The Confusion

"And you may answer in any mode you like, Monsieur Fatio, provided that you do answer."

"I suppose one would go by the numbers. Supposing that they were shelved in numerical order."

"Suppose they were. The numbers merely denote the order in which the Duke acquired, or at least cataloged, the volumes. They say nothing of the contents."

"Re-number them, then."

"According to what scheme? By name of author?"

"I believe it would be better to use something like Wilkins's philosophical language. For any conceivable subject, there would be a unique number. Write that number on the spine of the book and shelve them in order. Then you can go directly to the right part of the library and find all books on a given subject together."

"But suppose I am making a study of Aristotle. Aristotle is my subject. May I expect to find all Aristotle-books shelved together? Or would his works on geometry be shelved in one section, and his works on physics elsewhere?"

"If you look at it that way, the problem is most difficult."

Leibniz stepped over to an empty bookcase and drew his finger down the length of one shelf from left to right. "A shelf is akin to a Cartesian number-line. The position of a book on that shelf is associated with a number. But only one number! Like a number-line, it is one-dimensional. In analytic geometry we may cross two or three number-lines at right angles to create a multi-dimensional space. Not so with bookshelves. The problem of the librarian is that books are multi-dimensional in their subject matter but must be ordered on one-dimensional shelves."

"I perceive that clearly now, Doctor," Fatio said. "Indeed, I am begi

"Well played, sir. Consider the following: Suppose we assign the number three to Aristotle, and four to turtles. Now we must decide where to shelve a book by Aristotle on the subject of turtles. We multiply three by four to obtain twelve, and then shelve the book in position twelve."

"Excellent! By a simple multiplication you have combined several subject-numbers into one—collapsed the multi-dimensional space into a uni-dimensional number-line."

"I am pleased that you favor my proposal thus far, Fatio, but now consider the following: suppose we assign the number two to Plato, and six to trees. And suppose we acquire a book by Plato on the subject of trees. Where does it belong?"

"The product of two and six is twelve—so it goes next to Aristotle's book on turtles."

"Indeed. And a scholar seeking the latter book may instead find himself with the former—clearly a failure of the cataloging system."

"Then let me step once again into the rôle of Simplicio and ask you whether you have solved this problem."

"Suppose we use this coding instead," quoth the Doctor, reaching behind the bookcase and pulling out a slate on which the following table had been chalked—thereby as much as admitting that the conversation, to this point, had been a scripted demo'.