about his own concerns (elaborated in the next chapter), and they untie the horse and drive away from the fishing hole.