A physicist loose among the liberal arts

Category: trivial (Page 2 of 2)

A Contribution to the Mathematical Theory of Footnotes

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:

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:

U = –Σ 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.


The Monday meeting of the Defenders of Denethor is now in session. I commented over there, but I’m putting this here because Stephen’s got a serious discussion going on, and this gets less serious the more I think about it.

Where I think we both ended up is, Denethor is doing the right thing according to his reason. His proposed course of action is entirely defensible, all his priorities are well established, he’s acting within his authority, and if anything went wrong his CYA package was in order. Unfortunately, he’s operating outside the theater of reason alone. The circumstances require faith in Something much greater than the works of Men, which Denethor doesn’t have, or hasn’t found use for in government.

Here in our world, any christian (and large numbers from other religions) will tell you we have Scripture to tell us about that Something.  Nothing of the sort exists in Gondor.  I suppose the Steward could go ask Elrond, which is kind of what Boromir is doing at the Council.  Or he could ask Galadriel and Celeborn, since they were around for even more of the backstory.  Or Círdan would have an interesting perspective.  But these people all have their own interests, their own motives.  He’d never be sure they were telling him everything.  If you really want Denethor to take supernatural powers into his calculations, you’d have to give him something in writing.

What img_0159 if Denethor got hold of a copy of the Letters?  Would that have the same theological impact as the Epistles of St. Paul have in ours?  After all, Middle Earth has an omniscient creator (he’d say “subcreator”) who knows and sees all, and has a Plan for the world.

The book would contain the creator’s secret thought,  his intentions that didn’t make it into the obvious plot. It contradicts itself in some places, and is frustratingly silent when it gets to some things you really need to know. Some parts would make no sense at all to a character from LotR. It has all the trappings of the foundational text of a religion.

Somehow, though, I don’t see the Men of Gondor accepting it that way.

Take the Canon Quiz!

Brenton’s at it again. He has made a manageable list of canonical works, “for those who are inclined to soak in this great tradition.” I’m all for that. Let a hundred Harolds bloom. But it seems like missing the point of (a) lists of canonical works and (b) the World-Wide Web, if you make one without making a way to keep score.

Click the appropriate button for whether you’ve read the work (1 point), read part of it (half a point), or haven’t read it at all (no points).  I scored 12.5. I resisted the temptation to give bonus points for reading things in their original languages.

Read Partly Nope
Foundational Work (Theocratic Age)
The Iliad (Greek, 8th BCE)
The Odyssey (Greek, 8th BCE)
Virgil, The Aeneid (Latin, 29-19 BCE)
The Bible
Late Medieval and Renaissance (Aristocratic Age)
Dante Alighieri, Comedia/The Divine Comedy (Italian, 1308-1320)
Chaucer, Canterbury Tales (English, 1475)
Love’s Labour’s Lost (English, 1597)
Hamlet (English, 1603)
Othello (English, 1604)
King Lear (English, 1606)
Macbeth (English, 1611)
Miguel de Cervantes, Don Quixote (Spanish, 1605)
Moliere, The Misanthrope (French, 1666)
John Milton, Paradise Lost (English, 1667)
James Boswell, The Life of Samuel Johnson (English, 1791)
Johann Wolfgang von Goethe, Faust (German, 1772-1790)
19th Century+
William Wordsworth
• ‘The Ruined Cottage’ (English, 1800)
• ‘Tintern Abbey’ (English, 1798)
Jane Austen, Persuasion (English, 1818)
Walt Whitman,
Leaves of Grass (English, 1855)
• ‘Song of Myself’ (English, 1855)
Emily Dickinson, The Poems of Emily Dickinson (English, 1800s)
Charles Dickens, Bleak House (English, )
George Eliot, Middlemarch (English, 1874)
Henrik Ibsen, Peer Gynt (Norwegian, 1876)
Leo Tolstoy, Hadji Murad (Russian, 1896-1904)
Marcel Proust, In Search of Lost Time = Remembrance of Things Past (1913)
James Joyce, Ulysses (English, 1922)
Virginia Woolf
Orlando (English, 1928)
A Room of One ‘s Own (English, 1929)
Franz Kafka, The Blue Octavo Notebooks (German, 1917-1919)
Jorge Luis Borges, The Garden of Forking Paths (Spanish, 1941)
Pablo Neruda, Canto General (Spanish, 1938-1950)
Samuel Beckett
Endgame (English, 1957)
Murphy (English, 1938)
Waiting for Godot (English, 1953)

Brandy is too a Panacea

For anybody who’s laughing at the doctors in Bram Stoker’s Dracula who prescribe brandy for any illness: Remember that Elrond gave Gandalf a flask containing a cordial that was a sure-fire cure for hypothermia and squid attacks.

I’ll show myself out, thanks

While I’m thinking of multi-lingual puns:

The lake was too polluted for fishing or swimming, so the town council declared that the water was verboten.

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