mathematical understanding and the physical sciences

Available online at www.sciencedirect.comStudies in History

www.elsevier.com/locate/shpsa

Stud. Hist. Phil. Sci. 38 (2007) 667–685

and Philosophyof Science

Mathematical understanding and the physical sciences

Harry Collins

School of Social Sciences, Cardiff University, The Glamorgan Building, King Edward VIIth Ave, Cardiff CF10 3WT, UK

Abstract

The author claims to have developed interactional expertise in gravitational wave physics without engaging with the mathematical orquantitative aspects of the subject. Is this possible? In other words, is it possible to understand the physical world at a high enough levelto argue and make judgments about it without the corresponding mathematics? This question is empirically approached in three ways: (i)anecdotes about non-mathematical physicists are presented; (ii) the author undertakes a reflective reading of a passage of physics, firstwithout going through the maths and then after engaging with it and discusses the difference between the experiences; (iii) the aforemen-tioned exercise gives rise to a table of Levels of Understanding of mathematics, and physicists are asked about the level mathematicalunderstanding they applied when they last read a paper. Each phase of empirical research suggests that mathematics is not as central togaining an understanding of physics as it is often said to be. This does not mean that mathematics is not central to physics, merely that itis not essential for every physicist to be an accomplished mathematician, and that a division of labour model is adequate. This, in turn,suggests that a stream of undergraduate physics education with fewer mathematical hurdles should be developed, making it easier totrain wider groups of people in physical science comprehension.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Physics; Mathematics; Interactional expertise; Physics education; Mathematical literacy; Scientific literacy

When citing this paper, please use the full journal title Studies in History and Philosophy of Science

1. Introduction

Physics and the other physical sciences and technologiesare mathematical through and through. The very ‘idea ofphysics’ can be illustrated by the equations of planetarymotions; the predictions are so exact that tiny anomaliescan lead to the discovery of new planets or the confirma-tion of new theories. But the same kind of claim could bemade about experiment. Physics and the other physical sci-ences are experimental through and through and experi-ment is equally part of the very idea of physics. What is

0039-3681/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.shpsa.2007.09.001

E-mail address: [email protected] The quotations are taken from the internet: Soleimani (2006).

strange is that though experiment is just as much centralto physics as mathematics we do not draw the conclusionthat every physicist must be a good experimentalist whilewe often hear it said that every physicist must be a goodmathematician. People do not claim that you cannot ‘reallyunderstand’ physics unless you have done experiments wellbut they do claim that you cannot ‘really understand’ phys-ics unless you have worked through the mathematical der-ivations, a sentiment which informs educational practice inthe physical sciences and is expressed in the followingquotations:1

mailto:[email protected]

668 H. Collins / Stud. Hist. Phil. Sci. 38 (2007) 667–685

It’s obvious to anyone that’s worked with physics that‘understanding the mathematics’ is indeed integral tounderstanding the physical theory. It really is under-standing the physical theory, since the math is just theactual way of expressing the theory.

If you don’t understand the theory quantitatively thenyou don’t understand it well enough to know THE mostbasic thing about the theory—does it describe reality ornot.

Here I want to question the asymmetry in the way mathe-matics and experiment are treated when the essential com-ponents of physics and physicists are discussed. I will tryto establish the case that: first, mathematics is of less impor-tance in the day-to-day activity of physics than some phys-icists and most non-physicists think; second, that a goodgrasp of physics can be obtained without mathematics;and third, that not every physicist needs to be a good math-ematician although some do.2 Along the way we will dis-cover that, when they read their colleagues’ papers, onlyon rare and special occasions do physicists follow the math-ematical steps in detail. This discovery arises from a table of‘Levels of Mathematical Understanding’ (of a paper) whichemerges from the second empirical section of this paper. Fi-nally, I will argue that to increase the spread of scientific lit-eracy new kinds of routes for education in the physicalscience should be opened for those who want to be educatedin the physical sciences but are not mathematically gifted;these routes should lead all the way to university degrees.3

If the case for the relative unimportance of mathematicscan be established it will also have an significant implica-tion for the idea of interactional expertise in the physicalsciences—the starting point for the argument. In one exper-iment the author of this paper was shown to be capable ofpassing as a gravitational wave physicist in an imitationgame (Collins & Evans, 2007; Collins, Evans, & KES,2007). But the conditions of the test included a ban on alge-bra or calculations in the questions that the judges wereallowed to ask. Collins would certainly have failed the testif formulae or calculations were demanded; quite simply,Collins does not know the mathematics of the gravitationalwave field. The imitation game showed, nevertheless, that it

2 M. Gorman writes that physicists themselves disagree as to whether theory(see Miller, 1989). R. Crease writes that the mathematical and the quantitativeHe suggests, and cites Cliff Swartz and some of his own writings in support of tif they do not have a mathematical grasp. It seems to me that if back-of-thesomething in this: Later in the paper it will be suggested by physicists themselvepapers. But I still want to say that I can understand why Joe Weber’s cross-incompatible with known facts about neutrino absorption by asteroids and sothought I cannot prove it, this is something I understand, not just recite like

3 The physics education literature seems to show that there is no correlatiomathematical competence is not sufficient for high performance in physics. Estruggling in physics even while he is breezing through calculus’ (Halloun & Hestudents who have completed a year of college physics have incorrect mentaairplane. McCloskey recommends that physics teachers ‘discuss with studentsand how they differ from the views of classical physics’.

4 At one workshop where the example was presented, a physicist, seeing Co

is possible to answer question such as what effect would agravitational wave polarized in an unusual way have onan interferometer. The answers were given in a degree ofdetail which includes the directions of the source thatwould give the minimum and the maximum effects and thiswas enough to convince physicists that they could havebeen answered by a physicist. The answer, it turns out,can be worked out by thinking about the geometry of thedevice and the wave without doing any calculations. Towork out the exact function would, of course, have neededmathematics, but the principles and the general shape ofthe sensitivity diagram could be grasped qualitatively.4

Toward the end of this paper I will present evidence thatsuggests that this kind of qualitative thinking is the normalday-to-day stuff of physics. If this is true then an imitationgame test sans mathematics could be said to demonstratethat someone who passes it has interactional expertise inphysics. If, on the other hand, real physics always involvesmathematics, then imitation games that prohibit mathe-matics are less revealing than they might be. Well, actually,even if the latter is the case all is not lost; if it is the casethat the mathematically proficient and the non-mathemat-ically proficient are indistinguishable when asked questionsabout physics not cast in mathematical terms, then some-thing has been learned—that the mathematics does nothelp in grasping the conceptual structure of the physicalworld in so far as that conceptual structure can beexpressed in words.

As intimated, the implications of the argument for therelative unimportance of mathematics in understandingthe concepts of physical science go far beyond imitationgame experiments. One inference already touched on isthat physical science needs mathematicians only in thesame way that it needs experimentalists. There is a ‘divisionof labour’ in the physical sciences and if they are to flourishwe must nurture as many great experimentalists and asmany great mathematicians as we need, but not necessarilyany more. Those less gifted in the experimental arts, andthose less gifted in the mathematical arts, will find the helpthey need among their colleagues. It follows that we couldbe training some physical scientists, even to research level,without making them jump the mathematical hurdles theyare currently asked to jump. Even if the argument that we

ought to be based fundamentally on physical intuition or on mathematicsneed to be extricated from one another whereas I use them synonymously.he claim, that any physicist has to have a quantitative grasp on things even-envelope, ‘order of magnitude’, calculations are at issue there could be

s that they do such rough calculations in order to check mathematics-basedsection was wrong (see below) just by being told that the quantities wereon without having to work out the quantities myself, even roughly. Even

a parrot.n between mathematical skills and physical understanding and that ‘high

vidently, this explains the common phenomenon of the student who isstenes, 1985, p. 7). M. Gorman writes: McCloskey (1983) showed that evenl models of simple processes like the path of an object dropped from antheir naı̈ve beliefs, carefully pointing-out what is wrong with these beliefs,

llins’s answer, was initially sure that he must know the mathematics.

H. Collins / Stud. Hist. Phil. Sci. 38 (2007) 667–685 669

could produce research scientists this way is found uncon-vincing we could still give far more managers, teachers,journalists, and the like, fluency in the deep principles ofphysical science; many of these are currently discouragedby the mathematics-heavy higher-education syllabus inphysical science.

In what follows three kinds of evidence are presented.The first kind of evidence is anecdotal—some stories aboutbrilliant physicists who were poor mathematicians. Thesecond kind of evidence is a reflective account of theauthor’s own attempt to gain more understanding of apiece of physics by grasping its mathematical underpin-nings.5 Two things emerge out of the reflective investiga-tion. The first is that at the end of the mathematicalexercise the author did not feel that he understood the con-ceptual structure of the physics any better than he had atthe outset. This, of course, is a single case study but, inthe spirit of philosophical phenomenology, anyone can‘try it at home’ and see if they get the same outcome.The second outcome of the exercise, and it is this that jus-tifies the otherwise unnecessarily extended presentation ofthe example, is that levels of mathematical understandingcan be distinguished. It is this hierarchy of levels of under-standing—an eleven-point depth scale of understanding—that is used as the basis for the third kind of evidence:responses of audience members in physics departmentseminars to a question about the level of mathematicalunderstanding they used when they last read a physicspaper.

2. Anecdotes

I now present some anecdotes about brilliant physicistswho did not know much mathematics. Of course, theyknew a lot of mathematics compared to the average citizenbut I am not arguing for a physics without any mathemat-ics at all, just that the end point of mathematical trainingfor some physicists taking a new kind of undergraduatedegree might be equivalent of what is their starting pointtoday. The author of this paper, upon whose lack of math-ematical ability some of the argument turns, is not acomplete mathematical ignoramus—he has British (high)school-leaving qualifications in mathematics.6

The most well known case of a non-mathematical yetbrilliant physicist is Michael Faraday but perhaps physicshas moved on and a Faraday would be an anachronism

5 I initially called this a ‘phenomenological’ exercise, but a critic suggestedphilosophers. Therefore I have referred throughout to ‘reflection’ where I orig

6 Much more damaging to my argument in respect of physics educationmathematics training is comparable to US physics degree level. Certainly I leintegration, learned what a differential equation is, and may even have solved aargument in respect of education applies only to the UK. The saving grace migability to pretend to be a gravitational wave physicist.

7 These quotations are taken from Collins (2004a), p. 561. As explained therethey should not be taken to mean that either scientist is more talented or evenanalysis. Both scientists have made huge contributions to the LIGO project.

in the modern world. There are cases that suggest thismight not be so: in its early days, the Laser InterferometerGravitational-Wave Observatory (LIGO) project was besetwith conflict (Collins, 2004b). For a period the project wasmanaged by a ‘Troika’, consisting of Kip Thorne in thechair, with Ron Drever and Rai Weiss making up the othertwo team members. Drever and Weiss could agree on verylittle. Resolution of their differences was not helped by theirdifferent styles and abilities. A respondent who was presentduring these difficult times and had no reason to favourDrever’s approach, put the matter as follows:

The styles of the two, Ron and Rai, are so very differentthat it was just impossible for them to collaborate—theycouldn’t communicate, not even about technical things.If you look at them, Ron is extremely creative, very cle-ver, he’s a tremendously inventive person, but it tends tobe almost all pictorial in his mind—he’s very unmathe-matical . . . Rai tends to be encyclopedic in his knowl-edge of physics, extremely rigorous, and not terriblyintuitive. And so the two of them could never communi-cate. Ron would have a picture and Rai would say,‘Well, show me how this works out, show me—youknow—write down the equations,’ and Ron couldn’tdo that. And Rai would go away and do three pagesof calculations overnight, with Bessel functions, andthe whole business . . . you never approximate . . . youalways carry the full mathematics. And he’d hand thatto Ron and then say, ‘See, I’ve proven my assertion[for example, that the idea won’t work],’ and Ron wouldnot know what to do with it. So the two of themcouldn’t communicate about anything, so the two ofthem disagreed about almost everything.

This respondent went on to say:

[Ron’s approach is] highly effective in some areas. I willsay this: in most of the technical disagreements betweenRon and Rai, Ron was right more often than not. Therewere more occasions when Ron’s pictorial intuitionstood up against Rai’s mathematics than the otherway round in spite of the fact that Rai’s very good withthat kind of stuff.7

Drever’s claim to fame is as a great experimentalist. Evenmore extraordinary is the suggestion that one of the pre-eminent theoretical physicists of the Twentieth Century,Niels Bohr, was also a poor mathematician, yet a historianof the period makes this claim:

that it did not meet the criteria for phenomenology as understood byinally wrote ‘phenomenology’.is the suggestion, made by some US physicists, that UK (high) schoolarned to solve simple physics problems that required differentiation andsimple one. If US physics undergraduates do not go much further then myht be that I have forgotten nearly all of it so it does not bear much on my

, though the quotations nicely illustrate the point being made in this papermore intuitive than the other, just that one works without mathematical


Neils Bohr’s brother Harald, a great mathematician,once said that Neils could get along without mathemat-ics because of his great intuition. Heisenberg declaredthat ‘mathematical clarity in itself had no virtue forBohr.’ . . . There is ample evidence that Bohr’s masteryof mathematics was very limited . . . all the mathematicsof Bohr’s great 1918 paper on atomic theory was doneby Bohr’s assistant at the time . . . ‘Bohr had no ideahow to do the mathematical part. Even though the phys-ical ideas, so to say, he had.’ . . . The division of labourwas also clear during the Bohr–Rosenfeld lecturing tourto Russia—Bohr lectured on ‘matters of principle,’Rosenfeld of the ‘technical’ aspects.8

No doubt many more stories of this kind could be toldabout famous physical scientists who were not mathemati-cally gifted but we now move on to the second strand ofevidence.

3. Reflecting on mathematical understanding

I claim to have some reasonable level of interactionalexpertise in gravitational wave physics without any knowl-edge of the mathematics. What I would like to do to gaininsight into the nature of mathematical understandingwould be to compare my understanding of gravitationalwave physics as it currently stands with what it would belike if I spent a few years mastering the field mathematically.Unfortunately I cannot do this; quite apart from the ques-tion of time, I am probably not clever enough. Instead I willtry to do the equivalent but at a much lower level. I will lookat the way my own understanding of elements of an intro-ductory text on special relativity—David Mermin’s It’s

about time—changed as I read it, first, without taking upthe invitation offered by the author to engage in the highschool level algebra required to grasp it properly, and thenafter working through the algebra. (It turns out that adetailed analysis of only four pages of the book will beenough to exemplify the main point to be made here.)

I believed I could work through the high school levelalgebra because of my existing qualifications. This traininghad also given me the experience of the thrill and beauty ofcarrying through a mathematical proof, or of suddenlyseeing a mathematical idea. I have a sense, then, of the rich-ness of the experience of the mathematically accomplished.That there is such a thrill is not in question. To make thatclear I will start with some examples of what that richnessfeels like to me.

3.1. Two simple examples of mathematical (or is it

conceptual?) experience

Consider ‘mensuration’. In science fiction films ants aslarge as elephants sometimes stalk the screen but mensura-

8 These quotations and more along the same lines can be found in Beller (1

tion shows that they cannot exist. All large animals musthave thick legs and this is because the weight of an animalincreases with its volume whereas the strength of an animalincreases with the cross section of its bones. Increase thelinear dimension (the height or length) of an animal by10 keeping everything else in scale, and its volume willincrease by the ‘cube of 10’ whereas the cross section ofany of its parts, including its legs and the bones within, willincrease by only the ‘square of 10’ (just imagine the bodyand legs of the animal as boxes). So if you keep the animalin scale as it increases in size by 10 it will weigh 1000 timesas much but its legs will be only 100 times as strong.Increase the animal in scale and it will collapse. The onlyway to have big animals is to increase the thickness of theirlegs a lot faster than you increase the size of the rest ofthem. Immediately one can see why the animal and insectworld is as it is. There are no really big flying animals oranimals that are really fast or can jump really high for theirsize. Prodigious feats of gymnastic ability and flight withsmall wings are restricted to the insects. I now also havea simple rule for scaling creatures (or anything else). If Iwant make an elephant that has the abilities of an ant Ihave to make its bones and muscles stronger than thoseof the ant. If an ant is 1 mm high and an elephant is3000 mm high then to make an elephant-sized ant that iscomparable in shape and ability its bones, ligaments, andmuscles will have to be 3000 times as strong and powerful.Here, then, is an example of a consequential and satisfyinginsight that emerges from just a tiny area of algebraicunderstanding.

Whether mensuration should be described as ‘physicalintuition’ or mathematical reasoning I am not quite surebut it might be just the tip of an iceberg of thrills and insightsavailable to those with good mathematical understandingthat, for me, are obscured beneath the water. If mensurationcan stand for mathematics as a whole then a mathematicalunderstanding of gravitational wave physics would openup a much richer world than the world of gravitational wavephysics that I inhabit. But maybe mensuration is not typicalof the general use of mathematics in physics.

Another example of the rich world of what is eithermathematical or conceptual understanding is to beobtained from the very book which I am going to claimdid not improve my conceptual grasp of special relativity.In Chapter 1 of David Mermin’s book he provides an illus-tration, which I found almost literally ‘breathtaking’, ofwhat can be done by using what he calls ‘the principle ofrelativity’ (which is not the special theory). The problemis to understand how ideal balls bounce off one another.I’ll jump straight to his final example which has a ‘light’ball travelling at 5 feet per second, say, left-to-right if rep-resented on the page, toward the collision point while a‘heavy’ ball travels at 5 feet per second in the oppositedirection—right-to-left. What happens when they hit each

999), pp. 259–261.


other? These balls are ‘ideally bouncy’ which means that ifthey hit a fixed surface they will bounce back at the samespeed as they arrived. ‘Light balls’ experience ‘heavy balls’as fixed surfaces whereas ‘heavy balls’ do not notice theirimpact with ‘light balls’.

The most ‘natural’ answer, I am going to suggest, isobtained by the following reasoning. ‘If the heavy ballwas stationary the light ball would bounce back at thesame speed as it hit it—5 feet per second. But since theheavy ball is moving at 5 feet per second itself, the light ballwill gain an additional 5 feet per second from the heavyball and it will bounce back at 10 feet per second’. Nowlet us see how to do the sum according to Mermin.

The trick is that at these low speeds (speeds not compa-rable with the speed of light), the principle of relativity tellsus that everything will remain the same irrespective of our‘frame of reference’ so long as frames differ only in thatthey move at different steady speeds and no accelerationsare involved. Therefore we can think about things fromthe frame of reference of the heavy ball. To get into theframe of reference of the heavy ball we imagine ourselveson a train moving alongside it at 5 feet per second and thenwe forget we are moving; from the train it looks as thoughthe heavy ball is suspended in the air alongside us. What wethen see is a light ball coming towards us at 10 feet per sec-ond from the direction of the front of the train. 10 feet per

Before

Station

Before

Train

Station 5

5

10

5

5

5

Fig. 1. Light ball hitting heavy ball (after M

second is the speed of the light ball as originally presentedplus the speed of the train as originally presented. It hits theheavy ball and, according to the rule for light balls,bounces off at 10 feet per second in the opposite direction.We now see it receding from us and moving toward thefront of the train at 10 feet per second. The heavy ballremains where it is.

Now we switch back to the original frame of referenceby jumping off the train and coming to what we originallythought of as ‘a standstill’. At that moment the heavy ballseems to start receding from us at 5 feet per second travel-ing with the train. By the same token, the light ball is alsogoing an additional 5 feet per second away from us. So thelight ball is travelling at 15 feet per second right-to-left inthe original frame of reference, the speed of the bounceas viewed from the train (10 feet per second), plus the speedof the train! (see Fig. 1). The light ball, when it hits a heavyball moving at the same speed in the opposite direction,bounces off three times as fast as it arrives! This seems anon-obvious, even counter-commonsensical, result butthe logic is irresistible. And it is breathtaking because, asin the ant-elephant example, it can be proved by such asimple trick. Once more we are forced to ask the question,are we non-mathematicians missing things like this all thetime? Do the mathematicians really understand the worldso much better?

After

?

After

55

10

15

ermin, 2005, Fig 1.7, redrawn by HMC).


3.2. Two ways not to go

Before we go on let us note that there are several otherkinds of things we could say about the above examples.There is the mystery, pointed out in Lewis Carroll’s accountof ‘What the Tortoise said to Achilles’ involved in the‘irresistability’ of logical inference like that in the syllogism.We could start to ask whether the logic is really as irresist-ible as it seems. But that is not our business here—we startour inquiry wallowing in the experience of logical force.

Another reaction is to point out that what has beendescribed is not a physical world but a logical or mathe-matical world. There are no ideal balls which bounce offa fixed surface with exactly the same speed as they hit it,and there are no ‘light balls’ which bounce without affect-ing the ‘heavy balls’ that they hit nor ‘heavy balls’ whichare unaffected by the impact of ‘light balls’. Therefore thewhole thing could be said to be somewhat less than thrillingbecause it does not describe the real world. This analysistouches on the long debate about the mysterious relation-ship between the world and its geometrical/mathematicalrepresentation—the world and our ideal version of it.Indeed this is a lived problem in modern physics. Oncemore I turn to a remark by a scientist in the field of grav-itational wave physics.

LIGO Physicist: I was remembering an epiphany that Ihad. While I was a postdoc at MIT I audited a class inthe engineering department on modelling techniques.And the class was kind of a weird class and I kept feelinglike something of an outsider because I realised thatevery time the professor wanted an example of some-thing, and if you were a physicist he would have said‘let’s treat the simple case of a harmonic oscillator thisway’, the guy would always say ‘let’s treat an internalcombustion engine this way’.

Nevertheless I had one of my deepest revelationsabout the nature of science listening in his class oneday when the professor said ‘no model is complete’.And when I first heard it I thought this is just outra-geous—[I thought] this is just an argumentative thing.It was only later that I realised that not only was it cor-rect but it was deep, and it was the kind of deep state-ment that I’d never heard any physicist say. Thoughgood physicists know it, it’s never taught. And the kindof opposite thing is taught by implication in the way ourphysics courses are set up. And it is taught by saying‘assume you have an X’ and it lays out the model foryou and you solve the equations. And we seldom ask,in the real world—in this thing we care about—Is themodel right?

I think experimenters in general are good at this, andat least good experimenters are able to pose a bunch ofquestions, and list a bunch of worries, and start to ask‘OK, what should I include and what not?’ And so a

9 Subject to all the problems of replication discussed under the heading of ‘

lot of people in our community do this unconsciously,but I was still astounded by hearing it said for the firsttime and realising that I disagreed with it . . . sometimesconversations don’t make sense unless everyone sharesthe knowledge of ‘Are we talking about a model orare we talking about the real world?’ And I had myepiphany about that, or at least about how people tendto live more in one thing than the other, after a physicscolloquium at Syracuse some years ago on black holes. . . And I walked out of the room after the colloquiumwith my friend [. . .], and I asked him, ‘Do black holeshave such-and-such a property?’—I don’t rememberwhat it was.

And he turned to me and said ‘Do you mean in clas-sical relativity, in string theory, in quantum gravity, insuper gravity?’

And I said, [respondent’s heavy emphasis] ‘No, in thereal world do they have this property?’

And then he laughed and he said ‘I have no idea!’And at first I wanted to strangle him and then I realisedthere was a learning moment there. (Quoted in Collins,2004a, pp. 603–604)

But the tension between the real world and physical modelsis not the topic of this paper. Whatever the resolution, thefact remains that one just knows one has learned somethingfrom the non-real world demonstration that the light ballwill bounce off at three times the speed it arrived. It isthe quality and significance of what has been learned thatis the topic.

4. Algebra and relativity: pages 33–37 read intuitively and

then mathematically

Now we move to the heart of this part of the argument.This section of the paper reports an experiment. It is anexperiment that involves reflecting on the subjective experi-ence of one person, the author, as he reads some pages of abook in a variety of ways. The report is intended to be an‘objective’ account of a subjective experience. By this ismeant that anyone who repeats the exercise starting fromroughly the same subjective point should have roughlythe same experience. My report can, then, be checked foraccuracy by others—my experience should be replicable.9

One way to replicate it is for others to read the same pagesof the book in the same ways (preferably from the samestarting point), so I have to describe the ways I read it withsome care. But if the results are worth anything they shouldbe robust enough to be generalisable to different texts andslightly different starting points. Thus, readers who are farmore mathematically accomplished than I might have tocheck the experience by reading more difficult texts thanMermin’s chapter—texts that are as much on the edge oftheir particular abilities as Mermin’s text is on the edge

the experimenter’s regress’ (Collins, 1992 [1985]).


of mine. (I have not done any algebra of this kind for fortyyears.)

It may be that some readers can make a start immedi-ately. Thus, it is likely that a subset of those who have readthus far have not actually worked through the mathemati-cal/logical proof set out in the paragraph preceding Fig. 1(above), they have just read it and taken the solution ontrust. They might now like to consider their internal stateand then consider it again after working through the calcu-lation on paper without referring back to the earlier part ofthis paper.

I now describe the experimental reading of Mermin’sbook, mostly in real time and the present tense, the ‘present’being the time of my writing the first draft of this part of thechapter—January 2006. My starting point is that I haveread most of Mermin’s book already. I remember Chapter1 and its demonstration with great clarity and have repro-duced it in this paper with almost no reference back tothe source. I believe the reason I have managed to do thatis that when I first read the book I worked through theexample arithmetically, just as I have done here. Havingdone this amount of processing I, in a sense, came to‘own’ the proof to the extent that I could reproduce it.Unfortunately, or fortunately for the purposes of the exer-cise we are engaged in here, I was too lazy to work throughthe algebra of subsequent chapters when I first read thebook (about a month ago). As a result I cannot reproducethe arguments of the chapters here and now without the fur-ther reading which I will begin shortly. I remember onlythat as speeds approach that of the speed of light the simpleand invariant translation of frames of reference whichworked so neatly with the bouncing balls does not workso neatly. I know—and I was already familiar with popularaccounts of relativity—that as things move faster they aregoing to get more complicated. But I cannot rehearse whyin my head. And, having written accounts of the Michel-son–Morley experiment and its aftermath, I know all aboutthe significance for special relativity of the constancy of thespeed of light experienced by any observer.

Re-reading Mermin’s book I can see why I picked it forthis exercise. Mermin’s first paragraph of his ‘Note to read-ers’ (Mermin, 2005, pp. xiv–xv), reads:

Although the mathematical level of the book is elemen-tary—simple plane geometry and beginning high schoolalgebra—it cannot be read like a novel. I have scrupu-lously tried to adhere to the rule (widely attributed toEinstein) that the exposition should be as simple as pos-sible, but no simpler than that. It will therefore often benecessary for you to pause and reflect on a line of argu-ment, to examine a figure and contemplate its relation tothe accompanying text, and, in general, to participateactively in the process of thinking things through, ratherthan passively reading along.

It is the difference between passively and actively readingthe geometry, the algebra and the arithmetic that I am try-ing to explore. Already I have invited a subset of readers to

explore the difference between passively and actively read-ing the description of colliding balls that I have provided inthe paragraph preceding Fig. 1. Mermin’s book, then, is ni-cely designed for the exercise and, in particular, nicely de-signed so that those of us who are not very good atmathematics can carry it through.

4.1. The relativistic addition law

Now I have read Chapters 2 and 3 and begun Chapter 4and have decided that this is the chapter to concentrate on.I have just encountered the first formula over which I havea choice.

Mermin states on page 28, without explanation, that therule for adding velocities is not that given in earlier chap-ters—the ‘nonrelativistic’ rule where one velocity is simplyadded to another

w ¼ uþ v

but the following ‘relativistic’ rule (where c = speed oflight):

w ¼ uþ v=½1þ ðu=cÞðv=cÞ�He then says:

If u and v are both small compared with the speed oflight, then u/c and v/c are both small numbers. Theirproduct is then a small fraction of a small number—iea very small number—so the relativistic rule differs fromthe more familiar nonrelativistic rule only by dividingthe nonrelativistic result by a number that differs insig-nificantly from 1. If, on the other hand, u = c, the [therelativistic rule] requires w also to be c, whatever thevalue of v may be. (Check this for yourself! It’s an easyalgebraic exercise.)

We have reached the first ‘crunch point’. What differencedoes it make if we do check it ourselves?

First let us see what we get if we merely read the claimwithout doing any reflecting or checking. We get somethingwe probably already know—that relativistic calculationsreduce to non-relativistic calculations when the speeds con-cerned (u and v), are not close to the speed of light. We cansee this from the equation (at least I can), without hardlypausing at all. I can just ‘see’ that if c is large comparedto the other speeds then both u/c and v/c are approxi-mately 0 and that the bottom line of the right hand sideof the equation is just 1, and dividing by 1 makes no differ-ence to anything so the equation just ‘becomes’ the non-rel-ativistic w = u + v.

But, when I am asked about what happens when u = cthings are not so obvious. The first place I need to pauseis just before the bracketed ‘(Check this for yourself! It’san easy algebraic exercise.)’ in the quotation from thebook. The claim being made in the previous sentence ispretty extraordinary. It says that if one of the speeds inquestion is c then the sum of the speeds will be c irrespec-tive of the value of the other speed! That needs reflection. It


is surprising in two ways. Firstly it is surprising if it is takento represent the physical world, though perfectly familiarto those with any kind of ‘popular understanding’ of rela-tivity because it says that the speed of light is a constantirrespective of the speed of the source or the speed of theobserver (if the speed is c, then you cannot add a velocityto it or take a velocity away from it). That state of affairsis what the equation is supposed to represent. It is surpris-ing in an algebraic sense because it is hard to imagine thatsuch a strange state of affairs can be represented by such asimple equation. It is at this point that we need to askwhether we should accept Mermin’s invitation and ‘Checkthis for yourself’ or just take his word for it. Right now Ifeel that I am going to feel a bit more secure if I do checkit for myself. It is such a strange claim about the physicalworld that the fact that it can be represented by a simpleequation might just demystify it a bit—why I don’t know,but let’s try it. To try it I have to work it through.

(A) So let us start with u = c, as suggested, and v beingsmall. The denominator is 1 + v/c which is (c + v)/c. The numerator is c + v. Therefore the whole thingis (c + v)/(c + v)/c or [(c + v) · c]/(c + v). The answeris exactly c!

(B) What if v = c too? Then we have c + c as the numer-ator and 1 + 1 [1 + (c/c) · (c/c)] as the denomina-tor = 2c/2 = exactly c.

(C) What if v = 0.9c. Then we have numerator as c +.9c = 1.9c and denominator as 1 + 1 · .9 = 1.9.QED.

Results B and C are, I now see, implicit in result A, butthis is all so strange that it seemed worth trying the thingagain with high values of v. (Or more likely, it is so long sinceI did any algebra ‘in anger’ that I no longer grasp intuitivelyand immediately that result A implies results B and C). Wenow know/feel that this simple equation does represent a sit-uation where you cannot add anything to the velocity c. Ifeel just a bit more comfortable with the idea but I mustadmit that it is not that much more comfortable. It is a bitless ‘more comfortable’ than I thought it would be when Iwas just taking Mermin’s word for it. Right now I’m think-ing that all this shows is that someone out there has beeningenious enough to work out how to write an equation inwhich it is impossible to add anything to u if u = c. That’sjust algebraic cleverness, nothing more. It’s easy to see thisby inserting something physically impossible into the equa-tion. Suppose we leave u equal to c and make v = 3c, threetimes the speed of light, something that I know from my lay-person’s knowledge is physically impossible. w still equals c!Suppose I make u = c and make v = elephant! w still equalsc. So I have not gained a whole lot by working through thisequation for myself—certainly no understanding of thephysical world on a par with what I learned from mensura-tion or the colliding balls; I seem to be doing algebra notphysics. Disappointingly, the idea that the speed of lightremains a constant irrespective of the speed of the source,

or that of the observer, still seems just as strange. I suspect,however, that working through the equation will help meremember it if not understand it.

4.2. The train frame calculation

The next section of the chapter, the rest of the few pageson which I am going to concentrate, is about deriving therelativistic addition law equation—the one we have justbeen playing with. Let me switch out of reflective modeand jump ahead for ease of exposition. We will find thatthe equation is to be derived by solving some simultaneousequations. The equations themselves are to be obtained byrepresenting in symbolic form the journey of a ball and aphoton moving inside a train which is itself moving. Balland photon start from the back of the train at the sametime and move off toward the front. The photon reachesthe front of the train and bounces back, eventually tocollide with the ball a certain fraction of the way alongthe train. We work out the point along the train where theycollide in terms of that fraction. We do the calculation twoways, first from the ‘train frame’—what a passenger insidethe train sees—and second from the ‘track frame’—whatsomeone standing on the track sees. The photon movesat speed c irrespective of frame. The ball’s speed is differentaccording to the frame from which it is perceived. This waywe get two different expressions for the point in the trainwhere the collision happens but it must be the same phys-ical point in both cases. Therefore we can solve the simul-taneous equations and this will give us the relativisticaddition law. Now let us get back to the present tense.

I have little idea why Mermin is doing the calculationthis way but I suppose he has good reasons which I amready to take on trust. He does provide a bit of an expla-nation but this does not show whether or not there areall kinds of other ways of getting the same result; we justhave to assume this is the best way for expository purposes.

To go on with his argument, if the slow ball moves atspeed u and the photon moves at speed c and if the lengthof the train is defined as 1, and f is the fraction of the lengthof the train from the front at which ball and photon meetup again, then the photon has covered the distance 1 + f inthe same time as the ball has covered 1 � f. The relativespeed of ball and light is u/c where

u=c ¼ ð1� fÞ=ð1þ fÞMermin tells us that this equation can be converted to theequivalent:

f ¼ ðc� uÞ=ðcþ uÞAt this point Mermin once more advises us:

When I assert that two expressions are equivalent . . .you should convince yourself that I’m telling the truth. . . (p. 33)

So I do the algebraic manipulation which I find to my de-light I can manage. But what have I learned? I don’t feel I


have learned anything even though I have had some funproving to myself that I can still do elementary algebra.What is causing me discomfort is that the importance ofeverything I am doing rests on the claim that the speedof light is a constant but so far as I can see, this has no-where been proved. It is simply stated somewhere earlieron that Maxwell deduced the speed of light from his theo-retical considerations of electromagnetism and that theMichelson–Morley experiment did not show that it wasnot a constant. But all the elementary bits of proving Iam doing—I am told not to trust Mermin in this or that re-spect—in fact depend on my trusting him or the scientificcommunity in the much bigger matter of the constancy ofthe speed of light. Right now I feel as though I am beingtold that I am learning something about the physical worldwhereas I am really learning something about how alge-braic symbols can be cleverly manipulated. I am findingthat the wonderful promise of the first chapter, where I dis-covered that superb result about bouncing balls from theprinciple of relativity, is not being followed up in the wayI hoped it would be.

4.3. The track frame calculation

Now I am going to describe in still more depth my read-ing of pages 33–37. Mermin now advises us to forget allabout the calculation we have just done and do a calcula-tion as though we were standing on the track. In the firstcalculation the speed of the train did not enter into things;it was effectively zero since we had adopted the ‘trainframe’ as our own frame of reference. If we adopt the‘track frame’, however, the speed of the train will have toenter into things so its going to be a bit more complicated.

What I decide to do at this point, because I think it willhelp me understand what is going on, is to re-express whathas been done so far without mentioning light. In the trainframe thought experiment I will replace the photon bounc-ing off the front of the train with another ball, which hap-pens to travel faster than the first ball. So the first ball stilltravels with speed u but the second ball travels with speedA. We can now rewrite the equation we have above as

f ¼ ðA� uÞ=ðAþ uÞ

I can do this because the constancy of the speed of light hasnot so far been invoked in any way that is necessary for thealgebra so there is nothing special about the symbol c as itappeared in the original equation. I can replace c with any-thing that is faster than u and everything remains the same.

Now we go on with the second thought experiment—the‘track frame’ calculation but again I am going to replace cwith another symbol. In the case of the track frame I willreplace the photon travelling at c with a second ball movingat fast speed B.

The analysis in the track frame is more complicated andI won’t, and can’t, because I have not forced myself to,work it through without going back to the book. I followed

it on the page but have not tried to reproduce it. What we

get is the following equation (where I have replaced c withB).

f ¼ ðBþ vÞ=ðB� vÞ � ðB� wÞ=Bþ wÞwhere v is the speed of the train and w is the speed of theball as seen from the track.

Now the argument goes as follows. The two calculationsare for exactly the same physical event therefore the pointat which the balls meet must be the same point as it waswhen we did the calculation from the train frame. As Mer-min nicely puts it, we might imagine that the two ballsexplode when they meet and make a mark on the floorwhich can be inspected any time. That mark is in the sameplace whichever description of the experiment we are talk-ing about. Therefore,

ðA� uÞ=ðAþ uÞ ¼ ðBþ vÞ=ðB� vÞ � ðB� wÞ=Bþ wÞ½¼ f �

Now comes the relativistic crunch, the place of which, inmy view, is much more clear doing it my way up to thispoint with the faster balls travelling at A and B. We nowreplace the two fast balls with photons. The speed of pho-tons is constant which means we replace both B and A withc. The speed of the photon does not appear to changewhether we view it from the train or the track. Thereforewe have

ðc� uÞ=ðcþ uÞ ¼ ðcþ vÞ=ðc� vÞ � ðc� wÞ=cþ wÞMermin tells us that this can be reduced to the relativisticaddition law. (Some time later I try to reduce it but find,to my intense disgust, that I cannot make the algebra comeout. I have to follow the derivation which Mermin kindlyprovides at the end of Chapter 4.)

What has happened to my physical understanding nowthat the relativistic addition law has been derived? As faras I can feel, not much! The physical ‘crunch’ as I put itabove, was the point at which you take the speed of thephoton to be c irrespective of frame. But this is the point(p. 37) at which all the relativistic physics enters and itenters as an assumption. What we discover is that far fromthe algebra making it easy, presenting it this way showsthat it is really rather strange. It is simply very hard tounderstand why we should always treat the speed of thephoton as c irrespective of the frame. The algebra doesnot give us any clues at all. The algebra may be enoughto derive many of the consequences of the constancy ofthe speed of light but it is not enough to explain why onehas to assume it in the first place—and that is the key tophysical understanding as far as I can see. So we havereached the first conclusion: the pages we have analysedand the algebra we have done, quite unlike the bounc-ing balls example, have done nothing for my physicalunderstanding.

Let me make clear what my conclusion is not. My conclu-sion is not that I could pass physics examinations or becomea contributory expert in the theory of relativity with-out mathematics. It is that I can gain interactional expertise


in this part of special relativity without the mathematics.10

In other places it is argued that interactional expertisecan take one quite a long way in worlds that require physi-cal understanding. My conclusion from this part of theexercise is that, while mathematical understanding maysometimes yield physical understanding, sometimes it doesnot. Sometimes the physical understanding may be a sepa-rate matter from the mathematical understanding. Thisresult is not very robust as it depends on just my readingof four pages of Mermin’s book but it makes a reasonablestarting point for more discussion and for moreobservations.11

5. Types of mathematical understanding: relating the trackframe to the train frame

Back to the reflection: forgetting the speed of light, I findthat I do not ‘really’ understand even the algebraic calcula-tion. I am just going through the motions of repeatingMermin’s symbol manipulation but with no intuitive graspof, for instance, why his calculation for the track frame isso much more complicated than the calculation for trainframe. I realise that what I need to do to satisfy myselfin this respect is to show that, in a non-relativistic world,the second calculation is essentially the same as the first cal-culation. In other words I should be able to get the trainframe result from the track frame calculation by relatingA to B and U to w. It seems to me that B should simplybe A plus the speed of the train (A + v) and likewise wshould be u + v. I thought that if I substituted A + v forB and u + v for w in the track frame formula it should turninto the train frame formula and I would then be satisfiedthat nothing funny was going on. So I tried it and it didn’twork. I could not make the algebra come out and my dis-tant memory of ‘how algebra went’ told me that the uglyresult I was getting was never going to reduce to the simpleresult however much I moved the symbols around.

So I decided that this meant that I did not really under-stand what was going on after all and that I had betterwork through the derivation of the two formulae formyself—something I had avoided up till then, merely fol-lowing Mermin’s arguments on the page—they seemedpretty straightforward.

So, with pen and paper, trying to do as much as I couldmyself, and picking up clues from the page whenever I gotstuck, I repeated the derivations. The train frame deriva-tion is clever but simple. The track frame derivation, I dis-cover, is a bit more complicated than it looks. Really tounderstand it I have to redraw the helpful diagrams thatMermin provides but modify them in various ways. For

10 For the concepts of interactional and contributory expertise see Section 611 M. Gorman (private communication) writes: ‘it would be possible to run an

case, running a mental model with trains was better for understanding than dothe maths. HMC.] Students could be taught the concept either via mathematicgroups could be compared. It would be important to protocol the students asCharness, 1994.)

instance I take the balls out of the train and put them onthe track traveling along with the train, and I draw verticaldotted lines on the page to show where the train was awhile back, and so forth. I discover, in other words, thatthe seductive ease of Mermin’s explanation is an illusionto the extent that though it seems easy to follow on thepage I do not have had a hope of reproducing it myself.Step-by-step, by redrawing the diagrams, and deducingeach little increment for myself, I reach the point where Ido know what is going on. At last I feel I know what ishappening in the track frame calculation. But my formulastill comes out the same as Mermin’s and the substitutionof train frame speeds plus the speed of the train still doesnot reproduced the track frame formula that I think itshould.

After a while I discover that I have simply made a mis-take. It is that B is not always A + v because for some ofthe time in the track frame calculation the fast ball is goingin the opposite direction to the train so its speed is B � v.Having realised this I am anxious to try again to reachmy goal which is reducing the track frame formula to thetrain frame formula. First, however, I will try and repro-duce the track frame calculation on paper without any ref-erence to the book at all.

I manage it, at first with one elementary algebraic error,but then without error. Here is the sum—I reproduce ithere and now on the page without reference to the bookbut using a diagram that I have drawn myself withoutlooking back at the book (Fig. 2). This is like Mermin’sdiagram except that I have replaced c with B and, as men-tioned, I have added a series of vertical dotted lines andhorizontal two-headed arrows indicating the distancesbetween the dotted lines.

The train, of length L, moves with speed v. The fast ballmoves with speed B, the slow ball with speed w. The topbox represents the train at the starting point, the secondbox represents the train at the moment when the fast ballstrikes the front of it after time T0 has elapsed, and thethird box represents the train at the moment the fast ballhits the slow ball on its return toward the back of the trainafter a further time T1 has elapsed. The crucial things formy understanding are the vertical lines and the two-headedarrows that represent the distances between them. Theseare the distances that appear in the algebraic manipulationand I found I did not understand the manipulation until Ihad drawn these vertical lines and associated distances. Ithink readers should now look at each of the two headedarrows between dotted lines and satisfy themselves thatthe symbols in the middle of each of them do correctly rep-resent the distances indicated. The rest is easy.

of the Introduction to this special issue.experiment designed to replicate Collins’ observation that in the relativity

ing the algebra. [My claim is actually only that nothing is added by doings or via thought experiments, and the resulting understandings of the twothey worked, accessing their developing understandings’. (See Ericsson &

To my disgust, I find I can’t do it. I still cannot reproduce Mermin’s inventiveness in-

spite of having the advantage of having read his account and even re-worked it. I will

do the thing once more with Mermin’s book for reference and then do it myself

without the book.

w

B

fL

L

v

D

T0

T1

wT0

BT0

BT1

wT1

vT0

vT1

Fig. 2. The track frame calculation with faster ball moving at speed B.


5.1. The track frame calculation revisited

To do the track frame calculation we first find two waysof expressing D, the distance between the two balls at themoment when the fast ball hits the front of the train. Atthis point a time T0 has elapsed. A distance is a speed mul-tiplied by a time so all the distances can be expressed bysuch terms. By inspection of the figure one can see that

D ¼ BT0 � wT0

and

D ¼ BT1 þ wT1

Therefore

BT1 þ wT1 ¼ BT0 � wT0

which means that

T1=T0 ¼ ðB� wÞ=ðBþ wÞ

Now we find two ways of expressing L, the length of thetrain. Again by inspecting the diagram one can see that

L ¼ BT0 � vT0

and

fL ¼ BT1 þ vT1

Therefore

fðBT0 � vT0Þ ¼ BT1 þ vT1

which means that

T1=T0 ¼ fðB� vÞ=ðBþ vÞ

Putting both results for T1/T0 together we get

fðB� vÞ=ðBþ vÞ ¼ ðB� wÞ=ðBþ wÞ

or

f ¼ ðB� wÞ=ðBþ wÞ � ðBþ vÞ=ðB� vÞ

QED!

5.2. The track frame calculation repeated with train frame

speeds

Now I can go back to my ambition of reducing the trackframe calculation to the train frame calculation by substi-tuting the appropriate speeds without making a mistake.

W (=u+v)

B (= A+v)

fL

L v

D

T0

T0

(u+v)T0

(A+v)T0

(A-v)T1

(u+v)T1

vT0

vT1

Fig. 3. Track frame repeated with train frame speeds plus or minus speed of train.


I do this by reproducing the diagram with speeds substi-tuted (Fig. 3). Note that both occurrences of w are replacedwith u + v but B is replaced with A + v once and A � vonce—the later instance representing the period after thefast ball has bounced back from the front of the trainand the cause of my earlier mistake. Now let us redo thecalculation as before but with the substituted speeds.

D ¼ ðAþ vÞT0 � ðuþ vÞT0

and

D ¼ ðA� vÞT1 þ ðuþ vÞT1

Therefore

ðAþ vÞT0 � ðuþ vÞT0 ¼ ðA� vÞT1 þ ðuþ vÞT1

which means that

T1=T0 ¼ ½ðAþ vÞ � ðuþ vÞ�=½ðA� vÞ þ ðuþ vÞ�

Taking away the round brackets

T1=T0 ¼ ðA� uÞ=ðAþ uÞL ¼ ðAþ vÞT0 � vT0

12 David Mermin himself was in the audience when I presented this material adeserved an A+ for my algebra.

and

fL ¼ ðA� vÞT1 þ vT1

Therefore

f ½ðAþ vÞT0 � vT0� ¼ ðA� vÞT1 þ vT1

which means that

T1=T0 ¼ f ½ðAþ vÞ � v�=½ðA� vÞ þ v�

That is

T1=T0 ¼ f

Putting both result for T1/T0 together we get

f ¼ ðA� uÞ=ðAþ uÞ

This is the result for the train frame calculation and, oncemore this was what was to be demonstrated.

At last I think I understand what is going on in pages 33to 37 of Chapter 4 but it has taken me about three days.12

What is more, though I am now pretty satisfied with thealgebra, and have developed a degree of algebraic insight,or something like ‘inner certainty that the derivation is cor-

t Cornell University on 6 November 2006. He was kind enough to tell me I


rect, I still seem to have gained nothing at all in the way ofphysical insight. All the physical insight that emerges fromthe equations still turns on accepting that the speed of pho-tons is constant from wherever they are observed and noth-ing I have done shows me why that should be so.13

Now, it might be argued that in reading only four pagesof the book in this kind of depth I could not expect to gainphysical insight.14 It might be said that to gain physicalinsight I would need to read the rest of the book in similardepth. I think, however, that while it may be true thatworking through the rest of the equations would showme why such things as time dilation and mass increase atnear-light speeds are consequences of the algebra I wouldstill not be gaining physical insight from doing the algebrain the way I did it for the four pages. Here, once more, Iwill invoke the principle of division of labour, takingexperiment to be analogous to mathematics: thus, I getconsiderable physical insight from understanding theMichelson–Morley experiment. I can see why it is thatmeasurements with an interferometer could (and, to someextent, eventually did) lead scientists to accept that thespeed of light was not affected by the movement of theobserver and I can sense how extraordinarily counter-com-monsensical this must have been at first. In the same way, Ibelieve I can empathise with the intense excitement thatmust have been felt by physicists as they worked throughthe equations associated with constant light speed anddeduced the still more strange but yet unobserved conse-quences that followed mathematically if it was true. Yet Ido not have to repeat the Michelson–Morley experimentto get the experimentally based insight so why should Ihave to repeat the mathematical derivations to get themathematically based insights? Could it be a financialand logistical contingency that students of physics cometo understand the Michelson–Morley result without everrepeating it, and that professional physicists accept its out-come without ever doing it, while it still does not count asunderstanding relativity unless the sums have been done.The suggestive fact is that the Michelson Morley experi-ment takes a lot of time and money to repeat while themathematics costs little or nothing. Physicists are happyto take the results of the Michelson–Morley experiment,along with nearly all the other experimental results theyuse, on trust, whereas for some reason they are not contentto take the mathematical derivations on trust.15

13 This, of course, is a very simple piece of mathematics. The fact that it torustiness. But it is my very rustiness that I am exploiting here. To repeat theadvanced regions of rustiness.14 Mermin himself told me that the four pages was never intended to provid15 That, of course, is not really true; thus gravitational wave physicists take o

black holes on trust without feeling the need to reproduce the immensely comsubject of this part of the paper: what do we take on trust and what do we consiphysics?16 This is not to say that everyone who is not a physicist thereby understands

not understand it. I myself do not know if I understand it and have never been ecould be some people who understand it without being able to work through

Myself, I have to report, I am perfectly happy to takethese derivations on trust. Having re-read the remainingchapters of It’s about time, a bit slower than I first readthem, I have seen the way the equations unfold to producetime dilation and mass increase and the rest but I still feelthat nothing would be gained by my working through thealgebra in the same way as I worked through pages 33–37. I trust Mermin’s algebra as much as I trust the reportsof so many experimental findings that I cannot workthrough. I believe someone should be out there checkinghis algebra, just as someone should be checking every rad-ically new experimental result, but it does not have to beme. To repeat—I invoke the division of labour: there mustbe physicists who can work out all the consequences of theconstancy of light speed otherwise physics would be radi-cally impoverished, and there must be other physicistswho can check the results, but not all those who can be saidto understand the special theory of relativity need to havedone either the experiments or the mathematics.16

6. Levels of mathematical understanding and the way

physicists use them

In addition to what I expected to gain from the intro-spective reading of Mermin’s book a scale of mathematicalunderstanding when reading technical papers is beginningto emerge. The introspective reading shows that there aredifferent levels at which a mathematical exposition can beunderstood and they are surprisingly different. I will usethe painstaking reading of pages 33–37 of Mermin’s bookas the core of a ‘table of understanding’ but there are levelsthat both precede and follow the degrees of understandingthat emerged from the exercise. The first level hardlydeserves the accolade ‘mathematical understanding’ eventhough the mathematics plays a role. It is understandingthat something has been proved because it is known tohave been proved mathematically. An example from grav-itational wave physics can make the point.

In the late 1980s and early 1990s the Laser Interferom-eter Gravitational-Wave Observatory (LIGO) was fightingfor more than $200M of funds to build its installations. Atthe same time, Joseph Weber, the disaffected pioneer ofgravitational wave detection, was claiming that detectioncould be done for a few hundred thousand dollars. Weberhad published a new analysis of detector sensitivity

ok me three days rather than thirty minutes is something to do with myobservations that I have done others might have to find their own more

e physical insight into the relativistic addition law.n trust the wave forms calculated for inspiraling binary neutron stars and

plicated computer programs that generate them. Maybe this is the realder it necessary to work through in order to count as having been trained in

the theory of relativity. Many of those who think they do understand it doxposed to any test of my understanding. I merely make the point that therethe derivations in a deep way themselves.


suggesting that the cheap detectors he had built were nineorders of magnitude more sensitive than had been claimed.No one believed Weber’s claim for a moment because itswider implications were close to absurd but his calculationwas not obviously flawed and it took some theoreticalexpertise to counter it. The only refutation published in aphysics journal—Physical Review D—was by Leonid Gris-chuk.17 I asked Grischuk why he published his piece giventhat no one believed Weber’s result anyway. He replied:

[1996] When it became clear that it creates difficulties. Itcreates difficulties in the sense that, though I always sus-pected that he was wrong, it did not create too muchharm until he really started to lobby American Con-gressmen . . . So since I know that scientifically it cannotbe true, and since it became politically important, Idecided to write something . . .

Political motivation did play a role. It did play a rolebecause I notice that people are getting interested,people are getting confused, and if I do not clarify thisthey may wrongly assume that this is the correct for-mula and probably would decide not to build LIGOfor instance—you know—and so on.

Grischuk was not expecting American Congressmen toread his paper. What he expected was that other physicistswould be able to say to Congressmen who had been briefedabout Weber’s claims that they had been refuted in the lit-erature. All the Congressmen needed to know was thatpublication in Physical Review D carries the imprimaturof the scientific community. They could therefore feel com-fortable in rejecting the counter-claims of Weber or his rep-resentatives. The maths was needed to get the paperpublished in Physical Review D but there is every chancethat very few people gained anything from Grischuk’smathematics; certainly the Congressmen and women towhom it was directed gained from the fact that the mathshad been done, not from the maths itself.18

Level 1 of ‘mathematical’ understanding is what we cancall ‘authority-based understanding’. It has nothing to dowith the mathematics itself but only to do with the per-ceived authority of mathematics, the person who has donethe mathematics, and the place it is published. One cangain authority-based understanding by reading popularworks of science which will set out claims about who orwhat has authority in such matters.19

The next step is like authority-based understanding butLevel 2 might be attained by someone who has a broadconceptual grasp of the scientific domain rather than a

17 Grischuk (1992). For a full account of this and a number of related incidenthat concerning the refutation of the claimed detection of gravitational waves18 Grischuk stressed in his interview that he had tried to do an objective ana

analysis to be the only one to have explored the issues thoroughly and decisivel(which, incidentally, was not published in a journal).19 In the Periodic Table of Expertises (Collins & Evans, 2007, p. 14; Collin

located in the left hand side of the main two rows—it depends on ubiquitous20 Collins (2004a), p. 508.

mathematical grasp. Rather than accepting the importanceof a proof on the basis of authority—as with the putativepoliticians—one is able to make a similar judgement byunderstanding how a claim fits, or does not fit, with thegeneral way of thinking in a domain. For example, mybroad conceptual understanding of gravitational wavedetection was enough to enable me to make the occasionaltechnical judgement of this kind in the gravitational wavefield.20 This is a big step up from authority-based judge-ment but it has nothing to do with the mathematics thatunderlie the claims being made. We can call this level‘familiarity-based understanding’.

Level 3, which we will call ‘impressionistic understand-ing’, is the first level that involves the mathematics itself.Here the mathematics is read but rather than followingthe proof step-by-step the aim is to get a general impressionof what is being argued. I have found that this can beaccomplished even if the mathematics is well beyond one’scompetence so long as one knows the scientific field. This isthe level of understanding to which I aspire and sometimesachieve when I read a mathematical paper in the field ofgravitational wave physics. Sometimes I won’t be able toachieve it but even then it will sometimes, but not always,be that one of my respondents will explain it to me in a waythat enables me to ‘get it’. If I can ‘get it’, the fact that I canthen have an impression of the principles driving a deriva-tion gives me more confidence in the derivation thanmerely being told that the derivation had been accom-plished. Being able to attain impressionistic understandingis a huge step forward from not understanding at all. Iknow what not understanding at all is like because I expe-rienced it when I tried to do fieldwork on the theory ofamorphous semiconductors. In that area I could nevergrasp any of the principles. Indeed, my understanding ofthe field could never have got further than Level 1; it cer-tainly did not reach even Level 2.

Impressionistic understanding will be the level reachedby some of those readers of this paper who have notworked through the mathematics themselves. They cansee what is going on in the mathematical arguments andthose who have at least some conceptual ability of a math-ematical kind will have more reason to trust them thansomeone who is merely accepting the arguments on thebasis of their existing familiarity with special relativity—or even on the basis of trust in the current author’salgebraic abilities!

Four out of the next eight levels of understanding havebeen explored during the detailed reading of pages 33–37 of

ts that make the same point as we are making in this chapter (for examplefrom supernova 1987A), see Collins (2004b).lysis in spite of the quasi-political motivation, and that he considered hisy. He was not entirely happy with the one earlier analysis of the same thing

s, 2007, this issue) authority-based understanding comes from expertisestacit knowledge.


Mermin’s book. They are labelled 4a–4d. These plus Level4e can all come under the generic heading of ‘checking’,because they involve stepping through the derivation line-by-line in a way that has the potential to uncover errors.This is why they are presented as subdivisions of the gen-eric Level 4. The potential is markedly different as the levelsrise from low to high, however. This is a convenient pointto reassemble the whole list of Levels of MathematicalUnderstanding including four more levels which will be dis-cussed below.

6.1. Levels of Mathematical Understanding21

Level 1—Authority-based understanding: judgement onthe basis of whether a mathematical argument has beenpublished in a peer-reviewed journal or similar source with-out any understanding other than the reputation of theauthor.

Level 2—Familiarity-based understanding: acceptance, orotherwise, on the basis of the fact that the claim makessense or otherwise within the familiar and consensual con-ceptual structure of the scientific domain. The mathemati-cal exposition does not have to be read.

Level 3—Impressionistic understanding: the mathematicsis read to gain an impressionistic sense of the argument butit is not followed step-by-step.

Level 4—Checking: any level of reading that doesinvolve following the proof step-by-step. This, in turn.can be accomplished at a number of levels.

4a: stepping through the proof on the page without try-ing to reproduce it.4b: reproducing the proof with help from the page.4c: reproducing the proof without help from the page.4d: reproducing the proof while testing one’s ability bycarrying out some minor variations on the original.4e: doing brand new proofs of the same sort of whereyou already know the outcome.

Level 5—Innovatory understanding: doing brand newproofs of the same sort where you do not already knowthe outcome.

Level 6—Domain-specific ownership: ‘ownership’ of themathematical area in the manner of, say, a physicist, orwhichever domain scientist is using the mathematics.

Level 7—Mathematical ownership: ‘ownership’ of themathematical area in the manner of a mathematician.

In reading pages 33–37 Level 4d was reached by redoingthe proof with c replaced with A in the train frame and B in

21 A reader suggested that the five-level model of skill acquisition developed bysee Collins & Evans, 2007, pp. 24–26). It seems to me, however, that the Dreyfhere. The Dreyfus model shows that a person learning a physical activity, suchconscious rule following to unconscious, slick and efficient execution of the skillpresented here. Here the higher levels require more and more time and more22 But I once spent three days at the Perimeter Institute in Waterloo, Ontario,

clustered round blackboards writing symbols. I have no idea what they were

the track frame, showing how the latter could be reduced tothe former, and then replacing A and B with c once more.Level 4e would involve doing some more derivations of thesame kind, knowing only what the result should be. Thatwould be another big step up because it would involvethinking up what such a new problem would look like,something that I cannot do at the time of writing.

Level 5 seems like a small step beyond Level 4e but actu-ally it is a huge step as will be explained below. Level 6 is alevel of mathematical accomplishment attained by only afew physical scientists and Level 7 is there more for com-pleteness because the degree of logical purity demandedby mathematicians is not thought important by physicistsand the like.

7. How do physicists read mathematical papers?

Readers of this paper who are physical scientists mightnow like to think about how they read papers. Recently,when I have given presentations in physics departments Ihave been asking members of the audience at which levelthey approached the last journal paper they read in connec-tion with their work as professional physicists. Results areas reported in Table 1 with zeros left as blanks so that the‘shape’ of the distribution can be more easily seen.

As can be seen 60% of physicists reported that theyread their last paper only up to Level 3. Going up to Level4a—stepping through on the page—captures 85%. In otherwords, most of the time physicists read papers in not muchgreater mathematical depth than someone who mightknow little mathematics but plenty of physics. To avoidmisreading I have to repeat yet again that does not meanthat this is all there is to physics; it means that when phys-icists are trying to catch up on what their fellow physicistsare doing, they do not feel the need to follow each mathe-matical step along the way. They could get by with notmuch more mathematical ability than the most mathemat-ically weak of their colleagues such as Ron Drever, NeilsBohr, or perhaps even me. The new non-mathematicallyproficient physical scientists who I would like to see comingout of universities would be reading physics paper to get agood physical sense of what was in them; they too wouldnot need to read any deeper, mathematically speaking,than the majority of research physicists read most of thetime.22

The potential flaw in this argument is that even to readquickly in this way requires deep and tested mathematicalskills which are just not fully deployed on each occasion

the Dreyfus brothers should be mentioned at this point (for an expositionus model bears only the slimmest of relationships to what is being arguedas car-driving, progresses from stages that involve awkward and inefficient. I see no progression from rule-following to non-rule following in the tableand more careful execution.and noticed that the physicists there seemed to spend nearly all their time

doing.

Table 1

(1) UK Physics and

Astronomy Department

(2) US Theory of Gravity and

Stellar Structure Department

(3) UK Physics and


(4) US Physics and


Level TOTAL %

1

2 1 1 1%3 7 16 26 6 55 60%4a 1 5 15 1 22 24%4b 2 4 3 9 10%4c

4d 1 1 2 4 4%4e 1 1 1%5

6

7

Sum 9 25 47 11 92 100%


of reading. And I am not claiming that physics can beunderstood without enough mathematical experience toknow what a mathematical insight is like. But it seemsto me that the onus is on those who claim that physics ‘sim-ply is mathematics’ or the like, to demonstrate thatanything much more than high school mathematics isneeded in order to understand what a mathematical insightis like.

We can see from the table that a few physicists reportedreading their last paper to Level 4d or 4e, and we canassume that all, or nearly all, research physicists who cando it read to this depth sometimes if they have special rea-sons. There is a lucky gap in the table around 4c whichmakes the bimodality of the distribution (one big modeand one small one) more obvious. The distribution almostcertainly indicates what physicists said during the course ofthe little surveys: that they usually read to becomeinformed about what is going on but occasionally haveresearch-related reasons that demand that they go muchdeeper. One who reported a deeper reading in my quicksurvey admitted his reasons were ‘quirky’. He becameinterested in the technique but explained that the mathe-matics in the paper was particularly simple so it was nottoo demanding an exercise. One was a referee for a paperwhich contained controversial claims. He explained thateven when refereeing it was unusual to read to this degreeof depth and it was the controversial nature of the claimsthat made him feel he must do so on this occasion. Anotherphysicist remarked that given my survey method he wassurprised that I found so many cases at the 4d level sincethis level of reading was rare.

The reason why I do it is because if it’s related to my

research I want to reproduce that . . . Or [follow] closely

in some direction and that tells me where to go—this is a

useful step and I want to build on that step and then I

would do it. That happens very rarely . . . usually you’re

23 Collins (2004a), Chs. 19–21.24 Nearly all the physicists I spoke to justified their rejection of Weber’s ana

doing what you’re doing for 30 years. [Remark by mem-ber of the audience in dept. 4]

Other research physicists I have spoken to explainedthat even when they do feel the need to check the claimscontained in a mathematical paper they rarely do it bychecking the argument step-by-step. They might use‘dimensional analysis’—a simple technique that revealswhether the physical components of a formula could becorrect; they might do ‘order of magnitude’ calculations(does the formula come out right in a very approximateway when some rough numbers are inserted into it?); theymight ask if the formula covers some easily calculatedextreme cases; they might ask if the whole thing makesphysical sense?

This reflects what happened in the passage of gravita-tional physics referred to above when Joseph Weber devel-oped a new analysis of the ‘cross-section’ of a resonantmass gravitational wave detector. I explained above thatLeonid Grischuk’s motives for discovering the mistake inWeber’s derivation was motivated by the political necessi-ties associated with the funding of LIGO. In fact I foundthat while almost everyone in the gravitational wave phys-ics community was certain that Weber was wrong, almostno one found the flaw in his mathematical analysis.23 Asfor the mathematics, there was even a well known theoret-ical physicist, Giuliano Preparata, who initially published adisproof of Weber’s claim, but then changed his mind, pub-lishing a series of papers supporting Weber. There wereonly two mathematical disproofs. The most influential ofthese was by the prestigious theoretician, Kip Thorne,and was widely presented on the conference circuit andpublished much later in a conference proceedings volume.Grischuk’s proof was published in a refereed journal butwas less influential.24 The fact is, then, that only three phys-icists ever did the hard work of finding the flaw in Weber’sderivation and one of those changed his mind later.

lysis by reference to Kip Thorne’s disproof.


When I first wrote up this incident I thought I wasrevealing the crucial importance of trust in physics. Ithought I was revealing that the majority of physicists feltjustified in rejecting Joe Weber’s view because they trustedthe prestigious theoretical physicist Kip Thorne and felt noneed to do the work themselves. On further reflection, how-ever, I missed the deeper point that the analysis presentedhere reveals. This point is that disproof of a claim is notusually done by finding the flaw in the mathematics, it isdone on the basis of much more general considerations.For instance, in the case of Weber, his claims about thecross-section of solids to gravitational waves also hadimplications for their cross-section for neutrinos (out ofwhich he developed a new line of scientific work), but thisenhanced cross-section for neutrinos had the implicationthat astronomical bodies would no longer be transparentto them so the energetics and dynamics of planets, aster-oids, and so forth, would become incompatible with wellestablished observations. Physicists could feel quite justi-fied in rejecting Weber’s claims on this basis—a basis thateven I can understand—without finding the flaw in hismathematics. Finding the flaw in the mathematics was nec-essary for political reasons but it was not needed to justifyphysicists’ conclusion that they could continue theirresearch lives in an orderly way. As far as physics was con-cerned it was almost a matter of pedantry to find the flaw—it was a matter of getting rid of a piece of grit from theshoe. One could continue to walk the walk of physics per-fectly well on the basis of the broad conceptual rejection ofWeber’s claim but there would be a minor irritation untilthe flaw in the proof was uncovered.

Ironically, the real need to find the flaw quickly could beexplained in terms of the incorrect model of science held bynon-scientists. The politicians misunderstood the nature ofproofs in the physical sciences. An order of magnitude cal-culation, or a simple deduction that leads to an unaccept-able consequence, will do for physicists but not for thosewho think that physical proof and disproof is a kind ofquasi-logic. In so far as this example can be generalized,mostly, physics is actually done not as quasi-logic but asbroad conceptual understanding.

7.1. The exceptions

All that said there are occasions, in addition to referee-ing controversial papers, removing a mathematical stone inthe shoe, or satisfying politicians, when physical scientistswill want to do detailed mathematical checking of thepapers they read for the sake of the physical science. Onesuch occasion, as mentioned in the quotation above, iswhen a mathematical claim carries costly implications forthe way work, especially experiments, are to be done. If ascientist quickly reads a mathematical paper at, say, Level3, and discovers something so potentially important that itimplies a change of direction in a programme of experi-mentation or observation, even if it passes the order ofmagnitude or dimensional analysis tests, they are going

to want to check it in detail, or have it checked in detailby some trusted assessor. My guess is that, with exceptions,physicists in such a position are going to find they need torely on more than their own mathematical abilities. To dothe real work of checking they are going to make use of thedivision of labour and call in a Kip Thorne-like characteror characters who can really do the maths to help them.In other words when it is a matter of proving some claimthat might affect one’s life as a physicist in a major way,then the correctness of any mathematical part of the claimwill be something that will have to be checked too. A checkthat draws on the division of labour will still be goodenough, however. Even the necessity of mathematics undercircumstances like these does not show that every physicisthas to be a mathematician, only that some physicists haveto be mathematicians.

8. Mathematical understanding revisited

Before we reach a conclusion for action let us extractwhat more we can from the table of Levels of Understand-ing. There are three crucial breaks in the table. The first isbetween Level 1 and what comes below. As indicated Level1 depends purely on authority (and may well be based on athorough over-estimate of the importance of mathematicsin physics), while Level 2 rests on interactional exper-tise—a conceptual grasp of the domain.

The next important break is between Level 4a and whatcomes below, with 4d probably being the next crucialpoint. The difference between 4a and 4d is the seductivenessof the printed page. Following a proof on the page is fartoo easy and persuasive. The difference in experienced dif-ficulty between 4b and 4c makes the point and it becomestill more obvious when one goes to 4d. What seems easysuddenly becomes hard and one makes mistakes. In otherwords, one can read a proof and be convinced by it eventhough in reality one grasps it mathematically at a levelthat is in no way suited to checking it. The history of sci-ence is almost certainly filled with mistakes that haveresulted from following a proof on the page rather thanreally understanding it. To get to the point where even aphysicist can really begin to check a mathematical argu-ment, he or she must attain at least Level 4d and this is justtoo difficult and time-consuming for the majority of phys-icists the majority of the time.

Only after 4d do we begin to see anything that could becalled creative use of mathematics. The next crucial breakis between 4e and 5. So long as you know the answer thatyou are supposed to be getting you can work through aderivation over and over again, adjusting it, as I did inthe relativity example, until one gets the answer one hasbeen looking for. That is why even to be sure you haveunderstood something properly Level 4d may not beenough; the adjustments one makes may be too driven bythe answer one already knows.

Interestingly, in many mathematics examinations takenby students at schools and universities the student knows


whether the right answer has been achieved because thequestion is designed to have a simple outcome. So longas the apparent answer is ‘ugly’ the student can return tothe analysis and look for the mistake. This is not what realscience is like where initial results are mostly ugly and donot provide clues as to whether to go back again and workover the derivation.

What we are describing here is the problem for mathe-maticians analogous to that posed for experimentalists bythe experimenter’s regress (Collins, 1992 [1985]). So longas experimenters have an idea of what their experimentalresults should look like (as they usually do at school andfirst degree level and in most ‘normal science’), they knowwhether they have done the experiment well; if the indica-tions are that they have not done it well they can do itagain. When the right result is unknown, however (as inthe case of the first attempts to detect gravitational waveswith relatively insensitive detectors), then scientists tendto separate into groups, each championing different results,with no obviously ‘scientific’ way of knowing which onesare right; we have, in short, a ‘mathematicians regress’.

It is only when mathematics can be handled with the skillrequired to handle Level 5 confidently that creative workcan be done with it. It may even be that confidence at Level5 depends on confidence at Level 6. The point here is thatmathematical grasp at Level 4d might be positively mislead-ing when we get to understanding the meaning of a newresult. Just as the widespread meaning of experiment wasfor years taken from the experiments done at school andas an undergraduate—experiment is something that givesa clear and decisive answer—our image of the power ofmathematics may be misleadingly formed by repeatingmathematical derivations for which we already know theanswer—derivations which, therefore, always lead to a deci-sive answer. New mathematical derivations should, per-haps, be treated as being far less sure and certain than that.

The point, once more, is that to do mathematics reallywell is very hard. The lower levels are inadequate for check-ing others’ work or for doing creative work but mostly thatdoes not matter to physicists because work at the lessdemanding levels is quite good enough for what they haveto do most of the time.

9. Conclusion: how much mathematics do we need in anundergraduate physical science degree?

Physics, which we take to stand for all physical sciences,is mathematical through and through. Physics as we know

25 They need what has elsewhere been called ‘interactional expertise’ (see Collithis volume).26 At Cornell this discussion gave rise to a heated debate about the skills requi

contemporary mathematical science, can manage to do the job by developingwithout mastering the mathematics. The historian of the past may have nothingIt may be, then, that the typical historian of past physical sciences will need mothe historian to give a faithful account of his or her subjects’ scientific lives is

it could not go on unless a good proportion of physicistswere highly accomplished mathematicians. These mathe-maticians are needed, to write new mathematics-basedpapers, to develop new mathematical understandings ofthe subject, to referee controversial mathematical papers,and to check claims in detail when those claims imply ashift of resources into new areas. It is likely that thosewho have great mathematical ability will always find theirplace among the most prestigious physicists. None of thisis disputed here. What is claimed is that the discovery thata high level of mathematics is not used much of the time bymost physicists and none of the time by a few physicistsopens the way to a new understanding of what it meansto be a physicist. Existing undergraduate educational pro-grammes imply that to be a physicist is to be a mathemati-cian. Here the attempt has been made to show that thismight not be so. Rather, the matter should be thought ofin terms of the division of labour. There is a role withinthe physical sciences, even at the research front of the phys-ical sciences, for bona fide physicists who are not good atmathematics. There is a role because they can be bona fide

physicists at the research front without needing to do muchin the way of mathematics. Indeed, do away with all thepoor mathematicians and you would do away with someof our greatest physicists.

When we come to all the uses of the physical sciencesaway from the research front the point is still more undeni-able. Those who work with other physical scientists outsidetheir narrow specialism, those who manage physics, thosewho write about physics, those who make decisions in sci-ence-based industries, do not need mathematical under-standing to any depth in the science they are working with,managing, describing, or analysing. What they need is anunderstanding of the broad conceptual sweep and coherenceof the physical specialisms in question.25 Even more doesthis provide a justification for a route through a physical sci-ence higher education that is not driven by mathematics.26

Ironically, it is the very difficulty and time-consumingnature of doing mathematical proofs in detail that meansthat not much in the way of mathematical proofs will bedone in the day-to-day life of physicists or those with phys-ics-related roles. And there is no need for much of it to bedone because when it is necessary there are specialists whowill be able to do it. Not having that kind of specialist abil-ity does not put anyone at a detectable disadvantage in anyphysics-related roles except the roles normally taken byspecialists. Let us begin to develop a stream of physical sci-ence education that can exploit the wasted talent.

ns & Evans, 2007; Collins, Evans, & KES, 2007; and many of the papers in

red of a historian of science. It is possible that a sociologist, or historian ofinteractional expertise through immersion in the life of the respondentsbut journals to go on and these are expressed in mathematical formalisms.re mathematics than the physicists themselves! Whether or not this allowsan interesting question.


Acknowledgements

I am grateful to Robert Crease for his many perceptivecomments on this paper and for all the useful questions atthe places it has been presented, particularly the CardiffKES group, Cornell University Science and Technology Stud-ies seminar, and the four anonymised physics departmentswhere the survey was carried out. The physicists were particu-larly generous with their wise and sympathetic comments.

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mathematical understanding and the physical sciences

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