Understanding Special Relativity through History and Triangles (pt. 1)

We interrupt this usually-DH blog because I got in a discussion about Special Relativity with a friend, and promised it was easily understood using only the math we use for triangles. But I’m a historian, so I can’t leave a good description alone without some background.

If you just want to learn how relativity works, skip ahead to the next post, Relativity Made Simple [Note! I haven’t written it yet, this is a two-part post. Stay-tuned for the next section]; if you hate science and don’t want to know how the universe functions, but love history, read only this post. If you have a month of time to kill, just skip this post entirely and read through my 122-item relativity bibliography on Zotero. Everyone else, disregard this paragraph.

An Oddly Selective History of Relativity

This is not a history of how Einstein came up with his Theory of Special Relativity as laid out in Zur Elektrodynamik bewegter Körper in 1905. It’s filled with big words like aberration and electrodynamics, and equations with occult symbols. We don’t need to know that stuff. This is a history of how others understood relativity. Eventually, you’re going to understand relativity, but first I’m going to tell you how other people, much smarter than you, did not.

There’s an infamous (potentially mythical) story about how difficult it is to understand relativity: Arthur Eddington, a prominent astronomer, was asked whether it was true that only three people in the world understood relativity. After pausing for a moment, Eddington replies “I’m trying to think who the third person is!” This was about General Relativity, but it was also a joke: good scientists know relativity isn’t incredibly difficult to grasp, and even early on, lots of people could claim to understand it.

Good historians, however, know that’s not the whole story. It turns out a lot of people who thought they understood Einstein’s conceptions of relativity actually did not, including those who agreed with him. This, in part, is that story.

Relativity Before Einstein

Einstein’s special theory of relativity relied on two assumptions: (1) you can’t ever tell whether you’re standing still or moving at a constant velocity (or, in physics-speak, the laws of physics in any inertial reference frame are indistinguishable from one another), and (2) light always looks like it’s moving at the same speed (in physics-speak, the speed of light is always constant no matter the velocity of the emitting body nor that of the observer’s inertial reference frame). Let’s trace these concepts back.

Our story begins in the 14th century. William of Occam, famous for his razor, claimed motion was merely the location of a body and its successive positions over time; motion itself was in the mind. Because position was simply defined in terms of the bodies that surround it, this meant motion was relative. Occam’s student, Buridan, pushed that claim forward, saying “If anyone is moved in a ship and imagines that he is at rest, then, should he see another ship which is truly at rest, it will appear to him that the other ship is moved.”

Galileo's relativity [via]. The site where this comes from is a little crazy, but the figure is still useful, so here it is.
Galileo’s relativity [via]. The site where this comes from is a little crazy, but the figure is still useful, so here it is.
The story movies forward at irregular speed (much like the speed of this blog, and the pacing of this post). Within a century, scholars introduced the concepts of an infinite universe without any center, nor any other ‘absolute’ location. Copernicus cleverly latched onto this relativistic thinking by showing that the math works just as well, if not better, when the Earth orbits the Sun, rather than vice versa. Galileo claimed there was no way, on the basis of mechanical experiments, to tell whether you were standing still or moving at a uniform speed.

For his part, Descartes disagreed, but did say that the only way one could discuss movement was relative to other objects. Christian Huygens takes Descartes a step forward, showing that there are no ‘privileged’ motions or speeds (that is, there is no intrinsic meaning of a universal ‘at rest’ – only ‘at rest’ relative to other bodies). Isaac Newton knew that it was impossible to measure something’s absolute velocity (rather than velocity relative to an observer), but still, like Descartes, supported the idea that there was an absolute space and absolute velocity – we just couldn’t measure it.

Lets skip ahead some centuries. The year is 1893; the U.S. Supreme Court declared the tomato was a vegetable, Gandhi campaigned against segregation in South Africa, and the U.S. railroad industry bubble had just popped, forcing the government to bail out AIG for $85 billion. Or something. Also, by this point, most scientists thought light traveled in waves. Given that in order for something to travel in a wave, something has to be waving, scientists posited there was this luminiferous ether that pervaded the universe, allowing light to travel between stars and candles and those fish with the crazy headlights. It makes perfect sense. In order for sound waves to travel, they need air to travel through; in order for light waves to travel, they need the ether.

Ernst Mach, A philosopher read by many contemporaries (including Einstein), said that Newton and Descartes were wrong: absolute space and absolute motion are meaningless. It’s all relative, and only relative motion has any meaning. It is both physically impossible to measure the an objects “real” velocity, and also philosophically nonsensical. The ether, however, was useful. According to Mach and others, we could still measure something kind of like absolute position and velocity by measuring things in relationship to that all-pervasive ether. Presumably, the ether was just sitting still, doing whatever ether does, so we could use its stillness as a reference point and measure how fast things were going relative to it.

Well, in theory. Earth is hurtling through space, orbiting the sun at about 70,000 miles per hour, right? And it’s spinning too, at about a thousand miles an hour. But the ether is staying still. And light, supposedly, always travels at the same speed through the ether no matter what. So in theory, light should look like it’s moving a bit faster if we’re moving toward its source, relative to the ether, and a bit slower, if we’re moving away from it, relative to the ether. It’s just like if you’re in a train hurdling toward a baseball pitcher at 100 mph, and the pitcher throws a ball at you, also at 100 mph, in a futile attempt to stop the train. To you, the baseball will look like it’s going twice as fast, because you’re moving toward it.

The earth moving in the ether. [via]
The earth moving through the ether. [via]
It turns out measuring the speed of light in relation to the ether was really difficult. A bunch of very clever people made a bunch of very clever instruments which really should have measured the speed of earth moving through the ether, based on small observed differences of the speed of light going in different directions, but the experiments always showed light moving at the same speed. Scientists figured this must mean the earth was actually exerting a pull on the ether in its vicinity, dragging it along with it as the earth hurtled through space, explaining why light seemed to be constant in both directions when measured on earth. They devised even cleverer experiments that would account for such an ether drag, but even those seemed to come up blank. Their instruments, it was decided, simply were not yet fine-tuned enough to measure such small variations in the speed of light.

Not so fast! shouted Lorentz, except he shouted it in Dutch. Lorentz used the new electromagnetic theory to suggest that the null results of the ether experiments were actually a result, not of the earth dragging the ether along behind it, but of physical objects compressing when they moved against the ether. The experiments weren’t showing any difference in the speed of light they sought because the measuring instruments themselves contracted to just the right length to perfectly offset the difference in the velocity of light, when measuring “into” the ether. The ether was literally squeezing the electrons in the meter stick together so it became a little shorter; short enough to inaccurately measure light’s speed. The set of equations used to describe this effect became known as Lorentz Transformations. One property of these transformations was that the physical contractions would, obviously, appear the same from any observer. No matter how fast you were going relative to your measuring device, if it were moving into the ether, you would see it contracting slightly to accommodate the measurement difference.

Not so fast! shouted Poincaré, except he shouted it in French. This property of transformations to always appear the same, relative to the ether, was actually a problem. Remember that 500 years of physics that said there is no way to mechanically determine your absolute speed or absolute location in space? Yeah, so did Poincaré. He said the only way you could measure velocity or location was matter-to-matter, not matter-to-ether, so the Lorentz transformations didn’t fly.

It’s worth taking a brief aside to talk about the underpinnings of the theories of both Lorentz and Poincaré. Their theories were based on experimental evidence, which is to say, they based their reasoning on contraction on apparent experimental evidence of said contraction, and they based their theories of relativity off of experimental evidence of motion being relative.

Einstein and Relativity

When Einstein hit the scene in 1905, he approached relativity a bit differently. Instead of trying to fit the apparent contraction of objects from the ether drift experiment to a particular theory, Einstein began with the assumption that light always appeared to move at the same rate, regardless of the relative velocity of the observer. The other assumption he began with was that there was no privileged frame of reference; no absolute space or velocity, only the movement of matter relative to other matter. I’ll work out the math later, but, unsurprisingly, it turned out that working out these assumptions led to exactly the same transformation equations as Lorentz came up with experimentally.

The math was the same. The difference was in the interpretation of the math. Einstein’s theory required no ether, but what’s more, it did not require any physical explanations at all. Because Einstein’s theory of special relativity rested on two postulates about measurement, the theory’s entire implications rested in its ability to affect how we measure or observe the universe. Thus, the interpretation of objects “contracting,” under Einstein’s theory, was that they were not contracting at all. Instead, objects merely appear as though they contract relative to the movement of the observer. Another result of these transformation equations is that, from the perspective of the observer, time appears to move slower or faster depending on the relative speed of what is being observed. Lorentz’s theory predicted the same time dilation effects, but he just chalked it up to a weird result of the math that didn’t actually manifest itself. In Einstein’s theory, however, weird temporal stretching effects were Actually What Was Going On.

To reiterate: the math of Lorentz, Einstein, and Poincaré were (at least at this early stage) essentially equivalent. The result was that no experimental result could favor one theory over another. The observational predictions between each theory were exactly the same.

Relativity’s Supporters in America

I’m focusing on America here because it’s rarely focused on in the historiography, and it’s about time someone did. If I were being scholarly and citing my sources, this might actually be a novel contribution to historiography. Oh well, BLOG! All my primary sources are in that Zotero library I linked to earlier.

In 1910, Daniel Comstock wrote a popular account of the relativity of Lorentz and Einstein, to some extent conflating the two. He suggested that if Einstein’s postulates could be experimentally verified, his special theory of relativity would be true. “If either of these postulates be proved false in the future, then the structure erected can not be true in is present form. The question is, therefore, an experimental one.” Comstock’s statement betrays a misunderstanding of Einstein’s theory, though, because, at the time of that writing, there was no experimental difference between the two theories.

Gilbert Lewis and Richard Tolman presented a paper at the 1908 American Physical Society in New York, where they describe themselves as fully behind Einstein over Lorentz. Oddly, they consider Einstein’s theory to be correct, as opposed to Lorentz’s, because his postulates were “established on a pretty firm basis of experimental fact.” Which, to reiterate, couldn’t possibly have been a difference between Lorentz and Einstein. Even more oddly still, they presented the theory not as one of physics or of measurement, but of psychology (a bit like 14th century Oresme). The two went on to separately write a few articles which supposedly experimentally confirmed the postulates of special relativity.

In fact, the few Americans who did seem to engage with the actual differences between Lorentz and Einstein did so primarily in critique. Louis More, a well-respected physicist from Cincinnati, labeled the difference as metaphysical and primarily useless. This American critique was fairly standard.

At the 1909 America Physical Society meeting in Boston, one physicist (Harold Wilson) claimed his experiments showed the difference between Einstein and Lorentz. One of the few American truly theoretical physicists, W.S. Franklin, was in attendance, and the lectures he saw inspired him to write a popular account of relativity in 1911; in it, he found no theoretical difference between Lorentz and Einstein. He tended to side theoretically with Einstein, but assumed Lorentz’s theory implied the same space and time dilation effects, which they did not.

Even this series of misunderstandings should be taken as shining examples in the context of an American approach to theoretical physics that was largely antagonistic, at times decrying theoretical differences entirely. At a symposium on Ether Theories at the 1911 APS, the presidential address by William Magie was largely about the uselessness of relativity because, according to him, physics should be a functional activity based in utility and experimentation. Joining Magie’s “side” in the debate were Michelson, Morley, and Arthur Gordon Webster, the co-founder of the America Physical Society. Of those at the meeting supporting relativity, Lewis was still convinced Einstein differed experimentally from Lorentz, and Franklin and Comstock each felt there was no substantive difference between the two. In 1912, Indiana University’s R.D. Carmichael stated Einstein’s postulates were “a direct generalization from experiment.” In short, the American’s were really focused on experiment.

Of Einstein’s theory, Louis More wrote in 1912:

Professor Einstein’s theory of Relativity [… is] proclaimed somewhat noisily to be the greatest revolution in scientific method since the time of Newton. That [it is] revolutionary there can be no doubt, in so far as [it] substitutes mathematical symbols as the basis of science and denies that any concrete experience underlies these symbols, thus replacing an objective by a subjective universe. The question remains whether this is a step forward or backward […] if there is here any revolution in thought, it is in reality a return to the scholastic methods of the Middle Ages.

More goes on to say how the “Anglo-Saxons” demand practical results, not the unfathomable theories of “the German mind.” Really, that quote about sums it up. By this point, the only Americans who even talked about relativity were the ones who trained in Germany.

I’ll end here, where most histories of the reception of relativity begin: the first Solvay Conference. It’s where this beautiful picture was taken.

First Solvay Conference. [via]
First Solvay Conference. [via]
To sum up: in the seven year’s following Einstein’s publication, the only Americans who agreed with Einstein were ones who didn’t quite understand him. You, however, will understand it much better, if you only read the next post [coming this week!].

Bridging Token and Type

There’s an oft-spoken and somewhat strawman tale of how the digital humanities is bridging C.P. Snow’s “Two Culture” divide, between the sciences and the humanities. This story is sometimes true (it’s fun putting together Ocean’s Eleven-esque teams comprising every discipline needed to get the job done) and sometimes false (plenty of people on either side still view the other with skepticism), but as a historian of science, I don’t find the divide all that interesting. As Snow’s title suggests, this divide is first and foremost cultural. There’s another overlapping divide, a bit more epistemological, methodological, and ontological, which I’ll explore here. It’s the nomothetic(type)/idiographic(token) divide, and I’ll argue here that not only are its barriers falling, but also that the distinction itself is becoming less relevant.

Nomothetic (Greek for “establishing general laws”-ish) and Idiographic (Greek for “pertaining to the individual thing”-ish) approaches to knowledge have often split the sciences and the humanities. I’ll offload the hard work onto Wikipedia:

Nomothetic is based on what Kant described as a tendency to generalize, and is typical for the natural sciences. It describes the effort to derive laws that explain objective phenomena in general.

Idiographic is based on what Kant described as a tendency to specify, and is typical for the humanities. It describes the effort to understand the meaning of contingent, unique, and often subjective phenomena.

These words are long and annoying to keep retyping, and so in the longstanding humanistic tradition of using new words for words which already exist, henceforth I shall refer to nomothetic as type and idiographic as token. 1 I use these because a lot of my digital humanities readers will be familiar with their use in text mining. If you counted the number of unique words in a text, you’d be be counting the number of types. If you counted the number of total words in a text, you’d be counting the number of tokens, because each token (word) is an individual instance of a type. You can think of a type as the platonic ideal of the word (notice the word typical?), floating out there in the ether, and every time it’s actually used, it’s one specific token of that general type.

The Token/Type Distinction
The Token/Type Distinction

Usually the natural and social sciences look for general principles or causal laws, of which the phenomena they observe are specific instances. A social scientist might note that every time a student buys a $500 textbook, they actively seek a publisher to punch, but when they purchase $20 textbooks, no such punching occurs. This leads to the discovery of a new law linking student violence with textbook prices. It’s worth noting that these laws can and often are nuanced and carefully crafted, with an awareness that they are neither wholly deterministic nor ironclad.

The humanities (or at least history, which I’m more familiar with) are more interested in what happened than in what tends to happen. Without a doubt there are general theories involved, just as in the social sciences there are specific instances, but the intent is most-often to flesh out details and create a particular internally consistent narrative. They look for tokens where the social scientists look for types. Another way to look at it is that the humanist wants to know what makes a thing unique, and the social scientist wants to know what makes a thing comparable.

It’s been noted these are fundamentally different goals. Indeed, how can you in the same research articulate the subjective contingency of an event while simultaneously using it to formulate some general law, applicable in all such cases? Rather than answer that question, it’s worth taking time to survey some recent research.

A recent digital humanities panel at MLA elicited responses by Ted Underwood and Haun Saussy, of which this post is in part itself a response. One of the papers at the panel, by Long and So, explored the extent to which haiku-esque poetry preceded what is commonly considered the beginning of haiku in America by about 20 years. They do this by teaching the computer the form of the haiku, and having it algorithmically explore earlier poetry looking for similarities. Saussy comments on this work:

[…] macroanalysis leads us to reconceive one of our founding distinctions, that between the individual work and the generality to which it belongs, the nation, context, period or movement. We differentiate ourselves from our social-science colleagues in that we are primarily interested in individual cases, not general trends. But given enough data, the individual appears as a correlation among multiple generalities.

One of the significant difficulties faced by digital humanists, and a driving force behind critics like Johanna Drucker, is the fundamental opposition between the traditional humanistic value of stressing subjectivity, uniqueness, and contingency, and the formal computational necessity of filling a database with hard decisions. A database, after all, requires you to make a series of binary choices in well-defined categories: is it or isn’t it an example of haiku? Is the author a man or a woman? Is there an author or isn’t there an author?

Underwood addresses this difficulty in his response:

Though we aspire to subtlety, in practice it’s hard to move from individual instances to groups without constructing something like the sovereign in the frontispiece for Hobbes’ Leviathan – a homogenous collection of instances composing a giant body with clear edges.

But he goes on to suggest that the initial constraint of the digital media may not be as difficult to overcome as it appears. Computers may even offer us a way to move beyond the categories we humanists use, like genre or period.

Aren’t computers all about “binary logic”? If I tell my computer that this poem both is and is not a haiku, won’t it probably start to sputter and emit smoke?

Well, maybe not. And actually I think this is a point that should be obvious but just happens to fall in a cultural blind spot right now. The whole point of quantification is to get beyond binary categories — to grapple with questions of degree that aren’t well-represented as yes-or-no questions. Classification algorithms, for instance, are actually very good at shades of gray; they can express predictions as degrees of probability and assign the same text different degrees of membership in as many overlapping categories as you like.

Here we begin to see how the questions asked of digital humanists (on the one side; computational social scientists are tackling these same problems) are forcing us to reconsider the divide between the general and the specific, as well as the meanings of categories and typologies we have traditionally taken for granted. However, this does not yet cut across the token/type divide: this has gotten us to the macro scale, but it does not address general principles or laws that might govern specific instances. Historical laws are a murky subject, prone to inducing fits of anti-deterministic rage. Complex Systems Science and the lessons we learn from Agent-Based Modeling, I think, offer us a way past that dilemma, but more on that later.

For now, let’s talk about influence. Or diffusion. Or intertextuality. 2 Matthew Jockers has been exploring these concepts, most recently in his book Macroanalysis. The undercurrent of his research (I think I’ve heard him call it his “dangerous idea”) is a thread of almost-determinism. It is the simple idea that an author’s environment influences her writing in profound and easy to measure ways. On its surface it seems fairly innocuous, but it’s tied into a decades-long argument about the role of choice, subjectivity, creativity, contingency, and determinism. One word that people have used to get around the debate is affordances, and it’s as good a word as any to invoke here. What Jockers has found is a set of environmental conditions which afford certain writing styles and subject matters to an author. It’s not that authors are predetermined to write certain things at certain times, but that a series of factors combine to make the conditions ripe for certain writing styles, genres, etc., and not for others. The history of science analog would be the idea that, had Einstein never existed, relativity and quantum physics would still have come about; perhaps not as quickly, and perhaps not from the same person or in the same form, but they were ideas whose time had come. The environment was primed for their eventual existence. 3

An example of shape affording certain actions by constraining possibilities and influencing people. [via]
An example of shape affording certain actions by constraining possibilities and influencing people. [via]
It is here we see the digital humanities battling with the token/type distinction, and finding that distinction less relevant to its self-identification. It is no longer a question of whether one can impose or generalize laws on specific instances, because the axes of interest have changed. More and more, especially under the influence of new macroanalytic methodologies, we find that the specific and the general contextualize and augment each other.

The computational social sciences are converging on a similar shift. Jon Kleinberg likes to compare some old work by Stanley Milgram 4, where he had people draw maps of cities from memory, with digital city reconstruction projects which attempt to bridge the subjective and objective experiences of cities. The result in both cases is an attempt at something new: not quite objective, not quite subjective, and not quite intersubjective. It is a representation of collective individual experiences which in its whole has meaning, but also can be used to contextualize the specific. That these types of observations can often lead to shockingly accurate predictive “laws” isn’t really the point; they’re accidental results of an attempt to understand unique and contingent experiences at a grand scale. 5

Manhattan. Dots represent where people have taken pictures; blue dots are by locals, red by tourists, and yellow unsure. [via Eric Fischer]
Manhattan. Dots represent where people have taken pictures; blue dots are by locals, red by tourists, and yellow are uncertain. [via Eric Fischer]
It is no surprise that the token/type divide is woven into the subjective/objective divide. However, as Daston and Galison have pointed out, objectivity is not an ahistorical category. 6 It has a history, is only positively defined in relation to subjectivity, and neither were particularly useful concepts before the 19th century.

I would argue, as well, that the nomothetic and idiographic divide is one which is outliving its historical usefulness. Work from both the digital humanities and the computational social sciences is converging to a point where the objective and the subjective can peaceably coexist, where contingent experiences can be placed alongside general predictive principles without any cognitive dissonance, under a framework that allows both deterministic and creative elements. It is not that purely nomothetic or purely idiographic research will no longer exist, but that they no longer represent a binary category which can usefully differentiate research agendas. We still have Snow’s primary cultural distinctions, of course, and a bevy of disciplinary differences, but it will be interesting to see where this shift in axes takes us.


  1. I am not the first to do this. Aviezer Tucker (2012) has a great chapter in The Oxford Handbook of Philosophy of Social Science, “Sciences of Historical Tokens and Theoretical Types: History and the Social Sciences” which introduces and historicizes the vocabulary nicely.
  2. Underwood’s post raises these points, as well.
  3. This has sometimes been referred to as environmental possibilism.
  4. Milgram, Stanley. 1976. “Pyschological Maps of Paris.” In Environmental Psychology: People and Their Physical Settings, edited by Proshansky, Ittelson, and Rivlin, 104–124. New York.

    ———. 1982. “Cities as Social Representations.” In Social Representations, edited by R. Farr and S. Moscovici, 289–309.

  5. If you’re interested in more thoughts on this subject specifically, I wrote a bit about it in relation to single-authorship in the humanities here
  6. Daston, Lorraine, and Peter Galison. 2007. Objectivity. New York, NY: Zone Books.

Are we bad social scientists?

There has been a recent slew of fantastic posts about critical theory and discourse in the digital humanities. To sum up: hacking, yacking, we need more of it, we already have enough of it thank you very much, just deal with the French names already, openness, data, Hope! The unabridged version is available for free at an Internet near you. At this point in the conversation, it seems the majority involved agree that the digital needs more humanity, the humans need more digital, and the two aren’t necessarily as distinct as they seem.

The conversation reminds me of a theme that came at the NEH Institute on Computer Simulations in the Humanities this past summer. At the beginning of the workshop, Tony Beavers introduced himself as a Bad Humanist. What is a bad humanist? We tossed the phrase out a lot during those three weeks — we even made ourselves a shirt — but there was never much real discussion of what that meant. We just had the general sense that we were all relatively bad humanists.

One participant was from “The Centre for Exact Humanities” (what is that everyone else is doing?) in Hyderabad, India; many participants had backgrounds in programming or mathematics or economics. All of our projects were heavily computational, some were economic or arguably positivist, and absolutely none of them felt like anything I’d ever read in a humanities journal. Are these sorts of computational humanistic projects Bad Humanities? Of course the question is absurd. These are not Bad Humanities projects, they’re simply new types of research. They are created by people with humanities training, who are studying things about humans and doing so in legitimate (if as-yet-untested) ways.

Stephen Crowley printed this wonderful t-shirt for the workshop participants.

Fast forward to this October at the bounceback for NEH’s Network Analysis in the Humanities summer institute. The same guy who called himself a bad humanist, Tony Beavers, brought up the question of whether we were just adopting old social science methods without bothering to become familiar with the theory behind the social science. As he put it, “are we just bad social scientists?” There is a real danger in adopting tools and methods developed outside of our field for our own uses, especially if we lack the training to know their limitations.

In my mind, however, both the ideas of a bad humanist (lacking the appropriate yack) or of a bad social scientist (lacking the appropriate hack) fundamentally miss the point. The discourse and theory discussion has touched on the changing notions of disciplinarity, as did I the other day. A lot of us are writing and working on projects that don’t fit well within traditional disciplinary structures; their subjects and methods draw liberally from history, linguistics, computer science, sociology, complexity theory, and whatever else seems necessary at the time.

As long as we remain aware of and well-grounded in whatever we’re drawing from, it doesn’t really matter what we call what we do — so long as it’s done well. People studying humans would do well not to ignore the last half-century of humanities research. People using, for example, network analysis should become very familiar with its theoretical and methodological limitations. By and large, though, the computational humanities projects I’ve come across are thoughtful, well-informed, and ultimately good research. Whether it actually is still good humanities, good social science, or good anything else doesn’t feel terribly relevant.