August 8, 2002
On Thought Experiments:
Is There More to the Argument?

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
Pittsburgh PA USA 15260
jdnorton@pitt.edu

Thought experiments in science are merely picturesque argumentation. I support this view in various ways, including the claim that it follows from the fact that thought experiments can err but can still be used reliably. The view is defended against alternatives proposed by my co-symposiasts.
1. Introduction
	A scientist—a Galileo, Newton, Darwin or Einstein—presents us with some vexing problem. We are perplexed. In a few words of simple prose, the scientist then conjures up an experiment, purely in thought. We follow, replicating its falling bodies or spinning buckets in our minds, and  our uncertainty evaporates. We know the resolution and somehow we sense that we knew it all along. That moment of realization is exquisite; and it is difficult to resist the sense that something of profound epistemic moment has just transpired.
	My purpose here is to give strength to those who want to resist. My view of thought experiments is quite deflationary. I claim that they are just ordinary argumentation, disguised in some vivid picturesque or narrative form. As a result, they can do nothing more epistemically that can ordinary argumentation. I don't doubt that this picturesque clothing gives them special rhetorical powers, but they are not my concern. More precisely, my concern here is what I label:
The epistemological problem of thought experiments in the sciences. 
Thought experiments are supposed to give us knowledge of the natural world. From where does this knowledge come?
Since I claim that thought experiments are merely picturesque arguments, my solution to the problem is that this knowledge comes from premises introduced explicitly or tacitly into the thought experiment. That knowledge is then transformed, usually tacitly, through deductive or inductive argumentation to give the final result.
	 In Section 3, I will review the case made for this deflationary view of thought experiments. I will elaboration a recent development of that case that depends on the notion that thought experiments can err but nonetheless can be used reliably. In the guise of the "reliability thesis," I will urge that this is only possible if thought experimentation is governed by a very generalized logic; and evolutionary considerations suggest that these are the logics familiar to us from the literature on induction and deduction. In preparation for the discussion of reliability, in Section 2, I illustrate how thought experiments can err with an example.
	In Section 4, I will compare my view with those of my co-symposiasts. James Brown and Tamar Szabó Gendler both urge that thought experiments have more powers epistemically then mere argumentation—they say there is more to the argument. In Section 4.1-4.3 I will argue that these extra powers, if they exist, would be epistemically irrelevant, since they cannot be used reliably. James McAllister urges that epistemic powers are only accorded to thought experiments when certain narrow, historically contingent presumptions are satisfied. In Section 4.4 I will explain that the mere fact that a science seeks laws is probably sufficient in my view for it to license thought experimentation; and that in turn assures us of the near universal admissibility of thought experiments in science.
2. An Erroneous Thought Experiment
	While there is general agreement that thought experiments can deliver useful results in science, they are not infallible. Thought experiments can and often do yield false results. This can be seen most clearly in the existence of pairs of thought experiments that give contradictory results. I have described some of these in my (Norton, forthcoming), where I call them thought experiment – anti thought experiment pairs. We can have thought experiments that show the world finite or infinite; that there is an Absolute Space, or not; that the geometry of a rotating disk is Euclidean, or not; and that an infinite, resting rotor still supplies lift, or not. Here is another example of error. It is a variant of a well known thought experiment discussed in Brown (1991, pp. 38-40) that treats the Lorentz contraction in the length of moving bodies in special relativity.
	Imagine a flat metal plate out of which we carefully cut a long rod that just fits nicely into the slot created, as in Figure 1.

Figure 1. Rod cut from plate leaves slot of same size.
We transport the slotted plate and rod to a region of space remote from gravitation, where they float at rest. We locate the rod above the plate and to one side. We then set the rod into uniform motion towards the slot and in such a way that the rod remains aligned with the slot and parallel to it. (See Figure 2) Viewing the process from the plate and slot, we ask whether the rod will still fit through the slot. According to special relativity, the rod shrinks in the direction of its motion. Since part of that motion lies along the slot, the rod will become shorter than the slot and will pass through easily, as shown in Figure 2. At one moment it will be fully enclosed by the slot.
	We now view exactly the same process from the rod. According to that view, the rod remains at rest, but the slot moves uniformly towards it. Aside from the rod and slot switching places, the two cases are exactly the same. At rest the rod and slot are two objects of identical shape and size; all we change is that we move the slot instead of the rod. So this time the slot has shrunk and is shorter than the rod. Therefore the rod cannot pass through the slot. 
	We can see from the symmetry of the two views that it cannot pass. In the slot view, there comes a moment at which the contracted rod is fully enclosed in the uncontracted slot. By symmetry, in the rod view there must come a moment at which the contracted slot would be fully enclosed within the uncontracted rod—or there would if the material of the plate surrounding the contracted slot could penetrate the material of the rod. The inevitable collision happens when the material of the plate attempts to penetrate that of the rod.

Figure 2. Rod and slot in uniform, oblique motion
	Something has clearly gone very wrong, for these are two views of the same process. Either the rod passes through the slot or its doesn't. We cannot have both. There is a common and natural response to thought experiments like this one in special relativity. There is only one dependence of length on motion that won't produce problems in a rod-slot thought experiment and others like it. It is that the length of a rod does not alter with its motion after all. In that one case, if the resting rod fits the slot exactly when both are at rest, then they will always fit, whether we view a moving rod approaching a resting slot or a moving slot approaching a resting rod. That is our outcome: the relativistic contraction is not real after all. Moving rods and slots don't really shrink.
	However one approaches it, this thought experiment manifests error. Indeed Einstein began to fight this attempt at eliminating the Lorentz contraction as early as 1911. (Einstein 1911) How might we characterize the error? We might think of the rod view and slot view as two related thought experiments. Since they disagree on the outcome, at least one must err. They are a thought experiment – anti thought experiment pair since they deliver contradictory outcomes. Or, if one believes that standard special relativity provides a consistent account of moving bodies, the contradictory outcome of the combined thought experiment cannot be right.
3. Why Arguments?
3.1 Two Parts of the Thesis Justified
	I believe that thought experiments in science are merely picturesque argumentation. Elsewhere (Norton, 1991, 1996, forthcoming) I have tried to make my case by dividing the claim into two parts. In brief:
	First I urge that thought experiments in science can always be reconstructed as arguments based on explicit or tacit assumptions that yield the same outcome. One basis for this is empiricism. If thought experiments are to teach us about the world, empiricism tells us that they can only do so by drawing on our experience of the world. The result of a thought experiment must be the reformulation of that experience by a process that preserves truth or its probability, that is, by deductive or inductive argumentation. A second, independent basis is that I have found no thought experiment in science that cannot be reconstructed as an argument. Examples include those offered in the literature as resistant to such reconstruction. (See Norton, 1996.) 
	Second, I urge that the actual conduct of a thought experiment consists of the execution of an argument. Of course it is very tempting to forgo this second part of the thesis and hold some hybrid view that thought experiments exploit a mysterious mental power whose products can be validated separately by argumentation. That weaker view seems to me a loss of nerve. It amounts to a failure to accept the real significance of a remarkable fact: the epistemic reach of a thought experiment coincides exactly with that of an argument. Imagine an analogous case. We consult an oracle, but discover that the true predictions of the oracle coincide with what could be learned from ordinary experience. The oracle intones "Beware. The fire comes!" But after the recent tremors and lava flows, didn't everyone expect that the volcano was about to erupt?! The dreary truth is that this oracle has no special powers and merely infers predictions from ordinary experience, whether consciously or unconsciously. I propose the same verdict for thought experiments. Thought experiments in science have the same epistemic reach as arguments simply because they are arguments.
3.2 The Reliability Thesis
	In Norton (forthcoming) I seek to support my view that thought experiments are arguments by advancing:
Reliability thesis: If thought experiments can be used reliably epistemically, then they must be arguments (construed very broadly) that justify their outcomes or reconstructible as such arguments.
As I will explain below, the thesis draws on a notion of argumentation much broader than the one usually invoked in logic texts.
	In brief, the thesis proceeds from the fact that thought experiments can yield erroneous results. Nonetheless we do think that we can have confidence in the product of a thought experiment if the thought experiment is properly formulated. But how are we to know that the thought experiment is properly formulated? There must be some sort of mark of truth. The mark is not something external to the thought experiment. It may be comforting to know that, say, Einstein authored the thought experiment. But in principle it should stand alone. We should be able to know that the thought experiment is properly formulated merely by reading its text. So the mark must be something that we can recognize in the thought experiment itself. It cannot merely be that the thought experiment is found in an approved list; that would be an external mark. There must be some systematization—some common feature, some identifiable form, some common structure—in admissible thought experiments whose presence can be perceived by us. That form or structure need not fix all the details of the thought experiment. Indeed it cannot. Newton's bucket thought experiment is clearly unaffected by the choice of a wooden or a leather bucket. So a good thought experiment has two parts: a structure or form that assures its correctness and a portion we can alter without affecting that structure.
	That is, a good thought experiment is built from a template into which we are free to insert particular material of our choosing. A familiar example of the use of such templates is in logic: the schemas of a logic are the templates and the sentences or terms inserted into them the variable content. Thus I will say that a generalized logic is what governs any exposition that uses templates with variable content. The range of templates conceivable is enormous. Those employed in thought experiments, however, must be narrower. They must be such that they manifestly support the epistemic aspirations of the thought experiment. The schemas of logic satisfy this requirement in so far as they preserve truth or its probability. Perhaps also we should expect the templates of thought experiments to be sufficiently simple to allow their use to be tractable.
	What might the generalized logic of a good thought experiment look like if it is not a familiar logic? One possibility comes from the use of cognitive science to analyze thought experiments. (See Nersessian, 1992; Palmieri, forthcoming.) One portrays thought experiments as the manipulation of a mental model. At the center of the mental models are what amount to templates. For example, a template that encodes the spatial arrangement of three bodies A, B, C and D  in a square would be
A          B
C          D
Outcomes of cognitive activity that employs this template is warranted as long as this template correctly encodes properties of space.
	As indicated in Norton (forthcoming) I do not think that mental modeling can accommodate all thought experiments. Some, most notably in the physical sciences, depend upon explicit derivations of mathematical results within a physical theory. Those derivations are unequivocally arguments. I have also yet to see a thought experiment in science (as opposed to other instances of cognition) that cannot be reconstructed as an argument. So I persist in believing that thought experiments are arguments that exploit the familiar deductive and inductive logics.
	In part my confidence in the latter derives from a confidence in the ingenuity of logicians both deductive and inductive, whose profession seeks to extract and codify the schemas used in successful argumentation. Since the activity of thought experimentation has been prominent and important in science for a long time, my view is simply a vote of confidence that the logicians have succeeded in extracting the logics they employ. So I do not maintain that thought experiments are arguments because of some belief in the self discipline of thought experimenters to restrict themselves to a narrow canon of logic. Rather I have confidence that logic evolves to embrace any new, good argument forms that may emerge from the creative efforts of thought experimenters.
4. Epistemologies of Thought Experiments Compared
	My co-symposiasts offer different epistemologies of thought experiments. Two are more optimistic that mine over the epistemic powers of thought experiments; and a third more pessimistic. Before turning to the third, I will use the rod and slot thought experiment as test case to show why I do not share the optimism of the first two. I expect that a viable epistemology can give some account of why this thought experiment errs and how thought experiments can still be used reliably if such errors are possible. I will show my argument view can do this but the more optimistic views cannot.
4.1 The Argument View
	The error of the rod and slot thought experiment of the thought experiment is readily diagnosed and mended by looking at it as an argument. We simply find that it harbors a false assumption. Once that assumption is exposed and corrected, the contradiction with special relativity evaporates. The false assumption is manifested most vividly in Figure 2. We presumed that the rod and slot were parallel when seen in the slot view. The analysis tacitly but erroneously assumes that they remain parallel when we adopt the rod view. In the new view, they do not remain parallel. The change of view is effected by a Lorentz transformation and this transformation, it turns out, rotates bodies that are oriented obliquely to the direction of motion, such as the rod and the slot. In the corrected rod view, as shown in Figure 3, the rotated rod and slot are no longer parallel and the rotations allow the slot to pass the rod, even though the rod is longer than the slot. With the correction, both views now give the same result; the rod passes. (See Appendix for a simple account of how the Lorentz transformation leads to the rotation of oblique bodies.)

Figure 3. Corrected rod view
The argument for the original thought experiment  can be summarized as:
1. A rod moving towards the slot as in Figure 2 (slot view) is contracted and passes through. (Assumption from special relativity)
2. The process viewed from the rod is the same, but with rod and slot switched. (Symmetry assumption)
3. Under the symmetry of 2., if the rod passes in the slot view, it cannot pass in the rod view. (Established in text of thought experiment.)
4. The slot does not pass the rod. (From 1., 2. and 3.)
5. Contradiction. The rod passes (from 1.) and does not pass (from 4.).
A second argument reveals what is to be discarded to eliminate the contradiction.
6. There is a modified kinematics that gives consistent results for both views. (Assumption)
7. No dependence of length on motion is the one case which gives consistent results for both slot and rod views. (Assumption)
8. There is no length contraction with motion. (From 6., 7.)
What we now see is that there is no symmetry between the two views. If the rod and slot are parallel in one view, they will not be in the other. And this difference is the essential difference that allows both views to yield the same result. That is, to eliminate the contradiction, we should have discarded the Symmetry Assumption 2., not Lorentz contraction. The argument fails because of a false assumption, 2.
4.2 Brown's Platonism
	Elsewhere (Norton, 1993, 1996, forthcoming) I have described both my admiration for the boldness of James Brown's (1991, 1992, 1993, forthcoming) Platonic account of certain thought experiments and my misgivings about it. The latter are based in my own skepticism about the reality of a Platonic world of laws that might be accessed by thought experiments or by any other means. Certainly I do not believe that a viable epistemology of thought experiment requires it. The more immediate problem is how thought experiments can be used reliably if they are supposed to be the glimpsing of this Platonic world.
	Brown holds that only certain, specially favored thought experiments give us direct access to the Platonic world. These are what he calls "Platonic thought experiments." (1991, pp.43-45) They are distinctive in being simultaneously destructive and constructive; they destroy one view and simultaneously establish another. The rod and slot thought experiment is just such a case. It destroys the kinematics of special relativity, in so far as it shows the Lorentz contraction to be untenable. At the same time, it establishes its replacement, that there is no motion dependent contraction of bodies. Yet it errs. Unbeknown to us, our Platonic vision was clouded. So how are we to know which are the good Platonic thought experiments?
	Brown's response (1991, pp. 65-66; 1993) is to suggest that ordinary vision can sometime err as well without our wanting to dismiss all visual experience as illusion and that we can retain some confidence in visual experience even with only a rudimentary understanding of its mechanism. My response (Norton, forthcoming) is that we do not even have the most rudimentary account of the nature of Platonic perception. We might try to correct its products by noting that the outcome of the rod and slot thought experiment contradicts special relativity. But how can we be so sure that this isn't the thought experiment that seals the fate of special relativity after all? My answer is simple. We recognize that thought experiments are merely arguments and this particular one harbors a false assumption. Brown may want to point out the same false assumption to explain the failure of Platonic vision. But in doing this he is simply replicating the argument based analysis just given. Nothing in the diagnosis depends on thought experiments being anything more than arguments.
4.3 Szabó Gendler's Constructivism
	Tamar Szabó Gendler (1998) has elaborated a view of thought experiments that derives from Kuhn (1964). The principal notion is that a function of a thought experiments is to reveal lacunae in conceptual systems and to indicate how they are to be altered. I am very largely in agreement with her analysis. That function of thought experiments is both interesting and important. Where we differ, however, is in her insistence that their serving of this function shows thought experiments cannot be arguments. I am far more optimistic about the power of argumentation to reconfigure conceptual schemes. The argument of Russell's paradox led to the overthrow of the naïve conception of sets. The relative consistency proofs of the 19th century placed Euclidean and non-Euclidean geometry on equal footings as far as consistency is concerned. That eradicated millennia of belief that there is something logically defective about a geometry that violates Euclid's postulates. Or Gödel's proof of the incompleteness of arithmetic showed us that a concept of truth as derivability is inadequate. This is not a bad showing for mere argumentation.
	Szabó Gendler seems to allow that mere argumentation can reveal a contradiction. But she suggests that it cannot tell us what to adjust in our conceptual scheme in order to eliminate the contradiction. Here again I agree. The decision of what is refuted by a reductio ad absurdum seems to depend largely on the way the argument is set up and the predilections and larger interests of the scientist. However I have yet to see that thought experiments do any better in this task. The context of the thought experiment may powerfully suggest the target for the reductio and it may just be completely wrong. Such is precisely the case in the rod and slot thought experiment. We generate a contradiction, so we know we are mistaken somewhere. The setting of the thought experiment almost irresistibly draws us to the wrong target, the Lorentz contraction. The real culprit, the symmetry assumption, escapes because it is so natural even though it rests on well disguised assumptions about absolute simultaneity that are inadmissible in special relativity.
	If thought experiments have some extra power to reveal the right culprit when a contradiction emerges, what is the epistemic basis of this power? Szabó Gendler has mentioned Mach's idea of tacit knowledge assimilated from experience and talks of a "constructive participation" by the reader (pp. 414-15). How are we to know when the power of constructive participation is exercised well? How are we to distinguish it from those cases like the rod and slot in which the thought  experiment proves to be massively misleading in this aspect? In short, if thought experiments have such extra powers, how can they be exercised reliably?
4.4 McAllister and the Evidential Inertness of Thought Experiments in Science
	While Brown and Szabó Gendler are considerably more optimistic that I on the epistemic power of thought experiments, my co-symposiast James McAllister (1996) has developed an unwarranted pessimism. He observes correctly that thought experiments can only have persuasive power if they play by the right epistemic rules. The ritualistic reading of entrails, for example, does not play by these rules and thus has no epistemic power in science. However he thinks the epistemic license for thought experiments is narrow and time dependent. His sustained example is a "Galilean doctrine of phenomena" which licenses thought experimentation on accident free phenomena whose behavior is idealized as sufficiently regular to be subject to laws. The familiar example is bodies falling freely without the accidental complications of air resistance. Aristotelians concern themselves with real occurrences with all their accidents in place and thus, on McAllister's account, eschew Galileo's thought experiments on falling bodies.
	All I believe McAllister can show is that Aristotelians eschew particular thought experiments that may depend upon a notion they find illicit. He does not show they eschew thought experimentation in general. And he cannot. Thought experiments appear throughout Aristotle's corpus. Some are of cosmic moment and do even deal with the fall of bodies. Aristotle (On the Heavens, Bk. II, Ch. 14, 297a13-30) imagines the earth in formation and recounts how the motion of bodies to the center would yield a spherically shaped earth. Indeed, in a thought experiment of breathtaking ambition, Aristotle continues (297a33-297b13 ) to imagine that a weight many times that of the earth is added to one half of the earth's sphere and then traces out the resulting motions. 
	McAllister's concentration on the Galilean doctrine of phenomena as the grounding of thought experiment is entirely too narrow. Worse his insistence (1996, p. 242) that the evidential power of thought experiments is conferred "by the persuasive effort of scientists" confuses matters by conflating sociological facts about scientists with what is licit in their theories or epistemologies. McAllister alludes to "some alternative rationale" (p.248). I believe that they are available copiously. Thought experiments are just picturesque argumentation of a hypothetical or counterfactual nature. Essentially all that is needed is that the science admit hypothetical or counterfactual reasoning for it to admit thought experimentation. It is by no means assured that a field of inquiry will admit such reasoning. In many historical circles, counterfactual histories (such as I write in Norton, forthcoming (a)) are simply dismissed as illicit; historical scholarship does not support counterfactuals, we are told. However it is hard to imagine a science that does not admit such reasoning. All it must do is seek laws or even just projectible regularities; these support hypotheticals or counterfactuals; and these in turn support thought experiments. That doesn't mean the scientist must actually conduct thought experiments. They just might not be useful. But if the science supports counterfactuals, they are admissible. For example, the classification systems of botany sustain counterfactuals and thus admit thought experiments, although there seem to be few. Linnaeus, founder of modern botany, remarks (Smith, 1786, p.58) that the modern diversity of species of vegetables results from hybridization not from changes of soil, "else the plants would return to their original form, if removed again to their original situation." What of a very impoverished view that sees science merely as cataloguing occurrent fact and never going beyond it? Might that view preclude thought experimentation? Curiously Mach's own positivism came closest to this view, but even he did not develop it to the point that thought experimentation was precluded. His careful reflections on the topic established the modern literature in thought experiments.
Appendix: Why Does a Lorentz Transformation Rotate Oblique Bodies?
	The result may seem arcane, but is actually quite familiar once you see it. Special relativity tells us that bodies set in motion contract in the direction of motion; and that they expand in this direction when brought to rest. Apply this to a square with motion along its diagonal. We recover the changes of Figure 4.

Figure 4. Effect of the Lorentz transformation
Consider the effect of the transformation on the side OX of the square. It is rotated about O to the new position OX'. The mixes of rotation and length change of Figure 4 are the very ones that arise in the rod and slot thought experiment. To see this, imagine that the rod and the slot are each the sides of squares, as shown in Figure 5, and that the view is rotated in space so that the relative motion is horizontal. The figure shows a section through the rod and slot. The shaded elements are the slot view; the unshaded the rod view.

Figure 5. How the Lorentz transformation rotates rod and slot.

With a little patience, a complete qualitative analysis is recoverable from the figure. The ends of the rod are constrained to lie on the guide lines AA and BB as we switch between slot and rod view; similarly the ends of the slot are constrained to lie on guide lines CC and DD. It quickly follows that the rod passes through the slot in the slot view if and only if it passes through in the rod view.

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