Science in Against the Day: Vectors and Quaternions

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.

Stay tuned...

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

Apologies to any Pynchon readers for the delay...

I have to apologize for the fact that I haven't posted any new Pynchon material recently. I still am working on the essay/science primer for Against the Day, however the task has turned out to be more formidable than I first anticipated. The problem is this: there is so much material that could be included in a primer on vectors, quaternions, space-time, etc., much more than I have the time or, in some cases, expertise to cover. In order to keep things focused on the truly relevant science, I must finish a second reading of the book.

I'm well on my way through that second reading, furiously taking notes, and I'm working on various portions of the essay, but it will probably be at least a month before I finish and can post my results.

In the meantime, I highly recommend this book, which I'd bet was one of Pynchon's sources on quaterions and vector analysis (as you can imagine, there just aren't that many books out there on the history of vector analysis): A History of Vector Analysis. It looks like it's hard to find - I picked up a copy from my local university library. I'll leave you with a few choice excerpts from the book that relate to what Hamilton was trying to accomplish with quaternions, and how his efforts were viewed:


p. 37, from an 1857 review of Hamilton's work on quaternions:

"It is confidently predicted, by those best qualified to judge, that in the coming centuries Hamilton's Quaternions will stand out as the great discovery of our nineteenth century."

[In reality, quaternions were eclipsed by vector analysis only a few decades later.]

p. 23-24: From an essay published by Hamilton in 1837, a section called "On Algebra as the Science of Pure Time" (to understand this passage, recall that imaginary numbers are multiples of the square root of -1):

"The thing aimed at, is to improve the Science, not the Art nor the Language of Algebra. The imperfections sought to be removed, are confusions of thought, and obscurities or errors of reasoning; not difficulties of application of an instrument nor failures of symmetry in expression...

"For it has not fared with the principles of Algebra as with the principles of Geometry. No candid and intelligent person can doubt the truth of the chief properties of Parallel Lines, as set forth by EUCLID in his Elements, two thousand years ago... The doctrine involves no obscurity nor confusion of thought, and leaves in the mind no reasonable ground for doubt, although ingenuity may usefully be exercised in improving the plan of argument.

"But it requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine of Negatives and Imaginaries, when set forth (as it has commonly been) with principles like these: that a greater magnitude may be subtracted from a less, and that the remainder is less than nothing; that two negative numbers, or numbers each denoting magnitudes less than nothing, may be multiplied the one by the other, and that the product will be a positive number, or a number denoting a magnitude greater than nothing; and that although the square of a number, or the product obtained by multiplying that number by itself, is therefore always positive, whether the number be positive or negative, yet that numbers, called imaginary, can be found or conceived or determined, and operated on by all the rules of positive and negative numbers, as if they were subject to those rules, although they have negative squares, and must therefore be supposed to be themselves neither positive or negative, nor yet null numers, so that the magnitudes which they are supposed to denote can neither be greater than nothing, nor less than nothing, nor even equal to nothing. It must be hard to found a SCIENCE on grounds such as these..."

[Note in this upcoming section that Hamilton speculates that with geometry as the science of space, perhaps algebra could become the science of time:]

Hamilton asks "whether existing Algebra, in the state to which it has been already unfolded by the masters of its rules and of its language, offers indeed no rudiment which may encourage a hope of developing a SCIENCE of Algebra: a Science properly so called; strict, pure, and independent; deduced by valid reasonings from its own intuitive principles; and thus not less an object of priori contemplation than Geometry, nor less distinct, in its own essence, from the Rules which it may teach or use, and from the Signs by which it may express its meaning..."

He suggests that "the Intuition of TIME is such a rudiment... The argument for the conclusion that the notion of time may be unfolded into and independent Pure Science, or that a Science of Pure Time is possible, rests chiefly on the existence of certain priori intuitions, connected with that notion of time, and fitted to become the sources of a pure Science; and on the actual deduction of such a Science from those principles, which the author conceives that he has begun."

Hamilton here draws a comparison between Euclidean geometry as a pure science of space, and his efforts to make algebra a pure science of time. Historically, and in light of Against the Day, it is interesting to note that Hamilton wrote this before Non-Euclidean geometry became widely known (after 1860, according to Crowe), and long before experiments suggested there was anything wrong with our intuitive notions of time which Hamilton wanted to rely on. In essence, Hamilton's quaternions and Euclidean geometry are part of a classical world that came to an end during the time frame of Pynchon's book. (In the long run, vector analysis and quaternions themselves, instead of being a science of time, became an algebra dealing with space.)

The Science of Light, Space-Time, and Vectors in Thomas Pynchon's Against the Day

OK, Pynchon fans, this is the first draft of my first installment on the science in Against the Day. I have to preface this with two cautionary notes:

1. This is an early and still very rough draft.
2. There are many more connections to be drawn with the book. I'm on my second reading, taking notes as I go, so by the time I finish I'll be able to flesh this out much more.

This essay will come in 4 installments, one every week or so - if you like this, keep checking back.

Here we go:

Introduction
Thomas Pynchon is well known for the dense and obscure references to history, pop-culture, and especially science in his novels. His recent novel Against the Day is set during the turn of the 19th Century, a time when our understanding of space, time, and light, rooted in classical physics, was completely overturned and replaced by a revolutionary new perspective based on the theories of special and general relativity. Pynchon takes the science of this period and incorporates it deeply into the language and structure of Against the Day, more so perhaps than in any of his other novels. Against the Day is suffused with meditations on light, space, and time, and often plays with the tension between different perspectives in math and physics - classical physics versus relativity, or Maxwell's laws of electromagnetism described with the imaginary numbers of quaternions versus the real numbers of vector analysis. This material is not just filler - it's critical to the core of Against the Day, a fact which has been underappreciated in early reviews of the novel. One reviewer claimed that a new generation of writers has a "grasp of the systems that fascinate Pynchon -- science, capitalism, religion, politics, technology -- [that] is surer, more nuanced, more adult and inevitably yields more insight into how those systems work than Pynchon offers here." When it comes to science at least, this claim is not true - Pynchon's achievement in Against the Day proves that he is peerless as a poet who can mine the most abstract realms of very real science for gems of insight, and set them beautifully into the context of the humanity that is the ultimate concern of his novels.

My goal here is twofold: first, to illustrate how Pynchon goes beyond using science as simply a backdrop, or a way to show off his amazing erudition - he weaves scientific concepts into the language and structure of his book; and second, to lay out a primer on the basic scientific ideas so that readers of Against the Day can make their own discoveries about the novel. There are four main topics that I cover here: the Michelson-Morley experiment and the breakdown of classical physics, space-time and special relativity, the development of vector analysis and the eclipse of quaternions (I have a good guess at the identity of the 'Baedeker' that Pynchon 'looted' for his material on quaternions and vectors), and finally, Riemann surfaces. These four topics cover most of the scientific references in Against the Day. Pynchon, being a sucker for historical trivia, is mindful of the chronological development of these subjects, so I'll cover most of them from a historical perspective, including some famous, now-rejected explanations proposed for the negative result of the Michelson-Morley experiment. It is also important to note the science Pynchon did not include in the novel - other important advances were being made at the time, by some of the same characters, advances that Pynchon hardly mentions such as those in statistical mechanics (and yes, entropy) made by J.W. Gibbs. In this book, Pynchon has chosen to focus on space, time, and light.

The Michelson-Morley Experiment and the Failure of Classical Physics
The Michelson-Morley experiment (actually, a series of experiments performed over several years) was one of the definitive experiments providing physical evidence of the inadequacy of classical notions of space and time. The attempts to deal with the negative results of this experiment eventually led to Einstein's theories of special and general relativity, which are based on a completely different and very strange new way of viewing space and time. This new outlook has since been proven beyond any doubt by decades of experiments.

Early on in the book, Pynchon drops clues that the Michelson-Morley experiment is important for many of the themes that we'll find throughout Against the Day, and he even hints that the technical setup of the experiment itself is incorporates into the structure of Against the Day. A major character, Merle Rideout reads about the upcoming experiment and heads to Cleveland to learn more. He is encouraged by his friend, Yale professor Heino Vanderjuice, who tells him:

"Mr. Rideout, we wander at the present moment through a sort of vorticalist twighlight, holding up the lantern of the Maxwell Field Equations and squinting to find our way. Michelson's done this experiment before, in Berlin, but never so carefully. This one could be the giant arc-lamp we need to light our way into the coming century." (p. 58)

At the time this conversation takes place, in 1887, Michelson had improved on the design of his interferometer (the device used to carry out the experiment), so that it was easily sensitive enough to definitively answer the question he was posing. The negative outcome of this experiment was a major stepping-stone towards the development of a new understanding of light, space, and time. To see how this new understanding arose, we have to first understand the questions in classical physics that led to the Michelson-Morley experiment.

I. Classical Relativity
Albert Michelson began his famous series of experiments because the laws of classical physics, which in general were spectacularly successful, were running into trouble in one critical area: moving reference frames. Although we not may use the term 'reference frame' very often, we deal with moving reference frames in our everyday experience. As we have all experienced, Newton's laws of motion don't depend on whether we're on the ground or in a vehicle moving at a constant speed (constant here is an important qualifier). For example, you can play tennis on a steadily moving (that is, non-accelerating) cruise ship just as easily (or not so easily, in my case) as you play tennis on land - you handle yourself and the tennis ball the same way. Or, if you're on a steadily moving (again, non-accelerating) train, you can bounce a ball against the floor, and it behaves just as if you were bouncing it on the floor of your kitchen back home. The ball bounces straight up and down, keeping up with the train as you bounce it, as long as the train does not suddenly accelerate. From your perspective, or reference frame on the train, Newton's laws of motion, which govern the movement of the ball you are bouncing, are exactly the same as they would be if you were standing outside on the ground. Newton's laws apply equally well to moving and stationary reference frames (as the moving frame is not accelerating).

To someone standing on the ground outside of our hypothetical train, watching you bounce the ball as you go by, the situation looks a little different, but still completely in accordance with Newton's laws. This outside observer sees that the ball isn't going straight up and down; it's also moving forward with you and the train, nevertheless, the ball is also obeying Newton's laws from this outside perspective. All of this is common sense and intuitively obvious to us, but one can also show that it works out mathematically as well.

While this phenomenon is true of Newton's laws of motion, the classical laws of electricity and magnetism do not hold in different moving reference frames. To see what this means, we can use one key example: light. Maxwell's classical laws of electromagnetism imply that the speed of light is constant (in a given medium like a vacuum - light moves at different speeds in different media like water or air). If we simply treat light the way we treated Newton's laws in our above examples of the cruise ship or the train, the speed of light would not be constant for observers in different moving reference frames. If I'm standing on the ground watching a pulse of light go by, I would see that it's going at 300,000,000 meters per second (m/s). Someone on a train moving at 50 m/s, watching that same pulse of light go by, would perceive the light to be moving at 299,999,950 m/s. (This difference is of course, too small to be perceptible to unaided human senses). In this case, the speed of light, unlike Newton's laws, is not constant for observers in different reference frames, and thus the laws of electromagnetism would be different, depending on your frame of reference. So, while Newton's laws are the same whether you're playing tennis on the ground or on a cruise ship, this appears to not be true for Maxwell's laws of electromagnetism. Maxwell's laws would thus not be the correct laws to describe the behavior of light in moving reference frame. (At this point, we should be careful to remember that these are just theoretical considerations - we haven't discussed any actual experiments to really determine what happens with Maxwell's laws on a moving train or a cruise ship.)

Physicists in the late 19th century were well aware of this conundrum. They believed that there had to be one universal frame of reference where the speed of light was constant in any direction, in which Maxwell's laws of electromagnetism were perfectly valid; all other objects in the universe moved relative to the absolute space of this universal reference frame. This was called the aether frame of reference - aether was the stuff (although what kind of stuff, nobody knew) through which light supposedly propagated, much like sound must propagate through air or some other medium. The earth therefore moved relative to the stationary aether, much like a train moves relative to the 'stationary' earth. If this was in fact true, that the earth moves relative to the aether, then this movement should be detectable by experiment. And here is where Michelson and Morley come in.

II. Michelson and Morely Attempt to Measure the Absolute Speed of the Earth
Albert Michelson (who, like many characters in Against the Day, grew up in mining towns) and Edward Morley developed an extremely sensitive instrument to measure the speed of the earth relative to the hypothesized aether. The reasoning behind the experiment goes something like this: if light propagates at a constant speed through the aether (at 300,000,000 m/s), and if the earth moves at a certain speed relative to the aether (at, say, 30,000 m/s), then light moving in the same direction as the earth should appear to move more slowly to an observer on the earth - the speed of light in the aether, minus the speed of the earth. It's like driving on the highway - if you are going 60 mph and driving behind a car going 90 mph, then from your perspective the car in front is moving away from you at 30 mph. Michelson and Morley measured the speed of the earth relative to the aether and came up with a disturbing result - relative to the aether, the earth was not moving at all.

I'm won't describe exactly how the Michelson-Morley experiment worked - good explanations can be found in a physics textbook or a Google search. However, the basic setup of the experiment has a connection to some of the plot structure in Against the Day, as well as to the themes of double refraction and bilocation.

In the experiment, a light beam is split into two separate beams, which travel away from each other at a 90˚ angle, are reflected by mirrors, and then travel back and meet up, at which point they are either in phase or out of phase with each other (see the diagram below). If the earth is moving relative to the aether, the different light beams will travel different distances, and come back out of phase with each other.



Pynchon hints that there is a connection here with bilocation and double refraction, as well as human Michelson-Morley experiments (such as, possibly, when characters split up, go on long journeys and meet up again in or out of phase with each other in some way). There are some critical passages beginning on p. 61, where Merle "got the idea in his head that the Michelson-Morley experiment and the Blinky Morgan manhunt were connected." Blinky is referred to as a "human interferometer" or "A double-refractor, for that matter." (p. 62) In one of the earliest examples of bilocation in the book, Merle suspects that Morgan and Morely are the same person:

"... suppose when they split that light beam, that one half of it is Michelson's and the other is his partner Morley's, which turns out to be the half that comes back with the phases perfectly matched up - but under slightly different conditions, alternative axioms, there could be another pair that don't match up, see, in fact millions of pairs, that sometimes you could blame it on the Aether, sure, but other cases maybe the light goes someplace else, takes a detour and that's why it shows up late and out of phase, because it went where Blinky went when we was invisible, and-" (p. 62)

The connection between Iceland spar and the Michelson-Morley experiment is made more explicit later in the book, where the Cohen explains to Lew Basnight that his goal is to eventually be able to pass through Iceland spar, "which is an expression in crystal form of Earth's velocity as it rushes through the Aether, altering dimensions, and creating double refraction...." (p. 688)

Pynchon plays with this idea at multiple places throughout the book. He also includes elements of an aether culture - worshippers who show up for the Michelson-Morley experiment (p. 59-60), as well as hints of a whole science of aether weather, with which the Chums of Chance seem to be involved - a network of ships and balloons to monitor the ether is hinted at on p. 60. (Historically, people did in fact come up with elaborate ideas using putative aether behavior, such as vortices, to explain physical phenomena.)

Returning to Michelson and Morley - the results of their experiment were negative. They could detect no movement of the earth relative to the aether. Over the next several decades, scientists came up with various explanations for the negative result. One explanation, mentioned in Against the Day, is that the earth drags some of the aether along with it (and thus the earth isn't moving relative to the dragging layer of aether, so you don't detect a change in the speed of light). One of Pynchon's characters draws an analogy between this explanation and the dimples on a golf ball, and the lift of the golf ball through the air and the lift of the earth through the aether. [I'm sure Pynchon includes more, but I'll have to pick that up on a second reading.]

III. Revising Our Notions of Space and Time
One possible solution to handle the negative result of the experiment was to try to modify the laws of electromagnetism. Maxwell's equations were quite young compared to Newton's laws, so it seemed obvious that the problem was with Maxwell and not Newton. However, it became clear that Maxwell's laws were in fact correct, and eventually (as Einstein's theory of special relativity became accepted), that light traveled at a constant speed in any non-accelerating reference frame - not just in one universal aether reference frame. This means that whether you are standing on the ground or traveling at 100,000,000 m/s in a (currently fictitious) space ship, a light pulse will always appear to be traveling at 300,000,000 m/s. To go back to our highway analogy - you're going 60 mph behind a car going 90 mph, and instead of appearing to move away from you at 30 mph, the car appears to be going 90 mph away from you. The implication of this very weird phenomenon is that our everyday ideas about space and time are not correct.

Maxwell's equations weren't the problem - Newton's laws were. The physicist Henri Lorentz, between 1895 and 1905 proposed that objects in motion experience length contraction and time dilation, a proposal (in mathematical form) which could account for the results of the Michelson-Morley experiment. In other words, as you move faster and faster, space (from your frame of reference) shrinks in certain directions, and time slows down. If you go through Lorentz's math (known as the Lorentz tranformation), you see that space and time components, in one reference frame, get mixed together when you move to a different reference frame - space and time are not separate, they depend on how one is moving.

We can think about this by drawing an analogy with the video game Frogger In the game, you get a frog across the road by moving it left, right, up, or down. Now, imagine it this way - instead of having the frog face directly across the road, it's rotated at a 45˚ angle to the right, facing diagonally across the road - the frog's reference frame has rotated. 'Up' now (in this reference frame) is the equivalent of some 'up' and some 'right' in the original reference frame. 'Up' in one reference frame is a mixture of 'up' and 'right' in another.

The analogy isn't perfect (going over the math is the best way to look at it), but that's roughly what's going on with space-time. Instead of 'up' and 'left' in Frogger, we have three space dimensions and one time dimension, and how you perceive the combination of those dimensions depends on your frame of reference. (Unlike in our hypothetical game of Frogger, where we just changed the reference frame by rotating the frog, changing space-time reference frames depend on motion - the faster you move, the more slowly time passes, etc.)

So how we perceive space and time depends on how we are moving. (Does the eternal youth of the Chums of Chance somehow depend on their motion?) But no matter how we move, no matter what our reference frame, light moves at the same speed for all of us.

Pynchon makes a lot of this material, playing with both the language and the concepts. Stay tuned for a future draft, with much more complete references to the book.

In Part 2, I'll discuss special relativity and space-time, the new outlook that replaced the old worldview of classical physics.

For further reading:
Feynman Lectures on Physics, Vol. 1 chapter 15
Modern Physics for Scientists and Engineers, by John Taylor and Christopher Zafiratos.
The Fabric of the Cosmos, Brian Greene.