Here at last is the long-delayed next installment of my ongoing primer on the science in Thomas Pynchon's novel Against the Day. The draft of part 1 can be found here. Illness and major deadlnes put me back by months. I hope to have more installments out soon.
Anyway, here is part 2: Quaternions and Vectors in Against the Day:
Science and Against the Day
Part 2: Vectors and Quaternions
I. The need for algebra in more than one dimension
In Against the Day, Pynchon frequently refers to a relatively obscure conflict in the mathematics and physics community that took place in the early 1890's between advocates of quaternions and proponents of the newer vector analysis. This conflict is tied in to major themes in the book that emphasize the tensions between the old and the emerging world that culminated in the conflict of World War I, as well as the ability to perceive and describe the world in more than the three dimensions of Euclidean space. Quaternions, like the luminiferous aether discussed in Part 1 of this essay, became superfluous and obsolete, unnecessary in the efforts of physicists to describe the natural world after the advent of modern vector algebra and calculus.
To understand this conflict, it is important to understand what mathematicians and physicists were searching for when they developed first quaternions and then vector analysis. The most important aim of these mathematicians and physicists had in mind was the ability to do algebraic manipulations in more than one dimension.
All of us are familiar with the basic, one-dimensional operations which we learned in elementary school: addition, subtraction, multiplication, and division. By one-dimensional, I mean operations on combinations of single numbers; in other words, what we do in every day addition or multiplication. Each of these single numbers can all be represented on a one-dimensional number line, and each operation can be thought of as moving left or right along the line:
So for example, the operation 2 + 3 moves you to the right three units on the number line, from position 2 to position 5. I know that readers of Pynchon’s novels do not need a review of 1st grade math; the important point I’m trying to make is that these operations we’re all familiar with are one-dimensional operations on a number line.
These basic, one-dimensional operations have certain important properties, ones which most of us take for granted once we're out of elementary school. For example, two important properties are:
Associativity - when adding or multiplying more than two numbers, it doesn't matter how you group them:
(a + b) + c = a + (b + c) and (a x b) x c = a x (b x c)
Commutativity - when you add or multiply two numbers, it doesn't matter how you order them:
a + b = b + a and a x b = b x a
The challenge to mathematicians in the 18th century was to define algebraic operations such as addition and multiplication on pairs (or larger groups) of numbers - in essence, creating an algebra of more than one dimension. In order to be useful, these operations on pairs of numbers had to have at least some of the important properties for operations on single numbers; for example, the addition of number pairs should be associative and commutative.
Why are operations on paris or other groups of numbers important? One reason is that such definitions would represent an advance in pure mathematics, but another key reason is that higher dimensional mathematics would make it easier to work with the laws of physics in more than one dimension. To see what these means, let's take Newton's Second Law of Motion as an example. This law states that the force acting on an object is proportional to the mass of the object times the acceleration of the object produced by the force. Newton's second law can be written as this equation:
F = ma
But in three-dimensional space, Newton's Second Law is properly written with three equations, to account for the force and the acceleration in each dimension (each dimension represented by x, y, or z):
This means that when we make calculations using Newton's Second Law, we really have to perform our calculations on three equations if we want to deal with ordinary three-dimensional space. In a complicated situation where we want to add and subtract many different forces, we have to add and subtract the three components for each force. A system of analysis, where our operations of addition and subtraction could be performed on a set of three numbers at once, treating the three-dimensional force as one unit, would greatly simplify calculations using Newton's Second Law or the much more difficult laws of electromagnetism formulated by Maxwell.
Here is another way to see the problem. Scientists distinguish between the speed of an object, which is just a magnitude or a scalar quantity (such as ‘60 miles per hour’), and velocity, which is comprised of both a magnitude and a direction, and thus is a vector (such as ’60 miles per hour going northwest’). Adding speeds together is easy, but how do we add velocities? 18th and 19th century scientists could do this by breaking velocities down into their one-dimensional components (just as we did for Newton’s Second Law), but they realized that a better system was needed.
II. Complex numbers and two-dimensional math
Before tackling three dimensions, let’s start with just two. 19th century scientists already had a powerful system of analysis for dealing with pairs of numbers - complex numbers. Complex numbers are comprised of two parts, a real part and an imaginary part. The imaginary part consists of a number multiplied by the number i , which is the square root of -1:
This means ‘i squared’ is equal to -1:
Thus a complex number z looks like the following, where a and b are any numbers you choose:
Again, a is called the real part, and ib is known as the imaginary part.
Unlike our ordinary numbers on a number line, complex numbers can be represented on a two dimensional plane, called the complex plane. One axis of the plane is the number line for the real numbers, and the second axis is the number line for the imaginary numbers:
Instead of a point on a one-dimensional number line, complex numbers can be interpreted as points on the two-dimensional complex plane. For example, the complex number ‘6 + 3i’ is the point on the complex plane as shown below:
Using complex numbers, one can now describe two-dimensional operations like rotation. For example, multiplying a number by i is equivalent to a 90-degree rotation on the complex plane. Thus the operation:
is equivalent to this 90-degree rotation on the complex plane:
Instead of multiplying by i, one can multiply by any complex number to get a rotation of any angle other than 90-degrees. This subject comes up on p. 132 of Against the Day, where Dr. Blope talks about rotations, not in the two-dimensional space of the complex plane, but in the three dimensional space of quaternions:
“ ‘Time moves on but one axis, ‘ advised Dr. Blope, ‘past to future - the only turning possible being turns of a hundred and eighty degrees. In the Quaternions, a ninety-degree direction would correspond to an additional axis whose unit is √-1. A turn through any other angle would require for its unit a complex number.’”
This ability to use operations of complex numbers to describe two dimensional rotations and translations is an extremely important tool in math and physics.
Complex numbers have many other amazing properties, but most relevant to our discussion of Against the Day is that complex numbers can be manipulated with all of our basic operations - they can be added, subtracted, multiplied, and divided, with the kinds of useful properties mentioned earlier, such as associativity and commutativity. Thus, with complex numbers, we have a way to do algebra in two dimensions.
III. Extending complex numbers to three dimensions: Quaternions
In the mid-19th century, several mathematicians were looking for ways to extend the two-dimensional geometrical interpretation of complex numbers to three dimensions. One important figure was Hermann Grassman, whose system turned out to be closest to the yet-future vector analysis. Grassman is mentioned on occasion in Against the Day, but his role in the development of vector analysis was somewhat diminished by the fact that, compared to William Hamilton, Grassman was fairly unknown. It was William Hamilton, who was already famous for earlier work, who developed the most well-known immediate predecessor to vector analysis - quaternions.
William Hamilton had been struggling with the problem of how to generalize complex numbers to higher dimensions. While walking with his wife in Dublin, Hamilton discovered the fundamental relationship that could underlie such generalized complex numbers, which he called quaternions. This fundamental relationship is this:
Hamilton was so excited about the discovery that he carved this equation into the stone of Dublin’s Brougham Bridge. Readers of Against the Day will appreciate Hamilton’s own language describing this event (in a letter written to his son in 1865):
“But on the 16th day of the same month - which happened to be a Monday, and a Council day of the Royal Irish Academy - I was walking in to attend and preside, and your mother was walking with me, along the Royal Canal, to which she had perhaps driven; and although she talked with me now and then, yet an under-current of thought was going on in my mind, which gave at last a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough to distinctly communicate the discovery. Nor could I resist the impulse - unphilosophical as it may have been - to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols, i, j, k; namely
which contains the Solution of the Problem, but of course, as an inscription, has long since mouldered away.” (from Crowe, p. 29)
So what exactly are quaternions? It would be too difficult to explore their properties in any depth here. More thorough introductory references can be found at Mathworld, and also Roger Penrose’s book The Road to Reality, chapter 11. Briefly though, a quaternion is like a complex number, in that it is made up of multiple parts. It has four components, one scalar component and three vector components:
The three components of the vector portion of a quaternion are imaginary numbers, just like ‘i b’ is the imaginary number portion of a complex number. As we saw earlier, the imaginary number i is equal to the square root of -1, or:
The same holds true for j and k in quaternions:
Just as complex numbers can be used to algebraically describe rotations in the two-dimensional complex plane, quaternions can be used to describe rotations in three-dimensional space. That three dimensional space is defined by three imaginary axes, i, j, and k (instead of the x, y, and z we used earlier to describe our everyday, Cartesian, three-dimensional space).
Quaternions have most of the important algebraic properties of both real and complex numbers; for example, they have the associative property (i.e, a + (b + c) = (a + b) + c). However quaternions do not have one major property: multiplication is not commutative, that is i j ≠ j i. (To get an idea of how weird this is, imagine that 5 x 6 ≠ 6 x 5 !) Quaternions are actually anti-commutative, which means that: i j = -j i. (Again, imagine what it would be like of real numbers had this property - then 5 x 6 = -(6 x 5) - weird, but this kind of weirdness is an important property in quantum mechanics and other aspects of modern physics.)
Quaternions never caught on as widely as Hamilton had hoped, but they did have some very passionate advocates. A community of mathematicians and physicists put in a significant effort to show how quaternions could be useful for solving problems in physics. Maxwell’s law’s of electromagnetism (operating in three-dimensional space) could be written in terms of quaternions, but it still wasn’t clear that quaternions were the best tools for handling multi-dimensional problems in algebra and physics. As a recent paper put it, “Despite the clear utility of quaternions, there was always a slight mystery and confusion over their nature and use.” (Lasenby, Lasenby and Doran, 2000) Roger Penrose puts it this way:
“[Quaternions give] us a very beautiful algebraic structure and, apparently, the potential for a wonderful calculus finely tuned to the treatment of the physics and geometry of our 3-dimensional physical space. Indeed, Hamilton himself devoted the remaining 22 years of his life attempting to develop such a calculus. However, from our present perspective, as we look back over the 19th and 20th centuries, we must still regard these heroic efforts as having resulted in relative failure. This is not to say that quaternions are mathematically (or even physically) unimportant. They certainly do have some very significant roles to play, and in a slightly indirect sense their influence has been enormous, through various types of generalization. But the original ‘pure quaternions’ still have not lived up to what must have undoubtedly have initially seemed to be an extraordinary promise.
"Why have they not? Is there perhaps a lesson for us to learn concerning modern attempts at finding the ‘right’ mathematics for the physical world?” (Penrose, p. 200)
IV. "Kampf ums Dasein" - the struggle between quaternions and vector analysis
J. Willard Gibbs wrote a letter in 1888, in which he stated that “I believe a Kampf ums Dasein [struggle for existence] is just commencing between the different methods and notations of multiple algebra, especially between the ideas of Grassman & of Hamilton." (Crowe, p. 182) That struggle commenced in earnest in 1890, and lasted roughly four years. According to Michael Crowe, author of A History of Vector Analysis, the struggle involved eight scientific journals, twelve scientists, and roughly 36 publications between 1890 and 1894. (Crowe, p. 182) The following chronological outline is based on Michael Crowe’s extensive discussion of this struggle (chapter 6 of A History of Vector Analysis).
What was the argument about? Hamilton’s followers tried for years to bring what they perceived to be the quaternions’ untapped potential to fruition. They had not been as successful as they hoped, and a new competitor was emerging - the system of vector analysis developed simultaneously by Oliver Heaviside in England and J. Willard Gibbs at Yale. This new system was proving useful in a variety of contexts where quaternions had failed to live up to their promise. For instance, while Maxwell’s laws of electromagnetism had been at one point cast in quaternion form, Heaviside showed that Maxwell’s laws could be much more usefully presented in the form of vector calculus. Also, Gibbs had written a pamphlet laying out his system of vector analysis and argued its advantages in solving physics problems.
This competition riled the quaternionists. They began seeking support among mathematicians and physicists, trying to encourage their colleagues to join their effort to further develop quaternions into a useful tool. The leading quaternionist, successor to Hamilton (who had died in 1865), Peter Guthrie Tait argued in 1890 that quaternions were “transcendentally expressive” and “uniquely adapted to Euclidian [3-dimenesional] space.” Tait also launched what was basically the first shot in the struggle with the vectorists, when he wrote that Gibbs was “one of the retarders of Quaternion progress, in virtue of his pamphlet on Vector Analysis, a sort of hermaphrodite monster.”
Gibbs replied to Tait in an 1891 letter in the journal Nature. He argued that the scalar and vector products of his vector analysis had a fundamental importance in physics, but the quaternion itself (which, as discussed above, is a combination of both a scalar and a vector element) had little natural usefulness. (Briefly, the scalar and vector products are what we now call the ‘dot’ and ‘cross’ product of two vectors. Basically, Gibbs defined two types of multiplication for vectors: one could multiply two vectors to get a scalar quantity (the scalar, or ‘dot’ product); or one could multiply two vectors to obtain yet another vector (the vector, or scalar product). Both products are widely used today in physics.) Gibbs also pointed out that vector analysis could be extended to four or more dimensions, while quaternions were limited to three dimensions.
In his reply to Gibbs, Tait made the infamous comment (which crops up in Against the Day, p. 131) that “it is singular that one of Prof. Gibbs' objections to Quaternions should be precisely what I have always considered... their chief merit:- viz. that they are uniquely adapted to Euclidean space, and therefore specially useful in some of the most important branches of physical science. What have students of physics, as such, to do with space of more than three dimensions?” (Crowe comments wryly that “Fate seems to have been against Tait, at least in regard to that last point.”)
The arguments went back and forth for four years with little apparent progress. Gibbs repeatedly and calmly emphasized that the prime consideration in a system of analysis should be given to the fundamental relationships we wish to describe in the physical world. He wrote:
“Whatever is special, accidental, and individual [in these analysis systems] will die as it should; but that which is universal and essential should remain as an organic part of the whole intellectual acquisition. If that which is essential dies with the accidental, it must be because the accidental has been given the prominence which belongs to the essential...”
Other writers were not so calm as Gibbs. Several quaternionists were quite vitriolic, while Oliver Heaviside seemed to relish the battle when he wrote that “the quaternionic calm and peace have been disturbed. There is confusion in the quaternionic citadel; alarms and excursions, and hurling of stones and pouring of boiling water upon the invading host.”
After about 4 years, the arguments died down. Vector analysis began to be more widely adopted, not because of any arguments made in the ‘Kampf ums Dasein’, but because it became closely associated with the growing success of Maxwell’s theory of electromagnetism. Quaternions faded into a historical footnote, while a modernized version of the Gibbs and Heaviside vector analysis became what most students in physics, chemistry, and engineering learn to use today. Like the science of the luminiferous aether, which became obsolete after the work by Michelson and Morley, and the development of Einstein’s Special Relativity, quaternions are another largely abandoned subject of once-high 19th century hopes.
V. Speculations on quaternions in Against the Day
Why does Pynchon make such a big deal of quaternions and vectors in Against the Day? Possibly because they are so tied up with the changing notions of light, space, and time around the end of the 19th Century. An important theme in the history of science is that how we perceive our world is limited by how we can measure it, and what we can say about it (especially in terms of mathematics). The quaternionists’ views of space and time were limited by the mathematical formalisms they were working with. Some of them speculated that the scalar (or w ) term of a quaternion could be used somehow to represent time, while the three vector components covered 3-dimensional space, but this view treats time differently from how it would eventually be dealt with in the four-dimensional space-time of special relativity. For one thing, time as a scalar term would only have two directions, ‘+’ or ‘-’; that is, either forward or backwards, whereas in relativity individual observers can be rotated any angle relative to the time axis of four-dimensional space-time (recall the Frogger example from part I of this essay).
Characters in Against the Day speculate about the somewhat mysterious role of the w term of quaternions, suggesting that the ‘Quaternion weapon’ makes use the w term to somehow displace objects in time. As Louis Menand notes in his review of Against the Day, this book “is a kind of inventory of the possibilities inherent in a particular moment in the history of the imagination.” (I disagree with Menand’s claim that this is all the book is, and that it is just a rehash of what was done in Mason & Dixon. More on that in another installment of this essay.)
Spaces and geometries, those which we perceive, which we can’t perceive, or which only some of us perceive, are a recurring theme in Against the Day. As Professor Svegli tells the Chums about the ‘Sfinciuno Itinerary’, “The problem lies with the projection” of surfaces, especially imaginary ones beyond our three-dimensional earth. Thus ‘paramorphoscopes’ were invented to reveal “worlds which are set to the side of the one we have taken, until now, to be the only world given us.” (p. 249) To draw perhaps a too-crude analogy, the mathematical tools of physics are like paramorphoscopes - designed correctly, they can enable us to talk about worlds and imaginary axes that we would not have considered otherwise. And perhaps the by abandoning some of the tools once current in the 19th Century, we have closed off our perception of other aspects of nature that remain currently transparent to us. It turns out that Gibbs’ vector analysis itself was insufficient to handle important aspects of relativistic space-time as well as quantum mechanics, and physicists have since rediscovered important ideas in algebra developed by Hermann Grassman and William Clifford, whose 19th century work anticipated important 20th century developments better than quaternions or vector analysis.
There is much more that could be said. In future installments of this essay, I'll finish the science primer portion by covering Riemann surfaces and 4-dimensional space-time, and then hopefully move on to some interpretation and a reply to James Wood's claim that that Against the Day is just a massive Seinfeld episode - that is, a book about nothing.
For further reading:
Michael J. Crowe, A History of Vector Analysis (1969)
Roger Penrose, The Road to Reality, (2004) Chapter 11
The Feynman Lectures on Physics, Vol. 1, Chapters 11 and 22
Lasenby, et. al, "A unified mathematical language for physics and engineering in the 21st century", Phil. Trans. R. Soc. Lond. A (2000) 358, 21-39