Explaining, Seeing, and Understanding in Thought Experiments

James Robert Brown Department of Philosophy University of Toronto

In this paper, I analyze the relation between visualization and explanatory understanding in thought experiments. I discuss two thought experiments from special relativity. Both cases lead to a paradox, which can, however, be explained (away) as non-contradictory. The resolution in one of the two cases sheds light on the nature of thought experiments more generally by indicating what sort of thing it is that we see when we are observing events in the mind’s own laboratory. And it will in turn shed light on the notions of explanation and understanding as they arise in thought experiments. Theories often run into paradoxes. Some of these are outright contradictions, sending the would-be champions of the theory back to the drawing board. Others are paradoxical in the sense of being bizarre and unexpected. The latter are sometimes mistakenly thought to be instances of the former. That is, they are thought to be more than merely weird; they are mistakenly thought to be self-refuting. Showing that they are not self- contradictory but merely a surprise is often a challenge. Notions of explanation and understanding are often at issue. For instance, we might explain—or explain away—a paradox by invoking some mechanism pro- vided by the theory and showing how it does not really lead to a logically incoherent result. This is what I’m going to do here. I want to look at two paradoxes, both from special relativity. Neither are paradoxes in the sense of outright contradictions. But showing that they are merely paradoxes in the sense of being unexpected and bizarre is something of a challenge. In both cases the paradox arises in a thought I am grateful to the organizers and participants of the Leiden conference for a stimulating discussion and valuable comments on the talk that gave rise to this paper. I also thank Henk de Regt and an anonymous reader for providing several valuable suggestions for im- proving this paper. Finally, I am grateful to SSHRC for ongoing ªnancial support.

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experiment, which complicates the issue considerably, but also provides a valuable opportunity. The resolution in one of the two cases will, I claim, shed light on the nature of thought experiments more generally by indicating what sort of thing it is that we see when we are observing events in the mind’s own laboratory. And it will in turn shed light on the notions of explanation and understanding as they arise in thought experiments.

Special Relativity Einstein was perhaps the greatest thought experimenter ever. Galileo was arguably his equal, but none is his superior. Much of his theorizing was motivated by thought experiments. This was certainly true of both special and general relativity. To begin, let’s review some of the basics of special relativity, starting from the two standard postulates. And as we do so, let us keep in mind that special and general relativity contain several thought experiments, some of which are central to the creation of new principles and laws, while others play a more pedagogical role. Postulate 1: Laws of nature are the same in every inertial frame. Postulate 2: The speed of light is constant; it has the same value, c, in every frame.

There is probably no need to comment on these postulates, since special relativity is widely known, but we may need to be reminded of the deªni- tion of simultaneity.

Distant events e1 and e2 in frame F are simultaneous if and only if light from the two events meets at a spatial mid-point in F. Since the second postulate says in effect that light travels at a constant speed c, this deªnition should strike us a perfectly correct. We are in for a surprise, however, when we try to mix this concept of simultaneity with the ªrst postulate, the postulate that asserts that all inertial frames are equivalent. Consider a train running along a track. Suppose we have two frames, which we take to be attached to the train and to the track, respec- tively. We assume that the train is moving at velocity v in the track frame. We also assume that the observer in the track frame is midway between e1 and e2, which are separated events (light ºashes). We further assume that the observer receives signals from each event at the same time. Then by the earlier deªnition, events e1 and e2 are simultaneous in the track frame. Some time passes, however, while the light signals come to the mid- points—the train has moved forward. An observer midway on the train

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Figure 1. Events e1 and e2 are simultaneous in the track frame.

Figure 2. Event e2 is earlier than e1 in the train frame.

frame receives the signal from e2 before e1. So, the event e2 was earlier than e1 in the train frame. The upshot of this is simple but profound: the simultaneity of distant events is relative to a frame. (In case this seems too shocking, remember the equivalence of all frames and the fact that an event happens in every frame.) This is the ªrst of many profound discoveries in special relativity. From this conclusion and a bit of algebra we can derive the length contraction and time dilation formulae. (I will state them below but not derive them, since they can be found in any text on special relativity.) So far this is standard stuff and all part of the normal presentation of special relativity. Einstein developed the theory in the way I have pre- sented it (Einstein 1905) and most expositors follow him. The use of simple thought experiments to bring out the details of special relativity is clever but so far not uncommon or in any way exceptional. The philosophical lessons to be learned from these thought experiments will arise later.

Time Dilation and the Twin Paradox In Newtonian physics and in common sense, the time interval between two events is the same no matter what reference frame in which the interval is measured. Not so in special relativity. We’ll begin with the notion of proper time which applies to events happening at the same place in the same frame. First, we need a deªnition:

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The proper time is the temporal interval between those events that happen at the same spatial location in a given frame.

When a frame is in motion relative to another, time is dilated (stretched out, slowed down). The temporal interval between events is longer than the proper time. It is lengthened by a factor, known as the time dilation factor, which can be easily derived from the initial postulates of special relativity:

1 . 1 − vc22/

If a clock (which is at rest in frame F) registers an interval of time ⌬t, then a clock that is moving at velocity v will register an interval ⌬tЈϭ⌬t 1 − vc22/ , according to any observer in F. The limits should be noted: As the relative velocity goes to zero, the two clocks will coincide, that is, they will run at the same rate. And as the relative velocity increases to the speed of light, the moving clock will slow down and approach stopping when the limit, v ϭ c, is approached (though never reached). The important thing to take from this is that when v is not zero, then ⌬tЈϽ⌬t. The twin paradox grows out of this. Actually, there are two paradoxes. The ªrst is merely that sort of paradox that comes as a great surprise. Imagine a pair of twins, one stays at home while the other goes on a long trip by rocket at a very high speed, then returns home. Everything will be slower for the moving twin, all clocks on the rocket, all biological processes, and so on. When the travelling twin gets back, she will have aged, say, ªve years, while the stay-at- home twin will have aged most of a normal lifetime. This is a serious shock to common sense and a violation of pre- relativistic physics, but it seems to be correct. We may ªnd this stagger- ing, but we can and do get used to it. However, there might be a second paradox, the sort that we cannot live with, since it would be an outright contradiction. The second paradox—if it is indeed a paradox—arises from the following simple consideration. The principle of relativity (the ªrst postulate) demands the equivalence of all frames. That in turn suggests that the so- called travelling twin could be considered the stay-at-home and the stay- at-home could be considered the traveler. The same reasoning about moving things temporally running more slowly is now applied the other way; the twin who previously came back younger will this time be the older. Now we do seem to have an outright contradiction, because the theory

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seems to say that each twin is both older and younger than the other. The issue was hotly debated in the early years of special relativity. The paradox, however, is easily resolved. The alleged contradiction arises because of the assumed symmetry between the stay-at-home and the travelling twin. But this assumption is false. The one we called the travelling twin in the initial case occupied two distinct inertial frames, not a single frame as did the stay-at-home. That is a crucial difference between their two cases; they are simply not symmetric. The asymmetry is enough to block the paradox, but most people want a more detailed explanation. They rightly want to know what’s happening and how it ªts into the theory. Explaining a paradox or explaining it away isn’t quite the same as explaining a phenomenon, but it is related. We explain the paradox by showing how in appropriate detail to explain the phenomenon and do so in a non-contradictory way. I will give two explanations. Not only do they tell us something about the twin paradox, but they tell us something about explanation and understanding, too. The ªrst explanation is quite popular and one ªnds it in many texts on relativity. It points out that the travelling twin has to turn around. This requires an acceleration (e.g., the rocket engine is turned on) and that is what breaks the symmetry of the two cases, since the stay-at-home twin never accelerates. Sometimes it is added that since acceleration is involved, we have to move outside special relativity to general relativity to properly analyze the non-inertial motion of the travelling twin. The crucial aspect to the explanation is the invoking of a force, the source of the acceleration. As an explanation, it appeals to overarching principles such as the distinction between inertial and accelerated motion and to a mechanism, i.e., the rocket engine being ªred. It is the rocket engine ªring that provides the explanation for the asymmetry which in turn resolves the paradox. Unfortunately, this explanation is incorrect. This can be seen by slightly changing the problem. Instead of two twins, start with three clocks. One of these will be the stay-at-home clock. A second will come in from left inªnity at high velocity, then synchronize with the stay-at-home clock when they coincide, and then continue to move away inertially. A third clock coming in from right inªnity passes the second clock and syn- chronizes with it when they coincide, then continues to move inertially toward the stay-at-home clock. When the third reaches the stay-at-home clock, it will record much less time elapsed than the stay-at-home clock. The result is the same as the twin case and requires some explanation. But notice that there were no accelerations at any time in the overall process. This shows that the rocket engine (a force) played no role in explaining the time difference between the stay-at-home and the traveller in either the clock or the twin case.

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Then what does explain it? The answer is simply the geometry of spacetime. The paradox arises from purely kinematical considerations. Force and acceleration have nothing to do with it. A remarkable fact about Minkowski spacetime is the invariance of spacetime intervals. The “distance” between two spacetime points is the same regardless of the frame in which it is measured. Of course, this is no ordinary distance; Minkowski spacetime is very different from what we are used to. The interval between the two spacetime points is deªned in such a way that when the spatial component is small, then the temporal component will be large, so as to maintain the same value, regardless of the frame in which the interval is measured. Thus, the spatial change for the stay-at-home twin (or clock) is zero, since she is not moving. So the temporal component must be as large as possible (known as the proper time). The travelling twin (or clocks) will have a large spatial component, so the temporal component must be smaller to compensate. Thus, the clock reading is less than the stay-at- home clock. This is a geometric fact about Minkowski spacetime. As an explanation of the paradox, it uses overarching facts, namely, the facts about the Min- kowski geometry. As for providing a mechanism, this seems doubtful. If one considered spacetime a substance, a thing in its own right with causal powers, then it might provide a mechanism to explain the twin result, but we could not take it straightforwardly to be an efªcient cause. Spacetime or geometry do not seem like any other causal mechanism, e.g., atoms knocking one another about or gravity exerting a force. These latter examples are efªcient causes. Maybe the right way to think of the causal capac- ity of spacetime is in terms of a formal cause, or, perhaps, as a geometrical cause, which would be a close cousin to formal causation. Could either of those be a mechanism? Not as normally conceived, but I’ll come back brieºy to this question below.

Length Contraction and the Car-Garage Paradox High school algebra is all that is needed to understand special relativity. The new concepts of space and time—or rather, spacetime—present the real challenge, since we must radically reconceptualize our old way of thinking. From a pedagogical point of view, this is the hard part. One of the best ways to develop a feel for the theory is to present puzzles and paradoxes. One of the nicest of these is the car-garage example.1 But ªrst we need to say something about length contraction of moving objects. According to special relativity, a moving object will contract in the 1. I brieºy mentioned this example in Brown (2011). I develop it here in more detail and relate it to questions of explanation and understanding.

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Figure 3. The car and the garage in relative motion with velocity v.

direction of its motion. Suppose a meter stick has length L0 when at rest in an inertial frame. And suppose further that it is moving in some other frame (the rest frame) at velocity v. Then its length in the rest frame is:

=−22 LLv 0 1/ vc

This is the standard formula for length contraction, commonly known as Lorentz contraction. Notice—which you can readily see from the formula—that the faster an object moves in a given frame, the shorter it is. As it reaches the velocity of light, it contracts to zero length. Warn- ing: An object does not have a length, simpliciter. It has a length in a given frame and that length will vary from frame to frame, depending on the relative velocity. The meter stick, or any other object, exists in all frames, so has all velocities and all lengths between its maximum (the rest length) and arbitrarily short lengths when moving very close to (but never reaching) the speed of light. Now we can move to the car-garage puzzle, which is posed as a paradox: Can a moving car ªt inside a garage or not? It seems correct to say yes and it seems equally correct to say no. Let’s suppose that the car and the garage both have the same rest length, say, ªve meters, to be speciªc, and the car is moving toward the garage at a very high velocity v. The car will be Lorentz contracted, according to the garage frame’s point of view, because of its velocity. So it will be much shorter than ªve meters and thus will easily ªt inside the garage. Due to its high velocity, it will very soon crash through the rear wall of the garage, but for at least a brief time it will be wholly inside, which is all we want or need to establish. The equality of all inertial frames implies that the car may be considered the rest frame and the garage moving at velocity v within it. From the car’s point of view, the garage will be Lorentz contracted. This means that it will be much shorter than ªve meters, and consequently, the car will not ªt inside. This reasoning is just as correct as

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Figure 4. Top: In the garage frame the car is contracted, so it will ªt in the garage. Bottom: In the car frame the garage is contracted, so the car will not ªt in.

that in the previous paragraph. Now we have a paradox: the car will and the car will not ªt in the garage. It would seem that special relativity leads to an outright contradiction. The thought experiment only takes us this far, but note how very far this is. We could have calculated the result, but there was no need, since we can easily imagine qualitatively and visually how it goes and thus arrive at the paradoxical conclusion. The problem we now face is how to resolve the paradox? What went wrong with the reasoning? As a matter of fact, nothing is wrong with the reasoning in either case. In the garage frame, the car really does ªt in the garage and in the car frame the car really does not ªt in the garage. But we want to know how this could be true? The answer lies in the relativity of simultaneity, mentioned at the out- set. In the garage frame, the rear bumper of the car is in the garage before the front bumper has burst through the back wall of the garage. This means that it is indeed wholly inside—if only very brieºy. In the car frame the front bumper of the car breaks through the back wall of the garage before the rear bumper is inside the garage. The two frames (car and garage) disagree on the order of events. But both are right; it is an instance of the relativity of simultaneity. This is the resolution of the paradox: events that are simultaneous in one frame are not in another. It is indeed bizarre, but it is not a contradiction.

Appearance of Rotation In describing the car-garage example, I noted how easy it is to carry out as a thought experiment; no tricky computations were needed; it involves

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only simple, qualitative, pictorial reasoning. And yet, there is something wrong with this. There is a problem, but it is not with special relativity or with the setting up of the thought experiment, its analysis, and the solution of the paradox. Rather the problem is with thinking it is simple qualitative, pictorial reasoning, as if that is all we need consider. There is a complicating feature that will eventually lead us to a deeper understanding. As we saw above, rapidly moving objects are Lorentz contracted. But here is the surprise: they do not look contracted. In thought experiments we try to visualize things as realistically as we can. This grounds our intu- itions and improves the quality of our inferences when, for instance, we imagine Einstein chasing a light beam or Galileo dropping musket and cannon balls from the Leaning tower of Pisa. But being realistic won’t work in this case. The visual appearance of a rapidly moving object in special relativity is, surprisingly, not contracted. In fact, an object appears to be rotated, with the degree of rotation depending on the relative velocity. This was only discovered in the late 1950s and is still not widely known, even among physicists (see Penrose 1959; Terrell 1959; Weisskopf 1960). This means that in the garage frame, the car will, of course, be Lorentz contracted, but it would not look that way. Instead, it would look rotated. This means that we have a new and different kind of problem on our hands. In the earlier thought experiment, we could determine what was happening by simple visual inspection. In our imagination we could see the contracted car or the contracted garage and we reasoned from these simple images. But with rotation, our intuition would be confused. We would not see the paradox the way we should and we would have no hope of realizing how to solve it. The rotated car looks like it will crash side- ways into the garage; instead of ªtting directly inside. When I say “looks” I mean just that; this is what we would actually observe. (Well, perhaps not quite, but I’ll comment on that idealization below.)

The God’s-Eye View The metaphor of the god’s-eye-view is often associated with the idea of the one true description of reality. It’s tempting to say that in a thought experiment we see the god’s-eye-view. Realists (I include myself) should be willing to accept this. However, we are concerned with thought experiments here, so we must focus on the “view” part of the god’s-eye-view and take it rather seriously. Two other related metaphors also arise: the view from nowhere and the view from everywhere. These are frequently used in place of the god’s-eye view and convey the same idea: the one true, objective description of reality. Let’s consider each metaphor in turn, taking them

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Figure 5. The appearance of rotation of rapidly moving objects

rather literally, that is, taking “view” to mean an observer situated some- where in space. Rather than take the view from nowhere as no view at all, I will take it to treat the car-garage problem as one-dimensional. Why? One dimension is all that really matters to the physics of the situation; nothing is happening physically in the other two dimensions. Lorentz contraction is only in the direction of motion. In this one-dimensional setup there is no observer who is standing off to the side, which would require an additional dimension. So, in this one-dimensional version the viewer is located on the one dimension. Consequently, the Lorentz contraction of the car (or the garage) would not be seen at all. The only thing we could see is a point. The Lorentz contraction would have to be calculated, since it can’t be visualized.

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Figure 6. The rapidly moving car looks rotated in the garage frame

Rotation does not arise, since (as earlier mentioned), it would require a second dimension. We get the correct answer in this case (i.e., in the garage frame, the car ªts inside the garage, and in the car frame it does not). However, this version cannot be called a thought experiment, since we don’t make any observations that are crucial to the outcome. It differs from the standard thought experiment, where contraction would be actually visualized. Doing things this one-dimensional way is a departure from the standard way of presenting the car-garage paradox, the way that makes the paradox clear to anyone without needing to do a calculation. John Wheeler famously declared “Wheeler’s First Moral Principle: Never make a calculation until you know the answer.” (Taylor and Wheeler 1992: 20) Thought experiments often make this easy to do, but the view- from-nowhere interpretation (the one-dimensional account) would under- mine any hope of this. Next, let us try understanding the god’s-eye-view as the view from everywhere. This interpretation would treat the problem as two- dimensional. We could have more, but only two dimensions actually matter to the problem, since with two dimensions rotation would be visualized. However, since we normally (and rightly) ignore or deny rotation in the car-garage paradox, the view from everywhere cannot be the right way to capture the god’s-eye-view either. Some sort of idealization is involved, but what? We want something like a god’s-eye-view, but when we try to put ºesh on the bones, it comes to naught. What we need is something like this: There is a true description of what is going on, but rotation is not part of it. Though we rightly ignore rotation in the thought experiment, we are still not in a position to say why. We seem to have reached an impasse. So, let us try the time-honored technique of making distinctions.

Idealization within a Thought Experiment We normally distinguish theories from observations. But observation is not well understood. Of course, it is theory-laden, but that only begins to

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Figure 7. Data (left) and phenomena (right)

touch on the complexities. There is an intuitive distinction between the observable and the theoretical. It was the basis of a great deal of positivist philosophy of science. Streaks in cloud chambers are observable while electrons are not. One the one hand, physicists regularly talk about observing electrons, but on the other, they claim that quarks cannot be seen. This is because they cannot be separated within, say, a proton; any attempt to do so would require so much energy that the process would create new parti- cles and the quarks are still hidden. These issues are a long way from being settled. But a distinction has been introduced, which is growing in popularity and which sheds a lot of light on several related issues. It is the distinction between data and phenomena. Data are more or less raw sightings, while phenomena are idealizations of some sort that are constructed out of data, often under the inºuence of some theory. We talk about making observations within a thought experiment, so it should come as no surprise that issues concerning the nature of observation will be relevant. The distinction between data and phenomena is easily understood in the picture below (ªg. 7). Phenomena (an artist’s drawing, right) are constructed or abstracted out of data (a photo, left). Bogen and Woodward (1988) have developed this idea, as has James McAllister (1996, 2004). On this account, it is phenomena, not data, that theories explain and that are used to test those theories. No theory could hope to explain the chicken scratches on the left of ªg. 7. Here is a simple sugges- tion: Thought experiments involve phenomena. They certainly involve visualization, as do real experiments, but typically thought experiments

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are streamlined, cleaned up, and idealized in much the same way phenomena are. In the car-garage example we would see rotation, if we performed this experiment in reality. Actual rotation is part of the data. But in the standard presentation of the thought experiment we see only Lorentz contraction. Clearly, this is the phenomenon and it is quite different from the previous ways we interpreted the god’s-eye-view. But it may be a candi- date for the correct interpretation: God sees phenomena, not data. Phenomena are clearly linked to idealizations, but that in itself says lit- tle, since there are several ways in which idealization might take place. Specifying the nature of the idealization is important. One of the ways we might idealize is sometimes called Aristotelian. Such an idealization ignores some features, but will not ignore causal features that are thought to be truly at work. Aristotle would ignore the color of a falling body, but he would not ignore weight or air resistance. A second type of idealization is called Galilean. Here, one goes much further than Aristotle and actually falsiªes nature. Thus, Galileo would posit a vacuum, a frictionless plane, and so on. The difference, no doubt obvious, is very important. Aristotle could not allow himself to think about a body falling in a vacuum, since, by his lights, a vacuum is impossible. Ernan McMullin (1985) upholds Galilean idealizations for both real experiments and what he calls “subjunctive reasoning,” by which he clearly includes thought experiments. Galileo’s falling bodies thought experiment ªts this pattern nicely, since it not only ignores the color of the bodies, as an Aristotle idealization would do, but it ignores air resistance, as well. Let’s return now to the car-garage example? Does it involve idealizations in any interesting way? We make assumptions in the car-garage example that are outright dis- tortions; that is, they are as physically unrealistic as the assumption of a vacuum was for Galileo. A car moving at a velocity approaching the speed of light relative to a garage is highly fanciful. We could not actually see it; rather, it would at best be a mere blur. Could we actually discern whether it is at any time wholly within the garage, or is it going much too fast to tell? That we could probably do, but only with sophisticated measuring equipment. These problems seem, in principle, no different than ignoring air resistance. We stipulate what we want. There are limits to this; we can’t stipulate anything at all. But we would be going well past anything Aristotle might think acceptable. So far the car-garage example ªts the Galileo mold—with one crucial difference. The example differs in one important respect from the Galilean idealization framework. It does not merely falsify reality, the way Galileo would; it falsiªes appearances, as well. Lorentz contraction of the car is treated as objectively real, but rotation is not. This is a different type of

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idealization, one that applies only within thought experiments. I will call it Platonic idealization. Let’s work through the details. When we construct a thought experiment we do so by making it as close to a real experiment as we can. At this stage we could (and often do) idealize in the Aristotelian way. Thus, in the car-garage thought experiment, we ignore the color of the car. But when we ignore the appearance of rotation, we are doing something much more signiªcant. This is something similar to Galileo stipulating that there is no friction on an inclined plane or no air resistance that impedes a falling object, but it is not quite the same. We stipulate that in the thought experiment the car does not rotate; it only Lorentz contracts. We are not fal- sifying nature as Galileo does by stipulating no air friction; we are falsify- ing inside the thought experiment by stipulating no rotation. It is as if we said: “Visualize a situation in which a car is Lorentz contracted but is not rotated . . .” It is certainly possible to carry out this visualization, but it conºicts with the common view that thought experiments are imaginary versions of real experiments. Its unusualness, if not bizarreness, will be- come obvious when we consider it in detail. Galilean idealizations say, “Yes, there is air friction in reality, but we should ignore it and assume the object is moving in a vacuum. Now, having ignored it, how do things look?” At this stage we then draw the appropriate conclusion from what we see. Platonic idealization says something different: “Yes, the car will look rotated, but ignore it; assume that it does not rotate. Now, having ignored it, how does it look?” This borders on paradox, because we’re asking how things look when we ignore some aspects of how they look. However, it is not a contradiction. This sort of idealization should not be confused with exaggerations, which are commonplace in thought experiments. Einstein imagined running so fast he caught up with a light beam. This, of course, is biologically absurd, but there is no conºict with the thought experimental situation in thinking it. To imagine Lorentz contraction without the appearance of rotation is close to contradiction. Let’s be clear why. We see with light. It is the ªnite speed of light that leads to the Lorentz contraction and also to the appearance of rotation. It is not easily denied, the way biological facts about how fast we can run are easily denied in a physics thought experiment. And yet we do it. In some respects this is reminiscent of Aristotle’s position. Ignoring color, according to him, is harmless, but ignoring the atmosphere would be fatal. The upshot of ignoring the atmosphere is relative to the back- ground theory at issue. Galileo’s physics does not require an atmosphere, but Aristotle’s does, since in a vacuum an object would not know how to move. A rock or a ºame, says Aristotle, would not be able to detect its

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location, its natural place, and hence would not know which way to move to get to it. In some situations even Galileo could not ignore the atmosphere. For instance, he might note that air friction slows a moving body generally and thus conclude that air friction slows a cannon ball or a ºying bird. But without the air, the bird could not ºy at all. We don’t go quite that far, but we come close when we deny rotation, since we are implicitly assuming we can “see” by some means that does not involve light. This is really the key point. We see with the mind’s eye in a thought experiment. Photons are not needed for this sort of observation. As I earlier mentioned, the car-garage example is somewhat messy, since the example was discovered and solved long before the visual appearance of rotation was discovered. But this should not get in the way of a proper understanding. Now that we know about the appearance of rotation in rapidly moving objects, we must take it into account in relevant thought experiments. And the right way to take it into account is to stipulate that it does not occur. Then we visualize in the right way, that is, the way that rightly solves the problem. We see a contracted car (in the garage frame) and a contracted garage (in the car frame), and this gives rise to the initial paradox. We then solve it by appeal to the relativity of simultaneity. Empiricists would say that appearances give us reality, or that appearances are all the reality we can hope for. Scientiªc realists reject this and say that appearances are often misleading, even though they can and usu- ally do provide much of the evidence for our theories of reality. Typical thought experimenters might also say appearances give us reality, but it is the appearances within a thought experiment. They are experientially much richer than ordinary observations. A Platonist about thought experiments, that is, someone who champions Platonic idealization, would generally agree but deny that this is so in all cases. She would claim instead that some thought experimental appearances can be misleading and should be rejected. Roughly, a Platonist stands to other thought experimenters as a scientiªc realist stands to empiricists.

Explanation and Understanding I have so far been talking about explanation and understanding in rather general terms. I propose to go no deeper, except in one detail. There are two explanatory ideas, sometimes referred to as “top down” and “bottom up.” I have alluded to these already above, and I will elaborate now. Phi- losophers who stress uniªcation (for instance, Friedman 1974; Kitcher 1981, 1985) are in the former camp, while those who claim mechanisms are primary (for instance, Machamer, Darden, and Craver 2000) are in the

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latter. Wesley Salmon (1992) calls them the “two great traditions” in explanation. Before continuing, a word about understanding. I take explanation to be an objective feature of theories. Understanding is more subjective, a psychological feature of ourselves, at least in part. They often coincide, but not always. Electron spin can provide marvelous explanations of various quantum phenomena, yet spin deªes any sort of understanding or in- telligibility. As for thought experiments, explanation and understanding would seem to exactly coincide. A moment’s reºection makes this obvious. There is a joke going around about Galileo fudging the results of his thought experiments. It’s funny because it’s absurd. Real experimenters can lie about their results, but to understand a thought experimental result is to understand the thought experiment itself. The explanation in the thought experiment is inextricably linked to understanding. With this in mind, when I speak of explanation or of understanding, I should be taken to mean both. Rival explanations often exemplify the distinction between top down and bottom up. We saw two accounts of the twin paradox. One account explained the discrepancy in age as due to an acceleration of the travelling twin. This would count as a bottom-up or mechanistic explanation. The alternative, which was to appeal to the geometry of Minkowski spacetime, is a top-down explanation. I am inclined to call it a formal explanation (appealing to a formal cause), but this is of no serious consequence here, as long as we allow that it is not due to a physical, efªcient cause. Interestingly, Salmon claims that top-down and bottom-up explanations are often both correct, even in a single case.2 The following example is his: Imagine a helium-ªlled balloon inside an airplane that is about to take off. What will the balloon do? As the plane accelerates, will the balloon (relative to the plane) remain in place?, move forward?, or move back? Passengers all feel pushed back in their seats, so we might think the same must be so for the balloon; it, too, will move toward the back of the plane. Not so. There is a simple mechanistic (bottom-up) explanation: The air inside the plane will move to the back creating a differential pres- sure gradient that pushes the balloon forward. There is also a simple top- down explanation: Einstein’s principle of equivalence says that gravity and acceleration have the same effects. In a sea of air in a (vertical) gravita- tional ªeld, the balloon would rise, so when horizontal acceleration is sub- stituted for vertical acceleration, we get the same kind of motion as we

2. Salmon’s analysis is disputed by de Regt (2006). For my purposes the Salmon- de Regt debate can be ignored here. I need only assume that special relativity offers a top- down explanation. A related issue will come up below.

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would if gravity were present, that is, the balloon moves toward the front of the plane. Both explanations are correct in the balloon case, but, as we saw, this is not universally so. In the twin paradox there are top-down and bottom-up explanations, but only one is correct. I doubt there is any way to tell in ad- vance which sort of explanation is appropriate for some particular phenomenon. When initially faced with the twin paradox, it was not obvious how to solve it. One can only examine the rivals that come forward, de- clare a winner, then examine its top-down / bottom-up credentials. I strongly suspect, however, that when looking at explanations of phenomena within a thought experiment, we are much more likely to ªnd the top-down variety dominant. A slight detour will shed light on this. Einstein famously distinguished between two types of theories, principle and constructive. His distinction is not quite the same as top down and bottom up, but it is a certainly a close cousin. The latter type of theory, i.e., constructive, is any kind of hypothesis or conjecture that is put forward to explain a wide variety of facts. Notice that in Einstein’s account the explanation is mechanistic. [Constructive theories] attempt to build up a picture of the more complex phenomena out of the materials of a relatively simple formal scheme from which they start out. Thus the kinetic theory of gases seeks to reduce mechanical, thermal, and diffusional processes to movements of molecules—i.e., to build them up out of the hypothesis of molecular motion. When we say that we have succeeded in understanding a group of natural processes, we invariably mean that a constructive theory has been found which covers the processes in question. (Einstein 1919: 228) A principle theory, on the other hand, starts with something known to be true, such as that the speed of light in a vacuum is constant, and then forces everything else to conform to this principle. Unlike constructive theories which are speculative, explanatory, and attempt to unify diverse phenomena, principle theories, according to Einstein, never try to explain anything. The elements which form their basis and starting-point are not hy- pothetically constructed but empirically discovered ones, general characteristics of natural processes, principles which give rise to mathematically formulated criteria which the separate processes or the theoretical representations of them have to satisfy. Thus the science of thermodynamics seeks by analytical means to deduce neces- sary conditions, which separate events have to satisfy, from the

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universally experienced fact that perpetual motion is impossible. (Einstein 1919: 228) Einstein goes on to contrast these two types of theory and tells us which type relativity is. The advantages of the constructive theory are completeness, adapt- ability, and clearness, those of the principle theory are logical per- fection and security of the foundation....Thetheory of relativity belongs to the latter class. (Einstein 1919: 228) Einstein can’t be completely right in this distinction. There is no “security of foundation” for either special or general relativity. Both are seriously at odds with quantum mechanics and will very likely either give way completely or at least be seriously modiªed in the not too distant fu- ture. On the other hand, if we maintain a fallible version of principle theories, then we can see how they might loom large in thought experiments. The twin paradox and the car-garage examples are both within the framework of special relativity, a principle theory. Notice something interesting. When we do the car-garage thought experiment in its more realistic version, we see rotation. This is because we are introducing into the thought experiment the mechanistic causes of perception, namely, photons coming from the object. If we keep the actual mechanism of seeing out of consideration, we only see contraction, no rotation. In other words, the reason we see rotation in a rapidly moving object is in part due to the Lorentz contraction of the object and in part due to the light, which travels at a ªnite speed, coming from the back of the object.Lorentz contraction is clearly one of those grand unifying principles, as top down as they come. Having photons emitted from the rear of the car is just as clearly an instance of a bottom-up mechanism. This is how Einstein’s principle/constructive theory distinction ªts with the top- down/bottom-up explanation distinction. I have been assuming the standard way of interpreting special relativity. The central features of the theory can be derived a different way. Lorentz, for instance, understood the Lorentz contraction to be a physical effect, not a geometrical one. As a body moves through the ether, inter- molecular forces pull the body together, contracting it in the direction of motion. This is obviously a mechanistic or bottom-up explanation. Harvey Brown (2005) has recently advocated an approach to relativity that is in the spirit of Lorentz. Dennis Dieks (2009) won’t go that far, but insists that special relativity can be understood in either a top-down or a bottom-up fashion. Context and or practical purposes will decide which is appropriate. I won’t enter these debates—interesting and important

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though they are. I will, as indicated earlier, adopt the standard interpretation of special relativity and its top-down explanatory character. In the thought experiment we see the top-down explanation at work. Notice how remarkable this is. The main ingredients of top-down explanations are not the sorts of things we typically see. We may use Newton’s principle of universal gravitation, Einstein’s principle of equivalence, or Darwin’s natural selection to explain all sorts of phenomena, but we only see the phenomena, not the explanatory mechanisms themselves. To re- peat, in a real car-garage experiment, we would not see Lorentz contraction in its pure form. When we allow the normal mechanism of observation to be present, i.e., photons, we get the appearance of rotation. In some thought experiments it is not the mechanisms of efªcient causation and mechanical explanation that we see, but rather we peer into a more abstract realm, the home of formal causes and top-down explanations. This is the realm of phenomena (not data), which is why we can often learn so much from a brilliant single case.

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