It Must Be AbstractThomas Pynchon and Mathematics

‘All maps were useless.’

Thomas Pynchon, Against the Day

And so at what may indeed be considered the climax of Thomas Pynchon’s new novel, Against the Day, a map appears, purportedly of the Belgian Congo. Of course, nothing is as it seems and the map is encoded: is it the Belgian Congo? Or the Balkan Peninsula? And what is that straight line running east to west across the map? A railroad? And why is this important, the this being some sort of abstract un­defined comparison between the Belgian Congo and the Balkan Peninsula on Mr Pynchon’s part? Whether we view it from the vantage point of a mere geographical similarity or force ourselves to consider the similarities of colonialism in both places (for example, the importance of the railroad in establishing colonial rule…), what we come away with is a generalisation of both, a place which may be interpreted as either but which is by no means merely one or the other.

Now this abstraction is the realm of pure mathematics. Recall that at high school we say two triangles are congruent if they are the same shape but different sizes and then we deal with two congruent triangles as the same object. This is the realm of geometry. Then at university we find out that topologists consider squares and circles to be the same type of object since a square can be deformed into a circle if we allow ourselves to perform elastic deformations. 

As Pynchon informed the world in what was for him a rare blurb published on the net before the novel appeared, Against the Day has – 

[a] sizeable cast of characters [which] includes anarchists, ballooners, gamblers, corporate tycoons, drug enthusiasts, innocents and decadents, mathematicians, mad scientists, shamans, psychics and stage magicians, spies, detectives, adventuresses and hired guns.

With so many characters and storylines and 1,085 pages that put Gravity’s Rainbow (a mere 760 pages) to shame, where to begin? Why, with the mathematicians, of course: of the many characters with a scientific bent populating the novel, the two most important are Kit Traverse, a young man turned Vectorist from the Colorado mine fields, and Yashmeen Halfcourt, a Russian-born beauty whose two sole diversions are strange sexual acts and the Riemann Hypothesis, a still-as-yet unsolved problem concerning the zeroes of a particular complex-valued function, which is known to those in-the-know as the Riemann-zeta function. Kit is the youngest of four Traverse children, whose story is considered by most critics to be one of the main threads of the novel: their father, Webb Traverse, is a miner-come-anarchist-dynamiter who blows things up to draw attention to the plight of workers in the mining fields. Technocrat and wannabe Emperor of the World Scarsdale Vibe gets wind of this and has two hired guns kill Webb for him. Kit shows signs of a promising future in mathematics and Scarsdale takes him under his wing, while Kit’s brothers search for the killers. Kit, by contrast, pursues mathematical inquiry as opposed to inquiry into events. I shall return to this shortly.

Although Against the Day must surely be Pynchon’s ode to the world of pure and not-so-pure mathematics, the best introduction to the importance of mathematical concepts in Pynchon’s writing is to be found in Gravity’s Rainbow. The main narrative of Gravity’s Rainbow concerns Lieutenant Tyrone Slothrop, an American officer in London during the Second World War. Lieutenant Slothrop has a map of London on the wall next to his desk, on which he has placed a star wherever he has had sex. A certain British government official-of-sorts Roger Mexico has another map of London on which circles have been placed in the locations where the German V2 rockets have been falling. 

C’mon, then, where do you think this is going? ‘The slides that Teddy Bloat’s been taking of Slothrop’s map have been projected onto Roger’s and the two images, girl-stars and rocket-strike circles, demonstrated to coincide.’ But wait for the punch line: each and every time a rocket has fallen in London, it has been after our apparent stud Tyrone has been amorous there. Somehow Tyrone’s ejaculations cause rockets to fall in the exact same location, sometimes two days after, sometimes ten: the mean lag is about four-and-a-half days. What follows for the rest of the novel is Tyrone’s odyssey across Continental Europe to find out what is inside him that makes these rockets fall. We find him on the French Riviera saving a Dutch beauty named Katje (pronounced ‘Got ya!’) from an octopus named Grigori, who has been conditioned to capture her just so as to lure Slothrop into this very situation. We find him in post-war Berlin as a pseudo-superhero-comic-book-character called Rocketman, aka Racketmensch, due to the bizarre correlations between his body and the V2 rocket. We find him used and abused as a weapon and realise that he is pretty much a helpless tool for The Powers that Be, what Pynchon’s paranoia leads him to refer to as They. On the French Riviera, Tyrone discovers the presence of a rocket called the 00000 and a certain component of it called the Schwarzgerat (or S-Gerat) made from a plastic called Imipolex G. We find out that this plastic may or may not have been inserted into Tyrone’s penis as an infant, which in turn may or may not have resulted in the strange correlation between his fucking and rockets falling. Whatever the case may be, Tyrone travels across Continental Europe in search of the S-Gerat, in the process becoming a rocket-of-sorts himself – and in the end is entirely dismantled and fragmented as a rocket would be. 

Thus we have important aspect No. 1: the Rocket. Important Aspect No. 2 is Cinema: film plays an integral role in Gravity’s Rainbow. The novel is played out as if it were a movie: colours dissolve to black-and-white and characters return in blinding colour, the sections of each chapter are separated by sprocket-holes and we are in fact informed, as the final rocket descends to destroy Tyrone, and us, that we are all seated in a movie theatre upon which the rocket is descending. The cinema, of course, has to play a role in what is essentially a WWII novel. Not only this, for the Rocket and the Cinema are inextricably tied into one another. But how? 

Let me try and explain: first recall that the trajectory of a rocket is described by an inverted parabola. How do we study such objects? The first tool is calculus, which will allow us to measure either the area under the path travelled by the rocket (this is called integration) or the direction in which the rocket is travelling at any given time (this is differentiation). Let’s integrate. First we approximate the area under the curve by taking three points on the parabola as indicated below, constructing the two rectangles in Figure (a) and adding up their areas. This is what we call a bad approximation! 

We then take five points on the parabola and perform the same task. Now we have a better approximation. If we continue this process with nine points, fifteen points, twenty one points and so forth, our approximations will get ‘closer and closer’ to the real value of the area below the curve. In this sense we can take the ‘limit’ of all our successive approximations and retrieve the actual area.  

Finding the direction of the rocket at any given point (i.e. differentiating), is done in a similar manner. Now think about what happens when you watch a movie: what you actually see are twenty four still images each second and your brain fills in the gaps, transforms the successive images into a moving picture or, as Pynchon would ingeniously have you believe, performs the integration to make a sequence of still images move. The process of filling in the gaps, reaching the limit, bridging the discrete to give the appearance of a continuum is what really interests Pynchon here. At one point in the novel we have technicians watching ‘Askanian films of Rocket flights, frame by frame, delta-x by delta-y, flightless themselves…film and calculus, both pornographies of flight’. Whether we are watching a rocket fall on a cinema screen (which we are in effect doing by reading Gravity’s Rainbow) or trying to decipher its momentum or any other of its characteristics, we are performing the same task. 

In Gravity’s Rainbow, calculus appears both literally and metaphorically, this idea of passing to a limit taking on innumerable symbolic resonances: ‘the slices of time growing thinner and thinner, a succession of rooms each with walls more silver, transparent, as the pure light of the zero comes near…’ In this particular instance a woman named Leni Pokler is describing the feeling of revolutionary joy one experiences when passing from the fear of an oppressive police state to the realisation of being part of a collective and active will. This is a form of passing to a limit for Pynchon and the reader and it is no coincidence that it is couched in the context of film: the ‘walls more silver’ seem to be slivers of transparencies, as the ‘pure light’ is that which illuminates the film. But this is by no means all…For in this novel, light is a symbol of both life and death and however much calculus may be used, couched in the concept of a limit, the only un­deniable limit which exists is that of Death, the limit of Life itself, what happens when the rocket 00000 finally lands upon us all.

In Against the Day, as in all of Pynchon’s novels, there is a search, in this case two seemingly different searches, which manage to converge. With this search comes obsession; we see this in another of the main threads of narrative, the search for the mythical city of Shambhala undertaken by many characters, including adventurers The Chums of Chance. We have briefly discussed Kit. Now Yashmeen: leaving her weird sexual practices aside, let’s see if we can figure out what the Riemann-zeta business is, which is her obsession. Let’s start with the term ‘complex-valued’ function. We all know, at least intuitively, what a function is: it assigns to every real number another real number, for example y = x², our old friend the parabola. So what then is a complex-valued function? It’s exactly the same except now we don’t assign a real number to every real number, we assign a complex number to every complex number. The obvious question then: what is a complex number? Out of all the real numbers, there is no solution to the equation  x² = –1. There does not exist a square root of minus one. But let’s say there is a square root of minus 1, which we call i. To distinguish it from the real numbers, we call i an imaginary number. Now if i is going to be a bona fide number, we are going to have to be able to add to it and multiply it so that such things such as i+1 and 3i will also be bona fide numbers. What we end up with are the complex numbers, all numbers of the form a+ib, where a and b are real numbers and i is the square root of –1. Just as we can draw the real numbers as a straight line, aptly called the ‘real number line’, so we can draw the complex numbers as a plane, aptly and analogously called the ‘complex number plane’:

Now presume for an instant that you are standing on the real number line, which is contained in the complex number plane, but you are only able to look forward and backward along the real number line, entirely unaware of the world, the plane surrounding you. We can use this as a starting point for a discussion of Pynchon’s use of mathematics in Against the Day. 

Against the Day opens as a boys-own adventure novel, as we join the Chums of Chance, high above the earth’s surface in their hydrogen skyship Inconvenience, on their way to the Chicago World Fair; the year is 1893.  Although by this stage the Bessemer process had been introduced, allowing for the mass production of steel and for the existence of the skyscraper, the tall buildings around then were relatively short in comparison to what there is around us today, and the heights the Chums of Chance could achieve. Randolph St Cosmo and his Chums, along with their dog Pugnax (whom we first encounter reading Henry James’ the Princess Casamassima) fly high above the ‘surface dwellers’, who occupy what is in effect two-dimensional space. The boys who fly high above the earth are not constrained by the same natural laws as us ‘surface dwellers’.

One way to look at what happens in Against the Day is to think of those people inhabiting reality, what we and they consider to be reality – the ‘surface dwellers’ according to Pynchon’s terminology – as constantly finding themselves subjected to outside worlds intruding. This is why strange things happen when the 3rd dimension comes down upon us in the form of the Chums of Chance, and why time travellers are able to appear. This is due to the fact that time is widely regarded as the 4th dimension. Pynchon, however, does not quite buy this and forces other conceptions of the 4th dimension upon his characters and readers alike. On a lot of occasions this intrusion occurs less supernaturally and more metaphorically; Webb Traverse, as a miner in Colorado, is changed irreparably after going underground and the reality of the conditions he and his colleagues are subject to results in his becoming an anarchist dynamiter. Pynchon continually invokes his recurring concept of the individual as the unknowing victim or subject of History in this context, referring to History as ‘Time’s pathology’ and thus as the pathology of an extra dimension.

One of the five sections of Against the Day is titled ‘Bilocations’. A bilocation is a phenomenon whereby a single person is able to occupy two distinct locations at one point in time. Of course, outside the novel, the only reported occurrences of this occur in religious texts, most notably Christian doctrine. How does this work in the framework of hidden dimensions which Pynchon has set up in Against the Day? First think of a two-dimensional image in three-dimensional space, our space; given a mirror we are able to split the image so that there now exist two copies of the image. We may bring the mirror as close as we want to the original image and so make the images as close to one another as we want (Note that if we continue to move the images closer and closer, in the limit then there is only one image). Now, assuming we are three-dimensional creatures in four-dimensional space, to produce two copies of a human being all we would need is a four-dimensional mirror. Now what we have is not mere bilocation, but a slight displacement of the same person in time. Luca Zombini, magician-extraordinaire, is under the impression that he has created multiple copies of his wife Erlys, all of whom are now wandering around New York City. He is, however, informed by Professor Sveghli of the University of Pisa, that ‘the doubles you report having produced are actually the original subjects themselves, slightly displaced in time.’ This proves of no help to Erlys’ daughter Dahlia Rideout, who has been searching New York City for her mother, spotting what, as it becomes apparent, are her doubles.

Are we able, in this fashion, to explain the dual presences of the ‘rival University professors, Renfrew at Cambridge and Werfner at Gottingen, not only eminent in their academic settings but also would-be powers in the greater world’, who have vying interests in the Eastern question, their rivalry moving ‘well beyond the Balkans’ over the years, ‘beyond the ever-shifting borders of the Ottoman empire, to the single vast Eurasian landmass’? It is no coincidence that Werfner is referred to as Renfrew’s opposite number, and that they are both referred to as each other’s conjugate.  There is a particular mathematical definition of ‘conjugate’ which may be of use here: take any point z in the complex plane and reflect it in the axis of real numbers; the result is the conjugate of z. 

It is worth noting that any real number is the conjugate of itself. So, if Renfrew and Werfner are each other’s conjugates, then they are necessarily reflections of each other and indeed they are: look at their very names! Whether we are dealing with a reflection in the real axis of complex space or a reflection in the axis separating post-Treaty of Berlin Germany from England does not matter, they are one and the same here. One is led to consider the differences and similarities between the Anglo and German reactions to the Bosnian crisis of 1908 and everything which preceded and followed it. The reader is forced to do so more and more as the novel progresses and the two characters Renfrew and Werfner actually converge, and we realise that they are the same person although they might not know it themselves.

Of great import to the novel as a whole is Renfrew/Werfner’s interest in the Eastern question: the map mentioned at the very beginning of this essay is attributed to Renfrew and the horizontal line running across it attributed to Werfner in  a very different manner:

His plan…is – insanely – to install all across the peninsula, from a little east of Sofia, here, roughly along the Balkan Range and the Sredna Gora, coincident with the upper border of the former Eastern Roumelia, and continuing on, at last to the Black Sea – das Interdikt, as he calls it, two hundred miles long, invisible, waiting for certain unconsidered footfalls and, once triggered, irreversible–pitiless…

If the map is indeed representative of the Balkan Peninsula, then this line running across it represents Werfner’s explosive Interdikt. In Venice, the map is entrusted to three people: Yashmeen Halfcourt, Cyprian Latewood and Reef Traverse (yes, one of those Traverses). What they have to do is make their way to the Balkan Peninsula and dismantle das Interdikt. It becomes apparent that das Interdikt is actually a huge construction containing phosgene, a poisonous gas possessing two important characteristics. The first is that it gained its notoriety due to its use in chemical warfare in World War I, a direct consequence of the Bosnian Crisis. The second is the role light plays in the production of phosgene gas: a character called Vamos informs Cyprian and Reef that phosgene is produced by exposing a mixture of chlorine and carbon monoxide to light. 

‘Born of light,’ said Cyprian, as if about to understand something.
‘It seems this isn’t a gas weapon, after all…Phosgene is really code for light. We learned that it is light here which is really the destructive agent.’
[…]
‘A great cascade of blindness and terror ripping straight across the heart of the Balkan Peninsula. Like nothing that has ever happened. Photometry is still too primitive for anyone to say how much light would be deployed, or how intense – somewhere far up in the millions of candles per square inch, but there are only guesses – expressions of military panic, really.’

Now this really takes us to the heart of the heart of the matter, this idea of light as a weapon. One need not look past Against the Day’s title and its epigraph from Thelonius Monk to notice this: ‘It’s always night or we wouldn’t need light.’ Light is both a destructive and reviving force throughout Pynchon’s work; in this novel we encounter characters who eat only light and others who are destroyed by it. In Gravity’s Rainbow we encounter the pure light of the zero. In both novels light works on a very concrete level and also has many symbolic resonances. What I would be tempted to call the main event of the novel is the Tunguska Event, the only event linking up all our main characters, an explosion which occurred near the Stony Tunguska, a river in Siberia, between 7am and 8am on June 30, 1908, and killed animals and knocked down trees over a distance of more than 800 miles. (By comparison the Hiroshima blast had a radius of total destruction of four-and-a-half miles.) Eyewitness reports include mention of a light as bright as the sun descending, unbearable heat and massive shock waves. 

‘But look at the sky.’ It was certainly odd. The starts had not appeared, the sky was queerly luminescent, with the occluded light of a stormy day.

The Tunguska Event occurs in the main section of the novel and as it occurs we travel from the Stony Tunguska itself to the sky above Semipalatinsk to Vienna to Trieste to Venice to encounter the novel’s main characters at this critical juncture where the world becomes lighted for a month, time itself slowing down. This is nothing short of a cataclysmic event for all of Pynchon’s characters; the most readily discernible result being the revealing of Shambhala to the Chums of Chances by the Light of the blast:

For centuries the sacred City had lain invisible, cloaked in everyday light, sun-, star-, and moonlight, the campfires and electric torches of desert explorers, until the Event over the Stony Tunguska, as if those precise light-frequencies which would allow human eyes to see the City had finally been released.

This indeed allows the Chums to realise their goal, to find the fabled City, but they must pay what is in fact the greatest price: the Light of the Blast which makes Shambhala visible to them also makes them susceptible to what they refer to as the surface-world, ‘the great burst of light had also torn the veil separating their own space from that of the everyday world’. The light, then, a result of a concrete though un­certain-in-terms-of-detail event in history, occurs in Against the Day in at least two ways which are inextricably tied into one another: firstly as a physical phenomenon which the characters of the novel experience; and also in its many symbolic/metaphorical manifestations, such as the ability to make visible these hidden worlds which occupy so much of Pynchon’s work. This brings into view the hidden dimensions, indeed allows someone occupying real space to view complex space. This is why our knowledge of the final descent of the rocket in Gravity’s Rainbow flickers in front of us on the cinema screen as the rocket descends from above our heads, it is knowledge given us by light, light in this instance streaming through the slivers of film projecting the image on the cinema screen. This is not to mention the role of calculus in the final descent: the rocket eternally coming closer and closer, never quite reaching, except in the limit, when Life becomes Death.

Light as death and light as knowledge at the same time – this really does seem to be some sort of conceptual bilocation. We can unravel this by returning to Das Interdikt. On the abstract map our heroes receive, ‘the line-segment of interest seems to be labelled “Critical Line” – Yashmeen, isn’t that Riemann talk?’ These are the words of Cyprian Latewood and, yes, he is correct, the term ‘critical line’ is Riemann talk. Recall that the Riemann hypothesis has to do with where the zeroes of the Riemann-zeta function. The function of two variables f(x,y) = y – x³ has zeroes wherever y = x³, that is, on all the points on the graph. Similarly, since the Riemann-zeta function is a map from the complex numbers to the complex numbers, that is, assigns to each complex number another complex number, its zeroes will lie on the complex plane. The Riemann hypothesis states that all the zeroes of Riemann’s zeta function lie on the line Re(z)=¹/₂, that is, all complex number a+ib such that a=¹/₂. This line is called ‘the Critical line’!

In a novel essentially about searching, the Critical line in complex space is Yashmeen’s final goal; literally, the pursuit of mathematical knowledge, and the metaphor which follows directly from such knowledge is that of light. Thus the line is both a symbol of light and then light itself when realised in the context of das Interdikt as a luminary weapon. In the same way, Yashmeen’s Riemannian critical line is entirely destructive for her: she cannot see outside it. In fact, it becomes her real number line. Although surrounded by complex space, one cannot see outside the real except at those rare moments such as the Tunguska event; it’s just that Yashmeen’s real axis has been translated and rotated into Riemann’s Critical line. It is this inability to see through the falseness of the everyday that kills us, Pynchon seems to be saying, this inability to see everything hidden behind the real. This is why we need insight to descend upon us from above and this is indeed why we need learn the lessons of that fourth dimension, the past, the lessons of History. It is for this very reason that we need to notice trends, to believe and realise that there are connections binding this world together which we cannot yet see, in a word, to abstract, not to view the world as a series of discrete, disconnected objects and events but to recognise patterns, to see how ostensibly unrelated ideas are and must be connected. Wallace Stevens wrote in his Notes Towards a Supreme Fiction, ‘it must be abstract.’ Pynchon is merely using abstraction as a way to view the world, to make sense of it, to cherish it. To quote from his first novel, V.: ‘Perhaps if we lived on a crest, things would be different. We could at least see.’