UN CO RR EC TE D PR OO F Chapter 26 1 "New Water in Old Buckets: 2 Hypothetical and Counterfactual 3 Reasoning in Mach's Economy 4 of Science" 5 Lydia Patton 6 I owe a debt to Friedrich Stadler and to the Institute Vienna 7 Circle for inviting me to contribute a talk to the wonderful Ernst 8 Mach centenary conference, held in June 2016, and to 9 contribute a paper for the volume here. At the conference, I 10 received illuminating questions from Erik Banks, Don Howard, 11 Elisabeth Nemeth, Thomas Uebel, and others, which were 12 instrumental in subsequent improvements to the paper. Since 13 then, I gave this paper as the annual Joint Lecture of the 14 Departments of Philosophy and of HPS at Indiana University. 15 Comments from Amit Hagar, Jordi Cat, Jutta Schickore, Kirk 16 Ludwig, Gary Ebbs, Kate Abramson, Mark Kaplan, and Adam 17 Leite gave a new impetus to the project and helped me to find a 18 unifying thread, or so I hope. Sean Morris issued a kind 19 invitation to present the paper at Metropolitan State University, 20 at which Sergio Gallegos, Elizabeth Goodnick, Vijay 21 Mascarenhas, Daniel Krasner, and of course Sean Morris 22 himself pushed the narrative of the paper in novel directions 23 and identified threads of that narrative I hadn't considered. I am 24 grateful to all of those who participated in discussion and 25 analysis of the paper, and regret only that I haven't made better 26 use of their excellent suggestions. 27 Abstract Ernst Mach's defense of relativist theories of motion in Die Mechanik AQ1 28 involves a well-known criticism of Newton's theory appealing to absolute space, and 29 of Newton's "bucket" experiment. Sympathetic readers (Norton 1995) and critics 30 (Stein 1967, 1977) agree that there's a tension in Mach's view: he allows for some 31 constructed scientific concepts, but not others, and some kinds of reasoning about 32 unobserved phenomena, but not others. Following Banks (2003), I argue that this 33 tension can be interpreted as a constructive one, springing from Mach's approach 34 L. Patton () Virginia Tech, Blacksburg, VA, USA e-mail: critique@vt.edu © Springer Nature Switzerland AG 2018 F. Stadler (ed.), Ernst Mach – Life, Work, Influence, Vienna Circle Institute Yearbook 22, https://doi.org/10.1007/978-3-030-04378-0_26 UN CO RR EC TE D PR OO F L. Patton to scientific reasoning. Mach's "economy of science" allows for a principled 35 distinction to be made, between natural and artificial hypothetical reasoning, and 36 Mach defends a division of labor between the sciences in a 1903 paper for The 37 Monist, "Space and Geometry from the Point of View of Physical Inquiry". That 38 division supports counterfactual reasoning in Mach's system, something that's long 39 been denied is possible for him. 40 26.1 Debating Mach's Principle: Empiricism, 41 Counterfactual Reasoning, and Economy 42 The history of Mach's Principle has been tied closely to the history of the theory of 43 relativity, and thus, more broadly, to the history of relativistic theories of motion. On 44 relativistic theories, a body in motion must be in motion relative to some other body, 45 and not to absolute space. As Norton notes, acceleration often is a "stumbling block" 46 for relativistic theories.1 For instance, in the "two spheres" thought experiment, two 47 spheres are separated by a string. The spheres rotate around each other. When they 48 are rotating quickly with respect to each other, the string is taut. When the spheres 49 slow down, the string slackens. The string reflects the existence of a force. But on a 50 relativistic theory, since the two spheres are rotating only with respect to each other, 51 there is nothing with respect to which the entire two sphere system is rotating. But 52 if the system is not in motion with respect to any body outside it, then what is the 53 source of the tension on the string? 54 In Mach's comments on Newton's reasoning, in the famous 'bucket experiment', 55 Mach criticizes Newton's appeal to absolute space and time. Newton argues that 56 the centrifugal force in the bucket experiment, and the force on the string in the 57 two spheres experiment, result from the rotation of the bucket and the spheres with 58 respect to absolute space. Mach asserts three things in response: (1) that "no one 59 is competent to predicate things about absolute space and absolute motion", since 60 these are not facts but creatures of thought,2 (2) that the water in Newton's bucket, 61 or the two rotating spheres) could be rotating with respect to some more distant 62 inertial frame, such as the fixed stars; and (3) that the physicist's task in evaluating 63 these experiments is to analyze the facts before her, and not "arbitrary fictions of the 64 imagination".3 Mach concludes that "the phenomena of centrifugal forces compel 65 us not to postulate an absolute reference-system but to recognize the law of inertia 66 as a mere empirical generalization about the motions of bodies relative to the fixed 67 stars".4 68 1Norton, John. "Mach's Principle Before Einstein", in: Julian Barbour and Herbert Pfister (Eds.). Mach's Principle: From Newton's Bucket to Quantum Gravity. Einstein Studies, Vol. 6. Boston: Birkhäuser. 1995, pp. 9–57. 2Mach, Ernst. The Science of Mechanics. Chicago: Open Court. 1960 / 1883, p. 280. 3Mach, Science of Mechanics (1960), loc. cit., p. 284. 4DiSalle, Robert. "Carl Gottfried Neumann", Science in Context 6, 1, 1993, p. 350. UN CO RR EC TE D PR OO F 26 "New Water in Old Buckets: Hypothetical and Counterfactual Reasoning. . . On Norton's reading, Mach's principle is that "the motion of a body is caused 69 entirely by an interaction with other bodies" (op. cit., p. 10). While Mach himself 70 did not formulate this principle, scientists after him, most famously Einstein, took 71 it as a methodological or physical principle. Moritz Schlick appears to have been 72 among the first to formulate the principle. He "referred to Mach's general proposal 73 for a relativity of all motion, from which, Schlick noted, it follows that 'the cause 74 of inertia must be assumed to be an interaction of masses'".5 Thus, prima facie, 75 we may read Mach's principle either as a simple statement of a relativist theory of 76 motion, or as a physical statement about the cause of inertia. 77 Since its formulation, the principle has been controversial. One set of views 78 of its significance appeals to Mach's so-called "phenomenalism". On this reading, 79 which we can attribute to Howard Stein and others, Mach is guilty of an "abusive 80 empiricism" that sets unreasonable criteria for scientific reasoning. All conceptual 81 or formal notions used in science must be observable, a requirement Stein finds 82 unreasonable. 83 Another view, found in Norton (1995), is more sympathetic. On Norton's 84 reading, Mach's criticism of Newton is not intended to lay out a new physical 85 argument or mechanism. Rather, Mach intended only to redescribe Newton's 86 experiments without the term "space," which had an indeterminate meaning. He 87 did not intend to give a causal argument or mechanism. As Norton points out, this 88 reading leads to a puzzle. If Mach didn't intend to give a causal argument, why does 89 he accept without comment the many people who said, in his presence or in works 90 he read, that he did? 91 As Norton points out, though, what Mach meant by a "causal" argument was 92 simply a functional dependence between two observed variables (Norton, op. cit., 93 pp. 27–29). In the case of the bucket experiment, that functional dependence can be 94 captured by pointing out that centrifugal forces are produced by the relative rotation 95 of the water in the bucket with respect to "the masses of the earth and the other 96 celestial bodies".6 97 As Norton notes, even with this heroic save, the original puzzle remains. Einstein 98 wrote to Mach saying that "inertia has its origins in a kind of interaction of 99 bodies, quite in the sense of your reflections on Newton's bucket experiment" 100 (cited in Norton, op. cit., p. 29). But on Norton's reading, Mach's reflections quite 101 deliberately didn't amount to a general claim about inertia or its sources. Rather, 102 they amount to a re-description of the bucket experiment itself without using the 103 notion of "space" to explain the origins of inertia. As Norton points out, "Einstein's 104 notion of causal interaction extended well beyond the simple functional relations 105 of phenomena. It included relations on hypothetical and counterfactual systems of 106 precisely the type denounced by Mach" (op. cit., p. 29). 107 5Schlick, Moritz. "Die philosophische Bedeutung des Relativitätsprincips," Zeitschrift für Philosophie und Philosophische Kritik 159, 1915, p. 171; Norton, op. cit., p. 47. 6Cited Norton, op. cit., p. 29. UN CO RR EC TE D PR OO F L. Patton Here, there is an unexpected agreement between Stein's critical and Norton's 108 sympathetic reading. As DiSalle summarizes Stein's position, 109 Stein characterized Mach's philosophical opposition to Newton as "abusive empiricism"- 110 meaning by this not merely the prejudice against theories involving unobservable entities 111 or far-reaching counterfactual implications, but, more important, the absurd willingness to 112 accept empirically unmotivated hypotheses about cosmic geography, boundary conditions, 113 and so on, just to avoid theories of that sort.7 114 Stein makes a point closely related to Norton's: that Mach's pronouncements on 115 hypothetical reasoning seem inconsistent. 116 Mach's opposition to the kinetic-molecular theory is based upon the fact that, as he puts it, 117 atoms are "mental artifices." But what about perfectly ordinary objects? "Ordinary matter," 118 Mach says, is a "highly natural, unconsciously constructed mental symbol for a relatively 119 stable complex of sensational elements"; the only distinction he finds to the disadvantage of 120 atoms is that of the "natural unconscious construction" versus the "artificial hypothetical" 121 one. To conclude, as Mach does, on the basis of this distinction, that atomic theories should 122 eventually he replaced by some "more naturally attained" substitute is is very strange: not 123 only is the argument at right angles to Mach's view of the "economic" objective of science, 124 it actually accords a preference to the instinctive and unconscious over the conceptual and 125 deliberate mental processes.8 126 On the one hand, Mach seems to be arguing that "artificial entities" like "absolute 127 space", "atoms", and the like are to be eliminated from physical reasoning. On the 128 other hand, Mach appears willing to accept "empirically unmotivated" claims about 129 cosmic boundary conditions, the global conditions of matter, and the like, in order 130 to avoid absolute space and the atomic theory. 131 Moreover, as Norton observes, Mach does not object to Einstein's interpretation 132 of Mach's reasoning about the bucket experiment. But Einstein's interpretation has 133 it that Mach is making a general point about the origins of inertial forces, which 134 implies that Mach is making a strong causal claim beyond functional dependence. 135 That stronger claim involves hypothetical and counterfactual reasoning that is 136 supposed to be anti-Machian. 137 The accounts of Norton and Stein suggest a criticism of Mach. Mach uses his 138 well-known reasoning about the "naturalness" of concepts and the "economy" of 139 science to oppose concepts like absolute space and atoms. But when it is a matter 140 of his own preferred concepts or results, like ordinary matter, global boundary 141 conditions, or the origins of inertial force, Mach appears willing to violate his own 142 criteria. Those criteria then appear to be ad hoc, not scientific. That is why Stein 143 calls Mach's empiricism "abusive": from a Newtonian perspective, Mach is picking 144 and choosing which "conceptual monsters" he will allow within his system, and 145 7DiSalle, Robert. "Reconsidering Ernst Mach on space, time, and motion", in: David B. Malament (Ed.), Reading Natural Philosophy. Chicago: Open Court 2002, pp. 169–170. 8Stein, Howard. "Some Philosophical Prehistory of General Relativity", in: John Earman, Clark Glymour, and John Stachel (Eds.). Foundations of Space-Time Theories (Minnesota Studies in the Philosophy of Science, vol. VIII), Minneapolis: University of Minnesota Press, 1977, pp. 14. See also Stein, Howard. "Newtonian Space-Time", Texas Quarterly 10, 1967, pp. 174–200. UN CO RR EC TE D PR OO F 26 "New Water in Old Buckets: Hypothetical and Counterfactual Reasoning. . . violating his own criteria to do so. And, as we've seen, Norton agrees that Mach's 146 position appears internally inconsistent in this way. 147 26.2 A Division of Labor Within Mach's Economy of Science 148 If your friends and your foes agree that you have a fault, it may be that you do have 149 that fault. Or, it may be that, despite the heroic efforts even of your sympathetic 150 readers, there is still some misunderstanding. 151 To a certain extent, Mach has been misunderstood. But the misunderstanding 152 stems, not from his physics, but from his account of the "economy of science", found 153 in Mach's work The Development of Mechanics, Presented Historico-Critically.9 154 This work is the site of, and crucial context for, Mach's interpretation of Newton's 155 bucket experiment. 156 A thorough reading of The Development of Mechanics leads one to question the 157 usual reading of Mach on the economy of science. According to that reading, Mach 158 argues that the "principles" of mechanics, like the principles of least action or of 159 the straightest path, are "economical" in the sense that they allow for computational 160 efficiency or ease of memory. Many scientific results are encapsulated in the least 161 action principle, for instance, but humans don't have memories that can store and 162 recall all of those results with ease. But the least action principle takes a few minutes 163 to memorize. Moreover, it encapsulates more results. Thus, if the principle is used in 164 an inference, it allows for more computational power: we can more easily determine 165 what is derivable from what. 166 I don't disagree with this interpretation as a partial reading of Mach on the 167 economy of science. However, Mach's account is much richer than this. Mach 168 presents the economy of science, not just as a static set of principles that allow us 169 to compute more easily, but also as a system that develops over time. That system is 170 9The original impetus for this paper came from a discussion of that "economy" on the HOPOS listserv, with references provided from Erik Banks, Don Howard, Alan Richardson, and others, to the works of Banks himself, of Margaret Schabas, and of others. This discussion led me to look more carefully at Mach's work The Development of Mechanics, Presented Historico-Critically (Die Mechanik in ihrer Entwickelung historisch-kritisch Dargestellt). This work is usually mistranslated, especially the title. As Don Howard has emphasized in personal communication, the second part of the title, Presented Historico-Critically, responds to the nineteenth century tradition of Biblical criticism, and to the reception of hermeneutic, humanistic methods of Biblical criticism by historians including Karl-David Ilgen and Julius Wellhausen. Mach would have expected nineteenth century readers to be aware of this crucial context. His presentation of mechanics in this context implies strongly that the achievements of mechanics, like the doctrines of the Church, are products of human activity. The discussion in Nemeth, Elisabeth. "Freeing up one's point of view – Neurath's Machian Heritage Compared With Schumpeter's", in: Elisabeth Nemeth, Thomas E. Uebel, Stefan W. Schmitz (Eds.) Otto Neurath's Economics in Context. Vienna Circle Institute Yearbook 13. Dordrecht: Springer 2007, and in personal discussions that have been very illuminating, emphasize the role of "freeing up one's point of view" for Mach. I explore that notion in sect. 3, below, from a distinct but complementary standpoint. UN CO RR EC TE D PR OO F L. Patton dynamic. Certain ways of proceeding will promote the economy of science, while 171 certain others will detract from it. 172 Typology of the Economy of Science Methods that Promote Economy The Effects of These Methods "Instinctive knowledge" Transparency The method of differences Empirical fruitfulness The law of continuity of experience The ability to disregard details Relations of the whole Computational power Minimization Completion of experience 173 One can identify ten elements in The Development of Mechanics relevant to the 174 economy of science. The elements on the left hand side of the chart above, the 175 "methods", are ways to promote the economy of science. The elements on the right 176 hand side, the "effects", are results of employing these methods. Computational 177 power, according to this chart, should be seen not (just) as an absolute property of a 178 principle of science, but as the result of employing an economical method. 179 One example of an economical method is the "method of differences", which 180 Mach discusses in his lecture "On the Conservation of Energy". Experience teaches 181 us that some sensed elements of the world are interdependent pressure, volume, 182 and heat in a gas, for example. But there are differences as well as dependence. 183 Facts may be so nearly related as to contain the same kind of [elements], but the relation be 184 such that the [elements] of the one differ from the [elements] of the other only by the number 185 of equal parts into which they can be divided . . . if rules can be given for deducing from one 186 another the numbers which are the measures of these [elements], then we possess in such 187 rules the most general expression of a group of facts . . . This is the goal of quantitative 188 investigation . . . what we have found is that between the [elements] of a group of facts . . . 189 a number of equations exists. The simple fact of change brings it about that the number of 190 these equations must be smaller than the number of the [elements]. If the former be smaller 191 by one than the latter, then one portion of the [elements] is uniquely determined by the other 192 portion.10 193 Theories should be analyzed in terms of their observable "elements".11 The goal 194 of science is to show that some elements of a theory (variables such as pressure and 195 volume) are dependent on other elements of the theory (variables such as motion and 196 density). To show that the independent group is smaller than the dependent group 197 is to promote economy in science. Mach calls this the "method of differences". As 198 Banks notes, 199 The tension between the [observed] elements and their ordering into this general manifold 200 of space, time, and matter is a general problem in Mach's philosophy of nature. The divide 201 10Mach, Ernst. "On the Principle of the Conservation of Energy", in: Popular Scientific Lectures, Thomas McCormack (Ed.). Chicago: Open Court 1895, pp. 180–181. First published in The Monist (1894). 11See Banks, Erik. Ernst Mach's World Elements: A Study in Natural Philosophy. Dordrecht: Kluwer. 2003, for a wealth of discussion of Mach's "world-elements". UN CO RR EC TE D PR OO F 26 "New Water in Old Buckets: Hypothetical and Counterfactual Reasoning. . . falls between his heraclitean view that the elements are transitory unique events, arising 202 and vanishing and possessing always an individual existence, and his view that space, 203 time, and matter, however unreal they may be on a fundamental level, represent for Mach 204 economical permanencies that must be acknowledged as a task of science . . . Mach said that 205 he considered the real facts of nature to be the existence of "differences" or inequalities . . . 206 Mach's elements are the differences of state in the world and, by a careful tracking of 207 their effects on one another, the determinations of the rates and magnitudes of those effects, 208 Mach thought one could deduce the existence of independent potential sources and relations 209 of intensity from this raw data by finding orderings in it.12 210 The tension Banks notes, between "transitory unique events, arising and vanish211 ing and possessing always an individual existence", and "his view that space, time, 212 and matter, however unreal they may be on a fundamental level, represent for Mach 213 economical permanencies", is precisely the tension identified by Norton and Stein 214 as a problem for Mach. But Norton and Stein see the tension as a problem in a 215 negative sense: as an inconsistency, or as an un-scientific element in Mach's view. 216 On Banks's reading, we can see the tension as a "problem" in a positive sense: one 217 aim of science is to resolve the tension between transitory world-elements that are 218 experienced only once, and the "economical permanencies" of space, time, matter, 219 and the like. 220 Concepts like space, time, and matter can be economical in distinct ways. In The 221 Development of Mechanics, Mach provides a thoroughgoing, if disjointed, account 222 of how the economy of science is promoted. And this account can serve as a way 223 to explain why, for him, some concepts can be used to support hypothetical systems 224 and counterfactual reasoning, and some should not. Such an explanation will not 225 make all of Mach's stated positions consistent. But it will clear up the question, 226 asked by Norton, Stein, and others, of why Mach seemed to allow for such reasoning 227 in some cases and not in others. 228 Stein argues that 229 Mach says, in a famous and true remark, that the world is given to us only once, and he 230 concludes that it is "not permitted to us to say how things would be" if the world were to 231 be other than it is . . . But Mach does not make it a general rule for science that in every 232 statement based upon experience there should appear a list of all the circumstances over 233 which we have no control (the universe being given only once), in order to avoid seeming 234 to claim that we know that the statement would continue to be true even if these things were 235 otherwise. Such a rule would not only grievously violate Mach's "economy of thought", it 236 would make science impossible (op. cit. 1977, p. 15). 237 It's rare, fortunately, to have the dreadful responsibility of contradicting Howard 238 Stein. Even so, Mach does make it a general rule for science that we cannot 239 neglect the rest of the world even when concentrating only on two observed facts or 240 elements. 241 even in the simplest case, in which apparently we deal with the mutual action of only two 242 masses, the neglecting of the rest of the world is impossible. Nature does not begin with 243 elements, as we are obliged to begin with them. It is certainly fortunate for us, that we can, 244 12Banks 2003, p. 239. UN CO RR EC TE D PR OO F L. Patton from time to time, turn aside our eyes from the overpowering unity of the whole, and allow 245 them to rest on individual details. But we should not neglect ultimately to complete and 246 to correct our views by a thorough consideration of the things which for the time being 247 we left out of account . . . In fact, science can accomplish nothing by the consideration of 248 individual facts: from time to time it must cast its glance at the world as a whole. Galileo's 249 laws of falling bodies, Huygens's principle of vis viva, the principle of virtual velocities, 250 nay, even the concept of mass, could not, as we saw, be obtained, except by the alternate 251 consideration of individual facts and of nature as a totality.13 252 As we learn from Banks, the tension between the elements, the heraclitean world 253 "given only once," and the consideration of "nature as a totality" is a fundamental 254 problem for Mach's account of science. But it is a problem that Mach sees as one 255 we must try to solve, and he thinks that the framework of a solution must be built 256 into science itself. 257 Mach's "law of continuity of experience" is a crucial element of that framework. 258 This "law" is in fact more of a method or rule. It states that any principle of 259 mechanics must be considered, not as a universal and necessary proposition, but 260 as an assertion that is being checked constantly by experience. 261 The most important result of our investigations is that precisely the apparently simplest 262 mechanical principles are of a very complicated character, that these principles are founded 263 on uncompleted experiences, nay on experiences that never can be fully completed, that 264 practically, indeed, they are sufficiently secured, in view of the tolerable stability of our 265 environment, to serve as the foundation of mathematical deduction, but that they can by no 266 means themselves be regarded as mathematically established truths but only as principles 267 that not only admit of constant control by experience but also require it (op. cit., 1883, pp. 268 237–8). 269 The law of continuity is the basis of Mach's reasoning about the "completion" of 270 experience through science, one of the products of the economy of science. 271 The function of science, as we take it, is to replace experience. Thus, on the one hand, 272 science must remain in the province of experience, but, on the other, must hasten beyond 273 it, constantly expecting confirmation, constantly expecting the reverse. Where neither 274 confirmation nor refutation is possible, science is not concerned. Science acts and acts only 275 in the domain of uncompleted experience. Exemplars of such branches of science are the 276 theories of elasticity and of the conduction of heat, both of which ascribe to the smallest 277 particles of matter only such properties as observation supplies in the study of the larger 278 portions. The comparison of theory and experience may be farther and farther extended, as 279 our means of observation increase in refinement (op. cit., 1883, p. 490). 280 This passage alone contradicts the reading of Mach as a reductive phenomenalist 281 or as an "abusive empiricist" about science. Neither would suggest that science 282 "must hasten beyond" experience. Neither would suggest, either, that science "acts 283 and acts only in the domain of uncompleted experience". Certainly, Mach says that 284 our completion of experience is based on observed properties and relations. But, 285 immediately following this passage, Mach explains that those observed properties 286 and relations can be supplemented, and even can be hypothetical: 287 13Die Mechanik in ihrer Entwickelung historisch-kritisch Dargestellt. Leipzig: F. A. Brockhaus. 1883, p. 235. All translations cited as from the 1883 edition are by L. Patton for this essay. UN CO RR EC TE D PR OO F 26 "New Water in Old Buckets: Hypothetical and Counterfactual Reasoning. . . When we mentally add to those actions of a human being which we can perceive, sensations 288 and ideas like our own which we cannot perceive, the object of the idea we so form is 289 economical. The idea makes experience intelligible to us; it supplements and supplants 290 experience. This idea is not regarded as a great scientific discovery, only because its 291 formation is so natural that every child conceives it. Now, this is exactly what we do 292 when we imagine a moving body which has just disappeared behind a pillar, or a comet 293 at the moment invisible, as continuing its motion and retaining its previously observed 294 properties. We do this that we may not be surprised by its reappearance. We fill out the 295 gaps in experience by the ideas that experience suggests.14 296 From Mach's statements thus far, we can make a distinction fundamental to 297 his economy of science, between natural and artificial hypothetical reasoning. 298 The distinction is not found in Mach himself, but it is, I believe, a minor and 299 straightforward inference from his own account. 300301 Natural hypothetical reasoning "fills out the gaps in experience by the ideas that 302 experience suggests" with "sensations and ideas like our own". 303 Artificial hypothetical reasoning either: 304 (a) Does not merely fill in gaps in experience, but rather postulates a speculative kind of 305 experience, or 306 (b) Uses sensations and ideas that do not resemble our own to assemble speculative 307 systems. 308 The application of this distinction to Newton's bucket, and to the two spheres 309 experiment, is clear. If we appeal to the "fixed stars" or to global boundary 310 conditions that can be cashed out in terms of observations like our own, then we are 311 engaging in natural hypothetical reasoning. We are filling in the gaps of experience, 312 but with experiences that we ourselves could have had. On Mach's reading, if we 313 appeal to Newtonian absolute space and time, then we must construct a speculative 314 hypothetical system based on the sensorium of God. Such an appeal does not fill 315 in gaps in human experience – it goes outside any possible human experience.15 316 Newton's account of the bucket experiment is artificial hypothetical reasoning, 317 according to this distinction. 318 The first conclusion for which I want to argue is that, with a deeper understanding 319 of Mach's economy of science, the vicious tension identified by Stein and Norton 320 can become the more virtuous tension identified by Banks. The question, for Mach, 321 is how to fill out the gaps left by a mere description of observation. To move from 322 the punctiform mosaic of world-elements to a continuous, coherent physical system, 323 we must complete experience with some form of hypothetical reasoning. But that 324 14Mach, Science of Mechanics (1883), loc. cit., pp. 490–1. 15Newton did not appeal only to the sensorium of God in giving an account of absolute space and time. For instance, Newton argued that we could model absolute time using the notion of a "fluxion" of natural magnitudes from his calculus, a notion that is robustly observable (Newton, Isaac. "Methodus fluxionum et serriarrum infinitarum cum ejusdem applicatione ad curvarum geometriam", in: J. Castillion (Ed.). Opuscula mathematica, philosophica et philologica, Volume 1. Lausanne and Geneva. 1744.). UN CO RR EC TE D PR OO F L. Patton reasoning must be natural, in Mach's terms. It must complete experience using the 325 instruments that experience suggests. 326 26.3 Mathematics and Mach's Principle 327 The account in sect. 2 above appears to suggest the following way of reading 328 Mach. Mach does not rule out a certain kind of hypothetical reasoning, "natural" 329 hypothetical reasoning. And he does make a principled distinction between natural 330 and artificial hypothetical reasoning, which should clear him of the charge of 331 "abusive" empiricism. 332 Nonetheless, it seems that Mach still does not allow for counterfactual mathe333 matical reasoning, for which we might fault him. After all, Mach was among those 334 engaged in debate over non-Euclidean geometry in the nineteenth century, and he 335 was well aware of the work of Eugenio Beltrami, Nikolai Lobachevsky, Hermann 336 von Helmholtz, Bernhard Riemann, and Carl Friedrich Gauss. Mach's principle is 337 part of the background to relativity theory, but so are the advances in group theory, 338 continuous transformations, non-Euclidean geometry, and allied fields. But these 339 areas all involve robustly counterfactual reasoning – for instance, about the rigid 340 motions possible for a body in certain circumstances, or the paths a body can take 341 through space, or the different ways of determining metric relationships. 342 In the above section, we saw that Mach's account of scientific reasoning is 343 based on the tension between "transitory unique events, arising and vanishing and 344 possessing always an individual existence", and "his view that space, time, and 345 matter, however unreal they may be on a fundamental level, represent for Mach 346 economical permanencies".16 Moreover, as Banks remarks, the observed elements 347 for Mach are real, while spatiotemporal relationships are not (necessarily): "A full 348 appreciation for Ernst Mach's doctrine of the economy of thought must take account 349 of his direct realism about particulars (elements) and his anti-realism about space350 time laws as economical constructions".17 351 Twenty years after the publication of the first edition of The Development of 352 Mechanics, we find Mach engaging in an extended reflection on the distinction 353 between physiological, physical, and metric space, in an essay for The Monist, 354 "Space and Geometry from the Point of View of Physical Inquiry".18 355 Our notions of space are rooted in our physiological constitution. Geometric concepts are 356 the product of the idealisation of physical experiences of space. Systems of geometry, 357 finally, originate in the logical classification of the conceptual materials so gathered. 358 16Banks 2003, p. 239. 17Banks, Erik. "The Philosophical Origins of Mach's Economy of Thought", in: Synthese 139, 1, 2004, p. 23. 18The German version of this essay is reprinted as the twenty-second chapter of Erkenntnis und Irrtum (1905). The version in The Monist was translated by Thomas McCormack. UN CO RR EC TE D PR OO F 26 "New Water in Old Buckets: Hypothetical and Counterfactual Reasoning. . . All three factors have left their indubitable traces in modern geometry. Epistemological 359 inquiries regarding space and geometry accordingly concern the physiologist, the psychol360 ogist, the physicist, the mathematician, the philosopher, and the logician alike, and they 361 can be gradually carried to their definitive solution only by the consideration of the widely 362 disparate points of view which are here offered.19 363 Here, we find another aspect of Mach's "economy" of science. "Economy" can 364 refer to a way of minimizing or saving, as in effort, computation, or money. But it 365 can refer equally well to a managed economy, in which there is a division of labor, 366 for instance, within a society or community. As Schabas details, this latter notion of 367 an economy had a broad currency by the end of the nineteenth century.20 368 Mach refers to the division of labor in the epistemology of space and of geometry. 369 The physicist is concerned with what happens once: with the specific setup of 370 an experiment, and with the observed phenomena of the world before us. The 371 physiologist deals with how these Heraclitean sensations arise from an interaction 372 between our physical constitution and the events in question. The geometer is not 373 restricted to the private sphere of "individual intuitive space", but can move freely 374 from physiological to physical to metric space, which gives room for abstract and 375 even counterfactual speculation.21 Such speculation is not robustly physical – Mach 376 certainly spoke against it in some physical contexts. But in this essay, he argues that 377 the division of labor in the epistemology of space allows the physicist to appreciate 378 geometrical and hypothetical reasoning: 379 It little accords with the principles of a physicist to make suppositions regarding the deport380 ment of geometrical constructs in infinity and non-accessible places, then subsequently to 381 compare them with our immediate experience and adapt them to it. He prefers . . . to regard 382 what is directly given as the source of his ideas, which he considers applicable also to 383 what is inaccessible until obliged to change them. But [the physicist] too may be extremely 384 grateful for the discovery that there exist several sufficing geometries, that we can manage 385 also with a finite space, etc., grateful, in short, for the abolition of certain conventional 386 barriers of thought (Mach, "Space and Geometry", loc. cit., p. 1). 387 The person who took trouble to type the words above, and then send them to Paul 388 Carus for publication in The Monist, is not the Ernst Mach who allegedly opposed 389 all counterfactual or abstract mathematical reasoning. But he is the Ernst Mach who 390 wrote the chapter " n Experiments in Thought" in Knowledge and Error (1905). In 391 this chapter, Mach emphasizes that 392 the experiment in thought is a necessary precondition of the physical experiment. Each 393 experimenter, each inventor must have the arrangement that led there in mind, before he 394 translates it into action. Although [George] Stephenson also knew the train car, the track, 395 and the steam engine from experience, he still had to model in thought the combination of 396 a train car resting on the tracks driven by a steam engine before he could go on to execute 397 19Mach, Ernst. "Space and Geometry from the Point of View of Physical Inquiry", The Monist 14, 1, 1903, p. 1. 20Schabas, Margaret. The Natural Origins of Economics. Chicago: University of Chicago Press. 2005. 21See Mach 1903, p. 2, as cited below. UN CO RR EC TE D PR OO F L. Patton it. No less must Galileo see the arrangements of the investigation of falling motion in front 398 of him in imagination before he implements them.22 399 The physicist, the engineer, and the inventor must engage in robustly empirical 400 reasoning. The dramatic narrative of the first chapters of Mach's Development of 401 Mechanics follows the failure of the search for a perpetual motion machine. A 402 physicist – or an inventor! – who continues doggedly to build perpetual motion 403 machines will fail. Mach is quite clear that that failure must be accepted, and that 404 recognizing the fact of failure aids in the construction of future experiments and 405 instruments. 406 But Mach leaves room for truly imaginary counterfactual or hypothetical rea407 soning that makes no claim to be physical, or to work in practice. Galileo made 408 experiments, and Stephenson made steam engines, that worked. In both cases, 409 a dynamic mental model of how the experimental setup would be, or how the 410 invention would work, was a "necessary precondition" of the experiment itself. 411 But in "Space and Geometry" and in Knowledge and Error, Mach emphasizes 412 that mathematicians can help physicists by freeing their imaginations from mental 413 constraints that restrict the construction of new models, abolishing "conventional 414 barriers to thought", and clearing the way for new methods. No harm is done, 415 on Mach's view, as long as the physicist, the physiologist, and the mathematician 416 respect the epistemological division of labor. 417 It is possible that between Mach's pronouncements on Newton in 1883, and 418 his essay on geometry for The Monist twenty years later, Mach had warmed to 419 counterfactual and abstract reasoning. There are reasons why we might think so. As 420 DiSalle (2003, 1993) details, Mach had learned much more about the mathematics 421 of inertial frames and about relativity theory in the meantime, and he may have 422 accepted that some abstract or counterfactual reasoning in this area was productive 423 and even necessary. 424 But it is also possible that Mach's appreciation of one kind of counterfactual 425 reasoning was in place already in 1883.23 Most of the classic experiments in thought 426 22Mach, Ernst. Erkenntnis und Irrtum. Leipzig: Johann Ambrosius Barth. 1905, p. 184, emphasis in original. Thanks are due to Erik Angner, Erik Banks, Daniel Breazeale, Uljana Feest, Katrin Hohl, Don Howard, Robin O'Keefe, Flavia Padovani, Dirk Schlimm, January Simpson, Thomas Sturm, and Richard Zach for advice on this translation. "Arrangement" is translated "Anordnung". As several people pointed out, "Anordnung" could be translated "setup" as well, to refer to an experimental setup. 23We can make a distinction here similar to the one above between natural and artificial hypothetical reasoning, on the basis of Mach's texts from 1903 and 1905: Heuristic counterfactual reasoning frees the physicist from conventional barriers of thought, and is the contribution of mathematicians and metric thought to the epistemology of space. It is used to build models in thought and imagination that inform the setup of experiments that work. If successful, this allows us to understand our experience more fully using heuristic methods. Artificial counterfactual reasoning uses unnatural and non-empirical concepts to expand the range of models and theories beyond observed physical phenomena. It is not UN CO RR EC TE D PR OO F 26 "New Water in Old Buckets: Hypothetical and Counterfactual Reasoning. . . from Poincaré, Helmholtz, Beltrami, and others that probed the metric structure 427 of the universe (including Poincaré's expanding gas universe, Helmholtz's convex 428 mirror, and Beltrami's pseudosphere) were expressed in the 1860s and 1870s. In 429 the introductory passages of his 1903 essay, Mach cites Riemann's "Über die 430 Hypothesen, welche der Geometrie zu Grunde liegen", which was published in 1867 431 and given as a lecture in 1854; and letters from Gauss to Bessel from the 1820s and 432 1830s. Later he refers to Helmholtz's "Ueber die Thatsachen, welche der Geometrie 433 zu Grunde liegen", which he cites as published in the Göttinger Nachrichten in 434 1868.24 435 It is possible, of course, that Mach had been aware of these sources for decades 436 but had recognized their true import only much later. But if we respect Mach's 437 division of labor for the epistemology of space, we might try to argue that his 438 remarks in 1883 are coherent with the views expressed in 1903. 439 Making the effort to find common ground between the "Mach" of the two time 440 periods, the 1880s and the early 1900s, is instructive.25 We can try to reconcile the 441 two by recognizing that the Mach of the 1880s was reading the Newton of the bucket 442 and two spheres experiments as a physicist, and not as a mathematician. Mach's 443 moralizing about the vocation of the physicist assigns to Newton the task of finding 444 a way to implement his experiments in thought, just as Galileo and Stephenson had 445 done. But Newton's experiments in thought contain artificial elements that cannot be 446 made empirical. Stephenson's tracks, train cars, and engines can be made manifest; 447 Newton's absolute space drives no trains. 448 What's the difference between Riemann and Newton? Why does the Mach of 449 the 1900s approve of Riemann's investigations in thought, but not of Newton's? 450 Riemann took the object of his study to be the manifold itself. Thus, he was probing 451 a universal geometrical concept. As Mach writes, 452 By the recognition of permanency as coincident with spatial displacement, the various 453 constituents of our intuition of space are rendered comparable with one another,-at first 454 in the physiological sense. By the comparison of different bodies with one another, by the 455 introduction of physical measures, this comparability is rendered quantitative and more 456 exact . . . Thus, in the place of an individual and non-transmittable intuition of space are 457 substituted the universal concepts of geometry, which hold good for all men. Each person 458 has his own individual intuitive space; geometric space is common to all. Between the 459 used to understand experience more fully. Rather, it is intended to expand the reach of theories and models artificially, using constructed concepts that are never intended to be tested. Thus, it is similar to artificial hypothetical reasoning. 24David Hyder has emphasized to me that there is some question about the proper dating of this publication, but it certainly would not have been published after 1883. 25Mach's illness makes things much less simple. In 1898, he suffered what was probably a stroke, and was paralyzed on his right side. Gereon Wolters has questioned the authenticity of much of the work that postdates Mach's illness and subsequent paralysis, distinguishing between Mach I and Mach II. However, the work cited here in 1903, 1905, and later 1912 (see below) is consistent with, or at least a natural development of, Mach's earlier views. UN CO RR EC TE D PR OO F L. Patton space of intuition and metric space, which contains physical experiences, we must sharply 460 distinguish (Mach, "Space and Geometry", loc. cit., p. 2). 461 According to Mach, Riemann's metric space is based on comparison via physical 462 measurements.26 But it is also "universal" and common to all experiences of space, 463 and thus is a valid basis for abstract reasoning. Mach approves of Riemann because 464 he keeps such reasoning within the appropriate realm in the managed economy 465 of science. His arguments are mathematical – Riemann does not argue from the 466 properties of metric space to the existence of a physical force, for instance. 467 In the 1883 edition of The Development of Mechanics, Mach argues that Newton 468 had made two errors. He made assertions about absolute space, a space not derived 469 from comparison of observable phenomena and thus artificial. And he argued from 470 the properties of artificial absolute space to the existence of a physical force. 471 Later in his life, Mach modified his assessment of Newton himself, though not 472 his assessment of the "monstrous conceptions" of absolute space and time.27 The 473 Supplement to the third English edition of The Development of Mechanics contains 474 Philip Jourdain's transcription and translation of Mach's revisions to the seventh 475 German edition, published in 1912. Those revisions include a number of passages 476 that provide context for Mach's assessment of Newton's bucket and two spheres 477 experiments. 478 while Galileo, in his theory of the tides, quite naively chose the sphere of the fixed stars 479 as the basis of a new system of co-ordinates, we see doubts expressed by Newton as to 480 whether a given fixed star is at rest only apparently or really (Principia, 1687, p. 11). 481 This appeared to him to cause the difficulty of distinguishing between true (absolute) and 482 apparent (relative) motion. By this he was also impelled to set up the conception of absolute 483 space. By further investigations in this direction – the discussion of the experiment of the 484 rotating spheres which are connected together by a cord and that of the rotating water485 bucket (pp. 9, 11) – he believed that he could prove an absolute rotation . . . "But how we 486 are to collect," says Newton in the Scholium at the end of the Definitions, "the true motions 487 from their causes, effects, and apparent differences, and vice versa; how from the motions, 488 either true or apparent, we may come to the knowledge of their causes and effects, shall be 489 explained more at large in the following Tract." . . . do not the words quoted in inverted 490 commas give the impression that Newton was glad to be able now to pass over to less 491 precarious questions that could be tested by experience?28 492 In the section following, Mach suggests that Newton was guided by empirical 493 considerations in his postulation of absolute space. According to Mach, when 494 Newton read Galileo on the principles of mechanics, Newton rejected Galileo's use 495 of the postulate of an earth at rest in the explanation of the law of inertia. Newton 496 knew that the earth was rotating. Thus, instead, Newton "imagined a momentary 497 26Arguably, Mach's reading of Riemann takes Riemann to be much closer to Helmholtz than Riemann really was, because Helmholtz bases metric geometry much more on the observed facts [Tatsachen] than Riemann did. 27Mach, Ernst. The Science of Mechanics. Supplement to the third English edition, trans. and annotated by Philip Jourdain. Chicago: Open Court. 1915, p. xii. 28Citations from Newton, Isaac. Philosophiae Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy"), London, 1687. Main text Mach op. cit., 1915, pp. 33–34. UN CO RR EC TE D PR OO F 26 "New Water in Old Buckets: Hypothetical and Counterfactual Reasoning. . . terrestrial system of co-ordinates, for which the law of inertia is valid, held fast in 498 space without any rotation relatively to the fixed stars".29 He did so "in order to 499 have a generally valid system of reference" (Ibid.). Newton could assign a system 500 "any initial position and any uniform translation relatively to the above momentary 501 terrestrial system" (Ibid.). 502 Mach emphasizes that the postulate of absolute space was not necessary, for 503 Newton. In fact, as Mach observes, the reference system Newton constructs is 504 relativist: it's just that the target system is moving relative to the hypothetical 505 terrestrial coordinate system. In making this move, Mach urges, "Newton was 506 correctly led by the tact of the natural investigator. This is particularly to be 507 noticed, since, in former editions of this book, it was not sufficiently emphasised. 508 How far and how accurately the conjecture will hold good in future is of course 509 undecided".30 510 What we might call "the Mach-Newton conjecture" is the question of whether 511 all target systems can be defined relative to the terrestrial coordinate system defined 512 by Newton, and whether the law of inertia is valid for all motions relative to such 513 a reference system. It is an intriguing question whether this conjecture has been, or 514 can be, empirically or mathematically verified. 515 Mach concludes in this later edition that Newton was a good physicist, who gave 516 a physically meaningful characterization of the law of inertia in terms of coordinate 517 systems. The postulate of absolute space was not necessary for Newton. Instead, 518 Mach suggests, Newton should have used only the formal apparatus necessary 519 to give a "generally valid" characterization of inertial motion with respect to a 520 system of coordinates. This would satisfy Mach's law of continuity of experience, 521 according to which gaps in the explanation of experience should be filled by the 522 ideas suggested by experience. 523 Newton's use of a hypothetical terrestrial coordinate system is a nice example of 524 Mach's division of labor between physics and mathematics. When Newton is led to 525 postulate absolute space, Mach accuses him of using an artificial concept and thus 526 of engaging in artificial hypothetical reasoning. But the hypothesis of a terrestrial 527 system at rest with respect to the fixed stars is an empirically possible hypothesis, 528 and is a natural kind of hypothetical reasoning. 529 Mach's economy of science often is appealed to as a kind of minimization 530 argument, or as an economy of thought or of effort. The above should cast significant 531 doubt on any such reading of Mach on economy. Mach argues that reasoning that 532 allows us to extend what's observable and thus to promote the "continuity" of 533 experience is economical. 534 Mach allows even for hypothetical and counterfactual reasoning to contribute 535 to the progress of science. It extends the circle of what is observable by appeal 536 to what is accessible via well founded hypothetical reasoning. It is doubtful that 537 Mach's reasoning about the division of labor between mathematics, physics, and 538 29Mach, Science of Mechanics (Supplement 1915), loc. cit., p. 35. 30Mach Science of Mechanics (Supplement 1915), loc. cit., p. 36. UN CO RR EC TE D PR OO F L. Patton physiology can be supported in practice. But understanding Mach's economy of 539 science allows us to understand some of his more notorious arguments, including 540 his polemics against absolute space and time. 541 Finally, Mach's philosophical reasoning is part of the recognized background 542 to general relativity. The above account pushes against some of the more limited 543 readings of Mach's relativist theories of space, and urges a deeper understanding of 544 Mach's reception of Newton and of Newtonian methods.