Once upon a prime, p.14
Once Upon a Prime, page 14
When the French retreat from Moscow, they keep coming off worse in skirmishes with small groups of Russian troops, even though the French army is huge. This, says Tolstoy, seems to contradict conventional military wisdom that believes the strength of an army depends on its size alone. That, he says, is like claiming that momentum depends only on mass, when in fact it is the product of mass and velocity. In the same way, the strength of an army must be the product of its mass and some unknown x. Military science usually puts this unknown factor down to the genius of the commanders. But, says Tolstoy, the sweep of history is not decided by individuals. This x is rather the “spirit of the army, that is to say, the greater or lesser readiness to fight and face danger felt by all the men composing an army, quite independently of whether they are, or are not, fighting under the command of a genius.” Like any good math teacher, Tolstoy even gives us an example. Suppose ten men (or battalions, or divisions) defeat fifteen, sustaining four casualties. Then the winning side loses four to the fifteen lost by their opponents “and therefore Consequently ” This equation, as Tolstoy rightly points out, does not tell us what x and y are, but it does give us a ratio between them. Because =3.75, we can now say that the winning army has 3.75 times as much fighting spirit as the losing side. And, he concludes, “by bringing variously selected historic units (battles, campaigns, periods of war) into such equations, a series of numbers could be obtained in which certain laws should exist and might be discovered.” Ah, yes, the classic final sentence of every application ever made for renewal of grant funding: “More research is needed.”
If it’s a surprise to see Tolstoy detonate an equation in the middle of the Napoleonic battlefield, then wait until he brings the big guns out: he uses calculus as a metaphor for understanding the whole of human history. In War and Peace, he argues powerfully against the idea that the course of history can be altered by the actions of any one person. The French army, he says, does not retreat from Moscow toward Smolensk because Napoleon has ordered it. Rather, Napoleon gave the order to retreat because “the forces which influenced the whole army and directed it along the [Smolensk] road acted simultaneously on him also.”
So how do we make sense of these historical forces? Tolstoy begins by reminding us of that old puzzle of Achilles and the tortoise—known as Zeno’s paradox. Achilles runs ten times as fast as the tortoise, so he should win any race, even if he gives the tortoise a head start. But in the time it takes Achilles to catch up to where the tortoise started, the tortoise has moved a little farther forward. And by the time Achilles reaches that next place, the tortoise has moved again. It seems that Achilles can never overtake the tortoise—which is obviously ridiculous. The paradox, says Tolstoy, is caused by the fact that the movement of Achilles and the tortoise is being artificially divided into discrete, discontinuous parts, whereas in reality, the motion of both is continuous. Fortunately, there is a branch of mathematics that tells you exactly how to turn the discrete into the continuous.
Calculus was developed in the late seventeenth century by two of the all-time mathematical greats, Isaac Newton and Gottfried Leibniz. (There were bitter arguments about who had thought of it first.) It’s fantastic for solving problems that involve movement and change, like the motion of the planets, or objects accelerating under gravity (which is the other thing Newton is famous for). If something is moving at a fixed speed, we can work out how far it will travel—if it’s going 40 miles per hour, then after an hour, it has gone, well, 40 miles. But what if the speed is constantly changing? How can we work it out then? What we could try is measuring the speed every minute, say, and then assuming that’s the speed for the whole minute, working out the distance traveled in that minute, and adding up all those little distances. If we want to get more accurate, we could measure the speed every 30 seconds, or every second, or every nanosecond. Each time we are adding up ever tinier distances, summing over an ever larger number of these tiny changes, or “differentials.” But the challenge you face is that in the limit, you’d be trying to add an infinite number of zeros. Calculus is the technique that allows us to deal with these infinitesimal numbers rather than imposing an artificial division into separate units. It is one of the great achievements of mathematics.
Tolstoy explains that we need to do the same thing with history: “The movement of humanity, arising as it does from innumerable arbitrary human wills, is continuous. To understand the laws of this continuous movement is the aim of history. But to arrive at these laws, resulting from the sum of all those human wills, man’s mind postulates arbitrary and disconnected units,” such as particular events in isolation, or the actions of some king or commander. “Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.”
In War and Peace, Tolstoy is railing against the “great man” theory of history. It’s not the great leader who is the x factor for the victorious army, but the collective fighting spirit. It’s not the king or emperor who directs the course of events, but wider forces. He counteracts the theory with his fighting spirit equation and with the calculus metaphor we’ve just discussed. For him, mathematics is an emblem of logical rigor, a way to access the objective truth, and the only chance we have of understanding history.
* * *
War and Peace, with its mixture of history, philosophy, and narrative, was unlike any other novel. Tolstoy, in fact, said he didn’t think of it as a novel at all. I want to finish this chapter with a look at the mathematics in another uncategorizable book: James Joyce’s Ulysses.
I mentioned in the introductory chapter that Joyce had an admiration for mathematics, but if we think about what he is famous for, the stream-of-consciousness style of books like Ulysses and especially Finnegans Wake, he might be the author we would least think of associating with structure of any kind, never mind mathematics. And yet there’s a diagram from Euclid right in the center of Finnegans Wake. There’s a chapter full of calculations in Ulysses.
Geometry gets a mention in the first paragraph of the first page of Joyce’s first published book, Dubliners: “Every night as I gazed up at the window I said softly to myself the word paralysis. It had always sounded strangely in my ears, like the word gnomon in the Euclid and the word simony in the Catechism.” This mention of the gnomon is not a random allusion, either. The word is mostly used nowadays, if it’s used at all, to refer to the sticking-up part of the sundial that casts a shadow (as we saw in Moby-Dick), but its geometric meaning is a parallelogram with a smaller parallelogram cut out of it. This “shape with a missing part” is a good description of Dubliners. Sometimes the missing part in a story is around meaning—the language used is ambiguous and we cannot see the motivations of the characters. Other times it is parts of the action that are omitted. In one story we are with a young woman, Eveline, at home, until she stands up suddenly, and the narrative jumps to a scene somewhere entirely different. We aren’t party to her decision to leave the house, or where to go, or how she gets there.
Joyce had a reverence for, even an awe of, mathematics. Like Melville, he studied Euclid’s geometry at school. Though he was not such a star student as Melville, Joyce was certainly familiar with algebra and geometry, and his extensive notebooks reveal a fascination with mathematical ideas. He was curious about concepts of limits and infinity—there’s a page on which he writes things like 0 = 1 = ∞ = These represent limits, because if we divide 1 by ever larger numbers, we approach, but do not ever quite reach, zero, and the same is true for the characterization of infinity as Sometimes there is pseudomathematics too, as in the jotting which is a rather facetious “formula” about the Holy Trinity.
Writers on Joyce have sometimes used mathematical analogies to describe his work, but not perhaps for the reasons I might do so. The writer of this 1941 obituary, for example, doesn’t seem to have a very clear idea of what mathematics actually is:
Joyce was also the great research scientist of letters, handling words with the same freedom and originality that Einstein handles mathematical symbols. The sounds, patterns, roots and connotations of words interested him much more than their definite meanings. One might say that he invented a non-Euclidean geometry of language; and that he worked over it with doggedness and devotion.
I have some issues with this. Firstly, Einstein didn’t just say “Ooh, I think an m next to this c2 would look cool.” It’s not the handling of the symbols that Einstein was good at, it’s the meaning of the concepts. It reminds me of the time I was asked to prettify an equation for a newspaper article. Apparently, the graphic design department said it didn’t look exciting enough visually—could I zhuzh it up a bit? I told them, not if you want it still to be true. Secondly, what could a “non-Euclidean geometry of language” possibly be? The obituarist simply grabbed a clever-sounding term from mathematics to say that Joyce did something exciting and new.
In this century, non-Euclidean geometry, while still very exciting, is not new. Nowadays, we get told that Joyce invented fractals. (We’ll delve more into fractals in Part III.) I read an essay recently that posits the fractal (a new and exciting math concept circa 1980–2000) as “an active Joycean concept” and credits Joyce with “anticipating a fractal formalism that would not be officially discovered until well into the latter half of the twentieth century.” For me, this goes too far. We have to be very careful about crediting writers with this kind of fortune-telling. Let me give an over-the-top example to prove a point. The scientist Murray Gell-Mann described how James Joyce provided the name for a new kind of subatomic particle discovered in the 1960s: “In one of my occasional perusals of Finnegans Wake, by James Joyce, I came across the word ‘quark’ in the phrase ‘Three quarks for Muster Mark.’ … The number three fitted perfectly the way quarks occur in nature.” (For example, every proton contains three quarks.) Do we conclude from this that Joyce anticipated quantum physics? Of course not—and we shouldn’t go around saying he anticipated fractals either. It’s a shame, because as an analogy for what Joyce does in Ulysses, fractals are great. Zoom in as far as you like into the human experience, one might say, and the complexity is not diminished. The mind’s experience of a single day, a single hour, is as richly detailed as the memories of a lifetime. Notwithstanding this fact, James Joyce didn’t invent fractals. He doesn’t need to have done that to be brilliant.
So, what can a conversation between James Joyce and mathematics tell us? Is it just that the work of Joyce is so dense with both meaning and ambiguity that we can put any meaning we like into it? In the case for the defense, I bring to your attention Joyce’s own words: one entire chapter of Ulysses was, he said, a “mathematical catechism.” I want to explain that a little.
You might remember that Ulysses is loosely based on Homer’s Odyssey, an epic poem recounting the adventures of Odysseus, king of Ithaca, over ten years as he travels home after the Trojan Wars. The name Ulysses is the Latinized version of Odysseus. The action in Joyce’s book is transplanted to Dublin and describes the events of one fairly ordinary day in the life of one fairly ordinary middle-aged man, Leopold Bloom (Ulysses), a young man he meets, Stephen Dedalus (representing Telemachus—Odysseus’s son), and Bloom’s wife, Molly (Penelope). Each chapter is associated in some way with an episode from the Odyssey: Chapter 11 is known as “Sirens” and is full of singing and music; Chapter 17 is known as “Ithaca” because it describes Bloom returning home at the end of the day, accompanied by Stephen Dedalus; and the final chapter in the book is “Penelope,” with Molly Bloom’s famous stream-of-consciousness monologue as she falls asleep.
What does mathematics do for James Joyce in Ulysses? There are mathematical references scattered throughout the book, but “Ithaca” is the most overtly mathematical chapter. It is, says Joyce, a “mathematico-astronomico-physico-mechanico-geometrico-chemico sublimation of Bloom and Stephen … to prepare for the final amplitudinously curvilinear episode Penelope.” He goes further: it is best read by “someone who is a physicist, mathematician and astronomer and a number of other things.” The structure of “Ithaca” is a series of questions—a catechism—that parodies scientific certainty. The books of Euclid were a cornerstone of mathematical education in Jesuit schools, and they were held up for millennia as the apotheosis of pure logic. The joke in “Ithaca” is the attempt to apply this logic to things that definitely aren’t behaving rationally.
Stephen Dedalus and Leopold Bloom’s nocturnal wanderings around Dublin are given a pseudogeometric veneer of respectability here in the opening question and response:
What parallel courses did Bloom and Stephen follow returning?
Starting united both at normal walking pace from Beresford place they followed in the order named Lower and Middle Gardiner streets and Mountjoy square, west … they [crossed] the circus before George’s church diametrically, the chord in any circle being less than the arc which it subtends.
In other words, they took a shortcut across the circle, as it’s quicker than going around. When they arrive home, it is to “the 4th of the equidifferent uneven numbers,” which is Joyce’s way of saying that Bloom’s house number is seven. Bloom lights a fire using “irregular polygons” of coal. In the kitchen, there are “four square handkerchiefs folded unattached consecutively in adjacent rectangles,” suspended from a “curvilinear rope.” It reads like a crazy math problem. Joyce really goes to town with all this a few pages later. Stephen is younger than Bloom, and the disembodied questioner would like to know “what relation existed between their ages.” The answer to this is glorious:
16 years before in 1888 when Bloom was of Stephen’s present age Stephen was 6. 16 years after in 1920 when Stephen would be of Bloom’s present age Bloom would be 54. In 1936 when Bloom would be 70 and Stephen 54 their ages initially in the ratio of 16 to 0 would be as 17½ to 13½, the proportion increasing and the disparity diminishing according as arbitrary future years were added, for if the proportion existing in 1883 had continued immutable, conceiving that to be possible, till then 1904 when Stephen was 22 Bloom would be 374 and in 1920 when Stephen would be 38, as Bloom then was, Bloom would be 646 while in 1952 when Stephen would have attained the maximum postdiluvian age of 70 Bloom, being 1190 years alive having been born in the year 714, would have surpassed by 221 years the maximum antediluvian age, that of Methuselah, 969 years, while, if Stephen would continue to live until he would attain that age in the year 3072 A.D., Bloom would have been obliged to have been alive 83,300 years, having been obliged to have been born in the year 81,396 B.C.
It all reminds me of a mathematical puzzle posed by Gustave Flaubert (an author much admired by Joyce) in an 1841 letter to his sister Caroline: “Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?” You are getting a lot of information here, but none of it actually helps you to solve the problem. We are back to the data overdose of Captain Ahab.
The stream-of-consciousness style of much of Ulysses, and orders of magnitude more in Finnegans Wake, belies the fact that every word is nonetheless carefully chosen.5 Bloom’s inner monologue through the day is full, like everyone’s, of half-facts, snatches of quotations, fragments of misremembered science. The “Ithaca” chapter positions itself as authoritative, but Joyce inserts a huge number of errors that get in under the radar of the catechistic style. It reminds us that even sources like dictionaries and encyclopedias are not infallible. They are, after all, written by people. (My favorite dictionary definition of all time, by the way, is in the British Chambers Dictionary I have on my shelf, which defines an éclair as “a cake, long in shape but short in duration.”)
Like the scientific “facts” in “Ithaca,” many of the numerical calculations are incorrect. Some are incorrect on purpose, some probably aren’t. When Leopold Bloom sits down at the end of the day and tallies up his expenditure, the fact that he “forgets” to write down the money spent in a brothel is not an error in Joyce’s arithmetic. But there are also several miscalculations around Bloom’s and Stephen’s ages and the year Bloom would have had to be born to achieve the correct proportions. For example, for Bloom to be 1,190 years old (seventeen times Stephen’s age of seventy) in 1952, he would have been born in the year 762, not 714. We can see where the mistake comes from—if Bloom was born in the year 714, he would reach the age of 1,190 in 1904, when the book is set. But that would not preserve the 17:1 ratio of their ages. Even if these are deliberate mistakes, the number of corrections Joyce made to the calculation of Bloom’s budget over the course of several drafts and proofs of the novel is good evidence that he did have some difficulty in manipulating the numbers, in spite of having performed relatively well in arithmetic exams at school.
But arithmetic is not mathematics, just as spelling is not literature, and there is a lot more than just arithmetic in “Ithaca.” I want to show you a fun digression about powers because it resulted in a certain kind of number being named after James Joyce. Here is Leopold Bloom, thinking about the numbers involved in calculations about distances between the stars:
Some years previously in 1886 when occupied with the problem of the quadrature of the circle he had learned of the existence of a number computed to a relative degree of accuracy to be of such magnitude and of so many places, e.g., the 9th power of the 9th power of 9, that, the result having been obtained, 33 closely printed volumes of 1,000 pages each of innumerable quires and reams of India paper would have to be requisitioned in order to contain the complete tale of its printed integers of units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, hundreds of millions, billions, the nucleus of the nebula of every digit of every series containing succinctly the potentiality of being raised to the utmost kinetic elaboration of any power of any of its powers.
If it’s a surprise to see Tolstoy detonate an equation in the middle of the Napoleonic battlefield, then wait until he brings the big guns out: he uses calculus as a metaphor for understanding the whole of human history. In War and Peace, he argues powerfully against the idea that the course of history can be altered by the actions of any one person. The French army, he says, does not retreat from Moscow toward Smolensk because Napoleon has ordered it. Rather, Napoleon gave the order to retreat because “the forces which influenced the whole army and directed it along the [Smolensk] road acted simultaneously on him also.”
So how do we make sense of these historical forces? Tolstoy begins by reminding us of that old puzzle of Achilles and the tortoise—known as Zeno’s paradox. Achilles runs ten times as fast as the tortoise, so he should win any race, even if he gives the tortoise a head start. But in the time it takes Achilles to catch up to where the tortoise started, the tortoise has moved a little farther forward. And by the time Achilles reaches that next place, the tortoise has moved again. It seems that Achilles can never overtake the tortoise—which is obviously ridiculous. The paradox, says Tolstoy, is caused by the fact that the movement of Achilles and the tortoise is being artificially divided into discrete, discontinuous parts, whereas in reality, the motion of both is continuous. Fortunately, there is a branch of mathematics that tells you exactly how to turn the discrete into the continuous.
Calculus was developed in the late seventeenth century by two of the all-time mathematical greats, Isaac Newton and Gottfried Leibniz. (There were bitter arguments about who had thought of it first.) It’s fantastic for solving problems that involve movement and change, like the motion of the planets, or objects accelerating under gravity (which is the other thing Newton is famous for). If something is moving at a fixed speed, we can work out how far it will travel—if it’s going 40 miles per hour, then after an hour, it has gone, well, 40 miles. But what if the speed is constantly changing? How can we work it out then? What we could try is measuring the speed every minute, say, and then assuming that’s the speed for the whole minute, working out the distance traveled in that minute, and adding up all those little distances. If we want to get more accurate, we could measure the speed every 30 seconds, or every second, or every nanosecond. Each time we are adding up ever tinier distances, summing over an ever larger number of these tiny changes, or “differentials.” But the challenge you face is that in the limit, you’d be trying to add an infinite number of zeros. Calculus is the technique that allows us to deal with these infinitesimal numbers rather than imposing an artificial division into separate units. It is one of the great achievements of mathematics.
Tolstoy explains that we need to do the same thing with history: “The movement of humanity, arising as it does from innumerable arbitrary human wills, is continuous. To understand the laws of this continuous movement is the aim of history. But to arrive at these laws, resulting from the sum of all those human wills, man’s mind postulates arbitrary and disconnected units,” such as particular events in isolation, or the actions of some king or commander. “Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.”
In War and Peace, Tolstoy is railing against the “great man” theory of history. It’s not the great leader who is the x factor for the victorious army, but the collective fighting spirit. It’s not the king or emperor who directs the course of events, but wider forces. He counteracts the theory with his fighting spirit equation and with the calculus metaphor we’ve just discussed. For him, mathematics is an emblem of logical rigor, a way to access the objective truth, and the only chance we have of understanding history.
* * *
War and Peace, with its mixture of history, philosophy, and narrative, was unlike any other novel. Tolstoy, in fact, said he didn’t think of it as a novel at all. I want to finish this chapter with a look at the mathematics in another uncategorizable book: James Joyce’s Ulysses.
I mentioned in the introductory chapter that Joyce had an admiration for mathematics, but if we think about what he is famous for, the stream-of-consciousness style of books like Ulysses and especially Finnegans Wake, he might be the author we would least think of associating with structure of any kind, never mind mathematics. And yet there’s a diagram from Euclid right in the center of Finnegans Wake. There’s a chapter full of calculations in Ulysses.
Geometry gets a mention in the first paragraph of the first page of Joyce’s first published book, Dubliners: “Every night as I gazed up at the window I said softly to myself the word paralysis. It had always sounded strangely in my ears, like the word gnomon in the Euclid and the word simony in the Catechism.” This mention of the gnomon is not a random allusion, either. The word is mostly used nowadays, if it’s used at all, to refer to the sticking-up part of the sundial that casts a shadow (as we saw in Moby-Dick), but its geometric meaning is a parallelogram with a smaller parallelogram cut out of it. This “shape with a missing part” is a good description of Dubliners. Sometimes the missing part in a story is around meaning—the language used is ambiguous and we cannot see the motivations of the characters. Other times it is parts of the action that are omitted. In one story we are with a young woman, Eveline, at home, until she stands up suddenly, and the narrative jumps to a scene somewhere entirely different. We aren’t party to her decision to leave the house, or where to go, or how she gets there.
Joyce had a reverence for, even an awe of, mathematics. Like Melville, he studied Euclid’s geometry at school. Though he was not such a star student as Melville, Joyce was certainly familiar with algebra and geometry, and his extensive notebooks reveal a fascination with mathematical ideas. He was curious about concepts of limits and infinity—there’s a page on which he writes things like 0 = 1 = ∞ = These represent limits, because if we divide 1 by ever larger numbers, we approach, but do not ever quite reach, zero, and the same is true for the characterization of infinity as Sometimes there is pseudomathematics too, as in the jotting which is a rather facetious “formula” about the Holy Trinity.
Writers on Joyce have sometimes used mathematical analogies to describe his work, but not perhaps for the reasons I might do so. The writer of this 1941 obituary, for example, doesn’t seem to have a very clear idea of what mathematics actually is:
Joyce was also the great research scientist of letters, handling words with the same freedom and originality that Einstein handles mathematical symbols. The sounds, patterns, roots and connotations of words interested him much more than their definite meanings. One might say that he invented a non-Euclidean geometry of language; and that he worked over it with doggedness and devotion.
I have some issues with this. Firstly, Einstein didn’t just say “Ooh, I think an m next to this c2 would look cool.” It’s not the handling of the symbols that Einstein was good at, it’s the meaning of the concepts. It reminds me of the time I was asked to prettify an equation for a newspaper article. Apparently, the graphic design department said it didn’t look exciting enough visually—could I zhuzh it up a bit? I told them, not if you want it still to be true. Secondly, what could a “non-Euclidean geometry of language” possibly be? The obituarist simply grabbed a clever-sounding term from mathematics to say that Joyce did something exciting and new.
In this century, non-Euclidean geometry, while still very exciting, is not new. Nowadays, we get told that Joyce invented fractals. (We’ll delve more into fractals in Part III.) I read an essay recently that posits the fractal (a new and exciting math concept circa 1980–2000) as “an active Joycean concept” and credits Joyce with “anticipating a fractal formalism that would not be officially discovered until well into the latter half of the twentieth century.” For me, this goes too far. We have to be very careful about crediting writers with this kind of fortune-telling. Let me give an over-the-top example to prove a point. The scientist Murray Gell-Mann described how James Joyce provided the name for a new kind of subatomic particle discovered in the 1960s: “In one of my occasional perusals of Finnegans Wake, by James Joyce, I came across the word ‘quark’ in the phrase ‘Three quarks for Muster Mark.’ … The number three fitted perfectly the way quarks occur in nature.” (For example, every proton contains three quarks.) Do we conclude from this that Joyce anticipated quantum physics? Of course not—and we shouldn’t go around saying he anticipated fractals either. It’s a shame, because as an analogy for what Joyce does in Ulysses, fractals are great. Zoom in as far as you like into the human experience, one might say, and the complexity is not diminished. The mind’s experience of a single day, a single hour, is as richly detailed as the memories of a lifetime. Notwithstanding this fact, James Joyce didn’t invent fractals. He doesn’t need to have done that to be brilliant.
So, what can a conversation between James Joyce and mathematics tell us? Is it just that the work of Joyce is so dense with both meaning and ambiguity that we can put any meaning we like into it? In the case for the defense, I bring to your attention Joyce’s own words: one entire chapter of Ulysses was, he said, a “mathematical catechism.” I want to explain that a little.
You might remember that Ulysses is loosely based on Homer’s Odyssey, an epic poem recounting the adventures of Odysseus, king of Ithaca, over ten years as he travels home after the Trojan Wars. The name Ulysses is the Latinized version of Odysseus. The action in Joyce’s book is transplanted to Dublin and describes the events of one fairly ordinary day in the life of one fairly ordinary middle-aged man, Leopold Bloom (Ulysses), a young man he meets, Stephen Dedalus (representing Telemachus—Odysseus’s son), and Bloom’s wife, Molly (Penelope). Each chapter is associated in some way with an episode from the Odyssey: Chapter 11 is known as “Sirens” and is full of singing and music; Chapter 17 is known as “Ithaca” because it describes Bloom returning home at the end of the day, accompanied by Stephen Dedalus; and the final chapter in the book is “Penelope,” with Molly Bloom’s famous stream-of-consciousness monologue as she falls asleep.
What does mathematics do for James Joyce in Ulysses? There are mathematical references scattered throughout the book, but “Ithaca” is the most overtly mathematical chapter. It is, says Joyce, a “mathematico-astronomico-physico-mechanico-geometrico-chemico sublimation of Bloom and Stephen … to prepare for the final amplitudinously curvilinear episode Penelope.” He goes further: it is best read by “someone who is a physicist, mathematician and astronomer and a number of other things.” The structure of “Ithaca” is a series of questions—a catechism—that parodies scientific certainty. The books of Euclid were a cornerstone of mathematical education in Jesuit schools, and they were held up for millennia as the apotheosis of pure logic. The joke in “Ithaca” is the attempt to apply this logic to things that definitely aren’t behaving rationally.
Stephen Dedalus and Leopold Bloom’s nocturnal wanderings around Dublin are given a pseudogeometric veneer of respectability here in the opening question and response:
What parallel courses did Bloom and Stephen follow returning?
Starting united both at normal walking pace from Beresford place they followed in the order named Lower and Middle Gardiner streets and Mountjoy square, west … they [crossed] the circus before George’s church diametrically, the chord in any circle being less than the arc which it subtends.
In other words, they took a shortcut across the circle, as it’s quicker than going around. When they arrive home, it is to “the 4th of the equidifferent uneven numbers,” which is Joyce’s way of saying that Bloom’s house number is seven. Bloom lights a fire using “irregular polygons” of coal. In the kitchen, there are “four square handkerchiefs folded unattached consecutively in adjacent rectangles,” suspended from a “curvilinear rope.” It reads like a crazy math problem. Joyce really goes to town with all this a few pages later. Stephen is younger than Bloom, and the disembodied questioner would like to know “what relation existed between their ages.” The answer to this is glorious:
16 years before in 1888 when Bloom was of Stephen’s present age Stephen was 6. 16 years after in 1920 when Stephen would be of Bloom’s present age Bloom would be 54. In 1936 when Bloom would be 70 and Stephen 54 their ages initially in the ratio of 16 to 0 would be as 17½ to 13½, the proportion increasing and the disparity diminishing according as arbitrary future years were added, for if the proportion existing in 1883 had continued immutable, conceiving that to be possible, till then 1904 when Stephen was 22 Bloom would be 374 and in 1920 when Stephen would be 38, as Bloom then was, Bloom would be 646 while in 1952 when Stephen would have attained the maximum postdiluvian age of 70 Bloom, being 1190 years alive having been born in the year 714, would have surpassed by 221 years the maximum antediluvian age, that of Methuselah, 969 years, while, if Stephen would continue to live until he would attain that age in the year 3072 A.D., Bloom would have been obliged to have been alive 83,300 years, having been obliged to have been born in the year 81,396 B.C.
It all reminds me of a mathematical puzzle posed by Gustave Flaubert (an author much admired by Joyce) in an 1841 letter to his sister Caroline: “Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?” You are getting a lot of information here, but none of it actually helps you to solve the problem. We are back to the data overdose of Captain Ahab.
The stream-of-consciousness style of much of Ulysses, and orders of magnitude more in Finnegans Wake, belies the fact that every word is nonetheless carefully chosen.5 Bloom’s inner monologue through the day is full, like everyone’s, of half-facts, snatches of quotations, fragments of misremembered science. The “Ithaca” chapter positions itself as authoritative, but Joyce inserts a huge number of errors that get in under the radar of the catechistic style. It reminds us that even sources like dictionaries and encyclopedias are not infallible. They are, after all, written by people. (My favorite dictionary definition of all time, by the way, is in the British Chambers Dictionary I have on my shelf, which defines an éclair as “a cake, long in shape but short in duration.”)
Like the scientific “facts” in “Ithaca,” many of the numerical calculations are incorrect. Some are incorrect on purpose, some probably aren’t. When Leopold Bloom sits down at the end of the day and tallies up his expenditure, the fact that he “forgets” to write down the money spent in a brothel is not an error in Joyce’s arithmetic. But there are also several miscalculations around Bloom’s and Stephen’s ages and the year Bloom would have had to be born to achieve the correct proportions. For example, for Bloom to be 1,190 years old (seventeen times Stephen’s age of seventy) in 1952, he would have been born in the year 762, not 714. We can see where the mistake comes from—if Bloom was born in the year 714, he would reach the age of 1,190 in 1904, when the book is set. But that would not preserve the 17:1 ratio of their ages. Even if these are deliberate mistakes, the number of corrections Joyce made to the calculation of Bloom’s budget over the course of several drafts and proofs of the novel is good evidence that he did have some difficulty in manipulating the numbers, in spite of having performed relatively well in arithmetic exams at school.
But arithmetic is not mathematics, just as spelling is not literature, and there is a lot more than just arithmetic in “Ithaca.” I want to show you a fun digression about powers because it resulted in a certain kind of number being named after James Joyce. Here is Leopold Bloom, thinking about the numbers involved in calculations about distances between the stars:
Some years previously in 1886 when occupied with the problem of the quadrature of the circle he had learned of the existence of a number computed to a relative degree of accuracy to be of such magnitude and of so many places, e.g., the 9th power of the 9th power of 9, that, the result having been obtained, 33 closely printed volumes of 1,000 pages each of innumerable quires and reams of India paper would have to be requisitioned in order to contain the complete tale of its printed integers of units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, hundreds of millions, billions, the nucleus of the nebula of every digit of every series containing succinctly the potentiality of being raised to the utmost kinetic elaboration of any power of any of its powers.
