The Confusion

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 beginning to feel like the character of Simplicio in one of Galileo’s dialogs. So let me play that role to the hilt, and ask you how you intend to solve the problem.”

“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 role 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’.

???? 2 Plato

???? 3 Aristotle

???? 5 Trees

???? 7 Turtles

???? 2?5=10 Plato on Trees

???? 3?7=21 Aristotle on Turtles

???? 2?7=14 Plato on Turtles

???? 3?5=15 Aristotle on Trees

???? [etc.]

“Two, three, five, and seven-all prime numbers,” remarked Fatio after giving it a brief study. “The shelf-numbers are composites, the products of prime factors. Excellent, Doctor! By making this small improvement-assigning prime numbers, instead of counting numbers, to the various subjects-you have eliminated the problem. The shelf position of any book may be found by multiplying the subject-numbers-and you may be assured it will be unique.”

“It is a pleasure to explain it to one who grasps the principle so readily,” Leibniz said. “Huygens and the Bernoullis have both spoken highly of you, Fatio, and I can see that they were by no means insincere.”

“I am humbled to hear my name mentioned in the same sentence with theirs,” Fatio returned, “but since you have been kind enough to so favor me, perhaps you will indulge me in a question?”

“It would be my privilege.”

“Your scheme is a fine way to build a library. For the correct position of any book may be found by taking the product of the several primes that correspond to its subjects. Even when those numbers grow to several digits, that presents no great difficulty; and in any event it is well known that you have invented a machine capable of multiplying numbers with great facility, which I now perceive is just one element of the immense knowledge engine you have proposed to build.”

“Indeed, all of these are of a piece, and may be considered aspects of my Ars Combinatorica. Did you have a question?”

“I fear that your library, once built, will be difficult to understand. You are seeking the help of the Emperor in Vienna, are you not?”

“It cannot be accomplished without the resources of a great kingdom,” Leibniz said vaguely.

“Very well, perhaps you are in communication with some other great prince. At any rate, it would seem, then, that you wish to make your Knowledge Engine on a colossal scale.”

“Marshalling resources is a continuing problem,” the Doctor said, still treading gingerly.