Cognitive Science 32 (2008) 3–35 C 2008 Cognitive Science Society, Inc. All rights reserved. Copyright
ISSN: 0364-0213 print / 1551-6709 online DOI: 10.1080/03640210701801917

The Step to Rationality: The Efficacy of Thought Experiments in Science, Ethics, and Free Will Roger N. Shepard Department of Psychology, Stanford University, and The Arizona Senior Academy

Abstract Examples from Archimedes, Galileo, Newton, Einstein, and others suggest that fundamental laws of physics were—or, at least, could have been—discovered by experiments performed not in the physical world but only in the mind. Although problematic for a strict empiricist, the evolutionary emergence in humans of deeply internalized implicit knowledge of abstract principles of transformation and symmetry may have been crucial for humankind’s step to rationality—including the discovery of universal principles of mathematics, physics, ethics, and an account of free will that is compatible with determinism. Keywords: Thought experiments; Physical laws; Imagined transformations; Mental rotation; Symmetry; Rationality; Moral laws; The Golden Rule; Determinism; Agency; Free will

1. The problem Thought experiments are widely reported to have played a prominent role in the discoveries of physical laws. I begin by describing some specific thought experiments, similar to those that were—or, at least, may have been—carried out by Archimedes, Galileo, Newton, and Einstein and that appear to be sufficient to establish fundamental laws of physics, without carrying out any of these experiments physically. How is this possible? Where does such knowledge originate if not, as supposed by strict empiricists, from each individual’s own direct interactions with the physical world? The answer I propose grew out of my evolutionary perspective together with my cognitive psychological researches on mental transformations (Shepard & Cooper, 1982; Shepard & Metzler, 1971) and on generalization (Shepard, 1987; also see the related far-reaching developments subsequently achieved by Tenenbaum and others—e.g., Chater & Vitanyi, 2003; Feldman, 2000; Tenenbaum & Griffiths, 2001a, 2001b). Here, however, I focus primarily on Correspondence should be addressed to Roger N. Shepard, 13805 E. Langtry Lane, Tucson, AZ 85747. E-mail: [email protected]

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the role of mental transformations and an associated symmetry principle of invariance under transformation. I then venture to suggest how the same approach may be extendable, beyond the discovery of scientific laws, to shed light on the discovery of universal moral principles and to afford an account of free will that is compatible with determinism. I shall argue that the emergence in humans of the cognitive capabilities of discovering universal laws of science and principles of ethics, and of exercising free will are manifestation of a terrestrially unprecedented (if only partially achieved) “step to rationality.” A crucial component of all three of these cognitive capabilities is the ability to disengage from immediate self-interest and to imagine and to evaluate alternative events or actions with respect to explicitly represented criteria. I invite those engaged in cognitive modeling to think about how the kinds of abstract representational processes and symmetry principles I invoke here might be explicitly implemented in more concrete, detailed and even neuro-physiologically plausible ways.

2. Empirical science and mathematics Traditionally, a sharp distinction is maintained between the empirical sciences, on one hand, and mathematics and logic, on the other. Observations, measurements, and experiments on physical objects and phenomena are generally considered to be essential for the advancement of the empirical sciences. In contrast, mathematics is supposed to be concerned with what propositions are logically entailed by other propositions, regardless of whether these propositions correspond to anything in the physical world. Also, in physics what are taken to be the elementary or primitive objects are always provisional and subject to later reconceptualization—usually in terms of still more elemental entities. Thus, water, directly experienced as a continuous fluid, may be successively reconceived as composed of more fundamental entities: discrete molecules, then atoms, then electrons and protons, then quarks, and then perhaps modes of vibration in a convoluted high-dimensional manifold—with no definite end in sight. However, in mathematics the elementary or primitive objects are specified by the mathematician rather than by nature. Such objects as the integers of arithmetic or number theory or the points and lines of geometry are themselves completely transparent from the outset. Ordinarily, we do not suppose that a point or a line or that an integer such as 1, 2, or 3 (setting aside Frege’s and Russell’s abstract, set-theory of natural numbers) will be found to be composed of some previously unsuspected more elemental components. Instead, the advances yet to be made in number theory or in geometry are expected to concern what relations among such elementary numbers or points and lines will be found to be entailed by whatever axioms we have formulated for number theory or geometry. Nevertheless, mathematics and physics are alike in that we aspire, in both cases, to a consistent theoretical system of basic assumptions and derivable implications. But whereas in physics such a system is valued to the extent that it provides the simplest explanation or prediction of the widest range of what we observe or measure in the physical world, in pure mathematics the system may be valued for its own simplicity, elegance, symmetry, or beauty— independently of the extent to which it corresponds to what we observe or measure in the

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physical world. Nevertheless, results of pure mathematics that were originally valued for their intrinsic beauty are often later found to be most useful or even crucial for the development of physical theory. Notable examples include the non-Euclidean geometries of Minkowski and of Riemann for special and general relativity; the complex numbers, matrix algebra, and infinite-dimensional Hilbert spaces for quantum theory; and the theory of groups for both relativity and quantum mechanics. Moreover, the explicit formalizations of the laws of physics and of mathematics are, alike, creations of the human mind. Absent a comprehending mind, such equations would not exist and, even if by chance they did, they would exist only as meaningless configurations of physical matter. How do we humans come to formulate, to comprehend, and to evaluate the formalizations of mathematical and physical laws? When we try to do this for ourselves, we may find that the cognitive processes in the cases of physics and mathematics are often more similar than the traditional distinction between mathematical and the empirical sciences would suggest. To illustrate, I now present simple examples, first, from mathematics (specifically, geometry) and, then, from physics (primarily, mechanics).

3. Pythagoras’s theorem for right triangles The Pythagorean theorem relates the length of the hypotenuse (c) of a right triangle to the lengths of its two other sides (a & b) by the equation (a2 + b2 = c2 ). It is surely a theorem of mathematics. It may not hold exactly in the physical world. Long before Einstein, it clearly did not hold for large triangles on the spherical surface of the earth. (The equilateral spherical triangle formed by traveling due south from the North Pole to the equator then due east one quarter of the way around the equator and then due north back to the North Pole is in clear violation the Pythagorean theorem. Each pair of the three equal legs of this “triangular” journey forms a right angle.) Now, according to Einstein’s general theory of relativity, the Pythagorean theorem does not exactly hold more generally for triangles in three-dimensional physical space, more or less curved as that space is in the vicinity of massive bodies. Nevertheless, according to pure geometry, it does precisely hold for triangles in a flat plane. How can we convince ourselves of this latter, mathematical fact? There have been numerous intuitive demonstrations of the validity of the Pythagorean theorem. Fig. 1 illustrates one attributed to Pythagoras himself. A right triangle (with its 2 sides of arbitrary lengths a and b and hypotenuse of corresponding length c, as shown in A) is imagined to be replicated three times, yielding the identical copies (labeled “1,” “2,” and “3”) each rigidly rotated and/or flipped to fit within a square with sides of length a + b (as shown in B). One can immediately see that the two empty portions of that square (displayed as shaded in C) are both square and that one necessarily has area a2 and the other necessarily has area b2 . Now the four identical triangles are imagined to be rigidly translated and rotated within the same square to fit into the four corners of the square as illustrated in D. With this rearrangement of the four identical triangles within the same square, the remaining, empty portion of the square (again shaded, as shown in D) is immediately seen to be a tipped quadrilateral whose sides are each of length c (the length of the hypotenuse of the original triangle). Because the whole configuration (in D) is obviously invariant under 90◦ rotations, all angles of the

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Fig. 1. A proof of the Pythagorean theorem.

tipped quadrilateral must be identical and, hence, this quadrilateral is a square. Because its sides are of length c, the area of this square is c2 . Whatever area within the original larger square containing the four triangles is not occupied by those triangles must be invariant under non-overlapping rearrangements of those triangles within that larger square. Hence the sum of the areas of the two square shaded areas in C (i.e., a2 + b2 ) must equal the area of the single shaded larger rotated square in D (i.e., c2 ), Q.E.D. Fig. 2 illustrates another, quite different and in some ways simpler demonstration. (First brought to my attention by Douglas Hofstadter, personal communication.) As indicated in the figure, the Pythagorean theorem (a2 + b2 = c2 ) is equivalent to the statement that the sum of the areas of the squares constructed on the two shorter sides of the triangle equals the area of the larger, tilted square constructed on the hypotenuse. Again, the truth of the theorem can be confirmed by imagining a few simple operations. First, imagine a straight line constructed orthogonal to the hypotenuse and passing through the opposite vertex (as indicated in the figure by the dotted segment that thus divides the original triangle into the two areas labeled “A” and “B”). Second, imagine each constructed square (of area a2 , b2 , or c2 ) and its adjacent triangle (of area A, B, or A + B) rigidly rotated (and also flipped in the case of the largest square-plus-triangle) as a rigid unit to yield the three house-shaped objects lined up across the bottom of the figure. Third, from a consideration of the complementary angles of the original triangle, confirm that the three triangles at the tops of the three squares (lined up below that), of areas A, B, and A + B, are necessarily identical in shape. Finally, from this identity of shape, reach the conclusion for the proportional areas of the squares that a2 + b2 = c2 (QED). I suggest that the operations imagined in these two proofs of the Pythagorean theorem are of essentially the kind that my students and I investigated in our studies of imagined

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Fig. 2. An alternative proof of the Pythagorean theorem.

transformations, including “mental rotation” and apparent motion (Shepard & Cooper, 1982; Shepard & Metzler, 1971). Also crucial, here, is the realization that shape and size are invariant under these rigid transformations of translation, rotation, and horizontal flipping (which is just a rotation in depth). I have suggested, too, that the visual salience of symmetries arises from the implicit recognition that an object is invariant under these transformations of reflection, translation, rotation, or, more generally, screw displacement in three-dimensional space (as in the 4 illustrative examples displayed in Fig. 3). Such invariance under transformation also applies to the case of the mere exchange or permutation of identical objects, which will figure prominently in some of the ensuing examples (both from physics and from meta-ethics).

4. Archimedes’s law of the lever Unlike the theorem of Pythagoras, which belongs to pure mathematics, Archimedes law of the lever appears to be a law of physics. It can be stated as follows: Physical objects placed along a beam resting on a central fulcrum will balance if and only if the sum of the products of the weights and their distances from the fulcrum is equal for the objects on the left and for the objects on the right of the fulcrum. Archimedes may have verified this law by placing actual physical objects on actual physical balance beams. But he need not have done so. He may very well have seen that this law must hold by thought experiments and the principle of symmetry, as I illustrate in A through E of Fig. 4.

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Fig. 3. Symmetry as invariance under transformation.

If he had imagined the situation shown in A, Archimedes would immediately have seen by symmetry that the four identical weights must balance. Because the weights are identical, any permutation of them must leave the situation unchanged, with no reason for one side or the other to tilt down. Now imagine that without altering the weight of any part of the beam itself, it is modified to have a secondary fulcrum as shown in B. If the four identical weights are now placed as illustrated in C, everything will still balance as it did in A. Moreover, this remains true for any placement of the two weights on the secondary Beam 2 that is symmetrical around that secondary fulcrum. For any such symmetrical placement, the combined weight of those two objects is communicated to the primary beam, as before, at the location of that secondary fulcrum. Each of the cases exhibited (C, D, & E) thus satisfies Archimedes’s condition. As shown by the equation over each case, the negative sum of the products on the left of the fulcrum cancels the positive sum of the products on the right. More generally, beginning with any distribution of any number of weights along a beam, we can use the same symmetry principle to confirm the following: First, we note (from the preceding) that whether Archimedes’s condition holds or not (i.e., whether the beam balances or not), the fact of its balance or imbalance is preserved under any transformation of moving any two weights together at the midpoint between them. Second, as illustrated for a particular, arbitrary distribution of identical weights in Fig. 5, iteration of such transformations converges

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Fig. 4. Thought experiment leading to Archimedes’s law of the lever.

toward the situation in which all the weights are located at Archimedes’s centroid of the original distribution. At every stage of this process, the centroid will be at the fulcrum if and only if Archimedes’s condition holds and, hence, the beam balances—as it surely will if at the end all the weights are stacked directly over the fulcrum (as at the bottom of the figure). Thus, are we able to verify Archimedes law, without ever placing any actual weights on any actual beam. What appeared to be an empirical fact about the physical world turns out to be entailed by an abstract, mathematical principle of invariance under transformation—or, equivalently, of symmetry. We are able to verify the truth of this law of physics by thought

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Fig. 5. Confirmation of Archimedes’s law for an arbitrary distribution of weights.

alone, much as we did in the case of Pythagoras’s mathematical law concerning right triangles. In both cases the laws are universal under the conditions specified. The law of Pythagoras is universal, as we saw, only for right triangles in a flat plane. Likewise, the law of Archimedes is universal only for weights distributed on a beam that is rigid and balanced on the fulcrum prior to the weights being placed upon it.

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After choosing the mathematical example of Pythagoras and the physical example of Archimedes, I noticed a striking connection between them: Imagine that you wish to verify that three long physical rods of the same uniform diameter and rigid material would form a right triangle. But (a) the long narrow room in which you must do this is not wide enough to accommodate the triangle itself and (b) no ruler or measuring tape is available. I leave it as an exercise for the reader to use Archimedes’s law to verify the following statement: If a long (unmarked) beam and fulcrum are available, and the two shorter bars are laid side-by-side on one side of the beam extending from the fulcrum and the longest bar is laid on the other side extending from the fulcrum (as illustrated in Fig. 6), then if a triangle were formed by the three bars, it would be a right triangle if and only if the bars on the beam achieve a balance.

5. Galileo’s law of falling bodies Perhaps the most thoroughly analyzed thought experiment of all time is the one Galileo used to refute the claim attributed to Aristotle that falling bodies drop with speeds proportional to their weights (see Gendler, 1998). Presumably, Aristotle did not reach an erroneous conclusion by performing an actual experiment. If he reached it from a thought experiment, it evidently was one to which he did not devote enough thought. Possibly by imagining the hefting of a light object in one hand and a heavy object in the other hand he was led to the hasty conclusion that if the objects were released, the greater downward force he felt from the heavier object would manifest itself as a faster descent on the release of that object. But such a conclusion ignores inertia—something that Artistotle might have realized before Galileo if he had also thought about the greater force needed to accelerate a more massive object to the same speed as a lighter object (e.g., if both objects were resting on the slippery surface of a frozen pond). Galileo’s thought experiment was conclusive. It yielded the correct conclusion: The speed of descent is independent of the weight of the object (to the extent that air resistance is negligible). This implies that the greater downward force that Aristotle imagined he would feel from the heavier object is in fact exactly the force needed to accelerate the more massive object to the very same speed. I shall describe a thought experiment that is slightly different from the one actually offered by Galileo but that is, I believe, equally conclusive and more revealing of the relevance of the principle of symmetry. I imagine Galileo imagining himself at the top of the leaning tower at Pisa with three identical bricks. By the symmetry principle of invariance under permutation of identical objects, if the three bricks were dropped together, they should reach the ground at the same time. As in the case of Archimedes’s identical weights equally distant on each side of the

Fig. 6. A correspondence between the laws of Archimedes and Pythagoras.

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fulcrum, there is no reason for one to descend more than another. Now suppose two of these three bricks are glued together to form a single, twice-as-heavy brick. Surely, the virtually weightless film of glue used to make the two bricks into one would not cause that now heavier brick to fall twice as rapidly as the separate, third brick. Once again, the correct conclusion is reached without having to perform an actual experiment.

6. Newton’s law of action and reaction Newton’s inspiration for his universal law of gravitation is sometimes attributed to a thought experiment in which he imagined throwing an apple (according to legend, one that dropped on him from a tree under which he was sitting in an orchard). Newton would have realized that if he threw the apple with greater and greater force, the apple would fall to ground at a greater and greater distance from him. Knowing that the earth is a sphere, he would have concluded that if he were able to throw the apple with sufficient force, it would fall not to earth but around the earth. He might then conjecture that the moon was hurling with such speed as to be similarly ever falling around the earth. I now consider, instead, a different thought experiment from which Newton might have arrived at his Third Law of Motion—the law that every action has an equal and opposite reaction. I focus on this law for two reasons: First, of his three laws of motion, this is the one that was most original with Newton. Second, this thought experiment derives in a particularly simple and transparent way from the principle of symmetry. I imagine Newton imagining himself arched over the water with his feet on the gunwale of his boat and his hands on the gunwale of another boat of the same size that he is endeavoring to push away from his boat. From the obvious symmetry of the situation, Newton would realize that there is no way that he can push the other boat away from his without equally pushing his boat away from the other. He might also go on to realize that if he were stranded in the middle of a lake with an oar-less boat loaded, say, with apples, he could propel himself back toward shore by hurling the apples, one by one, to the rear with great force (just as space vehicles now accelerate through empty space by ejecting molecules of the gaseous products of combustion rearward at extremely high velocity).

7. Einstein’s theory of relativity Einstein, like Galileo, was a master of the thought experiment. He arrived at special relativity through thought experiments about how the same events would be experienced by observers in different states of motion. Galileo had asserted that events on a ship would appear the same to shipboard observers whether the ship were at rest or under full sail. In either case, he said, an iron ball dropped from the top of the mast would appear to drop straight down relative to the ship, and not (as some had supposed) further back on the deck if the ship were moving swiftly forward. Einstein proposed that the same principle should continue to apply even if the velocity of the vessel were to approach the speed of light. Fig. 7 illustrates a relevant thought experiment

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Fig. 7. Einstein’s thought experiment leading to the invariance of the speed of light.

in which Galileo’s ship is replaced by a train, and Galileo’s dropping ball is replaced by light signals propagating between passengers. (As indicated, the vertical axis represents time and the horizontal axis represents location in space.) A passenger in the middle of the train, B, is imagined to flash a light, which propagates both backward and forward. If the train is stationary (as shown at the top), the light should reach equidistant rear and the forward passengers (A and C) at the same time, hence at space–time locations A and C . If, instead, the train were moving forward at an appreciable fraction of the speed of light (as illustrated at the bottom—where c is the velocity of light), the arrival of the signals at the equidistant rear and forward passengers could still be simultaneous—but only if the space–time coordinates are subjected to the Lorenz transformation indicated by the large curving arrows. Such a transformation achieves the result that what is observed within a moving carrier of any kind is the same regardless of the velocity of that carrier. It also achieves the remarkable results (a) that the measured velocity of light is invariant for all observers regardless of their relative motions and (b) that nothing can be moved at a velocity that exceeds the velocity of light. Einstein’s general theory of relativity brings us back to my earlier theme of falling bodies. Einstein imagines himself inside an elevator that is in free fall (e.g., with severed supporting cable). Floating weightlessly in the falling elevator (like an astronaut in a vehicle free-falling around the earth), Einstein would experience no gravitational force. Because no force is acting within the elevator, a beam of light passing through the elevator would surely traverse a straight line. In the absence of gravitational force within the elevator, the symmetry between up and down removes any reason for the light to curve either upward or downward.

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Next Einstein imagines that he is, instead, an external observer standing on a stationary platform. He would now feel the force of gravity holding him down on the platform while the elevator accelerates by, thus breaking the symmetry between up and down. Clearly, a light beam that traverses a straight line relative to the observer in the accelerating elevator must traverse a downward curving (locally, parabolic) path to the stationary observer. That observer will thus attribute both his own experience of pressing down on the platform and the downward bending of the path of the light to the gravitational attraction of the massive earth. (It was the empirical confirmation of Einstein’s claim that light is bent in passing a massive body— obtained during Eddington’s expedition to measure such a deflection of starlight passing near the sun during a solar eclipse—that first brought Einstein instant and lasting celebrity around the world.) In his general theory of relativity Einstein dispensed with the Newtonian idea that massive bodies attract each other by a gravitational force that somehow acts across empty Euclidean space (an idea that Newton found necessary, but also troubling). Instead, Einstein proposed that massive bodies warp the neighboring space–time continuum itself in such a way that all other bodies simply traverse the straightest possible (i.e., geodesic) paths in this warped fourdimensional manifold. A rough intuitive idea of the difference between the Newtonian and the Einsteinian theories of gravitation can be gained by considering simplified two-dimensional diagrams with (again) time and space respectively represented vertically and horizontally. Diagram A in Fig. 8 represents the Newtonian picture in which a planetoid (of very small mass) is attracted to a massive star by that star’s force of gravitation. In accordance with Newton’s inverse-square law, the force of attraction (represented by the horizontal component of each vector along the path of the planetoid) increases as the planetoid approaches the star, yielding its parabolic free-fall into the star, as depicted. This picture would in no way be changed if represented with an (“extrinsic”) cylindrical curvature as shown in B. Now, however, in accordance with the local stretching of the fabric of space–time by the presence of the massive star’s world line, this (extrinsically curved) space–time manifold can be considered to take on an “intrinsic” negative curvature as depicted in C. There is now no need to invoke a “force” of gravity. Rather, the world line of the planetoid simply takes the straightest possible form in the space–time continuum, warped as it is in the vicinity of the world line of the massive star. Einstein’s special and general theories of relativity thus provide elegant mathematical formulations of the symmetries of invariance under transformations between observers that are moving relative to each other—either uniformly as in the special theory or in acceleration as in the general theory. Yet Einstein developed these revolutionary theories neither by performing physical experiments nor by studying empirical data collected by others. He developed them by mentally carrying out his own thought experiments.

8. Empiricists versus rationalists on the efficacy of thought experiments Here, I have considered just a few readily understood examples illustrating the effectiveness of thought experiments and symmetry principles in the discovery of physical laws. Elsewhere, I have discussed other, somewhat more complex examples—concerning the discovery of laws

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Fig. 8. Newtonian versus Einsteinian conceptions of gravitation.

of electromagnetism and the propagation of light (by Maxwell); laws of thermodynamics, statistical mechanics, and entropy (by Carnot, Clausius, and Boltzmann); laws of quantum mechanics (by Planck, Einstein, Bohr, Schr¨odinger, Heisenberg, Feynman, and others); and even (going beyond physics), principles of evolutionary biology (by Darwin). (See Shepard, 2001, 2003, for a few of the examples that I presented more fully at Harvard in my 1994 William James Lectures, “Mind and World.”) Still, deeply entrenched empiricist presuppositions motivate many to suggest that so-called thought experiments may either be re-imaginings the scientist’s own actual interactions with the world or else be disguised versions of deductive arguments from already accepted premises.

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The first possibility is surely implausible in cases like that of Einstein, who never traveled at anything like the speed of light or made observations from inside or outside a free-falling elevator. And the claim that thought experiments must be reducible to deductive arguments has been effectively countered by Gendler (1998, 2004). The dismissive stances taken toward thought experiments by strict empiricists may be motivated by their implicit “blank-slate” conception of the human mind. But such a conception has long been rendered untenable (a) by empirical evidence from evolutionary, developmental, linguistic, cognitive and brain sciences, which has established beyond any doubt that individuals come into the world already adapted to some important features of the world in which they must make their ways; and (b) by machine learning theory, which has provided conclusive mathematical proofs that there can in fact be no effective learning in the absence of principles of learning and generalization appropriate to the kind of world in which that learning or generalization is to occur. (For some relevant references concerning machine learning theory, see, e.g., Shepard, 2001, p. 712.) Skepticism about the effectiveness of thought experiments may also stem from a mistaken notion that any such effectiveness requires that we be innately endowed with explicit knowledge of specific facts and laws of nature. My claim is, rather, that the “knowledge” that natural selection has provided is, first, only implicit; second, typically of the most abstract and invariant features of our world. Some of these features are ones that we consider to be physical in nature—such as that space is three-dimensional and locally Euclidean (hence affording just 6 degrees of freedom of rigid motion). Other features may be even more abstract, mathematical principles, such as those of invariance under transformation and symmetry (Shepard, 1994, 2001). The value that Einstein placed on symmetry is attested by the symmetries he discovered—including the symmetries between relatively moving observers, between gravitation and acceleration, between matter and energy, and also between electric and magnetic fields (which, for brevity, I have not considered here). The major developments in theoretical physics can be understood as the construction of ever more general and internally consistent theories each of which subsumes preceding, more restricted theories as special cases (as diagrammed in Fig. 9). If asked why a law in any one of the theories in this hierarchy takes the particular form that it does, we may be able to explain that its form is dictated by constraints at the next higher level. Thus, Newtonian gravitational force falls off as the distance between massive bodies raised specifically to the power −2.0 (rather than to some other power such as −1.6 or −2.3) because the power −2 arises as a geometrical necessity from Einstein’s replacement of the idea of a gravitational “force” acting across empty Euclidean space by the idea of bodies merely tracing geodesic paths in a warped four-dimensional space–time continuum. But, clearly, a theory that is currently at the very top of the hierarchy has no higher level theory to explain the form of its laws. A noteworthy fact about the currently most general and successful theories of the physical world, namely, general relativity and quantum mechanics, seems to be insufficiently appreciated by strict empiricists. Each of these theories possesses such elegance and tight internal constraints of consistency and symmetry that it is extremely difficult to find any minor changes that can be made to either theory without disrupting its whole structure. In retrospect, it almost seems that if we had only been smart enough, we could have seen that each theory could not be otherwise. Yet, despite their enormous successes, these two theories are known to be

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Fig. 9. Hierarchy of the theories of physics.

inconsistent with each other (as I have metaphorically depicted in my drawing reproduced in Fig. 10). Hence, the discovery of a still more general theory that consistently subsumes both of them as limiting cases would be a towering achievement comparable to the discovery of each of the two component theories themselves—without having collected a single additional empirical datum. If there is such a thing as a final “theory of everything” (or, at least, of everything physical) we may still wonder why its laws take the particular form that they do. Three possible answers occur to me. The first is that the form of these laws is simply an arbitrary brute fact; there is no reason why they are this way rather than some other. The second is that these laws form the only mathematically possible self-consistent set. And the third is that the universe in which we reside is only one of infinitely many, and that the entire ensemble includes universes with laws of every mathematically possible form. The first of these alternatives seems to be implicitly presupposed by strict empiricists and the second by pure rationalists. The third possibility falls somewhere between. For, in accordance with the anthropic cosmological principle, the laws governing our universe are then constrained—perhaps quite tightly—by the requirement that they make possible the evolution of intelligent life and, hence eventually, the emergence of theorists capable of reasoning from abstract principles such as that of symmetry. Incidentally, with regard to the most rationalistic, second alternative just mentioned, one might ask: If physical laws are ultimately determined by mathematical necessity, what be-

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Fig. 10. The incompatibility between general relativity and quantum mechanics.

comes of the traditional distinction between empirical science and mathematics? One possible answer is that it may be only in our present state of relative ignorance—both about mathematical necessity and about physical reality—that we feel the need for such a distinction. Possibly, if we were to gain a sufficiently deep understanding both of mathematics and of physics we might begin to see (with Leibniz & Spinoza) how the same necessity governs both. In the meantime, these two realms will retain their respectively more empirical versus purely rational characters—including (as noted at the outset) the incompletely known, provisional nature of the currently hypothesized elemental constituents of physical theory as opposed to the completely characterized, transparent constituents of mathematics. At the same time, I confess that, at least for me, the question raised here tends to favor the third, many-universes alternative (together with its anthropic cosmological principle) over the second, purely rationalist alternative.

9. The principle of least action Whatever theory is finally deemed to subsume both general relativity and quantum mechanics (and, hence, all the other lower level theories included in the hierarchy depicted in the earlier Fig. 9) will presumably include as one of its most fundamental principles, the principle of least action—a principle that is consistent both with general relativity (as established by the mathematician David Hilbert) and with quantum mechanics (as established by the theoretical physicist Richard Feynman). Toward the end of his life, von Helmholtz (1886) had already judged it “highly probable” that the least-action principle governs “all processes in nature,” and Planck (1925/1993), who introduced the cornerstone of quantum mechanics with his quantum-of-action, characterized that principle as occupying the “highest position among physical laws” (p. 80). The relevance of the principle of least action, here, is that symmetry

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plays a significant role in that principle as well as in the thought experiments that led to the formulations of that principle and of its early precursors. In physics, action has the units of a product of energy and time. The original formulators of the idea that every process in nature is such as to minimize this action were Maupertuis (1690–1759) and (perhaps not independently of Maupertuis) Leibniz (1707–1783). The first to give it a correct mathematical formulation was Euler (1786–1818). The earliest precursors of the principle of least action were formulated as principles just of least time. Fig. 11 illustrates the kind of thought experiment that Hero of Alexandria (c. 125 BCE), who proposed that to minimize time, an object strives to move over the shortest possible distance, may have used to establish the equal-angle law of the reflection of light. Consider a farmer who must carry an empty bucket from his house to a stream to fill it with water and then must carry the filled bucket to the animals in his barn, at some distance from his house. What is the farmer’s shortest and, hence, least-time (as well, in this case, as least-energy) path to complete this task? As illustrated at the top of the figure, if one imagines a “virtual barn” located across the stream by the symmetry of reflection around the proximal edge of the stream, it is clear that the shortest path would be a straight-line path from the location of the farmer to the location of the virtual barn. But, if this path is re-reflected back to the actual barn, the angles of approach to the stream and departure from it must be equal, as shown. Now, as illustrated at the bottom of the figure, the situation of viewing an object reflected in a mirror is quite analogous. Thus, the least-time path of reflected light makes equal angles with the surface of a mirror. The great (amateur) mathematician Pierre de Fermat (1601–1665) may have employed a similar thought experiment to arrive at the law of refraction: In passing through different media, light takes the path that minimizes its time of transit. Fig. 12 depicts an analogous situation: that of a lifeguard who, on sighting a drowning swimmer, strives to take the leasttime path to that swimmer. Here, the straight-line path (A) is not the quickest. Because the

Fig. 11. Thought experiment leading to Hero’s least-time law of reflection.

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Fig. 12. Thought experiment leading to Fermat’s least-time law of refraction.

lifeguard can run across the sand faster than swim through the water, the quickest path to the swimmer goes farther over the sand in order to travel less far through the more resistant water (Path B, which accordingly must bend at the water’s edge). The water is here analogous to a medium through which light travels more slowly. By supposing light waves to be slowed and thus compressed in passing into a denser medium, Fermat anticipated Snell’s law that the ratio of the sin of the angle of incidence to the sin of the angle of refraction equals the ratio of the index of refraction of the second medium to the index of refraction of the first. The more general least-action principle in which not just time but the product of time and energy are minimized, as proposed by Maupertuis and successively refined and generalized by the mathematicians Euler, Lagrange, and Hamilton, attained its culmination with Feynman’s path-integral formulation, which applies to all quantum, as well as classical, systems (Feynman & Hibbs, 1965). Here, the ideas of Galileo and Einstein that the laws of nature are independent of the location, orientation, and state of uniform motion (or in mathematical terminology, that these laws are “covariant” with respect to displacements in space and time) can be seen as examples of the principle of invariance under transformation—that is, of symmetry. In Feynman’s culminating generalization, the symmetry emerges as the “democracy of histories” in which either light or a material particle virtually propagates over all possible paths with the least-action path emerging through mutual cancellation of competing alternatives (see Feynman, Leighton, & Sands, 1963, chap. 26).

10. The cognitive capabilities that empower thought experiments and “the step to rationality” It falls to us, as cognitive scientists, to elucidate the mental capabilities that make possible the discovery of the laws of nature and, as evolutionary psychologists, to suggest how these capabilities may have arisen through natural selection and individual learning. At the outset I noted that the ability to imagine identity-preserving transformations may be crucial. I have proposed that neuronal mechanisms that originally evolved in the service of the perceptual representation of the external world may have evolved additional capabilities. These may have included, successively, capabilities (a) for perceptual completion when the sensory input is brief, degraded, or incomplete; (b) for anticipation of probable ensuing or accompanying events; and, finally, (c) for the autonomous simulation of possible transformations in the

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sensory absence of such transformations or their objects. Possession of this last capability would presumably confer significant benefit in planning and problem solving. The emergence of the ability to construct explicitly articulated consistent theories about this world, however, presumably required additional developments. The following three may have been critical for the successful employment of thought experiments and abstract principles such as that of symmetry for this purpose: (a) a motivation, arising from our evolutionary branching into what has been termed the “cognitive niche,” to understand and comprehend the world; (b) the concomitant emergence of a capacity for holding immediate self interest in abeyance and for contemplating an explicit set of alternatives objectively with respect to explicitly chosen abstract criteria; and, of course, (c) the emergence of language, enabling the explicit communication, preservation, and critical analysis of results. In considering the emergence of cognitive capabilities sufficient for scientific discovery, I have focused on emergence through evolution more than through individual learning. Similarly, I have focused on rationality more than on empiricism. This is not to suggest that learning and empirical observation play no significant roles. Rather, this is partly to counteract what has seemed to me a prevailing bias in the opposite directions, and partly to call attention to the logically prior roles of evolution and rationality. As I have mentioned, in the absence of innate principles of learning and generalization that are already attuned to our world, no learning will be effective. Likewise, without rational guidance, the mindless accumulation of raw data yields no scientific law or theory. In practice, the processes through which an individual learns about the world and through which science advances depend, alike, on empirical observation and experimentation as well as on rational thought. In both cases, the resulting system of knowledge is successful (i.e., robust, powerful, and cognitively manageable) to the extent that its constituent facts, beliefs, and principles (a) have become sufficiently entrenched through past success and (b) have been determined to be consistent with each other. As new information arises from whatever sources, the system of knowledge or belief evolves through a process (ideally, consistent with Bayesian principles of inference) of striving toward mutual consistency and preservation of the system’s most securely established, successful constituents. The two cases of scientific theory and individual belief differ in that the importance of the criteria of explicitness, generality, internal consistency, and quantitative precision is more uniformly embraced in the realm of science than in that of everyday individual belief or behavior. The preceding remarks may help me address two interrelated questions frequently raised concerning the alleged effectiveness of thought experiments in science: First, does not the apparent conclusiveness of a given thought experiment depend on the validity of some unstated assumptions? And second, how is one to decide whether the conclusion of a given thought experiment is indeed conclusive? Regarding this first question, I readily acknowledge that every thought experiment (in mathematics as well as in physics) rests on a number of assumptions—in addition to (or in support of) the symmetry of invariance under transformation—that may not have been explicitly stated. The proofs offered for the Pythagorean theorem, assumed that the lines are straight and lie in a flat Euclidean plane in which any component assemblage of lines can be rigidly translated and rotated as a unit, preserving straightness, lengths, angles, areas, and so forth. Likewise, the thought experiment establishing Archimedes’s law of the lever, assumed

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that the masses of the weights and the rigidity of the balance arm, are conserved over time and translations or rotations in three-dimensional Euclidean space; and the thought experiment establishing Galileo’s law of falling bodies, assumed that the bodies being released are all of the same material, which (again) is conserved over time and translation in space, and that gravitation (in the absence of air resistance, electro-magnetic force, etc.) acts uniformly on this conserved material regardless of how it may be oriented, reshaped, separated, or fastened together. The validity of any of these background assumptions may of course be reconsidered in the light of newly obtained evidence or newly detected inconsistencies. But, to the extent that each such assumption is more securely established or entrenched than the law discovered through the thought experiment, that law has augmented our scientific knowledge. The answer I propose to the second question is already implicit in my answer to the first. One may provisionally take the conclusion of a thought experiment to be sound to the extent that all of the stated or unstated assumptions—when rendered explicit—are judged acceptable. The thought experiment of imagining hefting a heavier and lighter body that I supposed might have led Aristotle to the wrong conclusion (that the heavier must fall faster than the lighter) rested on an unstated (and false!) assumption; that is, the (inertia-less) assumption that motion (strictly, now, the acceleration) of a body is proportional to the force (or, for Aristotle, the “impetus”) applied, independent of that body’s mass. Clearly, then, a thought experiment does have something in common with a deductive argument. But I do not regard the process of constructing a scientific theory as the deductive erection of a system from categorically posited axioms. Rather, I regard it as an evolving process in which one continually seeks to render explicit what has been implicit, and in which all parts of the evolving system are subject to re-evaluation of their validity and mutual consistency. The process ultimately rests on assumptions about the uniformity of nature that can never be validated with absolute certainty but that have become so entrenched through past success that we continue to build upon them with confidence. There is no rational alternative. By “the step to rationality,” as I have termed it (e.g., Shepard, 2001, p. 741), I refer to the terrestrially unique emergence in humankind of such capabilities as I have been describing here: (a) to stand back from immediate self interest and to imagine alternative actions or transformations and their consequences; (b) to appreciate the symmetries of invariance under such transformations; and thus (c) to develop explicitly formulated principles, laws, or theories about the world. Also crucial has been the associated advent of an unprecedented motivation to understand the world. The step to rationality evidently has not yet been uniformly developed or exercised in the human population. Even among scientists, thinkers such as Archimedes, Galileo, Newton, and Einstein remain exceptional. Of course, biological evolution is a gradual process, taking countless generations to manifest appreciable change, as the continuities of bodily structure and behavior along evolutionary lines attests. Yet, the step to rationality—however incomplete it still remains—has sprung virtually discontinuously on the evolutionary time scale, presumably from relatively subtle and inconspicuous changes in the human brain. The uncertain but certainly enormous potential consequences of this development for the future of our species and our world motivates me to include, here, a provisional sketch of how the ideas I have been proposing about the cognitive grounds of science might be extended to illuminate the cognitive grounds of ethics.

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11. The cognitive grounds of moral principles Philosophers have long used thought experiments to evaluate alternative principles for moral action. In his A Theory of Justice, Rawls (1971) put forward what might be considered (even if not by Rawls) a thought experiment. He posed a hypothetical situation in which those endeavoring to frame rules for the just governance of society are blocked by a “veil of ignorance” from knowing what role each framer will have in that society. Such framers would thus be prevented from favoring rules that would be of special benefit to themselves by virtue of their accidental, inherited circumstances—for example, of race, gender, socioeconomic class, physical appearance, or political power. In terms of the more abstract, mathematical concepts I have been using, what is sought here is just the symmetry of invariance under permutation of individuals. Such a symmetry principle is basic to democracy and entails the “Golden Rule,” which may well be the candidate moral principle that comes closest to eliciting universal verbal assent (Wattles, 1996)—if not behavioral compliance. Following Hume (1739/1896, p. 469), I acknowledge that one cannot derive “ought” from “is.” I also understand why most scientists and many philosophers remain skeptical about the possibility of a universal, objective basis for ethics. But I am not ready to accept the alternative of a wholly biological/cultural relativism. Those content with such relativism might still investigate how what a human being in fact does or says is determined by the accidents of that individual’s genetic inheritance and cultural milieu. But the results of such research could never justify a conviction that human beliefs concerning moral principles—even if universally evolved or learned—are inherently, ultimately, and absolutely valid, independent of our anthropocentric standpoint. The meta-ethical objective, here, is not to solve any particular ethical dilemmas (which, in typical concrete situation, would require careful consideration of many complex, specific circumstances). Instead, the objective is to seek an ultimate, nonanthropocentric justification for our moral intuitions that prove most deeply entrenched and secure as we strive for an overall self-consistent system of beliefs. What I now regard as the best and perhaps only hope for avoiding nihilistic ethical relativism is to achieve a fully conscious grasp of the implications of the symmetry principle of invariance under permutation (Shepard, 2001, pp. 744–748). Such an abstract principle may seem too airy to have any significant moral force. Nevertheless, I suggest that just such a principle, at a deeply entrenched if unconscious level, underlies some of our most primitive and powerful emotions such, in particular, as the desire for retribution against someone who has broken the symmetry between him- or herself and another person by the gratuitous infliction of grievous injury, suffering, or death. “An eye for an eye, a tooth for a tooth,” has, at least, the virtue of symmetry. More generally, I take, as the most elemental thought experiment underlying moral intuition, that of imagining oneself in the situation of another. It is from the resulting empathic understanding that I know, as well as I know anything, that I ought not to abuse, torture, maim, or murder an innocent person. Moreover, I know this independently of any consideration of likely consequences for my own future freedom and well-being. For to commit any such act against another person is to violate not just a symmetry between physical things but to violate the symmetry between experiencing, teleological agents. It is not merely unaesthetic, undemocratic, or even wrong. It is, I want to say, evil.

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In short, the abstract principle of symmetry, which is so fundamental to the universal principle of least action in physics, may be equally basic to a universal principle of best action in ethics. If so, one (a) who has gained a degree of rationality sufficient to moderate and to bring into to a more harmonious system the often inconsistent emotions (including jealousy, greed, hatred, or revenge) that may have contributed to the reproduction and survival of our distant, pre-rational ancestors; and (b) who directly experiences the value his or her own life, freedom, and well-being, may come to know—as well as one knows anything—the binding force of the essential idea of the Golden Rule. We are left, however, with two daunting issues concerning exactly how the Golden Rule should be formulated. One issue concerns the class of beings to which the Golden Rule is supposed to apply. Should inclusion of a candidate being depend on that being’s susceptibility to feelings of pleasure, pain, and suffering (as advocated by utilitarian philosophers, beginning with Bentham, 1789/1946)? Or should inclusion depend on that being’s rationality (as advocated by Kant, 1785/1964)? I submit that these two different but overlapping classes of beings must play different roles in the formulation (Shepard, 2001, pp. 744–748). The ones who should be held responsible for acting in accordance with the Golden Rule are surely the rational ones. The ones who should be treated (by those rational beings) in accordance with the Golden Rule are primarily the sentient ones. These include, in addition to the rational ones, the less-than-fully-rational infants and mentally handicapped, as well as non-human animals—most of whom are to some extent susceptible to experiencing pleasure and pain or suffering. In addition, however, and irrespective of their degree of sentience, some non-rational beings should be treated in accordance with (an appropriately extended version of) the Golden Rule—to the degree that they have the potential for becoming rational. The human fetus, although not yet rational (or perhaps even sentient), obviously has the potential of growing into a rational (as well as sentient) being. Indeed, any animals, even the most primitive bacteria, have the potential (e.g., in case all higher forms of terrestrial life were to be wiped out by a cometary impact) of evolving, in the fullness of time, into rational beings. The suggestion, here, is that rationality and rational knowledge are good in themselves. True, we have our own anthropocentric hopes and strivings, and a justifiable belief that knowledge can contribute to our welfare and survival as human individuals. But we have as yet no certain knowledge of the ultimate destiny of the cosmos or of consciousness or of the objective grounds of good and evil. But we also have no hope of gaining any such further knowledge without the aid of our rationality. If there is any objective good to be known, then, should not the step to rationality required for knowledge of that good be itself judged as good? It is, anyway, onto these admittedly tenuous grounds that I might be forced to take my stand by any skeptic who demands a justification for my belief that the needless destruction of life is inherently bad. In any case, the considerations just preceding these eschatological speculations, indicate that the Golden Rule requires formulation in a way that takes account of degrees of (and even potential for) both rationality and sentience. In some early versions of the Golden Rule, the “others” whom we were enjoined to treat as we wished ourselves to be treated were defined as our “neighbors” or our “countrymen”—implying that all humans are not created equal and that individuals of foreign origin or appearance, including slaves, are somehow deficient in rationality or even sentience. Kant and his rationalist followers, including Rawls (1971),

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Gewirth (1978), and perhaps Jefferson (at least as implied in his contribution to the Bill of Rights), have formulated principles of morality or justice as applying to all humans or (in Kant’s writings) to all rational beings. Such formulations implicitly presuppose a sharp and well-defined boundary between (a) all rational human adults; and (b) the non-rational human infants, brain-damaged, demented, or psychotic patients, or our pre-homo-sapient ancestors. Yet, although we may have some abstract conception of an ideal, fully rational being, the step to rationality has, as I have noted, been only partially achieved. Even among the most rational thinkers of which we have concrete knowledge there have been lapses in engagement with the ethical implications of symmetry and consistency. Newton, in surreptitiously using his high position in the British Royal Society to denigrate both Hooke’s already partially correct insight regarding gravitational attraction and Leibniz’s independent invention of the calculus, violated the Golden Rule. And Kant, in his selfish and unsympathetic response to his long-time manservant Lampe’s disclosure of his intention to marry, and in Kant’s unending disregard of repeated heart-wrenching pleas by his own brother to be permitted to visit Kant, violated his own (related) “categorical imperative”—formulated, alternatively, as (a) always to treat another person as an end and never as a means to an end, or (b) always to act on that maxim that one can at the same time will to be a universal law (Kant, 1785/1964). Nevertheless, a person who aspires to being rational and just must ever endeavor to treat another person as one would be treated. But, if the concept of symmetry is, as I suggest, deeply internalized in human beings why do humans so often violate the Golden Rule? Why are heinous acts of abuse, rape, torture, and murder so prevalent in human societies around the world? Part of my answer is already implied by my observation that the step to rationality is not fully or uniformly achieved even in the human race. I believe that the torturer’s or murderer’s non-compliance with the Golden Rule is symptomatic of incompletely achieved internal consistency (and the associated presence of “logic-tight compartments”). Such inconsistency is manifested, for example, by ones (a) who justify murdering a doctor who has performed an abortion by citing the biblical commandment “Thou shalt not kill”; (b) who insist on the sanctity of life while advocating capital punishment; (c) who torture captives on the grounds that they are a threat to our liberty; and (closest to home) even (d) those who, in the interest of scientific research, justify the experimental ablation of portions of the brain of a non-consenting primate on two mutually incompatible grounds: The first is that the use of a primate is needed because it has the closest biological and behavioral similarity to humans (who are of greatest interest to us but who have passed human rights laws to protect themselves from such abuse). The second is that it is permissible to use a primate in this way because such an animal, being categorically different from humans (in rationality? or in sentience?), has no such rights. I conclude that the formulation of moral principles based exclusively on sentience or feelings (as proposed by Bentham, 1789/1946) is too simplistic, and that the particular rational approach developed by Kant (1785/1964) is too rigidly categorical. A satisfactory formulation of moral principles, including specifically the Golden Rule, must take explicit account of degrees of sentience and rationality. In addition, it must take explicit account of more pragmatic, reallife issues of degrees of biological or social relationship, connection, or proximity between individuals. For brevity, here, I merely mention a few simple examples of things that I suppose most people would regard as morally justified and that I believe can be rationally reconciled

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with moral principles such as the Golden Rule and even Kant’s categorical imperative—but only if those principles are significantly reformulated: 1. Giving priority to providing for one’s own young child or other loved one over providing for some stranger—even though that stranger may be as sentient or more rational than one’s own child. 2. Putting on one’s own oxygen mask before assisting one’s child or seatmate (e.g., on an aircraft that has lost cabin pressure)—even though one should not value one’s own life above that of others. 3. Lying to an armed intruder in order to protect an innocent loved one—even though this would be in violation of Kant’s categorical prohibition against lying under any circumstances. 4. Calling 911 to interrupt a theft—even though if we were in the thief’s full situation (e.g., of extreme hunger or drug-induced irrationality), we ourselves might desperately wish not to be caught. The necessary reformulation of the Golden Rule admittedly presents daunting challenges. There is, first, a technical or mathematical challenge. If we give up the convenient fiction that beings can be categorically classified as either sentient or non-sentient and as either rational or non-rational, we are no longer afforded the simple elegance of the symmetry of invariance under permutation of beings within a sharply bounded class. We must now seek a more complex formulation in which exchangeability is neither all-or-none nor, in general, strictly symmetric. Further complications arise if we accept the relevance of degrees of relationship, connection, or proximity between individuals. Here we encounter a further and still more daunting challenge: How do we establish quantitative metrics of degrees of class inclusion or degrees of relationship that are robust, practically implementable, universally valid, and morally justifiable? I see no way forward without confronting this challenge. Yet, the history of egregiously amoral, discriminatory abuses by those who claim moral and rational superiority but are, in fact, only more numerous or better armed—as in the crusades; the slave trades; the pogroms; the Nazi holocaust; and the forced displacements, mistreatments, and massacres of indigenous peoples everywhere— cautions us to proceed with the greatest possible thoughtfulness, care, and checks and balances as we venture onto these slippery slopes. Nevertheless, in recognition of the terrestrially unprecedented step to rationality of humankind, however incomplete, I submit that just as all normal humans have the unique potential for learning a fully expressive natural language, all normal humans have the latent potential for achieving a rational, self-consistent system of moral principles.

12. The cognitive grounds of deterministic free will Any effort to establish the cognitive grounds of meta-ethics would surely be pointless in the absence of an explication of an individual’s free will. For individuals who are not free to make their own moral choices cannot justifiably be held responsible for

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those choices. This consideration has moved me to attempt to reconcile (a) the need I thus fervently feel for an account of such a “freedom of the will” with (b) my unshakeable rational conclusion that every event arises deterministically from its causal antecedents except to the extent, if any, that it is affected by mere (e.g., quantum mechanical) chance. The reconciliation to which I arrived appears harmonious (as I later discovered) with reconciliations earlier proposed by Dennett (1984, 2003) and others. (See the recent overview of compatibilist proposals—especially the “reasons-responsive” proposals—in The Stanford Encyclopedia of Philosophy, McKenna, 2004, which came to my attention too late to be adequately addressed here.) My own approach grew more specifically out of consideration of the step to rationality— with its capacity for “holding immediate self interest in abeyance and for contemplating an explicit set of alternatives objectively with respect to explicitly chosen abstract criteria” (as earlier stated in Shepard, 2001, pp. 744–748). Invoking symmetry once again, I accordingly propose that one acts freely when one’s actual and alternative actions are symmetrical with respect to their potential for realization under appropriate circumstances (as I indicate, e.g., under Points 2, 5, and 6, below). Although one’s decision process could be strictly determined by preceding events such as those explicitly introduced in computational modeling, some of those preceding causes are of a kind that we also call reasons. What is required for one’s resulting choice to be free, in the intended sense, is just one’s own knowledge that one has cognitively represented the relevant set of alternatives, has evaluated or (as necessary) re-evaluated each alternative in that set, and finally has selected and acted on the alternative with the highest resulting evaluation—without interference (of the kinds I consider under Point 2, below). Simply stated, a free choice is one that has been made for one’s own reasons. Specifically, the evaluation of each alternative might be computed via utility theory, as the sum of the products of (a) the subjective probability and (b) the subjective value of each of the possible outcomes considered for that alternative. If so, that mental process can be seen as directly analogous to a more literal “weighing” of the alternatives on the Archimedian balance beam discussed earlier. The locations along the beam might then represent the (positive or negative) subjective values of the possible outcomes and the masses of the weights their corresponding subjective probabilities. Even today there are both philosophers and scientists who—like myself until but a few years ago—have regarded free will and determinism as being inherently incompatible with each other. Some determinists among these have accordingly proposed that our experience of free will is merely illusory. But this alternative is unacceptable to me, both because it removes the essential grounds for moral responsibility and because I do have knowledge (in a sense I shall try to explain) of my freedom that is as reliable as my knowledge of anything. On the other side, some libertarians have proposed that we reject determinism. But this alternative is equally unacceptable to me, because an event undetermined by any antecedents is not something that has been chosen by a person but something that has happened to that person. It is, accordingly, not a choice made for that person’s own reasons, or for which an assignment of praise or blame is justifiable. (Quantum mechanics does allow for undetermined events. But, despite the expressed hopes of some libertarians, even if some choice were determined

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by such a quantum-mechanical event, that event could be considered neither a reason nor a moral justification for the resulting choice.) Entire books have been written concerning the possibility of reconciling freedom and determinism (for references to some of these, see McKenna, 2004). For me, however, the following seven points seem sufficient: 1. One can represent to oneself the alternative choices in some small, well-defined set and know—as well as one knows anything—that one has considered and evaluated each of the alternatives with respect to the explicitly represented criteria associated with one’s current goal. Example: The familiar children’s game of tic-tac-toe (or “naughts and crosses”) serves as a trivially simple illustration, uncomplicated by the real-world uncertainties to which I return later. One can systematically consider each of the cells of the 3 × 3 array that are still unmarked and evaluate, for each such cell, whether the insertion of one’s own mark (whether “x” or “o”) in that cell will complete a (horizontal, vertical, or diagonal) row of three such marks (thus winning the game), or will block a potential row of three of the opponent’s mark (thus forestalling a win by the opponent). Indeed, one can contemplate the options that would remain for subsequent moves by one’s self and by one’s opponent, and thus arrive at a preference value for each empty cell into which one might insert one’s mark. (Moreover, the process can be significantly simplified by considerations, once again, of symmetry. For, although there are 9 cells to be evaluated in placing the first mark, equivalence under rotations and reflections of the array leaves only 3 distinguishable cases: a cell on any of the 4 corners, a cell in the middle of any of the 4 sides, and the one cell in the center.) 2. One can know—as well as one knows anything—whether one has, instead, been prevented from considering (or reconsidering) or from evaluating (or re-evaluating) any alternative, or whether one has been prevented from finally acting on the alternative to which one has assigned one’s highest evaluation. Only if one knows that one has not been prevented (internally or externally) from satisfactory completion of this process, does one know that one has chosen and acted freely. Counterexamples: One knows that one has not chosen freely if the process of considering and evaluating the alternatives does not converge to a consistent result, whether because of insufficient available time, because of recurring cognitive lapses (e.g., of attention, memory, or computation), or because of inner conflicts or compulsions (such as an irrational urge at variance with one’s rationally chosen criteria). And, one knows if one has not acted freely on one’s choice because of an external constraint (e.g., imposed by a gun held to one’s head). (The important case of conflicting goals, which I first discussed in Shepard, 1964, is deferred for special consideration in Point 7, below.) 3. If two or more of the represented alternatives are found to be equally valued by one’s chosen criteria, one knows that it does not matter which of the corresponding actions are taken. In this case, one can let the choice among those equally highest valued alternatives be made arbitrarily or at random. Examples: One could make such a choice by a flip of coin or roll of dice or, more simply and naturally, by leaving the choice to be determined by unknown processes (e.g., of thermal

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or chaotic fluctuations) within one’s own brain—just as one might allow an Archimedean beam on which the weights are perfectly balanced to tip to either side by a random gust of wind. (The unknown neuronal processes may, like gusts of wind, be strictly deterministic but, if the possible outcomes are equally valued, neither the outcome nor the process from which it arose matters.) 4. But what about the criteria one has chosen for the evaluation of the alternatives—have not these, too, been deterministically constrained by prior (e.g., genetic, cultural, and neuronal) circumstances? Of course, but those criteria, too, could have been selected freely, in the same sense, from a set of explicitly contemplated alternatives and a set of still higher-level criteria. If so, the determinants of those higher-level criteria are, again, both deterministic causes and cognitively represented reasons. Example: In playing tic-tac-toe with a young child who is just learning the game, one may deliberately adopt different criteria for motivating the child by letting the child win the game. One also knows that there is no necessary limit to the implied hierarchy of goals and their associated criteria. I agree with Eddy Nahmias (personal communication, September 7, 2007) that one who insists that the specter of an infinite regress vitiates an attempt to reconcile freedom with determinism is demanding something of freedom that it could not have and that we do not need. A decision made at any level of the hierarchy is itself still made freely at that level in the way I have explained, and one can be responsible for that decision at that level without being responsible for its sufficient condition at any higher level. (Again, important issues about choosing among conflicting goals are addressed in Point 7, below.) 5. To assert that one made a choice or took an action “freely” is to assert that one could have chosen or acted otherwise. (This is the already noted, essential symmetry of alternatives with respect to their potentials for realization.) Although such a possibility appears to be incompatible with determinism, reconciliation can be achieved through proper explications of what one actually means when one makes the claim, “I could have done otherwise.” Explications: (a) The choice made was from the equivalence class of those alternatives for which one had computed the highest value according to one’s adopted criteria. (b) One knows, as well as one knows anything, that one was not compelled by any extraneous constraint (internal or external) to choose a lesser-valued alternative. (c) One also knows that if a choice were to have been made from the same equivalence class of one’s equally top-rated alternatives on a different occasion, a different choice might have been made (e.g., automatically, by neuronal processes of the same deterministic sort but in a somewhat different initial state) and, further, that such a different choice would have been equally acceptable because the neuronal process was constrained to meet the same instantiated criteria. (d) One also knows that a different choice could have—and, indeed, would have— been made if a different goal, with its different evaluative criteria, had been instantiated, as would be likely under different circumstances (e.g., with access to additional or different information). 6. But can one really be certain that all of the alternatives in the intended set have been considered and properly evaluated in accordance with the intended criteria? Is it not possible that one’s memory of which alternatives have already been checked and what evaluations made is in error (possibly, as in the thought experiment in the first of Descartes’

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1641 Meditations, because some evil demon is playing tricks on one’s mind)? Of course, one can never be absolutely certain; and, in practice, may become doubtful when the alternatives become too numerous or ill-defined. But in the simplest cases of choosing between only a few well-defined alternatives, with a clear head and ample time, I think one can “know as well as one knows anything” that one has chosen freely (on the basis of the available information). Thus, accepting a Chomskian “competence–performance” distinction, I believe this is sufficient to establish the possibility of free will of the only kind that makes any sense. What skeptical doubts remain are of the speculative philosophical kind that can always be raised about any of one’s own cognitive functions of perception, memory, or thought. Examples (of kinds of “knowledge” that can be, but usually are not, doubted): one’s knowledge that one has proceeded correctly in any process of reasoning or calculation or (at a deeper epistemological level) one’s knowledge of the uniformity of nature, or of the very existence of an external world, of a past or a future or of other consciousnesses beyond one’s own. 7. Additional issues are raised by cases in which one’s own goals come into conflict with each other. The process of working toward internal consistency in one’s hierarchy of goals and sub-goals, which has concerned me for many decades (beginning with the second and the final sections in Shepard, 1964), is essentially like the process that (as I have more recently been suggesting) may work toward a consistent system of (scientific or personal) beliefs about the world. But, despite all efforts toward consistency, the flux of personal circumstances gives rise to conflicting goals that I believe to be central to the disputes between those who do and those who do not believe that free will is compatible with determinism. Examples (of such conflicting goals): Everyone experiences conflicts between such ordinary goals as those of (a) continuing a project in which one is enjoyably engaged, (b) tending to some problem that has developed with the house or yard (e.g., as requested by one’s spouse), (c) taking a break with some friends (e.g., who are looking for a fourth for tennis), or (d) furthering one’s professional career (e.g., by agreeing to take on an onerous task with a looming deadline). Some may additionally find themselves torn between a desire for immediate gratification or release from a relentlessly increasing physiological craving, on one hand, and their own resolution to abstain from the substance to which they had become addicted, on the other hand. In the most extreme cases, one may experience agonizing conflicts that are forced upon one externally (e.g., by a gun-wielding intruder who demands to know the whereabouts of one’s most valued possessions or loved ones). It is interesting to note that, as has been documented by Nahmias, Morris, Nadelhoffer, and Turner (2004), there is a systematic difference between compatibilist and libertarian philosophers in what they present as paradigmatic examples of free will. The compatibilists tend to cite “confident” decisions, whereas the libertarians tend to cite the more difficult “close-call” decisions. This difference is understandable in that the antecedent causes that determined the outcome are more obvious in confident decisions, where these causes are also consciously accessible reasons, than for close-call decisions, where these causes may be unknown neural events. I argue, however, that with a little more explication, the compatibilist

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account that I have proposed—with its provision both for determination by reasons, in the more “confident” cases, and determination by causes that are not reasons, in the “close-call” cases—can handle both types of cases. Moreover, it can do so without resorting to the unintelligible claim of some libertarians that a decision that is not determined by any antecedent events is somehow also a decision for which one is morally responsible. It also does so without requiring the unintelligible claim of some other libertarians that the “cause” of the decision is not an antecedent event but the enduring agent him- or herself. What is unintelligible about this latter claim is how an entity enduring over time (the agent) can cause an event (the decision) to occur at a particular time. (In the next section, I propose a replacement of such “agent causation” with a different analysis of agency that is, I believe, both intelligible and consistent with my deterministic account of free will.) For fuller clarification of the differences between libertarians and compatibilists, however, we must recognize that the close-call cases vary widely from the trivially easy to the agonizingly difficult, depending on uncertainties as to the probabilities and the magnitudes of the likely consequences of the alternatives. At the easy end, one who is indifferent between vanilla and chocolate ice cream may be content to leave the choice of which to order to unknown inner events (which are, accordingly, only causes and not reasons). But at the agonizing end, one threatened by an intruder holding a gun to one’s head will desperately try to weigh the reasons favoring different courses of action; and one who is tempted to break one’s own resolution about avoiding a substance to which one has become addicted may endeavor to reinforce that resolution by reviewing the rational reasons adduced for it. The varieties of kinds of deliberation thus range between not two, but three paradigmatic extremes. These may be illustrated by corresponding beam balance analogues as follows: (a) A confident deliberation corresponds to the case in which the sum of the products of the weights and their distances from the fulcrum is unmistakably greater on one side of the fulcrum, ensuring that that side will tip down. (b) An easy close-call deliberation corresponds to the case in which weights are distributed relatively close to the fulcrum and so that the sum of the products is equal for the two sides, leaving the direction of tilting to be determined by an extraneous vibrations or currents of air. (c) An agonizingly difficult deliberation corresponds to the case in which the weights are distributed to positions that are far from the fulcrum on both sides that are also uncertain and in flux, so that the final direction of tilt may primarily depend on the (arbitrary) moment during this flux at which action is (or must be) finally taken. The instabilities inherent in deliberations of this last, difficult type are also characteristic of conflicts among goals, as I long ago argued in writing about “selections among multi-attribute alternatives” (Shepard, 1964). In the case of such agonizing decisions, the person who must make the decision is still responsible for the reasons adduced in favor of each of the alternative actions. If we judge the reasons to have been adopted carelessly, foolishly, or as a consequence of a life style resulting from habitually poor choices, we will be inclined to hold the person responsible for the outcome. But if, on imagining ourselves in the position of that person, we conclude that the outcome was determined not from bad reasons but from irreducible uncertainties concerning the values and probabilities of the outcomes, we will (in accordance with the Golden Rule) charitably hold that person less responsible for the ensuing outcome.

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The external constraints or internal compulsions affecting subjectively difficult deliberations may be experienced as limiting one’s freedom. Yet (as noted by Nahmias et al., 2004), libertarians regard just such cases as particularly demonstrative of free will. Perhaps this is because such cases call for a triumph of “will power” over opposing forces. My claim, however, is that those cases, like all the others I have considered, can be explained solely in terms of antecedent causes—some of which are also reasons, others of which may be essentially random. If the addict succeeds in holding to a resolution to abstain, this “triumph of will” arises from a triumph of rationality. If the one threatened by a gun-wielding intruder is successful in thinking of a reason why one of the alternative courses of action is more promising than the others, that reason is also the antecedent cause of the resulting decision. If that threatened individual is not successful in coming up with a reason for favoring any one alternative, the ensuing decision, however agonizing, still had antecedent causes but they may not qualify as reasons.

13. Self-knowledge and the cognitive grounds of agency What I have so far written here about the grounds of ethics and free will may appear to suffer from an inconsistency between (a) the claimed sufficiency of a completely deterministic account and (b) my frequent locutions about a “one” who “can evaluate the alternatives in a represented set,” who “knows that one has chosen an alternative with the highest evaluation,” and so on—locutions that may seem to presuppose the existence of an agent who, independently of the world’s deterministic causal network, is autonomously doing the knowing, evaluating, and choosing. I believe, however, that the implied agent can be explicated as an integral (although very special) part of that causal network. Hume (1739/1896) reported that his own introspections revealed sensory impressions, memory images, and ideas but never a “self” that is the experiencer of these impressions, images, or ideas. But Hume did not indicate how a self, if one were to exist, could reveal itself through introspection, if not in the form of further impressions, images, or ideas. My proposal is that just as one has the concept of another person as an experiencing, choosing, and acting agent, one also has the concept of one’s self as such an experiencing, choosing, and acting agent. Moreover, one’s concept of one’s self, just as one’s concept of another person’s self, undoubtedly consists of both images and ideas. These images and ideas are of (a) the physical body, (b) the conscious experiences, and (c) the behavioral dispositions of that self. (An early example of a computational model for such behavioral dispositions may be found in the “production systems” of Allen Newell, 1973.) Indeed, one can represent (within one’s self) another self’s representation of one’s own self, or even of one’s own self representing that other self. The potential depth of such reciprocal relationships between selves reinforces our deepest intuitions about the symmetry of invariance under permutation of persons and thereby contributes to our confidence in the existence of other minds, to empathy, to love, and to the normative force of the Golden Rule. In any case, the recognition of the special character of such selves and of the reciprocity between them seems to me essential for the attribution of moral responsibility.

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To “know thyself,” then, is (a) to know the existence of a robust (if unconscious) internal system of one’s own representations of alternatives, goals, and conditional predispositions for action; and (b) to know that relevant subsets of those representations are accessible for (conscious) evaluation and adjustment—in essentially the way that I have already described for representing and choosing among alternatives. Just as one knows the existence of the external, material world through what J. S. Mill (1865) called the “permanent possibility of sensations,” one knows of one’s self as the enduring possibility of access to one’s unique set of stable memories, beliefs, and predispositions. Moreover, to write, as I have been doing, of one’s self having knowledge of alternatives, of evaluating alternatives, of acting on such evaluations, and of being morally responsible for one’s actions, is not incompatible with determinism. Returning, finally, to the importance of thought experiments, consistency, and symmetry, I suggest that the most effective way of gaining such knowledge of one’s self may be by performing thought experiments on how one would respond to various situations, imagined as vividly and concretely as possible. In a kind of Socratic dialog with one’s self one may then be able to work toward mutual consistency among candidate moral principles (much as a scientist works toward a consistent system of laws governing the external world). Mutual consistency of one’s principles for deciding between alternatives is necessary for unity of one’s self. A person who operates with “logic-tight compartments” that are inconsistent with each other is like a dysfunctional family of selves.

14. Concluding remarks The issues I have sought to address here—concerning mathematics, the discovery of physical laws, the grounds of moral principles, the freedom of the will, and the nature of personal agency—may have appeared inordinately heterogeneous for a single article in cognitive science. I, nevertheless, hope that readers will recognize that a core of central ideas has underlain my approach to all of these issues. I conclude with a brief recapitulation of what I regard as perhaps the most central of these core ideas: 1. Humans (apparently uniquely among terrestrial animals) are in the process of taking (however partially) an unprecedented “step to rationality.” 2. Essential for this step is a capability to hold immediate self-interest in abeyance, to consider and to evaluate (with respect to explicit, objective criteria) the alternatives in a mentally well-represented set. 3. A deeply internalized (although initially unarticulated) wisdom about the most pervasive and abstract features of our world—concerning, for example, space, time, transformation, invariance, and symmetry—has made possible the mental simulation of different alternative actions and their likely consequences via “thought experiments.” 4. Such thought experiments, together with the guiding symmetry principle of invariance under transformation, have enabled some individuals to arrive at explicit knowledge of universal laws of mathematics and physics, and even of moral principles that may hold for all rational beings.

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5. The same rational capabilities for the evaluation of alternatives in terms of their likely consequences also provides for an explication of how “free will,” far from being incompatible with determinism, makes personal responsibility possible through deterministic antecedents of the kind we call “reasons.”

Acknowledgments I thank the Robert J. Glushko and Pamela Samuelson Foundation and the Cognitive Science Society, as well as Bob Glushko and the other members of the Rumelhart Prize Committee for affording me this opportunity to present some of my thinking about the remarkable effectiveness of thought experiments and for awarding me the 2006 David E. Rumelhart Prize—a very special honor for me, long an admirer of my former colleague David’s brilliant contributions to cognitive science. I thank Joshua Tenenbaum for his role in organizing the symposium in my honor at the 2006 meeting of the Cognitive Science Society and in coordinating the preparation of this special issue of Cognitive Science. I am indebted to Josh, Nick Chater, Jacob Feldman, and Tamar Szabo Gendler for their inspiring presentations at that meeting, and to Thomas Griffiths and all the other contributors to this special issue. I also thank Paul Smolensky for his beautifully expressed and well-informed introduction to my presentation at that meeting. In preparing the present, expanded version of my presentation there, I have benefited enormously from insightful and clarifying suggestions provided by Tamar, by Eddy Nahmias, and by two anonymous reviewers.

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