Tom Hillman is shaking his head at editors who footnote things that can’t conceivably need footnotes. When we discussed this on Twitter, Tom tried to cast it in the form of a natural law:

@virginicus we need the Hillman-Hoffman Law of fn utility, their probability being inversely proportional to the square of their utility.

— tom hillman (@alas_not_me) December 3, 2016

That got me wondering what the utility of a footnote might be. As with any form of communication, it must be related to the difference in knowledge of the writer and the reader. Let’s suppose there’s a set of intended readers. Some of them know the facts in question; others do not. For any fact *i*, define *R(i)* as the fraction of the audience that knows it. This knowledge is measured at the point where the footnote is marked. Define *A(i)* as the author’s knowledge of the fact on a scale from 0 to 1, where 0 is perfect cluelessness and the total knowledge of all relevant facts is normalized to 1.

Now, we can use a result from information theory called the information gain between two probability distributions. In place of the two distributions, we use *R(i)* and *A(i)*, and the utility of a footnote is:

*R(i)*log[

*R(i)/A(i)*]

where the sum is over all facts

*i*. (Sorry, Tom — in Digital Humanities, the logarithms just keep coming.)

Using the example in the linked post, the text contains two facts: A. E. Houseman’s name and the title of the book *A Shropshire Lad*. *R*(1) = 1 and *R*(2) =1; all the readers know these things because C.S. Lewis just told us. Since R=A, the contribution to the utility from these terms is log(1) = 0. The footnote adds a third fact, that the book was published in 1896. How useful is that? Let’s suppose that Idiosophers are typical readers of this book. Before reading that line, I’d have said *A Shropshire Lad* was published in the 1890’s. The number 1896 has 11 bits of information in it, of which I knew 8. (8/11)*log(8/11)=-0.33, where the answer is in bits because I took the log to the base 2.

The total utility of this footnote is therefore U=0+0+0.33, or one third of a bit of information. (For purposes of comparison, a *useful* footnote might contain tomorrow’s winning Pick-3 lottery numbers, which is 10 bits of information.) This footnote is therefore almost worthless, so by the Hillman-Hoffman law, its appearance in the book was almost inevitable.

*Nota Bene*: the utility formula goes to infinity if the author does not know what he’s talking about and the readers do, i.e. if *A(i)* = 0 when *R(i)*>0. This is the case for student papers, which implies that footnotes there are of infinite utility. There is no way to have too many footnotes in a thesis.