Psychological correctness

In this meeting, we read the crucial section of “Literate Programming,” J, in which Knuth lays out his theory of ‘programs as webs’. This section justifies WEB in the most general possible terms: all programs are already webs, Knuth argues. All WEB does is allow the programmer to clearly express the structure that is already there.

Knuth uses the ‘web’ metaphor in an interesting way. Those of us who are used to the Internet and the World Wide Web may think of a ‘web’ as an inherently dynamic or unstable system. We browse or surf the web. We hop from page to page, from app to app, using hyperlinks. We summon up fragments of the web using search engines, or allow recommendation algorithms to summon up fragments for us as we scroll the feeds of our favourite platforms.

This is not what Knuth means by web. For Knuth, a web is a static, well-ordered structure, like the delicately woven web of a spider, or a narrative tapestry whose threads are chronological. A web is for reading from start to finish. A web has a finite set of components, which have been joined carefully by the weaver of the web. Of course, you may use an index to jump to particular joins on the web. You may use cross references to travel along particular strands. But the web itself is single and entire, with a beginning, middle and end.

Structures and structures

A hierarchical structure is present, but the most important thing about a program is its structural relationships. (p. 107)

Knuth argues WEB accomodates both ‘top-down’ and ‘bottom-up’ programming, or rather, it transcends these two approaches. The WEB programmer can start with a top-level description of a program, or they can start by defining subroutines, or they can mix both freely. This freedom to decide between top and bottom at will frees the programmer from the ‘hierarchy’ of the program. Of course, in the end, ‘[a] hierarchical structure is present’: a program must be a single object the computer can execute, comprising smaller parts that lie within it. But the WEB approach allows the programmer to reveal the ‘structural relationships’ of the program: the logical and intellectual links between different parts of the program.

For example, perhaps some global variables are manipulated by subroutine X, and others are manipulated by subroutine Y. From a hierarchical perspective, each global variable and each subroutine is a separate part of the program, on the same level, while the code inside each subroutine is at the next level down, nested within the subroutine. Using WEB, however, the programmer can explicitly reveal the relationships between the variables and the subroutines, for example by declaring the variables next to the subroutines that matter to them, or by building up the subroutines in parts that a clearly related to other global aspects of the program.

There is an interesting slippage in Knuth’s argument. There is the ‘hierarchical structure’ on the one hand, and the ‘structural relationships’ on the other. Both of these are structur(e|al). What makes them different? How are they related?

Knuth implies that there is no single description of a program that is the right one. Programs have many parts, which combine to form the entire program. These parts have many possible relationships: the orderly hierarchy of their execution by the machine is only one set of relationships. The human reader of a program may observe many other sets of relationships in the program that matter to them.

We could think of this in practical terms. A human might use a profiler, observing how and when different parts of the program are called in practice. They might use a flowchart tool to visualise the control flow. They might write out mathematical theorems that characterise the invariants of parts of the program. They might observe the way that the program models the problem domain, the user, the machine itself. There are (possibly) infinitely many ‘structures’ in a program. Knuth’s aim with WEB is to let the programmer structure their program in whatever way will maximise human comprehension of the code.

Psychological correctness vs. [personal] style

Knuth’s theory of coding style is simulataneously aesthetic, cognitive and functional. Code written in WEB should be aesthetically pleasing, according to literary criteria; it should be easily comprehended (or congnised); and it should function correctly.

These three aims don’t always go together, according to Knuth. He gives an example on page 108. Imagine a programmer is writing a function that does a simple data update, but it needs to check the user input for errors. If the programming language obliges the programmer to put the error-checking code first, then they may feel the urge to shrink the error-checking. The error-checking code is tangential to the function: what really matters is the code at the end, which actually performs the data update for which the function is being written. If there are dozens of lines of error-checking code, which make up virtually the whole function, the programmer may find the function aesthetically repulsive. It would be like designing a pencil with a grip so enormous and contorted that you can no longer clearly see the barrel and tip of the pencil itself. In this case, aesthetics pulls against both cognition and functionality. To make the function seem less ugly, the programmer will try to write the error-checking code as concisely as possible, which may mean it is terse and difficult to understand. They will also be tempting to omit error checks, potentially impairing the functionality of the code.

Knuth demonstrates how WEB resolves the contradiction between aesthetics, cognition and functionality. By giving the programmer complete control over the presentation of the code, and the ability to add labels or commentaries to any part of it freely, WEB allows the programmer to achieve any functionality they like without compromising on either aesthetics or cognition.

There is a tight link, and nonetheless a tension between aesthetics and cognition in Knuth’s theory. Knuth argues that the best way to present a program is in the “psychologically correct” order. But he also argues that programmers can and should develop a personal “style” of programming. If there is a “correct” way to present the program, how is there room for individual “style”?

Knuth’s theory of “psychological correctness” is highly individualised. He argues that a program should represent the programmer’s “stream of consciousness” (p. 107)—that is, the program should be written in the order that the programmer conceived of it. He insists throughout the essay that in his own experience, he only ever envisages a program in one order. There is an order in which the program occurred to him, and this is the order in which it must be written. He argues that when he reads another programmer’s code, he can understand their stream of consciousness easily: the program he presents on pages 98-102 of the article is actually no Knuth’s own stream of consciousness, but Edsger Dijkstra’s.

There is a commonsense aspect to this. If the programmer builds up the program logically, then they can communicate this logical process to the reader, who will hopefully find it easer to comprehend what is going on. Knuth does occasionally modify his theory, admitting that the programmer should not simply regurgitate their actual “stream of consciousness,” but shape the program text with the reader in mind.

But Knuth nonetheless presents the idea of “psychological correctness” in such a stark way that its implications are thriling and extreme. Is it true that every program Knuth writes appears to him in exactly one way? Is this a universal experience of programming? We felt in the group that perhaps Knuth is not accounting for his own extreme level of skill and learning—most programmers probably fumble around, and need to experiment, much more than this most famous computer programmer needs to when he writes software. Is it true that we can understand one another’s thought-processes so easily? Many in our group found the presentation of the primes program on pages 98-102 extremely difficult to follow. The program makes many assumptions about the prior knowledge and discursive competence of its readers. Does Knuth believe that there is a single programming literacy that all programmers share, such that the reader of any program can be assumed to be the same kind of person with the same kind of consciousness?

There is something deeply Kantian about Knuth’s views. He seems to believe in a universal rationality, which extends to the task of aesthetic judgment, and which links cognition to the feeling of beauty. As an unreconstructed Romantic, I find this point of view to be very attractive, even if our experience in this very group demonstrates (for the millionth time) that rationality is more contingent and culturally determined that Kantians may like to admit.

We recommence next time partway through section K, on page 108.

Simplicity and Neglect

There have been some interruptions to our group—and the blog—but we are back today. We read slowly through sections H and I of Knuth’s “Literate Programming,” which contained some suggestive clues to Knuth’s aesthetic philosophy.

Simplicity: for whom?

Simplicity is Knuth’s justification for one-parameter macros:

Again, I did this in the interests of simplicity, because I noticed that most applications of multiple parameters could in fact be reduced to the one-parameter case. (p. 104)

We discussed this argument for some time, because there is a fascinating lacuna in Knuth’s argument: simplicity for whom? Knuth could mean, simplicitiy for the implementor of WEB (namely, for Donald Ervin Knuth), or he could mean, simplicity for the user of WEB. Knuth’s example doesn’t clarify the situation. He presents this example of a two-parameter macro:

mac(#1, #2) == m[#1*r+#2]

and shows how it could be rewritten using only one-parameter macros like so:

mac_tail(#) == #]
mac(#) == m[#*r+mac_tail

At first blush, the rewritten version looks considerably more complicated. It takes more lines of code. It reverses the order of the [] symbols, requiring the reader to put them back in the correct order by mentally substituting mac_tail for #]. Most of us in the group recoiled at first from Knuth’s example. Why not allow two-parameter macros, and the apparently more elegant initial example?

On close inspection, some points in favour of Knuth’s decision emerged. One of the key simplifications of the rewritten version is that the parameters (#) no longer need to be numbered. This is presumably much easier for TANGLE to handle, and therefore easier for Knuth to implement. It also eliminates certain possible errors, such as inconsistent labelling, or too many parameters being passed to the macro, or too few.

The main criterion of ‘simplicity’ for Knuth seems to be: paucity of primitive elements and means of combination. The one-parameter macro comprises one macro label (e.g. mac_tail) and one parameter (#). There is a simple substitution: replace every # on the right-hand side with the passed value of #. The two-parameter macro introduces an additional primitive element: the number of the parameter (e.g. the 2 in #2). It also introduces an additional substitution rule: first match the parameters on the left and right hand sides using their numbers, then proceed with the substitution.

Programmers often want to ‘reason’ about their programs. This is probably the activity Knuth hopes to support by allowing only macros of one parameter. It is easier for me to see what a macro will do, because it is so constrained in what it can do. I can construct a more complex macro by combining several macros together. While this may seem complex at first, the advantage is that each individual piece is extremely simple, and I can understand it perfectly with little effort.

In this way, there is a happy marriage between the impelementor and the user, both of whom benefit from the same kind of simplicity. How often do programmers and computer scientists aim for this kind of happy marriage? Is the user’s simplicitiy alway reducible to the implementor’s?

Of course in Knuth’s case, the users and implementors were often the same people. As he explains in section I, on “Portability,” installing WEB was no easy matter in the 1980s. This importability¹ extended to programs written in WEB, such as TeX. In the days when software was distributed as source code, the user often had to modify the code in order to get the software working. Ease of implementation and ease of use are hard to distinguish in such a context.

We spoke for sometime about the culture of programming in the 1980s. All this fuss about macros seems otiose to digital humanists raised on dynamically typed intepreted languages like R, Python and JavaScript. There is simply no need for all this mucking about! There are two tracks through Knuth’s arguments that are difficult to disentangle:

1. Knuth’s attempt to overcome the particular limitations of PASCAL, which requires the programmer to write their program in a certain order using certain sections.
2. Knuth’s attempt to devise a new form of writing, of general application, which will make software more enjoyable to read and write.

In the context of (1), the discussion of macros is necessary. As Knuth himself demonstrates, some programming tasks are basically just torture in PASCAL without WEB’s macros. In the context of (2), however, the discussion seems to wander down the garden path. Is this detail necessary for me to understand the acts of WEAVING or TANGLING code?

Neglect: bootstrapping the authorial persona

The WEB system caters to system-dependent changes in a simple but surprisingly effective way that I neglected to mention when I listed its other features. (p. 106)

The word ‘neglected’ evoked a range of responses in the group. There is an obvious fictionality to it. Knuth could of course edit his essay, and introduce this feature earlier on. His essay is obvioulsy artful, and to ascribe the ordering of its contents to ‘neglect’ is misleading—on the literal level.

But of course Knuth and his readers are well aware of the fictionality of essayistic neglect. The word indicates two aspects of Knuth’s writing:

1. His cultivation of Socratic humility.
2. His belief in the linearity of text.

Knuth’s humility is an expression of mastery. He can afford to ‘neglect’ a topic, because he knows when is the right time to introduce it. His writing is replete with such self-effacement. It is easy to imagine his smiling presence in the classroom, as he gently introduces students to theorems that he ‘neglected’ to tell them earlier on. I personally like this authorial persona—as I like its obvious antecedent, the Socrates of Plato’s dialogues. But perhaps his ingratiating humility and pedantic specificity are not to all reader’s tastes.

Knuth’s belief in linearity expresses his deeper ideas about literacy. For Knuth, the model reader is one who reads a text from the start to the end. The whole WEB system is designed to make this possible for software generally. And indeed, how nice it would be if all programs did have a single reading path, so that a new programmer could take a guided tour of the software before they start hacking on it. Knuth is essentially a teacher, and he views the writer’s task as pedagogical. Take the reader through the content in the order that it makes sense to human cognition. Avoid complexity and digressions. ‘Neglect’ what is not necessary to explain until the opportune moment to introduce it.

In this way, Knuth’s humility and linearity converge in a common ideal. Both his humility and linearity serve to make the text transparent and open. There is nothing hidden from the reader, at least not intentionally.

There may also be a sly, and thoroghly computational, humour behind Knuth’s avuncular language. As one member of the group observed, Knuth himself knows that he hasn’t really ‘neglected’ the topic of change files. Of course he had to neglect it! The contents of the essay have to be in some order! It’s not possible to write the entire essay in the first sentence! This ‘neglect’ represents the bootstrapping of the authorial persona. The author writes himself into existence. Knuth didn’t ‘neglect’ the topic of change files until he introduced the topic of change files on page 106. Like TANGLE.WEB, Knuth’s authorial persona is self-hosted.

We resume next time at section J. Programs as Webs, where Knuth’s implicit poststructuralism becomes explicit.

The loop is broken

The inner loop

In this session we concluded the central section of the central section of “Literate Programming”. In sections 22-26 of the “woven output,” Knuth presents the “inner loop” of his program to print the first 1000 primes. The inner loop checks each candidate number $j$ to see if it is prime. It is the kernel of the program, the part that consumes the most computation time, and which performs the function closest to the program’s ultimate goal. And it does it all without performing a single multiplication or division…

We noticed again some common ticks in Knuth’s rhetoric. Once again we encounter “the remaining task” (which we had already met in section 11). Once again the program is “quite simple” and “straightforward” (as have been most parts of the program). Knuth’s program text unrolls like a function being optimised. It relentlessly converges on its solution, following a chain of logic whose links are joined each to each in an intuitive way. By this point in the program, the whole group were baffled by the mathematics. For this group of humanists, the refrain of straightforwardness had become rather humerous.

For my part, I find Knuth’s authorial persona amusing, his program elegant, and his presentation of it poetic—to the extent I was able to understand any of them. But others in the group found his authorial persona “judgey,” and his presentation of the program intolerably taxing on the powers of memory. What do all these auxiliary variables mean again? What exactly is the structure of that table? How are those different numbers combined? Who exactly was Eratosthenes, and what does his “sieve” have to do with any of this?

Way back in section 3, Knuth informed us that the program should be “reliable, well motivated, and reasonably fast.” We were all amused by the program’s final motivation:

Let’s suppose taht division is very slow or nonexistent on our machine. We want to detect nonprime odd numbers, which are odd multiples of the set of primes $\set{p_2, …, p_{ord}}$.

We have discussed at many points the exemplary nature of the program. Knuth does not intend to provide either a useful program listing (e.g. a prime number solver for use in production), or an exercise in programming style (e.g. an example of structured programming). He intends to provide a well-documented program using WEB, and one gets the impression at many points that the program has been made needlessly complicated simply to justify the need for WEB’s features. In this case, the computer’s lack of division requires explanation, which requires documentation, which requires—WEB.

A maze of strands

We took a moment, upon completing the program itself, to reflect on it as a literary work. Most in the group agreed that the WEB program has a remarkably “tangled” structure—to misuse Knuth’s own metaphor. You could say that the program is deeply intratextual. It constantly refers to itself. No part of the program can very easily be detached, and viewed independently on its own. To understand any part, it is necessary to recall the whole structure of variables and constants that govern its behaviour. The text of the explanation typically refers to code that has not been presented yet. To understand the “motivation” for each programming decision, you need to already understand the program, or at least programs like it. The text is topsy-turvey, round-about, splayed-across and liketty-split. I personally enjoyed the ride, but well-oiled labyrinths are not for everyone!

Upon reading section 27, the index, some of the group were dismayed. (Not really.) If only we had kept the index open the entire time, it might have been possible to keep track of all the quantities and arrays that comprise the program code! But as one member of the group observed, Knuth’s preference for mathematical symbolism makes the index unreadable on it’s own. What do $c$ and $cc$ mean? You already need an intimate knowledge of the program to understand that they mean “current column” and “columns per page.”

Another in the group observed the difficult role that time plays in “the plot of the code.” The program text has its own narrative time, marked out sequentially by the section numbers. It also refers to the serial time of the computation, and to the “parallel time” of certain “auxilary variables” which evolve in lockstep with the main variables of the program. On top of this, the reader is constantly aware of the way the program loops back on itself, providing a structure for WEB to reorder all the code as the PASCAL compiler demands. And then there is the overall evolution of the algorithm as a process when the program is run. Knuth’s program has the complex temporality of a Virginia Woolf novel, and a Woolfian quality of internal reflection. Perhaps for someone whose daily life is number theory, the program would also have those Woolfian qualities of sacredness and care that we were, for the most part, unable to draw from the text.

We also discuss the personality of the text. Knuth argues from the start that “[p]rogramming is a very personal activity,” and suggests that the programmer’s personality should shine through the code. If you can see the person behind the code, you can understanding the reasoning behind the code, and therefore can understand the code itself. One member of the group suggested that this contradicts the common ideal of “egoless programming,” which is important for engineers working on complex projects in large teams over many years. How “egotistical” is Knuth’s idea of “personality”? Knuth himself is a master-craftsman. Perhaps for him there is no link between personality and ego, for the master-craftsman loses herself in the craft, and can bear any criticism of her work so long as it is in the cause of programming elegance. But perhaps ego and personality cannot so easily be separated.

We recommence next week in the fourth paragraph of D. How the example was specified.

A scripture language

Cultural literacy

A moment’s thought makes it clear that ord changes in a simple way when j increases, and that another variable square facilitates the updating process. (p. 101)

Literate programming requires literate programmers. Readability is relative. Readers must become literate before they can successfully decode a text, and enter the free world of interpretation.

In this quotation, Knuth once again draws attention to the rationality of his program, its unfolding according to agreed principles of thought. What kind of rationality is this? What principles have been agreed? What does the reader have to know in order to grasp after a “moment’s thought” that j, ord and square are related in the way Knuth implies?

Kunth presupposes a certain level of mathematical knowledge in his readers. We discussed the reasonableness of this assumption. Presumably The Computer Journal primarily has a readership of academic computer scientists, whose education includes at least one course in number theory. If we presume this audience, then Knuth’s gestures of understanding—”A moment’s thought makes clear”—are consistent with the literacy of his readers.

Number theory of course remains fundamental to programming in the twenty-first century. The entire edifice of internet communcation rests on encryption algorithms which derive from number theory. Students who undertake degrees in Computer Science surely still acquire this literacy.

But does this assumption square with Knuth’s aims? He seems to want literate programs to become part of general literature. He suggests that the program text should make explicit the programmer’s thought-process. How much of this thought-process can be left implicit, in a program destined for a general readership?

The audience of programmers is much wider than the audience of computer scientists. In 1984, the concept of “computer literacy” was in the air. Many governments, educational institutions, and visionary programmers believed it was essential for the general public to learn to program. The UK’s Computer Literacy Project was in full swing. Millions of Britons tuned into televisions programs such as The Computer Programme and Making the Most of the Micro, which taught viewers how to code in BASIC. More than 150,000 bought 30-Hour BASIC, a coding course that could be taken independently or via correspondence. Aside from BASIC, other programming languages such as Logo and Smalltalk had been developed for children and hobbyists to use. Number theory, it needn’t be stressed, was not a key component of 1980s “computer literacy.” It is notable that this notion of “computer literacy” seems not to have informed Knuth’s notion of “literate programming.”

Modern professional developers, of course, are often data scientists or web developers with little training in mathematics. A front-end developer trying to position a <div> correctly is unlikely to know many theorems about the cardinal numbers. Knuth’s program arguably makes incorrect assumptions about the professionalisation of programmers, even if it makes correct assumptions about the literacy of The Computer Journal’s readers.

It is probably unfair to criticise this one program of Knuth’s. He has selected a particular program which will appeal to a particular readership so he can demonstrate WEB. One could also make a Romantic/modernist argument in favour of Knuth’s text: there is a place in literature too for complex texts that address an intellectual few. The world can accomodate Ulysses as well as Treasure Island. But then, Knuth did seem to set out for the kind of crossover appeal of a Jane Austen…

One member of the group compared Knuth’s use of number theory to an eighteenth-century gentleman’s use of the Bible. It is simply assumed that the references will make sense. Where does the problem lie… Should we all know number theory? Or should we change religions?

On complexity

The main aim of sections 11-21 of Knuth’s program is to reduce the “time complexity” of the algorithm, as one member of the group noted. The naive algorithm for filtering the primes consumes far more time than is necessary. Knuth steps us through the derivation of Djikstra’s prime-number algorithm to explain how it will reduce the complexity of the calculation.

“Complexity” is a word that is used so often in so many different contexts that it is virtually meaningless. Knuth, it must be said, does not actually use the word in this paper.

The naive way to find the primes would result in a more “complex” program, because the program would need to perform more calcuations. But the naive algorithm would result in a much simpler program text: it could be written in fewer lines, and would require much less effot on the part of the reader to understand.

Knuth is a pioneer in the analysis of algorithms, which focusses on the space- and time-complexity of algorithms: that is, how simple they are for the computer to execute. There are other kinds of complexity that computer scientists care about, such as the Solomonoff–Kolmogorov–Chaitin complexity, which is related to how “big” the program is as a set of instructions.

Neither kind of complexity quite explains the readerly experience of programs. The simple-complex distinction is only useful in certain domains. The distinction established by Erich Auerbach between parataxis and hypotaxis as literary styles may be more important. Knuth’s text can be hard for a nonspecialist to follow because it is highly hypotactic. Each part of the program radically depends on every other. He uses WEB to manage this hypotaxis, by putting interdependent parts of the program near each other. But if he had found a paratactic style for his program, that required the reader to keep less information in their heads, perhaps our little group of readers may have had an easier time navigating his program text.

We recommence in a fortnight at section 24.

All that remains

The three-act structure of the literate program

The remaining task is to fill table p with the correct prime numbers.

Halfway through Knuth’s program, we find the algorithm for generating primes. To this point, his program has focussed on controlling the machine. He has defined the global parameters for the program, and some complicated routines for placing numbers in a printed table. Thus far, we have read about variables such as row_offset, being the number of rows that have already been printed, and constants such as cc, the number of columns that can be accomodated on a single page of the table.

In the group, we spoke about the relationship between the programmer and the machine. In Digital Humanities, we only rarely interact with the machine in this direct and geometric way. Gone is the age of the early microcomputer, when the computer in ‘text mode’ would allow the user to display a certain number of characters on a screen of fixed column and row dimensions. Today we take flowing text for granted. Gone also are the days when printing a simple heading at the top of each page would require six statements:

begin print_string(`The First `);
print_integer(m);
print_string(` Prime Numbers --- Page `);
print_integer(page_number); new_line; new_line;

In Python, this could be accomplished with string interpolation:

print(f"The First {m} Prime Numbers --- Page {page_number}\n")

To complicate things, Knuth’s heading-printing code is actually written using a series of macros defined in a previous section. When the macros are applied, the resulting PASCAL code is as follows (see Figure 3 in Knuth’s text):

WRITE('The First '):WRITE(M:1);
WRITE(' Prime Numbers --- Page ');
WRITE(PAGENUMBER:1);WRITELN;WRITELN;

Only after several pages of this complicated programming does Knuth arrive at the ‘task’ of actually generating the first m prime numbers, which would seem to be the point of the program. Although of course, generating the first m primes is not the point of this program. The point of this program is to illustrate the concept of ‘literate programming’ and to demonstrate the WEB system. The inner purpose, to generate the first m primes, is a fictional purpose. Knuth is telling a story about a fictional version of himself, who needs to generate the primes, and wishes to write a readable program for doing so.

In this context, his heading code is very good. It gives him scope to demonstrate WEB’s macro system. It gives him scope to interleave code and explanation. It gives him scope to reflect on the proper narrative flow of a literate program.

In Knuth’s view, the program should have a three-act structure. Act 1, declare the basics. Act 2, do the housekeeping. Act 3, implement the algorithm. He is a Shandyan narrator, who includes many digressions—e.g. how to print a heading—on the way towards his ultimate goal. The digressions have caused some level of frustration in our group, but perhaps this is the point. Act 2 creates suspense. Knuth establishes the fictional purpose of his program in the first block of code, and then waylays the reader, whetting their appetite for the ‘real’ programming to come. It was some relief in our group to finally reach section 11, when the ‘remaining task’ of generating the primes recurred. Act 3 begins with a bang.

In this article, the programmer is like an eighteenth-century stage manager. The script of the play is only secondary. What matters is the scenery, the special effects, the footlights and the songs. We watch the play from the wings, as the stage manager summons up the mechanisms of the theatre. Flood the stage! Bring on the horses! Swing in the heroine on the trapeze! Knuth is a poet-hacker of the old school, the plot of whose text is the unfolding of the machine. In his hands, the abstract purity of Djikstra’s algorithm for finding the primes becomes concrete. As one in the group said: you can’t have a table of numbers if you haven’t got a way of printing the table. Knuth may well agree.

Conventions

〈 Output a line of answers 8 〉≡
  begin for c ← 0 to c - 1 do
    if row_offset + c * rr ≤ m then
      print_entry(p[row_offset + c * rr]);
  new_line;
  end

A second theme of our discussion this week was literary convention. Few in the group find Knuth’s naming conventions easy to understand. Knuth likes his code to look like algebra, and uses a host of unstated convention when he names variables. To someone with a background in discrete mathematics, it must seem obvious that n is how many primes we currently have, and m is the maximum number we’d like to reach. It must seem normal that j is the current number we are testing, and k is the index of the current prime number in the table. It must seem normal too that r, c, and w mean the current row, column and word position, while rr, cc and ww mean the maximum number of rows, columns and word positions. But in our group we don’t all have this background, and these choices of name are not so obvious!

Many in the group would maybe have preferred the above code if it looked like this:

〈 Output a line of answers 8 〉≡
  begin for column ← 0 to column - 1 do
    current_prime_index ← row_offset + column + num_rows;
    if current_prime_index ≤ max_number_of_primes then
      print_entry(primes[current_prime_index]);
  new_line;
  end

Most of us have done most of our coding in Python, where it is common to label variables explicitly as above. Indeed, pylint will highlight single-letter variable names like c or r, recommending the coder to expand them. Most of us do our coding in a humanistic context, where algebra is an uncommon acquirement. Mathematical formulae tend to look forbidding rather than familiar, even in Digital Humanities.

This is all to say that conventions are strong in coding as well as in other kinds of writing. These conventions are generally used unconsciously: even Knuth, the most self-aware of programmers in this most self-aware program, doesn’t note the algebraism of his own style!