Bring a large saucepan filled with salted water to a boil. Follow package directions, add the spaghetti so that it is ready just as the tomatoes are cooked.
Pour two tablespoons of the olive oil into a large, flameproof casserole or heavy saute pan. Lay out the anchovy fillets, side by side, in the oil and place the pan on the lowest possible heat. The anchovies will melt in the warm oil, they should not fry.
Meanwhile, in a large frying pan, warm the remaining 3 tablespoons of olive oil over high heat. Add the garlic cloves and when they begin to sizzle but before they brown, add tomatoes and salt to taste. Be careful not to burn garlic. Toss the tomatoes until their liquid evaporates, about 5 minutes; the tomatoes should not cook to a puree. Add the olives and empty the frying pan into the pan with the anchovies.
Grind over some pepper and stir in the basil.
Drain the spaghetti and add the anchovy sauce. Toss together with a fork and spoon.
Serve in heated pasta dishes or soup plates, accompanied with a dish of Parmesan and a dish of butter to add as desired.