EXPERIMENTS AND THOUGHT EXPERIMENTS IN NATURAL SCIENCE

DAVID ATKINSON

Abstract

My theme is thought experiment in natural science, and its relation to real experiment. I shall defend the thesis that thought experiments that do not lead to theorizing and to a real experiment are generally of much less value that those that do so. To illustrate this thesis I refer to three examples, from three very different periods, and with three very different kinds of status. The first is the classic thought experiment in which Galileo imagined that he had, by pure thought, demolished Aristoteles’ dogma that heavier bodies fall more quickly than light ones. I will show that he was mistaken. The second is the Einstein-Podolsky-Rosen paper purporting to show that quantum mechanics must be incomplete in its domain of application. This thought experiment is a very good one, not because its conclusions are correct, but precisely because it was fruitful, leading to theory and, above all, to a real experiment. Finally I discuss the modern string theory of everything, which, while it is regarded as a physical theory by its instigators, shares some properties of the least successful sort of thought experiment. 

1  Falling Bodies

Let us begin with Aristotle and Galileo Galilei. In some places, Aristotle claims merely that heavier bodies fall more quickly than lighter ones:
 
The mistake common to all those who postulate a single element only is that they allow for only one natural motion shared by everything. … But in fact there are many things which move faster downward the more there is of them. [Aristotle, De Caelo, Book III/v/304b].  

We shall call this the weak Aristotelian dogma: it is the qualitative statement that heavier bodies fall faster than lighter ones. Moreover, in other places Aristotle maintains that the natural motion of a body is proportional to its weight. Here we have to understand what he meant by natural motion, or perhaps which property of a falling body comes closest to the ancient notion of natural motion, before we can reasonably consider the question of inconsistency. Be that as it may, here is another passage that is less susceptible of interpretational uncertainty:   

If a certain weight move a certain distance in a certain time, a greater weight will move a same distance in a shorter time, and the proportion which the weights bear to one another, the times too will have to one another, e.g. if the half weight cover the distance in x, the whole weight will cover it in x/2. [Aristotle, De Caelo, Book I/vi/274a].    
We shall call this the strong Aristotelian dogma: it is the quantitative statement that times of fall, from a given point to a lower point, of bodies of differing weights, that are alike in other ways, are inversely proportional to their weights. 
The great contributions made by Galileo to physics have little to do with his claims, via his spokesman Salviati, that the Aristotelian dogma, whether in its weak or its strong version, implies a logical inconsistency. It is sufficient to point to a physical situation in which Aristotle’s dogma, even in its strong form, is empirically correct. Since an inconsistent argument points at nothing at all, but Aristotle’s argument does in fact indicate a realizable configuration, it follows that the dogma cannot be internally inconsistent. The case which gives Galileo the lie is that of bodies falling in a fluid (such as air or water) at their terminal velocities in the case of laminar fluid flow. Consider this quotation:
We see that bodies which have a greater impulse either of weight or lightness, if they are alike in other respects, move faster over an equal space, and in the ratio which their magnitudes bear to one another. … In moving through plena it must be so; for the greater divides them faster by its force. For a moving thing cleaves the medium either by its shape, or by the impulse which the body that is carried along or is projected possesses. [Aristotle, Physica, Book IV/viii/216a].    
Under the restriction of laminar flow, and for bodies of identical size and shape, as specified by the Stagyrite, the viscous forces are proportional to the velocities, and so the terminal rates of fall are proportional, the times of fall inversely proportional, to the weights. It is not part of my thesis that Aristotle espoused, or could have espoused, this detailed interpretation, nor that Galileo excluded, or might have excluded, the particular case of terminal motion with laminar flow. I maintain simply that the possibility of a realizable model of the strong Aristotelian dogma frees it of any possible logical inconsistency. 
Galileo’s own resolution of the imagined inconsistency in the doctrine that different bodies fall at different rates (as implied by the weak dogma) was that all bodies must fall at the same rate (Galileo 1638). Moreover, he presents this as a truth about the world that is accessible to reason, rendering experiment unnecessary. 

Salviati: But, even without further experiment, it is possible to prove clearly, by means of a short and conclusive argument, that a heavier body does not move more rapidly than a lighter one, provided both bodies are of the same material, and in short are such as those mentioned by Aristotle. … If then we take two bodies whose natural speeds are different, it is clear that on uniting the two, the more rapid one will be partly retarded by the slower, and the slower will be somewhat hastened by the swifter. Do you not agree with me in this opinion? 
Simplicio: You are unquestionably right. 
Salviati: But, if this is true, and if a large stone moves with a speed of, say, eight, while a smaller stone moves with a speed of four, then when they are united, the system will move with a speed less than eight; but the two stones when tied together make a stone larger than that which before moved with a speed of eight. Hence the heavier body moves with less speed than the lighter; an effect which is contrary to your supposition. Thus you see how, from your assumption that the heavier body moves more rapidly than the light one, I infer that the heavier body moves more slowly. … We infer therefore that large and small bodies move with the same speed, provided they are of the same specific gravity. [Galileo Galilei 1638, p. 108.]

Galileo was in imagination arguing with Aristotelians and the above quotation may be regarded as being largely a polemical device. As an argument, it is disappointing in both its destructive and constructive aims: not only must the criticism of Aristotle’s dogma be defective, as we have just seen, but the new Galilean dogma concerning free fall is itself a non sequitur. Again we will not immediately demonstrate this, but we rather draw attention to the fact that a physical model exists in which different bodies fall at different rates, even in vacuo. In a nonuniform gravitational field, as in the terrestrial situation, the rate of fall is a function of the distance from the centre-of-mass of the earth: a body at a higher elevation falls less quickly than one at a lower elevation. Moreover, the rate of fall can depend, in special circumstances, on other parameters too, such those defining magnetic or electric fields, etc. 
Thus the insufficiency of Galileo’s reasoning has been shown indirectly by means of physical models in which both the destructive and the constructive aspects of his reasoning fail. More directly, the reason for the failure is Galileo’s insertion into the argument, as if it were self-evident and not in need of empirical testing, the following supposition: 
(S1) Natural speeds are mediative
By this it is meant that if two bodies with different natural speeds are bound together, the natural speed of the composite lies between those of the constituents. Whether this is true or not depends of course on the meaning attached to the notion of natural speed. For Aristotle it had a significance bound up with the notion of the natural places of earth, fire and so on. If we reinterpret natural speed as acceleration, following Galileo’s lead, then indeed it is true that two bodies, with different accelerations, if bound together, will thenceforth have an acceleration lying at an intermediate value: accelerations are mediative, as we shall prove by using Newton’s laws. However, if, as indicated above, we interpret natural speeds as terminal velocities in a fluid with laminar flow, the situation is more complicated. In the first place, if we consider simply binding together two bodies that have different terminal velocities, the terminal velocities will generally be mediative. Other possibilities are open, according to which the terminal velocities, even of composite objects, could be proportional to the weights and thus be additive rather than mediative. This would be the case if the bodies were placed one at a time, or both together, in an impermeable container of negligible weight. The situation is even more complicated when the terminal velocity is reached in a condition of turbulent fluid flow, as is often the case in practice. 
In short, Galileo’s thesis [S1] is anything but self-evident, being in fact in some circumstances empirically true and in others empirically false. Any claim that his argument offers a glimpse into a Platonic realm of truth, rendering empirical testing unnecessary, is demonstrably false.  
Such a claim is indeed made by J.R. Brown (1991). According to Brown, Galileo’s thought experiment is a classic in the field, since, so it is claimed, an old belief has been destroyed by pure thought, and replaced by new knowledge concerning the world, without the need for a real experiment, that is, without extra empirical input. For Brown, the landscape of empirico-theoretical truths (and also that of purely mathematical truths) is something that is there to be observed by a sufficiently cultivated inner eye. Brown’s world view is denied by J.D. Norton (1996), for whom thought experiments are disguised arguments. On the basis of a careful study of the epistemology of thought experiments (as opposed to, for instance, their impact on the scientific community), Norton concludes that thought experiments ”can do no more than can ordinary thinking with its standard tools of assumption and argument”. T.S. Gendler (1998), on the other hand, opposes both Brown and Norton. She argues that thought experiments are ”guided contemplations”, i.e. arguments with a particularly strong persuasive power, their ”justificatory force”. Apparently taking her inspiration from Mach, Gendler ascribes this justificatory force to the fact that, in a thought experiment, instinctive and hitherto unarticulated empirical knowledge suddenly becomes organised and manifest. For all her astute analysis, I do not agree with Gendler, if she means to suggest that Galileo did not need to make real experiments, but merely to articulate what before his time was inarticulate. Galileo performed, and needed to perform, real experiments. In the appendix we provide further technical analysis of the mechanics of falling bodies.  

2. EPR Thought Experiment  

In quantum mechanical theory, an observable, like the position or velocity of a particle, is represented by an operator, not just by a number. One may like to think of this as a sort of table of all the possible results of measurements of the observable. What the result of such a measurement will be in a given case depends on the state of the system to be measured. For example, suppose that the system is an excited calcium atom, which decays into its ground state, emitting two light quanta, or photons, in the process. Quantum mechanics is silent about the direction of propagation of the photons, and thus about the result of a measurement of the position, at a given time, of one of them. However, if a definite result of a position measurement of one of the photons has been obtained, that of the other can be inferred from theory — successive measurement results are correlated, that is the prediction. The classical, or common sense interpretation of this fact would be that the positions of both photons, as functions of time, exist, and are indeed correlated, but that one has insufficient information about the detailed nature of the decaying calcium atom to make unconditional predictions. Such a situation is common enough in classical theory. The Copenhagen dogma asserts, however, that the positions of the photons do not exist prior to a measurement of one of them, and that after a measurement of one photon’s position, and its consequent entification, the state of the other photon is immediately changed, its position being entified by the act of measurement performed on the other. 
A similar story may be told about the momenta of the photons: no unconditional prediction of the outcome of a momentum measurement is possible, but if the momentum of one photon is measured, that of the other is thereby also known. But there is more. A measurement of the momentum of one photon gives knowledge about the momentum of both photons, but it gives no information about the position, either of the photon that is subjected to the measurement, nor about that of the other. The Copenhagen interpretation claims that a particle, whose momentum has just been measured, does not have a position, that is, it is not simply that we do not know what the position is, but there is no position to know. Moreover, this applies equally for the second photon, the one that has not been subjected to measurement. The idea that a measurement of one property of a localized system might well change the value of another property of the same system is easily understandable, but that this other property should, as a result of the invasive nature of the measurement, fail to exist, is counterintuitive, to say the least. That the ontological status of the second photon, with respect to the existence of its position or its momentum, should depend on, and be changed by, what is done to the first photon, which is out of contact with, and indeed may be greatly separated from, the first photon, seemed to Einstein, Podolsky and Rosen (EPR 1935) to be an absurdity. In Einstein’s view, such a holistic interdependence of different parts of reality would nullify the physicist’s profession. There is a sous-entendu that, given the manifest empirical successes and technical applications of the physicist’s trade to date, such a nullification would be intolerable. At any rate Einstein, to the end of his days, would not tolerate it. 
For Bohr and Heisenberg, the new quantum ontology seemed to be enforced by the disjunction of 
1	The mathematics of Heisenberg’s matrix mechanics, and 
2	The evangelical conviction that the new quantum mechanics is, in its domain of application, complete.
The latter conviction was strengthened by von Neumann’s flawed proof (Neumann 1932) that hidden variables, which might ‘complete’ quantum mechanics and restore classical ontology, are logically impossible. The reason that simultaneous existence was not accorded to incompatible observables like position and momentum is that the Copenhagen dogma proclaims an observable, represented by an operator, to be well-defined in a given state, and by an insidious epistemic slide, to be, only when the vector that represents this state is an eigenvector of the operator. Since noncommuting operators like those representing position and momentum do not possess common eigenvectors, it follows, given the epistemic slide, that a photon, or any other system, cannot possess both a position and a momentum. 
Einstein and Bohr agreed that the quantum ontology, as sketched above, was inescapable if one assumes quantum mechanics to be complete. For Einstein, this was tantamount to a proof that quantum mechanics cannot be complete, and this is the burden of the EPR paper. Bohr accepted, nay created and exulted in the new ontology; and he welcomed any proof, defective or not, that hidden variables could not be tacked on to the fledgling discipline. The stand-off between Einstein and Bohr was complete, and it continued to their deaths, seemingly unaffected by unfolding events. 
The first significant event was theoretical, when David Bohm created a new version of quantum mechanics that was empirically indistinguishable from the Copenhagen canon, but which implemented the unimplementable (Bohm 1952). Hidden variables, in the form of his quantum potential, were inserted into the interpretation of the wave-function of quantum mechanics. It was left to John Bell to put his finger on the weak point of von Neumann’s no-go theorem, namely the assumption that the measurement outcome of an observable that is represented as the sum of two operators is necessarily the sum of the separate outcomes of measurements of the observables corresponding to those operators (Bell 1987).
It was also David Bohm who initiated the second important step in the development by retooling the EPR effect in terms of the components of spin of the two particles, instead of their positions and momenta. This reformulation paved the way for John Bell, who derived an inequality that must be satisfied by correlation coefficients between spin measurements performed on the two particles in the EPR configuration, on condition that local, noncontextual hidden variables exist, and that stochastic independence in the sense of Kolmogorov is equivalent to physical independence. Eighteen years were to elapse between the derivation of the inequality and the demonstration of its experimental violation by Alain Aspect et al. at Orsay (Aspect 1982) 
Some exegeses of the EPR paper speak of a ‘paradox’, others of a thought experiment or theorem, but its true significance for the development of physics lies in its development from the stand-off of a thought experiment and of two competing world views (Einstein versus Bohr) via theoretical insights (Bohm and Bell) to a genuine experiment (Aspect). Had this genuine experiment not been performed, the EPR thought experiment would have remained a fruitless stand-off.  
I once asked Bell whether he had thought, after he had derived his inequality, that it would be violated by nature. ”Well”, he said, ”I expected it would be, but I hoped it would not be!”. He added with a smile, ”then I would have been famous.” In the period that experiments were being planned to test the inequality, and, by extension, quantum mechanics itself, the emotions of the community of physics were divided between the aspirations of the realists and the confidence of those who practiced quantum methods on a day-to-day basis. The impact of Aspect’s work was not merely negative, in destroying the hidden-variable assumptions of Bell, but also constructive, for it verified in detail the numerical predictions of quantum mechanics.
For our purposes, the lesson to be learned from the ascent 
1	EPR thought experiment
2	Bohr-Einstein stand-off
3	Bohm reformulation of EPR
4	Bell inequality
5	Aspect experiment  
is that a thought experiment which supports two contradictory intuitions can fruitfully point the way forward, by stimulating theoretical development and suggesting a real experiment, to a resolution of the dilemma. As a thought experiment alone, the value of EPR was limited to the challenge it made to Bohr to sharpen his distinction between quantum object and classical measuring instrument, but he was never able to convince Einstein that his interpretation of quantum mechanics was viable. 

3. String Theory

The modern candidate for a Theory of Everything is string theory, according to which the known fundamental particles, photons, electrons, quarks and so on, with their associated fields, are nothing more nor less than different frequencies of vibration of the postulated string. In other words, the building blocks of our universe are notes on a cosmic string, rather than autonomous elementary particles. The theory aims at a definitive unification of all known forces, including gravitation. According to string theory, the gravitational attractive force between two particles should increase more rapidly, as the particles approach one another, than do the other forces, until gravity is as strong as all these other forces. However, to test string theory experimentally, one would have to penetrate to impracticably tiny distances, or equivalently to accelerate particles to impossibly high energies before allowing them to collide with one another. In this sense, string theory looks more like a recipe for a thought experiment than a genuinely physical theory that can be put to the test of experiment. 
To give a rough idea how impractical it would be to perform a real experiment to test string theory, let us consider briefly the history of particle accelerators at CERN, in Geneva. The first machine, a proton synchrotron (PS), came on-line in 1959 and accelerated protons to an energy of 28 GeV (the unit, the giga-electron-volt, corresponds roughly to the energy that would be produced if one proton were to be annihilated, turning all its rest-mass into energy). In 1980 the SPS (S for ‘super’) produced protons and antiprotons at 170 GeV per particle, and in 2005 the Large Hadron Collider (LHC), a machine with a diameter of 27 km, is expected to produce protons of energy 7000 GeV. An American project to build an even bigger machine in Texas was killed by the former president Clinton; and there are no plans anywhere to build bigger accelerators than LHC. The economic limit seems to have been reached. In forty-five years the maximum attainable energy per particle has risen by a factor of 250. To test string theory adequately one would have to produce energies that are ten to the power sixteen (ten thousand million million) times higher than those that LHC will produce in 2005. It seems safe to say that we will never be able to produce energies anywhere near this value, and that string theory can never be confronted with the crucial test of experiment. 
Is string theory truly a scientific theory? String theory could be tested in principle, and in this it differs from unscientific world systems. But it will never be testable in practice, and in this it differs radically from Newton’s or Einstein’s theories of gravitation. Suppose that string theory can be completed in a consistent manner; and suppose that it accommodates the Standard Theory of elementary particles, as well as Einstein’s General Theory of Relativity, as a low-energy approximation. This future Theory of Everything would postulate certain new properties of gravity at very high energies, where these new properties de facto cannot be tested.  
Edward Witten, the most prominent proponent of string theory, once said, on being asked about experimental support for string theory: ”Things fall.” The chain of implication may be reconstructed as follows: string theory contains Einstein’s general relativistic theory of gravitation, Einstein’s theory contains Newton’s theory as an approximation, and Newton’s theory describes quantitatively the falling of ‘things’, like planets, moons and apples. The weakness of the answer is apparent if one reverses the order, and considers the nested inductions: Newton’s explanatory theory is subsumed in Einstein’s curved space-time explication-explanation, and this is further seen as a property of multidimensional strings. As for Einstein’s inductive leap, it predicts not only the precession of Mercury’s quasi-elliptical orbit, but does so with good numerical accuracy. This, and other successful predictions, support the realist’s conviction that Einstein’s leap was at least in the right direction. But what now of the string theorists’ specific claims, as for example that space-time has ten dimensions (of which we can perceive only four)? Perhaps all that could be said, in the most favourable case, is that string theory is, or may come to be, one of the possible unifying logical systems relating General Relativity and the Standard Model. In this sense one might then claim it to be a scientific theory. 
Ed Witten once said that, while General Relativity and presumably the Standard Model are included in string theory, they can hardly be claimed as predictions of that theory. At best one might call them postdictions (Witten, 1999). But even that claim would be too generous. For although string theory predicts the particles of the Standard Model, it predicts also the existence of a host of other particles, of which there is no sign. 
On the other hand, he did say that supersymmetry is a genuine prediction of string theory, and suggested that, if supersymmetric partners of ordinary particles were to be found in future high-energy experiments, that would constitute a confirmation of string theory. This claim is however overly enthusiastic. In the first place, supersymmetry antedates string theory, and its experimental observation, while being good news for string theorists, who need supersymmetry to avoid causal paradoxes involving tachyons, would not specifically favour string theory above the earlier supersymmetric descriptive programs. In the second place, since string theory does not place any bounds on the masses of the postulated supersymmetric partners, a failure to detect any of them in experiments at LHC, for example, could always be shrugged off with the claim that the masses must then be higher, out of reach of the new machine. In short, string theory’s prediction that supersymmetry exists is not falsifiable.
String theorists do not think of themselves as mathematicians, and certainly not as philosophers (it is still the case that, for many scientists, ‘philosophizing’ is put on a par with daydreaming or sloppy reasoning). No, string theorists think of themselves as full-blooded physicists: they really want to say something about the furniture of the world, and they hope that their mathematics provides a glimpse of the world as it is. Could we honour such aspirations; and, if so, does this not imply that string theory differs somewhat from an explicatory program? The latter can explicate a theory, for example the Copenhagen Interpretation is an explicatory program that aims at expounding the notion of complementarity, and clarifying the distinction between classical observer and quantum object. On this view string theory is not concerned with explication, in the sense of making a certain intuition in an explanatory program clear (cf. Kuipers 2001). Rather, it aims at unification, that is the bringing together of apparently different explanations into one coherent logical or mathematical framework. If no new predictions can in practice be empirically tested, however, then we have to do here with a new sort of metaphysics: different empirically confirmed explanations could be underpinned by a mathematical theory whose essentially new ontological claims cannot be tested in the crucible of experiment.

4.  Envoi

To summarize, we have considered three theoretical systems, from three different historical periods, with three very different kinds of status. Galileo’s thought experiment, designed to destroy the Aristotelian dogma and instate his own in its place, has been shown to be logically deficient in both its negative and in its positive aspects. As a thought experiment, it leaves something to be desired, to say the least; that its major conclusion is in the case of free fall approximately correct, however, is not an accident: it is the result of real experiments performed by Galileo with steel balls and inclined planes. The EPR thought experiment threw Einstein’s a priori convictions about the nature of physical reality into sharp relief, but they made no impression at all on Bohr’s equally dogmatic espousal of the new natural order of things. However, unlike Galileo’s thought experiment, which merely pretended to take the place of, and to render superfluous, real experimentation, the idea of Einstein, Podolsky and Rosen, transformed by Bohm and Bell, led to a real experiment, the result of which would have disturbed Einstein, but which has had ramifications which have not yet been exhausted. Finally, the string theory of everything purports to be scientific, but it seems not to be susceptible to any serious experimental test, although it arrogates to itself all the empirical successes of the partial theories that it subsumes.   

Appendix 

Gendler analyzes Galileo’s thought experiment with especial acumen. A generalized version of Gendler’s analysis, couched in modern language, is that the following three statements are not consistent with one another:
[S1]	Accelerations of falling bodies are mediative. 
[S2]	Weight is additive.
[S3]	Accelerations of falling bodies are proportional to their weights. 

Here [S3] is an interpretation of what we called the strong form of the Aristotelian dogma. It implies, for instance, that if two falling bodies B(1) and B(2) have weights W(1) and W(2), with W(1) < W(2), then B(1) accelerates less than does B(2): a(1) < a(2). To show the inconsistency of [S1] — [S3], suppose that [S2] and [S3] are true. Then  
a(1) / W(1) = a(2) / W(2) = a(12) / W(12)  
and hence 
a(12) =  a(1) W(12) / W(1) =  a(1) [W(1) + W(2)] / W(1) > a(1)
a(12) =  a(2) W(12) / W(2) =  a(2) [W(1) + W(2)] / W(2) > a(2) 
which is inconsistent with [S1], that is, 
[S2] & [S3]    ¬ [S1]
By rejecting [S3], and by stating that the acceleration of all falling bodies of the same material is the same (thereby making [S1] trivially true), Galileo succeeds in avoiding a contradiction. 
As Gendler explains at length, an Aristotelian who wishes to parry the destructive force of Galileo’s argument, while retaining [S3], would have to deny [S1] or [S2], or both. He might for instance, against all reasonable common sense, postulate an essential difference in mechanical behaviour between bodies that are merely united (i.e. tied, or glued together) and bodies that are unified (i.e. truly one, whatever that means). He could for instance claim that weights and accelerations are additive for unified bodies, but that both are mediative for composite bodies whose pieces are merely tied together. Gendler’s view is that even a dyed-in-the-wool Aristotelian, on being confronted with these exotic ways of saving the Master’s theory, would recant and deny any reality to such a distinction between union and unification.
Thus Galileo is right in accepting [S1] and [S2] and rejecting [S3]. However, the fact is that [S1] and [S2] are true, that [S3] is false, but that nevertheless Galileo’s conclusion is also false. The statement that objects of the same material always fall at the same rate, Galileo’s escape from the Aristotelian contradiction, is not true. In other words, with the caveat that natural speed has been replaced by acceleration, the destructive part of Galileo’s thesis is correct, but the constructive part is incorrect. Such, at least, is the verdict given by Newton’s laws of motion and gravitation. To find out where exactly the flaw is situated, we will first scrutinize, from a Newtonian point of view, [S1] and [S2] successively.
The gravitational forces acting on bodies B(1) and B(2), i.e. their weights, are W(1) and W(2). Let their inertial masses be m(1) and m(2), respectively. From Newton’s second law of motion, F = ma, the accelerations are thus given by a(1) = W(1)/m(1) and a(2) = W(2)/m(2). It is part of Newton’s theory that:
[N1a] Forces, and hence in particular gravitational forces, are additive.² 
[N1b] Inertial masses are additive.

According to [N1a] and [N1b], the force acting on the composite made by tying bodies B(1) and B(2) together is W(1) + W(2), while the inertial mass of the composite is m(1) + m(2). The acceleration of the composite is 
 a(12) = [W(1)+W(2)] / [m(1)+m(2)] = [m(1) a(1)+m(2) a(2)] / [m(1)+m(2)]
If we suppose that   a(1) < a(2) ,  then the expression on the right is increased if a(1) is replaced by a(2), whereas it is decreased if a(2) is replaced by a(1). It follows therefore immediately that  a(1) < a(12) < a(2). In other words, Newton’s laws [N1a] and [N1b] imply [S1], i.e. accelerations are indeed mediative. 
As far as [S2] is concerned, the additivity of weights, one might at first think that it is equivalent to [N1b]. But this is not so, for in Newton’s system weight is a function of a body’s gravitational mass³ and the local gravitational field. However, since weight is a force, [S2] is implied by [N1a]. Conclusion: Newton underwrites both [S1] and [S2]. Thus, on pain of falling into an Aristotelian contradiction, [S3] must be wrong. 
This is all of course exactly in accordance with Galileo’s reasoning. So where did Galileo go wrong? It is precisely in the constructive part of his thought experiment, where he replaces [S3] by the statement that all weights fall with the same acceleration. For this statement is neither the only possibility to escape from the Aristotelian contradiction, nor is it correct for bodies falling to the surface of the earth. The reason is that the earth’s gravitational field is not uniform, since the earth is a spheroid and the weight of a body depends not only on its gravitational mass, but also on how high it is above the surface of the earth.
The acceleration of a falling body at two earth-radii from the centre of the earth is only one quarter what it is when the body is close to the earth’s surface. As a matter of fact, in this case the acceleration is precisely proportional the weight, but the weight is a function of the distance between the centre of mass of the earth and that of the body. The rate at which the body’s velocity increases is independent of its gravitational mass, but dependent on its position. Galileo’s solution is only correct in a uniform gravitational field, and that the earth does not have such a uniform field is a brute empirical fact.
Of course, the differences in accelerations are very small for a difference in elevation of a few metres. Be that as it may, the size of the effect is not at issue here. In the presence of a homogeneous gravitational field (and in the absence of air), different bodies would indeed fall at the same rate. However, this is not an a priori statement about the way bodies fall; indeed, given that the earth’s gravitational field is inhomogeneous, it is not even an accurate statement. Logically speaking, Galileo’s claim that the failure of Aristotle’s dogma implies that all bodies fall at the same rate is a non sequitur; and moreover empirically it is in fact not the case that all bodies fall at the same rate. 
At this juncture, a modern apologist for Galileo might remark that the inhomogeneity of the gravitational field could be seen as a disturbing factor, on a par with air friction. It requires after all little effort to postulate a homogeneous gravitational field in order to reinstate Galileo’s thought experiment in all its pristine splendour. Our answer to this apologist would be that one can only postulate such a field within the framework of a theory of gravitation. For one can only identify the inhomogeneity of a gravitational field as a disturbing factor if one knows enough about it. Galileo lacked such knowledge. He did not have a theory of gravitation, at least not one in which gravitational forces (i.e. weights) drop off as the inverse square of the distance from the centre of the earth. Such a theory was only invented a generation later by Newton, who was able to test it quantitatively with the help of his calculus; he was able to compare the motions of falling apples with that of the moon, as it ‘falls’ endlessly in its month-long orbit around the earth.
The same goes of course for any other putative disturbing factor: one needs a theory to be able coherently to postulate conditions under which it would be absent. It would be circular to require all unspecified disturbing factors to be absent, so that Galileo’s law of falling bodies be correct. Since physical laws are tested by their empirical implications, it is not only that thought experiments are subordinate to the theories which they inspire (and by which they are inspired), but it is also the case that disturbing factors must likewise be considered, controlled and rendered manageable within a theoretical framework. 
In short, Galileo’s conclusion that all bodies fall from the same height at the same rate, if air friction is negligible, and there are no appreciable electrostatic or magnetic forces at work, is approximately correct; but this fact is not a consequence of pure logic, despite Salviati’s ease in discomfiting the Aristotelian straw-man, Simplicio. What is more, a good case can be made that Aristotle was interested in falling bodies in situations where fluid viscous forces are important: 

… in the ratio which their magnitudes bear to one another. … In moving through plena it must be so. (vide supra)  

In this case Newton’s law of motion reads 
ma = mg – κ v
m being the mass (strictly speaking, the inertial mass on the left and the gravitational mass on the right), g the acceleration due to gravity (assumed constant),  v the instantaneous velocity, and κ the frictional coefficient due to viscous drag. This equation has the following solution for the velocity: 
v = [ 1 – exp (– κ t /m ) ] mg / κ
and this yields, for the terminal velocity, 
v(term) = W/ κ
where W = mg is the weight. Evidently, if we interpret Aristotle’s natural speed as terminal velocity in a medium (in plena), then Aristotle is right that the speed of a body is proportional to its weight! But what has happened then to the Galilean contradictio? The three statements [S1] – [S3] are replaced by 

[S1']	Terminal velocities of falling bodies are additive (not mediative).
[S2']	Weights are additive.
[S3']	Terminal velocities of falling bodies are proportional to their weights. 

Due to the change [S1]  [S1'], there is now no inconsistency. Twin sisters suspended from one parachute fall twice as quickly as one sister. The Stagyrite is vindicated! 
It is not my contention that Aristotle had the above interpretation in mind when enunciating what I have called the strong dogma, only that such an interpretation is possible, and it serves, among other things, to throw further doubt on the worth of Galileo’s thought experiment. The conclusion, of course, applies only in very special circumstances, namely for two bodies of the same shape and size (and then only if the fluid motion is laminar rather than turbulent).  Aristotle did write

We see that bodies which have a greater impulse either of weight or  lightness, if they are alike in other respects, move faster over an equal space, and in the ratio which their magnitudes bear to one another. (Vide supra, my italics)

But the situation can be more complicated. Consider, for example, skydiving. In this sport one jumps from an aeroplane but does not immediately open a parachute. Now the twins fall at the same terminal speed as does one sister alone! And how fares the child of one of the sisters, if he is pushed out of the plane too? He does not fall as quickly, of course. But if he is fastened with a rope to his mother? Will not the mother, the fast body, be partially braked by her son, the slow body? Do you not agree with me in this opinion? 
Simplicio: You are unquestionably right. 
How do we explain this? Clearly we have to replace v(term) = W/ κ by 
V(1) = W(1) / κ(1)      V(2) = W(2) / κ(2)      V(12) = W(12) / κ(12)
where V, W and κ are the terminal velocities, the weights and the friction coefficients, respectively, for the bodies (1) (the mother), (2) (the son) and (12) (the system comprising the mother and son, fastened together). The weights are proportional to the volumes of the bodies, but the frictional coefficients are proportional to their effective cross-sections, which increase approximately as the power 2/3 of the weights. In this more complicated scenario, we have in place of the statements [S1] — [S3] the following  
[S1'']	Terminal velocities of falling bodies are (again) mediative. 
[S2'']	Weights are additive.
[S3'']	Terminal velocities of falling bodies are functions of weights and of cross-sections. 
The last form, vague as it is, can be extended to the case of bodies falling under turbulent conditions.  

University of Groningen, The Netherlands. 
(Email: atkinson@phys.rug.nl)

ENDNOTES

¹ Bell added a characteristic footnote to Einstein’s famous dictum, ”Raffiniert ist der Herrgott, aber boshaft ist er nicht”, to the effect that, had he been still alive in 1982, the year of Aspect’s experiment, Einstein might well have had reason to suspect malice on the part of the Almighty. 
² In general, forces obey the rules of vector addition, but in the present case of forces that are parallel to one another, vector reduces to scalar addition.
³ The gravitational mass of a body may be defined as the coefficient of proportionality between the body’s weight and the gravitational field in which it is situated. This is conceptually different from the inertial mass of the same body, which is the coefficient of proportionality between a force acting on the body (for example its weight) and its resulting acceleration. That the two kinds of mass are numerically equal has been experimentally tested to high accuracy. This equality was built into the very foundations of Einstein's general theory of relativity. Because of the equality, two bodies of different gravitational masses, if placed in the same gravitational field (with no retarding forces), will experience the same acceleration, precisely because the ratio of the two bodies’ inertial masses is the same as the ratio of their gravitational masses. (Atkinson and Peijnenburg 2002)

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