Whence function notation?

September 28th, 2015

I begin — in continental style, unmotivated and, frankly, gratuitously — by defining Ackerman’s function \(A\) over two integers:

\[ A(m, n) = \left\{ \begin{array}{l}
n + 1 & \mbox{ if $m=0$ } \\
A(m-1, 1) & \mbox{ if $m > 0$ and $n = 0$ } \\
A(m-1, A(m, n-1)) & \mbox{ if $m > 0$ and $n > 0$ }
\end{array} \right. \]

… drawing their equations evanescently in dust and sand…
…drawing their equations evanescently in dust and sand…
Image of “Death of Archimedes” from Charles F. Horne, editor, Great Men and Famous Women, Volume 3, 1894. Reproduced by Project Gutenberg. Used by permission.

You’ll have appreciated (unconsciously no doubt) that this definition makes repeated use of a notation in which a symbol precedes a parenthesized list of expressions, as for example \(f(a, b, c)\). This configuration represents the application of a function to its arguments. But you knew that. And why? Because everyone who has ever gotten through eighth grade math has been taught this notation. It is inescapable in high school algebra textbooks. It is a standard notation in the most widely used programming languages. It is the very archetype of common mathematical knowledge. It is, for God’s sake, in the Common Core. It is to mathematicians as water is to fish — so encompassing as to be invisible.

Something so widespread, so familiar — it’s hard to imagine how it could be otherwise. It’s difficult to un-see it as anything but function application. But it was not always thus. Someone must have invented this notation, some time in the deep past. Perhaps it came into being when mathematicians were still drawing their equations evanescently in dust and sand. Perhaps all record has been lost of that ur-application that engendered all later function application expressions. Nonetheless, someone must have come up with the idea.

… that ur-application…
…that ur-application…
Photo from the author.

Surprisingly, the origins of the notation are not shrouded in mystery. The careful and exhaustive scholarship of mathematical historian Florian Cajori (1929, page 267) argues for a particular instance as originating the use of this now ubiquitous notation. Leonhard Euler, the legendary mathematician and perhaps the greatest innovator in successful mathematical notations, proposed the notation first in 1734, in Section 7 of his paper “Additamentum ad Dissertationem de Infinitis Curvis Eiusdem Generis” [“An Addition to the Dissertation Concerning an Infinite Number of Curves of the Same Kind”].

The paper was published in 1740 in Commentarii Academiae Scientarium Imperialis Petropolitanae [Memoirs of the Imperial Academy of Sciences in St. Petersburg], Volume VII, covering the years 1734-35. A visit to the Widener Library stacks produced a copy of the volume, letterpress printed on crisp rag paper, from which I took the image shown above of the notational innovation.

Here is the pertinent sentence (with translation by Ian Bruce.):

Quocirca, si \(f\left(\frac{x}{a} +c\right)\) denotet functionem quamcunque ipsius \(\frac{x}{a} +c\) fiet quoque \(dx − \frac{x\, da}{a}\) integrabile, si multiplicetur per \(\frac{1}{a} f\left(\frac{x}{a} + c\right)\).
[On account of which, if \(f\left(\frac{x}{a} +c\right)\) denotes some function of \(\frac{x}{a} +c\), it also makes \(dx − \frac{x\, da}{a}\) integrable, if it is multiplied by \(\frac{1}{a} f\left(\frac{x}{a} + c\right)\).]

There is the function symbol — the archetypal \(f\), even then, to evoke the concept of function — followed by its argument corralled within simple curves to make clear its extent.

It’s seductive to think that there is an inevitability to the notation, but this is an illusion, following from habit. There are alternatives. Leibniz for instance used a boxy square-root-like diacritic over the arguments, with numbers to pick out the function being applied: \( \overline{a; b; c\,} \! | \! \lower .25ex {\underline{\,{}^1\,}} \! | \), and even Euler, in other later work, experimented with interposing a colon between the function and its arguments: \(f : (a, b, c)\). In the computing world, “reverse Polish” notation, found on HP calculators and the programming languages Forth and Postscript, has the function symbol following its arguments: \(a\,b\,c\,f\), whereas the quintessential functional programming language Lisp parenthesizes the function and its arguments: \((f\ a\ b\ c)\).

Finally, ML and its dialects follow Church’s lambda calculus in merely concatenating the function and its (single) argument — \(f \, a\) — using parentheses only to disambiguate structure. But even here, Euler’s notation stands its ground, for the single argument of a function might itself have components, a ‘tuple’ of items \(a\), \(b\), and \(c\) perhaps. The tuples might be indicated using an infix comma operator, thus \(a,b,c\). Application of a function to a single tuple argument can then mimic functions of multiple arguments, for instance, \(f (a, b, c)\) — the parentheses required by the low precedence of the tuple forming operator — and we are back once again to Euler’s notation. Clever, no? Do you see the lengths to which people will go to adhere to Euler’s invention? As much as we might try new notational ideas, this one has staying power.


Florian Cajori. 1929. A History of Mathematical Notations, Volume II. Chicago: Open Court Publishing Company.

Leonhard Euler. 1734. Additamentum ad Dissertationem de Infinitis Curvis Eiusdem Generis. In Commentarii Academiae Scientarium Imperialis Petropolitanae, Volume VII (1734–35), pages 184–202, 1740.

The lot is cast into the lap,
but its every decision is from the Lord.
(NIV Proverbs 16:33)

…“Lux et Veritas”…
…“Lux et Veritas”…
Seal of Yale University image from Wikimedia Commons.

The seal of Yale University shows a book with the Hebrew אורים ותמים (urim v’thummim), a reference to the Urim and Thummim of the Old Testament. The Urim and Thummim were tools of divination. They show up first in Exodus:

Also put the Urim and the Thummim in the breastpiece, so they may be over Aaron’s heart whenever he enters the presence of the Lord. Thus Aaron will always bear the means of making decisions for the Israelites over his heart before the Lord. (NIV Exodus 28:30; emphasis added)

Apparently, the Urim and Thummim worked like flipping a coin, providing one bit of information, a single binary choice communicated from God.1 Translations of Urim as “guilty” and Thummim as “innocent” indicate that the divination was used to determine guilt: “Thummim you win; Urim you lose.” An alternate translation has Urim “light” and Thummim “truth”, hence “Lux et Veritas” in the Yale seal’s banner.

Saul, King of Israel and father-in-law of King David, uses the binary choice provided by Urim and Thummim divination in 1 Samuel to unmask the party who violated the king’s oath:

Then Saul prayed to the Lord, the God of Israel, “Why have you not answered your servant today? If the fault is in me or my son Jonathan, respond with Urim, but if the men of Israel are at fault, respond with Thummim.” Jonathan and Saul were taken by lot, and the men were cleared. Saul said, “Cast the lot between me and Jonathan my son.” And Jonathan was taken. (NIV2 1 Samuel 14:41-42)

Saul executes a small (and highly unbalanced) binary search. He first divides the population into two parts. He and his son Jonathan are assigned Urim and all the rest get Thummim. God responds with Urim. Then to decide between Saul and Jonathan, the process is repeated, and Jonathan is fingered as the guilty party. (The method apparently works; the preceding verses of 1 Samuel give the story of Jonathan’s transgression.)

The universality of binary as an information conveying method has a longer history than one might have thought.

  1. The Old Testament Urim and Thummim should not be confused with the “higher bandwidth” device of the same name that Joseph Smith claimed to use to receive the translation of a now lost 116 pages of the Book of Mormon. This device purportedly resembled a pair of spectacles with transparent rocks for lenses, a kind of “oraculus rift”. For the extant Book of Mormon, Smith changed his method to scrying with a “seer stone” placed in his hat. A photograph of the stone, coincidentally, has just recently been released by the LDS church.
  2. The use of a Septuagint-based version of the Bible, here the New International Version, is important, as this verse is considerably shortened in versions such as the King James based on the Masoretic text, leaving out the use of Urim and Thummim to make a binary decision.

Becoming tin men

August 3rd, 2015

From the 2015 introduction to the 1965 novel The Tin Men by Michael Frayn:

“I hadn’t in those days heard of the Turing Test—Alan Turing’s proposal that a computer could be said to think if its conversational powers were shown to be indistinguishable from a human being’s—so I didn’t realise that what I was suggesting was a kind of converse of it: a demotion of human beings to the status of machines if their intellectual performance was indistinguishable from a computer’s, and they become tin men in their turn. The William Morris Institute is about to be visited by the Queen for the opening of a new wing, and I realise with hindsight that I’ve used a similar idea quite often since: the grand event that goes wrong, and deposits the protagonists into the humiliating gulf that so often in life opens between intention and achievement. My characters at the Institute could have written a story programme for me and saved me a lot of work. I’ve become a bit of a tin man myself.”

…“blockhead” argument…
…“blockhead” argument…
“Blockhead by Paul McCarthy @ Tate Modern” image from flickr user Matt Hobbs. Used by permission.

Alan Turing proposed what is the best known criterion for attributing intelligence, the capacity for thinking, to a computer. We call it the Turing Test, and it involves comparing the computer’s verbal behavior to that of people. If the two are indistinguishable, the computer passes the test. This might be cause for attributing intelligence to the computer.

Or not. The best argument against a behavioral test of intelligence (like the Turing Test) is that maybe the exhibited behaviors were just memorized. This is Ned Block’s “blockhead” argument in a nutshell. If the computer just had all its answers literally encoded in memory, then parroting those memorized answers is no sign of intelligence. And how are we to know from a behavioral test like the Turing Test that the computer isn’t just such a “memorizing machine”?

In my new(ish) paper, “There can be no Turing-Test–passing memorizing machines”, I address this argument directly. My conclusion can be found in the title of the article. By careful calculation of the information and communication capacity of space-time, I show that any memorizing machine could pass a Turing Test of no more than a few seconds, which is no Turing Test at all. Crucially, I make no assumptions beyond the brute laws of physics. (One distinction of the article is that it is one of the few philosophy articles in which a derivative is taken.)

The article is published in the open access journal Philosophers’ Imprint, and is available here along with code to computer-verify the calculations.

The two Guildford mathematicians

February 18th, 2015

…the huge ledger…
…the huge ledger…
Still from Codebreaker showing Turing’s checkout of three Carroll books.

The charming town of Guildford, 40 minutes southwest of London on South West Trains, is associated with two famous British logician-mathematicians. Alan Turing (on whom I seem to perseverate) spent time there after 1927, when his parents purchased a home at 22 Ennismore Avenue just outside the Guildford town center. Although away at his boarding school, the Sherborne School in Dorset, which he attended from 1926 to 1931, Turing spent school holidays at the family home in Guildford. The house bears a blue plaque commemorating the connection with Turing, the “founder of computer science” as it aptly describes him, which you can see in the photo at right, taken on a pilgrimage I took this past June.

…the family home…
…the family home…
The Turing residence at 22 Ennismore Avenue, Guildford

And this brings us to the second famous Guildford mathematician, who it turns out Turing was reading while at Sherborne. In the Turing docudrama Codebreaker, one of Turing’s biographers David Leavitt visits Sherborne and displays the huge ledger used for the handwritten circulation records of the Sherborne School library. There (Leavitt remarks), in an entry dated 11 April 1930, Turing has checked out three books, including Alice in Wonderland and Through the Looking Glass, and What Alice Found There. (We’ll come back to the third book shortly.) The books were, of course, written by the Oxford mathematics don Charles Lutwidge Dodgson under his better known pen name Lewis Carroll. Between the 1865 and 1871 publications of these his two most famous works, Carroll leased “The Chestnuts” in Guildford in 1868 to serve as a home for his sisters. The house sits at the end of Castle Hill Road adjacent to the Guildford Castle, which is as good a landmark as any to serve as the center of town. Carroll visited The Chestnuts on many occasions over the rest of his life; it was his home away from Christ Church home. He died there 30 years later and was buried at the Guildford Mount Cemetery.

…through the looking glass…
…through the looking glass…
Statue of Alice passing through the looking glass, Guildford Castle Park, Guildford

Guildford plays up its connection to Carroll much more than its Turing link. In the park surrounding Guildford Castle sits a statue of Alice passing through the looking glass, and the adjacent museum devotes considerable space to the Dodgson family. A statue depicting the first paragraphs of Alice’s adventures (Alice, her sister reading next to her, noticing a strange rabbit) sits along the bank of the River Wey. The Chestnuts itself, however, bears no blue plaque nor any marker of its link to Carroll. (A plaque formerly marking the brick gatepost has been removed, evidenced only by the damage to the brick where it had been.)

…The Chestnuts…
…The Chestnuts…
The Dodgson family home in Guildford

Who knows whether Turing was aware that Carroll, whose two Alice books he was reading, had had a home a mere mile from where his parents were living. The Sherborne library entry provides yet another convergence between the two British-born, Oxbridge-educated, permanent bachelors with sui generis demeanors, questioned sexualities, and occasional stammers, interested in logic and mathematics.

But there’s more. What of the third book that Turing checked out of the Sherborne library at the same time? Leavitt finds the third book remarkable because the title, The Game of Logic, presages Turing’s later work in logic and the foundations of computer science. What Leavitt doesn’t seem to be aware of is that it is no surprise that this book would accompany the Alice books; it has the same author. Carroll published The Game of Logic in 1886. It serves to make what I believe to be the deepest connection between the two mathematicians, one that has to my knowledge never been noted before.

…Carroll’s own copy…
…Carroll’s own copy…
Title page of Lewis Carroll, The Game of Logic, 1886. EC85.D6645.886g, Houghton Library, Harvard University.

After watching Codebreaker and noting the Game of Logic connection, I decided to refresh my memory about the book. I visited Harvard’s Houghton Library, which happens to have Carroll’s own copy of the book. The title page is shown at right, with the facing page visible showing a sample card to be used in the game. The book was sold together with a copy of the card made of pasteboard and counters of two colors (red and grey) to be used to mark the squares on the card.

The Houghton visit and the handling of the game pieces jogged my memory as to the point of Carroll’s book. Carroll’s goal in The Game of Logic was to describe a system for carrying out syllogistic reasoning that even a child could master. Towards that goal, the system was intended to be completely mechanical. It involved the card marked off in squares and the two types of counters placed on the card in various configurations. Any of a large class of syllogisms over arbitrary properties can be characterized in this way, given a large enough card and enough counters, though it becomes unwieldy quite quickly after just a few.

… marked off in squares…
… marked off in squares…
The game card depicting a syllogism. Lewis Carroll, The Game of Logic, 1886. EC85.D6645.886g, Houghton Library, Harvard University.

(The photo at right shows the card and counters that came with the book. I’ve placed the counters in such a way as to depict the syllogism:

No red apples are unripe.
Some wholesome apples are red.

∴ Some ripe apples are red.         )

To computer scientists, this ought to sound familiar. Just six years after checking out The Game of Logic from his school library, Turing would publish his groundbreaking paper “On computable numbers”, in which he describes a system for carrying out computations in a way that is completely mechanical. It involves a paper tape marked off in squares, and markings of at least two types placed on the tape in various configurations. Any of a large class of computations over arbitrary values can be characterized in this way, given a large enough tape and enough markings, though it becomes unwieldy quite quickly. We now call this mechanical device with tape and markings a Turing machine, and recognize it as the first universal model of computation. Turing’s paper serves as the premier work in the then nascent field of computer science.

Of course, there are differences both superficial and fundamental between Carroll’s game and Turing’s machine. Carroll’s card is two-dimensional with squares marked off in a lattice pattern, and counters are placed both within the squares and on the edges between squares. Turing’s tape is one-dimensional (though two-dimensional Turing machines have been defined and analyzed) and the markings are placed only within the squares. Most importantly, nothing even approaching the ramifications that Turing developed on the basis of his model came from Carroll’s simple game. (As a mathematician, Carroll was no Turing.) Nonetheless, in a sense the book that Turing read at 17 attempts to do for logic what Turing achieved six years later for computation.

I have no idea whether Lewis Carroll’s The Game of Logic influenced Alan Turing’s thinking about computability. But it serves as perhaps the strongest conceptual bond between Guildford’s two great mathematicians.

Update February 25, 2015: Thanks to Houghton Library Blog for reblogging this post.

The Turing moment

November 30th, 2014

Ed Stoppard as Alan Turing in Codebreaker
…less histrionic…
Ed Stoppard as Alan Turing in Codebreaker

We seem to be at the “Turing moment”, what with Benedict Cumberbatch, erstwhile Sherlock Holmes, now starring as a Hollywood Alan Turing in The Imitation Game. The release culminates a series of Turing-related events over the last few years. The centennial of Turing’s 1912 birth was celebrated actively in the computer science community as a kind of jubilee, the occasion of numerous conferences, retrospectives, and presentations. Bracketing that celebration, PM Gordon Brown publicly apologized for Britain’s horrific treatment of Turing in 2009, and HRH Queen Elizabeth II, who was crowned a couple of years before Alan Turing took his own life as his escape from her government’s abuse, finally got around to pardoning him in 2013 for the crime of being gay.

I went to see a preview of The Imitation Game at the Coolidge Corner Theatre’s “Science on Screen” series. I had low expectations, and I was not disappointed. The film is introduced as being “based on a true story”, and so it is – in the sense that My Fair Lady was based on the myth of Pygmalion (rather than the Shaw play). Yes, there was a real place called Bletchley Park, and real people named Alan Turing and Joan Clark, but no, they weren’t really like that. Turing didn’t break the Enigma code singlehandedly despite the efforts of his colleagues to stop him. Turing didn’t take it upon himself to control the resulting intelligence to limit the odds of their break being leaked to the enemy. And so on, and so forth. Most importantly, Turing did not attempt to hide his homosexuality from the authorities, and promoting the idea that he did for dramatic effect is, frankly, an injustice to his memory.

Reviewers seem generally to appreciate the movie’s cleaving from reality, though with varying opprobrium. “The truth of history is respected just enough to make room for tidy and engrossing drama,” says A. O. Scott in the New York Times. The Wall Street Journal’s Joe Morgenstern ascribes to the film “a marvelous story about science and humanity, plus a great performance by Benedict Cumberbatch, plus first-rate filmmaking and cinematography, minus a script that muddles its source material to the point of betraying it.” At Slate, Dana Stevens notes that “The true life story of Alan Turing is much stranger, sadder and more troubling than the version of it on view in The Imitation Game, Morton Tyldum’s handsome but overlaundered biopic.

Of course, they didn’t make the movie for people like me, that is, people who had heard of Alan Turing before. And to the extent that the film contributes to this Turing moment — leading viewers to look further into this most idiosyncratic and important person — it will be a good thing. The Coolidge Corner Theatre event was followed by commentary from Silvio Micali and Seth Lloyd, both professors at MIT. (The former is a recipient of the highest honor in computer science, the Turing Award. Yes, that Turing.) Their comments brought out the many scientific contributions of Turing that were given short shrift in the film. If only they could duplicate their performance at every showing.

Those who become intrigued by the story of Alan Turing could do worse than follow up their viewing of the Cumberbatch vehicle with one of the 2012 docudrama Codebreaker, a less histrionic but far more accurate (and surprisingly, more sweeping) presentation of Turing’s contributions to science and society, and his societal treatment. I had the pleasure of introducing the film and its executive producer Patrick Sammon in a screening at Harvard a couple of weeks ago. The event was another indicator of the Turing moment. (My colleague Harry Lewis has more to say about the film.)

To all of you who are aware of the far-reaching impact of Alan Turing on science, on history, and on society, and the tragedy of his premature death, I hope you will take advantage of the present Turing moment to spread the word about computer science’s central personage.

With few exceptions, scholars would be better off writing their papers in a lightweight markup format called Markdown, rather than using a word-processing program like Microsoft Word. This post explains why, and reveals a hidden agenda as well.1

Microsoft Word is not appropriate for scholarly article production

Old two pan balance
Old two pan balance” image from Nikodem Nijaki at Wikimedia Commons. Used by permission.

Before turning to lightweight markup, I review the problems with Microsoft Word as the lingua franca for producing scholarly articles. This ground has been heavily covered. (Here’s a recent example.) The problems include:

Substantial learning curve
Microsoft Word is a complicated program that is difficult to use well.
Appearance versus structure
Word-processing programs like Word conflate composition with typesetting. They work by having you specify how a document should look, not how it is structured. A classic example is section headings. In a typical markup language, you specify that something is a heading by marking it as a heading. In a word-processing program you might specify that something is a heading by increasing the font size and making it bold. Yes, Word has “paragraph styles”, and some people sometimes use them more or less properly, if you can figure out how. But most people don’t, or don’t do so consistently, and the resultant chaos has been well documented. It has led to a whole industry of people who specialize in massaging Word files into some semblance of consistency.
Backwards compatibility
Word-processing program file formats have a tendency to change. Word itself has gone through multiple incompatible file formats in the last decades, one every couple of years. Over time, you have to keep up with the latest version of the software to do anything at all with a new document, but updating your software may well mean that old documents are no longer identically rendered. With Markdown, no software is necessary to read documents. They are just plain text files with relatively intuitive markings, and the underlying file format (UTF-8 née ASCII) is backward compatible to 1963. Further, typesetting documents in Markdown to get the “nice” version is based on free and open-source software (markdown, pandoc) and built on other longstanding open source standards (LaTeX, BibTeX).
Poor typesetting
Microsoft Word does a generally poor job of typesetting, as exemplified by hyphenation, kerning, mathematical typesetting. This shouldn’t be surprising, since the whole premise of a word-processing program means that the same interface must handle both the specification and typesetting in real-time, a recipe for having to make compromises.
Because Microsoft Word’s file format is effectively proprietary, users are locked in to a single software provider for any and all functionality. The file formats are so complicated that alternative implementations are effectively impossible.

Lightweight markup is the solution

The solution is to use a markup format that allows specification of the document (providing its logical structure) separate from the typesetting of that document. Your document is specified – that is, generated and stored – as straight text. Any formatting issues are handled not by changing the formatting directly via a graphical user interface but by specifying the formatting textually using a specific textual notation. For instance, in the HTML markup language, a word or phrase that should be emphasized is textually indicated by surrounding it with <em>…</em>. HTML and other powerful markup formats like LaTeX and various XML formats carry relatively large overheads. They are complex to learn and difficult to read. (Typing raw XML is nobody’s idea of fun.) Ideally, we would want a markup format to be lightweight, that is, simple, portable, and human-readable even in its raw state.

Markdown is just such a lightweight markup language. In Markdown, emphasis is textually indicated by surrounding the phrase with asterisks, as is familiar from email conventions, for example, *lightweight*. See, that wasn’t so hard. Here’s another example: A bulleted list is indicated by prepending each item on a separate line with an asterisk, like this:

 * First item
 * Second item

which specifies the list

  • First item
  • Second item

Because specification and typesetting are separated, software is needed to convert from one to the other, to typeset the specified document. For reasons that will become clear later, I recommend the open-source software pandoc. Generally, scholars will want to convert their documents to PDF (though pandoc can convert to a huge variety of other formats). To convert file.md (the Markdown-format specification file) to PDF, the command

 pandoc file.md -o file.pdf

suffices. Alternatively, there are many editing programs that allow entering, editing, and typesetting Markdown. I sometimes use Byword. In fact, I’m using it now.

Markup languages range from the simple to the complex. I argue for Markdown for four reasons:

  1. Basic Markdown, sufficient for the vast majority of non-mathematical scholarly writing, is dead simple to learn and remember, because the markup notations were designed to mimic the kinds of textual conventions that people are used to – asterisks for emphasis and for indicating bulleted items, for instance. The coverage of this basic part of Markdown includes: emphasis, section structure, block quotes, bulleted and numbered lists, simple tables, and footnotes.
  2. Markdown is designed to be readable and the specified format understandable even in its plain text form, unlike heavier weight markup languages such as HTML.
  3. Markdown is well supported by a large ecology of software systems for entering, previewing, converting, typesetting, and collaboratively editing documents.
  4. Simple things are simple. More complicated things are more complicated, but not impossible. The extensions to Markdown provided by pandoc cover more or less the rest of what anyone might need for scholarly documents, including links, cross-references, figures, citations and bibliographies (via BibTeX), mathematical typesetting (via LaTeX), and much more.For instance, this equation (the Cauchy-Schwarz inequality) will typeset well in generated PDF files, and even in HTML pages using the wonderful MathJax library.\[ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) \](Pandoc also provides some extensions that simplify and extend the basic Markdown in quite nice ways, for instance, definition lists, strikeout text, a simpler notation for tables.)

Above, I claimed that scholars should use Markdown “with few exceptions”. The exceptions are:

  1. The document requires nontrivial mathematical typesetting. In that case, you’re probably better off using LaTeX. Anyone writing a lot of mathematics has given up word processors long ago and ought to know LaTeX anyway. Still, I’ll often do a first draft in Markdown with LaTeX for the math-y bits. Pandoc allows LaTeX to be included within a Markdown file (as I’ve done above), and preserves the LaTeX markup when converting the Markdown to LaTeX. From there, it can be typeset with LaTeX. Microsoft Word would certainly not be appropriate for this case.
  2. The document requires typesetting with highly refined or specialized aspects. I’d probably go with LaTeX here too, though desktop publishing software (InDesign) is also appropriate if there’s little or no mathematical typesetting required. Microsoft Word would not be appropriate for this case either.

Some have proposed that we need a special lightweight markup language for scholars. But Markdown is sufficiently close, and has such a strong community of support and software infrastructure, that it is more than sufficient for the time being. Further development would of course be helpful, so long as the urge to add “features” doesn’t overwhelm its core simplicity.

The hidden agenda

I have a hidden agenda. Markdown is sufficient for the bulk of cases of composing scholarly articles, and simple enough to learn that academics might actually use it. Markdown documents are also typesettable according to a separate specification of document style, and retargetable to multiple output formats (PDF, HTML, etc.).2 Thus, Markdown could be used as the production file format for scholarly journals, which would eliminate the need for converting between the authors’ manuscript version and the publishers internal format, with all the concomitant errors that process is prone to produce.

In computer science, we have by now moved almost completely to a system in which authors provide articles in LaTeX so that no retyping or recomposition of the articles needs to be done for the publisher’s typesetting system. Publishers just apply their LaTeX style files to our articles. The result has been a dramatic improvement in correctness and efficiency. (It is in part due to such an efficient production process that the cost of running a high-end computer science journal can be so astoundingly low.)

Even better, there is a new breed of collaborative web-based document editing tools being developed that use Markdown as their core file format, tools like Draft and Authorea. They provide multi-author editing, versioning, version comparison, and merging. These tools could constitute the system by which scholarly articles are written, collaborated on, revised, copyedited, and moved to the journal production process, generating efficiencies for a huge range of journals, efficiencies that we’ve enjoyed in computer science and mathematics for years.

As Rob Walsh of ScholasticaHQ says, “One of the biggest bottlenecks in Open Access publishing is typesetting. It shouldn’t be.” A production ecology built around Markdown could be the solution.

  1. Many of the ideas in this post are not new. Complaints about WYSIWYG word-processing programs have a long history. Here’s a particularly trenchant diatribe pointing out the superiority of disentangling composition from typesetting. The idea of “scholarly Markdown” as the solution is also not new. See this post or this one for similar proposals. I go further in viewing certain current versions of Markdown (as implemented in Pandoc) as practical already for scholarly article production purposes, though I support coordinated efforts that could lead to improved lightweight markup formats for scholarly applications. Update September 22, 2014: I’ve just noticed a post by Dennis Tenen and Grant Wythoff at The Programming Historian on “Sustainable Authorship in Plain Text using Pandoc and Markdown” giving a tutorial on using these tools for writing scholarly history articles.
  2. As an example, I’ve used this very blog post. Starting with the Markdown source file (which I’ve attached to this post), I first generated HTML output for copying into the blog using the command
    pandoc -S --mathjax --base-header-level=3 markdownpost.md -o markdownpost.html

    A nicely typeset version using the American Mathematical Society’s journal article document style can be generated with

    pandoc markdownpost.md -V documentclass:amsart -o markdownpost-amsart.pdf

    To target the style of ACM transactions instead, the following command suffices:

    pandoc markdownpost.md -V documentclass:acmsmall -o markdownpost-acmsmall.pdf

    Both PDF versions are also attached to this post.

Turing Test
…that’s not Turing’s Test…
Turing Test” image from xkcd. Used by permission.

There has been a flurry of interest in the Turing Test in the last few days, precipitated by a claim that (at last!) a program has passed the Test. The program in question is called “Eugene Goostman” and the claim is promulgated by Kevin Warwick, a professor of cybernetics at the University of Reading and organizer of a recent chatbot competition there.

The Turing Test is a topic that I have a deep interest in (see this, and this, and this, and this, and, most recently, this), so I thought to give my view on Professor Warwick’s claim “We are therefore proud to declare that Alan Turing’s Test was passed for the first time on Saturday.” The main points are these. The Turing Test was not passed on Saturday, and “Eugene Goostman” seems to perform qualitatively about as poorly as many other chatbots in emulating human verbal behavior. In summary: There’s nothing new here; move along.

First, the Turing Test that Turing had in mind was a criterion of indistinguishability in verbal performance between human and computer in an open-ended wide-ranging interaction. In order for the Test to be passed, judges had to perform no better than chance in unmasking the computer. But in the recent event, the interactions were quite time-limited (only five minutes) and in any case, the purported Turing-Test-passing program was identified correctly more often than not by the judges (almost 70% of the time in fact). That’s not Turing’s test.

Update June 17, 2014: The time limitation was even worse than I thought. According to my colleague Luke Hunsberger, computer science professor at Vassar College, who was a judge in this event, the five minute time limit was for two simultaneous interactions. Further, there were often substantial response delays in the system. In total, he estimated that a judge might average only four or five rounds of chat with each interlocutor. I’ve argued before that a grossly time-limited Turing Test is no Turing Test at all.

Sometimes, people trot out the prediction from Turing’s seminal 1950 Mind article that “I believe that in about fifty years’ time it will be possible to programme computers, with a storage capacity of about \(10^9\), to make them play the imitation game so well that an average interrogator will not have more than 70 per cent. chance of making the right identification after five minutes of questioning.” As I explain in my book on the Test:

The first thing to note about the prediction is that it is not a prediction about the Test per se: Turing expects 70 percent prediction accuracy, not the more difficult 50 percent expected by chance, and this after only a limited conversation of five minutes. He is therefore predicting passage of a test much simpler than the Test.

Not only does the prediction not presuppose a full Turing Test, but it could well be argued that it had already come to pass with the advent of Joseph Weizenbaum’s Eliza some thirty-five years early. Weizenbaum developed a simple computer program that attempted to imitate the parroting conversational style of a Rogerian psychotherapist…. Although the methods used were quite simple – repeating the user’s question after adjusting some pronouns, throwing in a stock phrase every now and then – the result was, in its own way, extremely convincing.

Second, “Eugene Goostman” uses the very techniques that began with Weizenbaum’s  “Eliza” program from the 1960’s. We see the same tricks – repeating the judge’s statements with simple substitutions, keyword-triggered responses, falling back on vague or unresponsive replies, and the like. Those tricks are no more successful than they have been in the two decades of runnings of the Loebner Prize Competition, another ill-conceived attempt at running a Turing-like test. And there too, entrants used the trick of having their programs emulate humans with built in excuses.  “Eugene Goostman” purports to be a non-English-fluent child, rather than a fluent adult.1

In fact, anyone with the slightest familiarity with chatbots, the modern incarnations of “Eliza”, could unmask “Eugene Goostman” quickly. (The judges at the recent competition were apparently not so familiar.) Scott Aaronson has provided a transcript of his own interaction with “Eugene Goostman”, which shows the telltale signs as soon as the computer’s first reply, which blatantly ignores an utterly trivial question and tries to change the subject:

Scott: Which is bigger, a shoebox or Mount Everest?

Eugene: I can’t make a choice right now. I should think it out later. And I forgot to ask you where you are from…

Here’s an appeal to cute, vague language:

Scott: How many legs does a camel have?

Eugene: Something between 2 and 4. Maybe, three? :-))) By the way, I still don’t know your specialty – or, possibly, I’ve missed it?

And here’s the repetition of a canned response:

Scott: No, I need to know that you’re not a chatbot. Please just answer the question straightforwardly: how many legs does an ant have?

Eugene: Something between 2 and 4. Maybe, three? :-))) Oh, what a fruitful conversation;-)

Here’s a canned keyword-triggered response, nonresponsive as usual:

Scott: In your opinion, does your existence demonstrate any idea or principle that wasn’t demonstrated just as convincingly by ELIZA, Joseph Weizenbaum’s chatbot from the 1960s?

Eugene: Eliza was a break-thru. All the bots after it were nothing but weak parodies, claiming to have “revolutionary improvements”.

And here’s “Eliza”‘s trick of rewriting the judge’s utterance after pronoun replacement and (poorly executed) synonym substitution:

Scott: OK, I’ll count that as the second sensible thing you’ve said (grading generously). Don’t you think that a more convincing chatbot could be created, by using Google to trawl the entire web for plausible-sounding answers (analogous to what IBM’s Watson did)?

Eugene: Yes, I do think that a more convincing chatbot can be built by using google to trawl the all web for plausible – sounding – I hope I have right to have my own opinion. Maybe, let’s talk about something else? What would you like to discuss?

Literally every one of “Eugene”‘s responses reflects its “Eliza”-like programming. It would be amusing, if it weren’t so predictable.

In summary, “Eugene Goostman” is not qualitatively superior to other chatbots, and certainly has not passed a true Turing Test. It isn’t even close.

  1. In a parody of this approach, the late John McCarthy, professor of computer science at Stanford University and inventor of the term “artifical intelligence”, wrote a letter to the editor responding to a publication about an “Eliza”-like program that claimed to emulate a paranoid psychiatric patient. He presented his own experiments that I described in my Turing Test book: “He had designed an even better program, which passed the same test. His also had the virtue of being a very inexpensive program, in these times of tight money. In fact you didn’t even need a computer for it. All you needed was an electric typewriter. His program modeled infantile autism. And the transcripts – you type in your questions, and the thing just sits there and hums – cannot be distinguished by experts from transcripts of real conversations with infantile autistic patients.”
'Halves' by flickr user Julie Remizova
…how to split a cupcake…
Halves” image by flickr user Julie Remizova.

Why is gerrymandering even possible in a country with a constitutional right to equal protection?:

No State shall make or enforce any law which shall…deny to any person within its jurisdiction the equal protection of the laws.

By reshaping districts to eliminate the voting power of particular individuals, as modern district mapping software allows, some persons are being denied equal protection, I’d have thought. And so have certain Supreme Court justices.

It’s hard to know what to do about the problem. Appeals to fairness aren’t particularly helpful, since who decides what’s fair? It would be nice to think that requirements of “compact districts of contiguous territory” (as Chief Justice Harlan put it) would be sufficient. But this reduces the problem of districting to a mathematical optimization problem; James Case proposes something like minimum isoperimetric quotient tessellation of a polygon. But such purely mathematical approaches may yield results that violate our intuitions about what is fair. They ignore other criteria, such as “natural or historical boundary lines”, determined for instance by geographical features like rivers and mountains or shared community interests. These boundaries may not coincide with the mathematical optima, so any mathematical formulation would need to be defeasible to take into account such features. This leads us right back to how to decide in which cases the mathematical formulation should be adjusted: who should decide what is fair?

A comment at a ProPublica article about gerrymandering from “damien” caught my attention as a nice way out of this quandary. In essence, he proposes that the parties themselves choose what’s fair.

The first solution to gerrymandering is to have a fitness measure for a proposed districting (e.g. the sum of the perimeters), and then to allow any individual or organisation to propose a districting, with the winner having the best fitness value.

What “damien” is proposing, I take it, is the application of an algorithm somewhat like one familiar from computer science (especially cryptography) and grade school cafeterias known as “cut and choose”. How do you decide how to split a cupcake between two kids? One cuts; the other chooses. The elegance of cut-and-choose is that it harmonizes the incentives of the two parties. The cutter is incentivized to split equally, since the chooser can punish inequity.

Cut-and-choose is asymmetrical; the two participants have different roles. A symmetrical variant has each participant propose a cut and an objective third party selecting whichever is better according to the pertinent objective measure. This variant shares the benefit that each participant has an incentive to be more nearly equal than the other. If Alice proposes a cut that gives her 60% of the cupcake and Bob 40%, she risks Bob proposing a better split that gives her only 45% with him taking the remaining 55%. To avoid getting taken advantage of, her best bet is to propose a split as nearly equal as possible.

In the anti-gerrymandering application of the idea, the two parties propose districtings, which they could gerrymander however they wanted. Whichever of the two proposals has the lower objective function (lower isoperimetric quotient, say) is chosen. Thus, if one party gerrymanders too much, their districting will be dropped in favor of the other party’s proposal. Each party has an incentive to hew relatively close to a compact partition, while being allowed to deviate in appropriate cases.

A nice property of this approach is that the optimization problem doesn’t ever need to be solved. All that is required is the evaluation of the objective function for the two proposed districtings, which is computationally far simpler. (In fact, I’d guess the minimum isoperimetric quotient optimization problem might well be NP-hard.)

There are problems of course. The procedure is subject to gaming when the proposal-generating process is not private to the parties. It is unclear how to extend the method to more than two parties. Of course, the obvious generalization works once the eligible parties are determined. The hard part is deciding what parties are eligible to propose a redistricting. Most critically, the method is subject to collusion, especially in cases where both parties benefit from gerrymandering. In particular, both parties benefit from a districting that protects incumbencies for both parties. The parties could agree, for instance, not to disturb each other’s safe districts, and would benefit from observing the agreement.

Nonetheless, once districting is thought of in terms of mechanism design, the full range of previous algorithms can be explored. Somewhere in the previous literature there might be a useful solution. (Indeed, the proposal here is essentially the first step in Brams, Jones, and Klamler’s surplus procedure for cake-cutting.)

Of course, as with many current political problems (campaign financing being the clearest example), the big question is how such new mechanisms would be instituted, given that it is not in the incumbent majority party’s interest to do so. Until that’s sorted out, I’m not holding out much hope.

Aaron Swartz’s legacy

January 13th, 2013

Government zealotry in prosecuting brilliant people is a repeating theme. It gave rise to one of the great intellectual tragedies of the 20th century, the death of Alan Turing after his appalling treatment by the British government. Sadly, we have just been presented with another case. Aaron Swartz committed suicide at his apartment in New York this week in the face of an overreaching prosecution of his JSTOR download action. I never met him, but I understand from those who knew him well that he was a brilliant, committed person who only acted intending to do good in the world. I’m on the record disagreeing with the particulars of the open access tactic for which he was being prosecuted, on the basis that it was counterproductive. But I empathize with the gut instinct that led to his effort. I hope that it will inspire us all to redouble our efforts to eliminate the needless restraints on the distribution and use of scholarship as Swartz himself was trying to achieve.