There really aren't all that many physical objects in this story. Actually, we can only think of one (aside from all those books, of course): the letter that the lawyer writes renouncing the bet money. It's a ranty, semi-crazed letter, and its meaning is not entirely clear, so let's focus instead on the actual piece of paper thing that the letter is—the "sheet of paper on which there was something written in fine handwriting" (2.10).
For one thing, this little sheet of paper saves the lawyer's life. There the banker is, all ready to whack the guy, when he catches sight of the thing out of the corner of his eye. The info in the letter shows there is no reason for the murder anymore, because it communicates the lawyer's newfound belief system. So at this point, the letter is a substitute for speech. It's basically the paper version of the guy cowering before a hit man begging "please don't kill me, you can have your money." In other words, it's connected pretty totally to the lawyer.
But what happens when the lawyer leaves and the letter stays behind? The banker "took from the table the writing in which the millions were renounced, and when he got home locked it up in the fireproof safe" (2.20).
Now suddenly, the letter isn't really speech, but instead evidence. In the hands of the banker, it becomes yet another in a series of proofs that the banker keeps demanding for the big discussions and decisions in his life.
First, he turns an abstract chat into a physical challenge when he starts up the bet. Then, when he is planning on killing the lawyer, he puzzles out that "I have only to take this half-dead man, throw him on the bed, stifle him a little with the pillow, and the most conscientious expert would find no sign of a violent death" (2.11). And finally, he takes the letter as proof that he doesn't owe the lawyer squat.
In the end, the banker is really all about the material world, and evidence, in all its forms, is yet another example of his need for actual proof in metaphorical pudding.