2/22/2016 Adaptive Complexity: Science in Against the Day: Vectors and Quaternions

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Sunday, June 10, 2007Science in Against the Day: Vectors andQuaternionsHere at last is the longdelayed next installment of myongoing primer on the science in Thomas Pynchon's novelAgainst the Day. The draft of part 1 can be found here.Illness and major deadlnes put me back by months. I hopeto have more installments out soon.

Anyway, here is part 2: Quaternions and Vectors in Againstthe 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 relativelyobscure conflict in the mathematics and physics communitythat took place in the early 1890's between advocates ofquaternions and proponents of the newer vector analysis.This conflict is tied in to major themes in the book thatemphasize the tensions between the old and the emergingworld that culminated in the conflict of World War I, as wellas the ability to perceive and describe the world in morethan 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 effortsof physicists to describe the natural world after the adventof modern vector algebra and calculus.

To understand this conflict, it is important to understandwhat mathematicians and physicists were searching forwhen they developed first quaternions and then vectoranalysis. The most important aim of these mathematiciansand physicists had in mind was the ability to do algebraicmanipulations in more than one dimension.

All of us are familiar with the basic, onedimensionaloperations which we learned in elementary school: addition,subtraction, multiplication, and division. By onedimensional,I mean operations on combinations of single numbers; inother words, what we do in every day addition ormultiplication. Each of these single numbers can all berepresented on a onedimensional number line, and eachoperation can be thought of as moving left or right along theline:

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So for example, the operation 2 + 3 moves you to the rightthree units on the number line, from position 2 to position 5.I know that readers of Pynchon’s novels do not need areview of 1st grade math; the important point I’m trying tomake is that these operations we’re all familiar with areonedimensional operations on a number line.

These basic, onedimensional operations have certainimportant properties, ones which most of us take for grantedonce we're out of elementary school. For example, twoimportant properties are:

Associativity when adding or multiplying more than twonumbers, 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, itdoesn'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 todefine algebraic operations such as addition andmultiplication on pairs (or larger groups) of numbers inessence, creating an algebra of more than one dimension. Inorder to be useful, these operations on pairs of numbers hadto have at least some of the important properties foroperations on single numbers; for example, the addition ofnumber pairs should be associative and commutative.

Why are operations on paris or other groups of numbersimportant? One reason is that such definitions wouldrepresent an advance in pure mathematics, but another keyreason is that higher dimensional mathematics would makeit easier to work with the laws of physics in more than onedimension. To see what these means, let's take Newton'sSecond Law of Motion as an example. This law states thatthe force acting on an object is proportional to the mass ofthe object times the acceleration of the object produced bythe force. Newton's second law can be written as thisequation:

F = ma

But in threedimensional space, Newton's Second Law isproperly written with three equations, to account for theforce and the acceleration in each dimension (eachdimension represented by x, y, or z):

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This means that when we make calculations using Newton'sSecond Law, we really have to perform our calculations onthree equations if we want to deal with ordinary threedimensional space. In a complicated situation where wewant to add and subtract many different forces, we have toadd and subtract the three components for each force. Asystem of analysis, where our operations of addition andsubtraction could be performed on a set of three numbers atonce, treating the threedimensional force as one unit, wouldgreatly simplify calculations using Newton's Second Law orthe much more difficult laws of electromagnetismformulated by Maxwell.

Here is another way to see the problem. Scientistsdistinguish between the speed of an object, which is just amagnitude or a scalar quantity (such as ‘60 miles per hour’),and velocity, which is comprised of both a magnitude and adirection, and thus is a vector (such as ’60 miles per hourgoing northwest’). Adding speeds together is easy, but howdo we add velocities? 18th and 19th century scientists coulddo this by breaking velocities down into their onedimensional components (just as we did for Newton’s SecondLaw), but they realized that a better system was needed.

II. Complex numbers and twodimensional math

Before tackling three dimensions, let’s start with just two.19th century scientists already had a powerful system ofanalysis for dealing with pairs of numbers complexnumbers. Complex numbers are comprised of two parts, areal part and an imaginary part. The imaginary part consistsof a number multiplied by the number i , which is the squareroot of 1:

This means ‘i squared’ is equal to 1:

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Thus a complex number z looks like the following, where aand b are any numbers you choose:

Again, a is called the real part, and ib is known as theimaginary part.

Unlike our ordinary numbers on a number line, complexnumbers can be represented on a two dimensional plane,called the complex plane. One axis of the plane is thenumber line for the real numbers, and the second axis is thenumber line for the imaginary numbers:

Instead of a point on a onedimensional number line,complex numbers can be interpreted as points on the twodimensional complex plane. For example, the complexnumber ‘6 + 3i’ is the point on the complex plane as shownbelow:

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Using complex numbers, one can now describe twodimensional operations like rotation. For example,multiplying a number by i is equivalent to a 90degreerotation on the complex plane. Thus the operation:

is equivalent to this 90degree rotation on the complexplane:

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Instead of multiplying by i, one can multiply by any complexnumber to get a rotation of any angle other than 90degrees.This subject comes up on p. 132 of Against the Day, whereDr. Blope talks about rotations, not in the twodimensionalspace of the complex plane, but in the three dimensionalspace of quaternions:

“ ‘Time moves on but one axis, ‘ advised Dr. Blope, ‘past tofuture the only turning possible being turns of a hundredand eighty degrees. In the Quaternions, a ninetydegreedirection would correspond to an additional axis whose unitis √1. A turn through any other angle would require for itsunit a complex number.’”

This ability to use operations of complex numbers todescribe two dimensional rotations and translations is anextremely important tool in math and physics.

Complex numbers have many other amazing properties, butmost relevant to our discussion of Against the Day is thatcomplex numbers can be manipulated with all of our basicoperations they can be added, subtracted, multiplied, anddivided, with the kinds of useful properties mentionedearlier, such as associativity and commutativity. Thus, withcomplex numbers, we have a way to do algebra in twodimensions.

III. Extending complex numbers to three dimensions:Quaternions

In the mid19th century, several mathematicians werelooking for ways to extend the twodimensional geometricalinterpretation of complex numbers to three dimensions. Oneimportant figure was Hermann Grassman, whose systemturned out to be closest to the yetfuture vector analysis.Grassman is mentioned on occasion in Against the Day, buthis role in the development of vector analysis was somewhatdiminished by the fact that, compared to William Hamilton,Grassman was fairly unknown. It was William Hamilton, whowas already famous for earlier work, who developed themost wellknown immediate predecessor to vector analysis quaternions.

William Hamilton had been struggling with the problem ofhow to generalize complex numbers to higher dimensions.While walking with his wife in Dublin, Hamilton discoveredthe fundamental relationship that could underlie suchgeneralized complex numbers, which he called quaternions.This fundamental relationship is this:

Hamilton was so excited about the discovery that he carvedthis equation into the stone of Dublin’s Brougham Bridge.

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Readers of Against the Day will appreciate Hamilton’s ownlanguage describing this event (in a letter written to his sonin 1865):

“But on the 16th day of the same month which happened tobe a Monday, and a Council day of the Royal Irish Academy I was walking in to attend and preside, and your mother waswalking with me, along the Royal Canal, to which she hadperhaps driven; and although she talked with me now andthen, yet an undercurrent of thought was going on in mymind, which gave at last a result, whereof it is not too muchto say that I felt at once the importance. An electric circuitseemed to close; and a spark flashed forth, the herald (as Iforesaw, immediately) of many long years to come ofdefinitely directed thought and work, by myself if spared,and at all events on the part of others, if I should even beallowed to live long enough to distinctly communicate thediscovery. Nor could I resist the impulse unphilosophical asit may have been to cut with a knife on a stone ofBrougham Bridge, as we passed it, the fundamental formulawith the symbols, i, j, k; namely

which contains the Solution of the Problem, but of course, asan inscription, has long since mouldered away.” (fromCrowe, p. 29)

So what exactly are quaternions? It would be too difficult toexplore their properties in any depth here. More thoroughintroductory references can be found at Mathworld, and alsoRoger Penrose’s book The Road to Reality, chapter 11.Briefly though, a quaternion is like a complex number, inthat 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 quaternionare imaginary numbers, just like ‘i b’ is the imaginarynumber portion of a complex number. As we saw earlier, theimaginary number i is equal to the square root of 1, or:

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The same holds true for j and k in quaternions:

Just as complex numbers can be used to algebraicallydescribe rotations in the twodimensional complex plane,quaternions can be used to describe rotations in threedimensional space. That three dimensional space is definedby three imaginary axes, i, j, and k (instead of the x, y, andz we used earlier to describe our everyday, Cartesian,threedimensional space).

Quaternions have most of the important algebraic propertiesof both real and complex numbers; for example, they havethe 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 anidea of how weird this is, imagine that 5 x 6 ≠ 6 x 5 !)Quaternions are actually anticommutative, which meansthat: i j = j i. (Again, imagine what it would be like of realnumbers had this property then 5 x 6 = (6 x 5) weird,but this kind of weirdness is an important property inquantum mechanics and other aspects of modern physics.)

Quaternions never caught on as widely as Hamilton hadhoped, but they did have some very passionate advocates. Acommunity of mathematicians and physicists put in asignificant effort to show how quaternions could be useful forsolving problems in physics. Maxwell’s law’s ofelectromagnetism (operating in threedimensional space)could be written in terms of quaternions, but it still wasn’tclear that quaternions were the best tools for handling multidimensional problems in algebra and physics. As a recentpaper put it, “Despite the clear utility of quaternions, therewas always a slight mystery and confusion over their natureand use.” (Lasenby, Lasenby and Doran, 2000) RogerPenrose puts it this way:

“[Quaternions give] us a very beautiful algebraic structureand, apparently, the potential for a wonderful calculus finelytuned to the treatment of the physics and geometry of our 3dimensional physical space. Indeed, Hamilton himselfdevoted the remaining 22 years of his life attempting todevelop such a calculus. However, from our presentperspective, as we look back over the 19th and 20thcenturies, we must still regard these heroic efforts as havingresulted in relative failure. This is not to say thatquaternions are mathematically (or even physically)unimportant. They certainly do have some very significantroles to play, and in a slightly indirect sense their influence

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has been enormous, through various types of generalization.But the original ‘pure quaternions’ still have not lived up towhat must have undoubtedly have initially seemed to be anextraordinary promise.

"Why have they not? Is there perhaps a lesson for us tolearn concerning modern attempts at finding the ‘right’mathematics for the physical world?” (Penrose, p. 200)

IV. "Kampf ums Dasein" the struggle between quaternionsand vector analysis

J. Willard Gibbs wrote a letter in 1888, in which he statedthat “I believe a Kampf ums Dasein [struggle for existence]is just commencing between the different methods andnotations of multiple algebra, especially between the ideasof Grassman & of Hamilton." (Crowe, p. 182) That strugglecommenced in earnest in 1890, and lasted roughly fouryears. According to Michael Crowe, author of A History ofVector Analysis, the struggle involved eight scientificjournals, twelve scientists, and roughly 36 publicationsbetween 1890 and 1894. (Crowe, p. 182) The followingchronological outline is based on Michael Crowe’s extensivediscussion of this struggle (chapter 6 of A History of VectorAnalysis).

What was the argument about? Hamilton’s followers tried foryears to bring what they perceived to be the quaternions’untapped potential to fruition. They had not been assuccessful as they hoped, and a new competitor wasemerging the system of vector analysis developedsimultaneously by Oliver Heaviside in England and J. WillardGibbs at Yale. This new system was proving useful in avariety of contexts where quaternions had failed to live up totheir promise. For instance, while Maxwell’s laws ofelectromagnetism had been at one point cast in quaternionform, Heaviside showed that Maxwell’s laws could be muchmore usefully presented in the form of vector calculus. Also,Gibbs had written a pamphlet laying out his system of vectoranalysis and argued its advantages in solving physicsproblems.

This competition riled the quaternionists. They beganseeking support among mathematicians and physicists,trying to encourage their colleagues to join their effort tofurther develop quaternions into a useful tool. The leadingquaternionist, successor to Hamilton (who had died in 1865),Peter Guthrie Tait argued in 1890 that quaternions were“transcendentally expressive” and “uniquely adapted toEuclidian [3dimenesional] space.” Tait also launched whatwas basically the first shot in the struggle with thevectorists, when he wrote that Gibbs was “one of theretarders of Quaternion progress, in virtue of his pamphleton 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 vectoranalysis had a fundamental importance in physics, but thequaternion itself (which, as discussed above, is acombination of both a scalar and a vector element) had littlenatural usefulness. (Briefly, the scalar and vector products

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are what we now call the ‘dot’ and ‘cross’ product of twovectors. Basically, Gibbs defined two types of multiplicationfor vectors: one could multiply two vectors to get a scalarquantity (the scalar, or ‘dot’ product); or one could multiplytwo vectors to obtain yet another vector (the vector, orscalar product). Both products are widely used today inphysics.) Gibbs also pointed out that vector analysis could beextended to four or more dimensions, while quaternionswere 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 issingular that one of Prof. Gibbs' objections to Quaternionsshould be precisely what I have always considered... theirchief merit: viz. that they are uniquely adapted to Euclideanspace, and therefore specially useful in some of the mostimportant branches of physical science. What have studentsof physics, as such, to do with space of more than threedimensions?” (Crowe comments wryly that “Fate seems tohave been against Tait, at least in regard to that last point.”)

The arguments went back and forth for four years with littleapparent progress. Gibbs repeatedly and calmly emphasizedthat the prime consideration in a system of analysis shouldbe given to the fundamental relationships we wish todescribe in the physical world. He wrote:

“Whatever is special, accidental, and individual [in theseanalysis systems] will die as it should; but that which isuniversal and essential should remain as an organic part ofthe whole intellectual acquisition. If that which is essentialdies with the accidental, it must be because the accidentalhas been given the prominence which belongs to theessential...”

Other writers were not so calm as Gibbs. Severalquaternionists were quite vitriolic, while Oliver Heavisideseemed to relish the battle when he wrote that “thequaternionic calm and peace have been disturbed. There isconfusion in the quaternionic citadel; alarms and excursions,and hurling of stones and pouring of boiling water upon theinvading host.”

After about 4 years, the arguments died down. Vectoranalysis began to be more widely adopted, not because ofany arguments made in the ‘Kampf ums Dasein’, but becauseit became closely associated with the growing success ofMaxwell’s theory of electromagnetism. Quaternions fadedinto a historical footnote, while a modernized version of theGibbs and Heaviside vector analysis became what moststudents in physics, chemistry, and engineering learn to usetoday. Like the science of the luminiferous aether, whichbecame obsolete after the work by Michelson and Morley,and the development of Einstein’s Special Relativity,quaternions are another largely abandoned subject of oncehigh 19th century hopes.

V. Speculations on quaternions in Against the Day

Why does Pynchon make such a big deal of quaternions andvectors in Against the Day? Possibly because they are so

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tied up with the changing notions of light, space, and timearound the end of the 19th Century. An important theme inthe history of science is that how we perceive our world islimited by how we can measure it, and what we can sayabout it (especially in terms of mathematics). Thequaternionists’ views of space and time were limited by themathematical formalisms they were working with. Some ofthem speculated that the scalar (or w ) term of a quaternioncould be used somehow to represent time, while the threevector components covered 3dimensional space, but thisview treats time differently from how it would eventually bedealt with in the fourdimensional spacetime of specialrelativity. For one thing, time as a scalar term would onlyhave two directions, ‘+’ or ‘’; that is, either forward orbackwards, whereas in relativity individual observers can berotated any angle relative to the time axis of fourdimensional spacetime (recall the Frogger example frompart I of this essay).

Characters in Against the Day speculate about the somewhatmysterious role of the w term of quaternions, suggestingthat the ‘Quaternion weapon’ makes use the w term tosomehow displace objects in time. As Louis Menand notes inhis review of Against the Day, this book “is a kind ofinventory of the possibilities inherent in a particular momentin the history of the imagination.” (I disagree with Menand’sclaim that this is all the book is, and that it is just a rehashof what was done in Mason & Dixon. More on that in anotherinstallment of this essay.)

Spaces and geometries, those which we perceive, which wecan’t perceive, or which only some of us perceive, are arecurring theme in Against the Day. As Professor Svegli tellsthe Chums about the ‘Sfinciuno Itinerary’, “The problem lieswith the projection” of surfaces, especially imaginary onesbeyond our threedimensional earth. Thus‘paramorphoscopes’ were invented to reveal “worlds whichare set to the side of the one we have taken, until now, tobe the only world given us.” (p. 249) To draw perhaps a toocrude analogy, the mathematical tools of physics are likeparamorphoscopes designed correctly, they can enable usto talk about worlds and imaginary axes that we would nothave considered otherwise. And perhaps the by abandoningsome of the tools once current in the 19th Century, we haveclosed off our perception of other aspects of nature thatremain currently transparent to us. It turns out that Gibbs’vector analysis itself was insufficient to handle importantaspects of relativistic spacetime as well as quantummechanics, and physicists have since rediscovered importantideas in algebra developed by Hermann Grassman andWilliam Clifford, whose 19th century work anticipatedimportant 20th century developments better thanquaternions or vector analysis.

There is much more that could be said. In futureinstallments of this essay, I'll finish the science primerportion by covering Riemann surfaces and 4dimensionalspacetime, and then hopefully move on to someinterpretation and a reply to James Wood's claim that thatAgainst the Day is just a massive Seinfeld episode that is,a book about nothing.

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Newer Post Older Post

Stay tuned...

For further reading:

Michael J. Crowe, A History of Vector Analysis (1969)Roger Penrose, The Road to Reality, (2004) Chapter 11The Feynman Lectures on Physics, Vol. 1, Chapters 11 and22Lasenby, et. al, "A unified mathematical language forphysics and engineering in the 21st century", Phil. Trans. R.Soc. Lond. A (2000) 358, 2139

Posted by Mike White at 4:15 PM

Labels: science in literature, Thomas Pynchon

3 comments:

Hank said...It's good stuff, though it took me parts of two days to read itall.

I don't see why you wouldn't put that over on sb too.

The categories we have are just for where your nameappears in the index. It's not a prison!! :)8:14 AM

Mike said...I can do that I hadn't done it before because the piece isn'tselfcontained, and I never got to the next installment onRiemann surfaces.

I've also been meaning to finish a piece on Jack Kerouac andbeing on the road in molecular biology never finished thateither, and I missed the 50th annivesary of On The Road.11:43 AM

Reverend Lowell said...Mike...Waiting for Part Three. Enjoying the heck out of yourwriting so far. Hope your health continues to improve. Oh,I'm writing this less than one mile from Jack'sBirthplace.......sometimes, when the wind is right, I can hearthe sounds of scruffy little children playing ball at DracutTigers Field....12:13 AM

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