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SCIENTIFIC CONFIRMAT ION AND NATURALIZED MATHEMATICAL REALISM By Andrew Steele A Thesis Submitted to the Division of Humanities New College of Florida i n partial fulfillment of the requirement for the degree Bachelor of Arts Under the sponsorship of Aron Edidin Sarasota, Florida May 2010
ii Table of Contents Pg. iii Abstract Pg. 1 Introduction Pg. 3 Naturalized Argument for Mathematical Realism Pg. 23 Pg. 51 Chapter 3: Quine, Hacking, and the Scientific Confirmation of Entities Pg. 92 Bibliography
iii SCIENTIFIC CONFIRMAT I ON AND NATURALIZED M ATHEMATICAL REALISM Andrew Steele New College of Florida 2010 ABSTRACT My thesis addresses the issue of naturalized mathematical realism and the possibility of upon the adoption of a naturalized approach to philosophy wherein scientific methods and standards are taken as authoritative, we must subsequently fully accept the existence of mathematical objects. I argue for the possibili ty and legitimacy of rejecting mathematical realism while accepting just such a naturalistic starting point. Specifically, I argue against the ion concerns the nature of scientific standards and procedure. It asserts that the knowledge of the existence of entities regularly manipulated in the course of physical experimentation, and the knowledge of the causal natures of these entities that allows for this manipulation, receives an especially strong degree of confirmation and stability. Further, Hacking believes that we might legitimately regard scientific knowledge not so described with an attitude of scientific anti realism. I argue that, as sci entists cannot manipulate mathematical
iv naturalized nominalism wherein one would accepted the experimental knowledge already described while rejecting the remaining scient ific theory. It is this latter body of scientific theory which Quine argues is most truly committed to mathematical entities. An appendix sketches an extension of this argument to the naturalized mathematical realism of John P. Burgess and Gideon Rosen. I approve this abstract ___________________ Aron Edidin Aron Edidin Humanities
1 Introduction Mathematical realism is the doctrine that those objects which are the special concern of pure mathematics numbers, sets, functions, relations and so on all really exist. While I will generally refer to the position as mathematical realism, it is often known as mathematical platonism as well. Naturalism, broadly, is the position that properly scientific methods and con clusions should be authoritative, even when approaching issues in a philosophical setting. This has been an important and influential view in philosophy for at least the last century. On the basis of naturalism, a number of philosophers have argued for mat hematical realism. W. V. Quine, perhaps, is the most influential of these thinkers. Many such arguments, and certainly mathematical entities. In this essay, I ma intain a basically naturalistic outlook and focus on the question of the scientific support for mathematical realism. Ultimately, from these considerations, I argue that it is possible to reject mathematical realism while maintaining a position of naturali sm as well. In confirmation of entities. Specifically, he emphasi zes the scientific importance of experimental mathematical realism while, at the same time, maintaining a broadly Quinean naturalism. This is mathematical entities as systematically less well confirmed than many other scientific entiti es
2 from the same naturalized outlook. The finer points of this position will, of course, be made clear as we proceed. ve also included an appendix that involves the sketch of an extension of my response to Quine to a related naturalized argument for mathematical realism present by the writing team of John P. Burgess and Gideon Rosen, however.
3 Naturalized Argument for Mathematical Realism Quine is a somewhat unusual case in the history of philosophy of mathematics. He may well have been one of the first mathematical realists to have been reluctant and unenthusiastic in his realism. That he had to be a realist is evident enough, however. Many of the most important features of his philosophy converge to the conclusion. These include his naturalism, his epistemic holism, his criteria of ontological commitment and the acknowledged failure of the pro ject of reconstructive nominalism using resources of which he approved. I shall examine argument for mathematical realism. For individuals raised as members of a linguistic and epistemic community, i.e. for the presenting themselves as beliefs. These beliefs weigh in on what exists, what does not, who we are, where we are, the nature of reality and so on. They also specify methods and standards for learning more about the world as well as for assessing evidence and new information as it content of the common sense of different communities may be significantly different, however. If a community has survived, developed, and thrived, it is likely because its c ommon sense is up to the task of getting its members through daily life and navigating its various challenges. There is more to be hoped for than simply this common knowledge however.
4 Knowledge can be pursued beyond what is needed for these everyday demand s. Expansion of common sense begins with the pursuit of knowledge by means of the standards for the acquisition and accreditation of knowledge already going in common sense. As this activity is continually pursued, the total knowledge of the community expa nds as does the variety, systematicity, and rigidity of the standards used to pursue it. In the course of this concerted sense may even come to be challenged or repl aced. In our own community, this kind of activity has taken on a life of its own and has allowed every greater understanding and control of the Important to note about this expanded portion of common sense is that it does not represent a distinct break with the common sense from which it originated. It represents a continuous organic growth from this body of knowledge and it is in virtue of this that they we might call the we realize that the distinction it implies is not a truly hard and distinct one. Quine puts this of common sense. The scientist is indistinguishable Ways of Paradox 233) Naturalism, in this context, is the idea that this science and its methods are the best and m neither distinct, nor distinctly strong. Philosophy becomes a continuous outgrowth of science proper. Its conclusions must square with those of other going scien ces. It differs only in the
5 general nature of the questions with which it deals. Quine describes naturalism as the Theories and Things 67). This term is of limited use in summarizing several different ways throughout his career. If we are to speak of an overall naturalism in regard to Quine, we might just as well talk about a naturalistic spirit or inclination that guided his philosophy as realism and this spirit heavily informs many of the philosophical conclusions that lead Quine to this position. Of course, what this naturalistic spirit even means will be infor med by the of elaborating on what science says, or is, in certain situations. We should note at the outset engagement and sympathy with the empiricism of his day. itself most clearl naturalized epistemology primarily involves finding a firmer footing for science overall than the one provided by the evidential resources it employs already. sympathies we mentioned earlier, Quine sees the empiricist project as the most promising one. This project would involve the demonstration, with perhaps the aid of deductive logic and mathematical logical auxiliaries, of all of scientific knowledge from actual sensory experience ( Ontological Relativity and Other Essays 72).
6 Quine believes this project has been a demonstrated failure since th e 18 th century when Hume wrote ( Ont. 75). Quine takes the failure of its most promising avenue of success as enough reason to abandon this project of extrascientific justification entirely. When the project of justification is abandoned, it ceases to be ci rcular to employ scientific conclusions about cognition in our epistemological investigations. The naturalized epistemological project Quine proposes in wake of the old one begins by assuming the truth of these scientific beliefs and the authority of scien tific methods. It then involves the construction of an account of how we acquire these beliefs and methods, inasmuch as this is possible, as dictated by them and to be justified by their authority ( Ont. primarily that epistemology primarily becomes a branch of empirical psychology. Quine takes it upon himself to produce such an account. While he has abandoned the old empiricist project of demonstration of science from truths of experience, he tells us that Ont. 75). But if sensory evidence cannot entail scientif ic conclusions, then how can it ( Pursuit of Truth 1, 2). Needed, t hen, is a general account of how exactly theory can be said to predict experience. Quine believes the relevant relation of theory to experience is one of deductive implication ( Pursuit 9). Quine believes the method of empirical testing, then, is uniformly characterized by the hypothetico deductive method. Here then, we have the sense of
7 of an inventory and description of reality is testability of its observable c onsequences in the time honored hypothetico The hypothetico deductive method (henceforth H D) begins with the framing of a theoretical hypothesis. The source of the hypothesis, in most accounts, is irrelevant. Then, we derived prediction P from it. This relation of derivation proves the conditional H P. An experiment is then conducted to test P. If P fails the test, (that is, i f we observe that P) then by the deductively valid principle of modus tollens, we can conclude that H. A successful test of P shows far less. In such a case, deductive logic tells us only that such a test does not show that H is false, but H may still be false. If we are to talk about positive empirical support, we will need also to involve some kind of inductive reasoning principle. Quine seems to allow for such a thing, but never attempts to offer one. Perhaps the above presentation gave the impression that the hypotheses that were traditionally subject to such tests were rather short, perhaps about a sentence long. This impression is not unfounded; when conducting an experiment, scientists generally do so with some rather brief physical law or assertio n as to the unobservable constitution of things in mind. Quine believes that this appearance is deceptive. In general, he says, individual sentences of science do not have a store of empirical consequences to call their own ( Ont. 79). Rather, only relative ly large bodies of theory are capable of producing empirical predictions of the kind we have been describing, and typical individual sentences only have such consequences by being part of these larger bodies. With this insight, we have the germ of what is
8 In terms of H D testing, this insight significantly alters the meaning of empirical results. is a number between 1 and n) is required to produce an empirically testable consequence labeled, P. This means that a test that proves P, along with modus tollens, proves only the falsity of the antecedent conjunction. Deductive logic, then, indicates only that at least one of the conjuncts is false. In the case of actual scientific theory this indeterminacy persists. In the case of a failed prediction, it means that any one of a large number of different portions of the theory of varying sizes involved in the implication of that assertion must be rejected to stop the failed implication. On the other hand, when dealing with large bodies of theory like this, we are never required to reject any one piece of theory but, if we liked, could hold any of them true no matter what; as Ways 254). Even these large portions of empirically significant theory are not entirely self contained and isolated from our broader world theory. If we chose to alter a certain portion of theory to eliminate a false prediction, then we must perform similar revisions to other portions of theory that are logically related to it in the relevant ways. This is in order to avoid subsequent contradictions and tensions. These portions of theory can vary significantly as to the amount of subsequent revision that will need to be done to accommodate their rejection. While empirical evidence alone does not distinguish among revisable portions of theory, broader scientific practice demands that the amount of revision involved in our compensatory alteration of theory
9 Pursuit 14). In other words, we must revise that portion of theory which demands the least amount of subsequent alteration. An example will help us get a feel for how this situation goes. Let us look at the the refraction of a ray of light as it passes through two adjacent mediums. It says that sin sin = where and are the angles of incidence of a light ray in each of the two mediums and A and the ratio A/B to be, in this instance, C. By solving the equation, we conclude that the angle of experiment, however, b is not the angle we measure. What do we do? wanted. If we did this, though, we would have to revise our overall theory of optics inasmuch as it assumes or implies this law holds in perfect generality. More modest options would involve rejecting the assertion that our instruments worked in our determ ination of C, or of the angle of incidence a. We might conclude that the second substance was not glass. This suggestion would involve revising all our beliefs that had led us to believe it was. This might include, for example, the belief that the individu al that produced it was telling the truth. More extreme in this regard is the possibility that the process that was used to create this object, previously thought to invariably produce glass, does not actually do so. We could make even more extreme revisio ns. We might suggest that the experimental result is an elaborate illusion. We might even reject the mathematics that had lead us to believe that the equation did not hold and leave everything else the same. By this, I mean
10 that we could still accept that a, b, and C were all accurate, we would simply reject that sin sin This, of course, would mean massive revisions throughout mathematics as many unattra ctive one. All the theory, and more, that could have been revised in light of this failed prediction how revision in light of such a failure, and the subseque nt revision it would necessitate, might go. We have been focusing on the holistic meaning of a failed prediction. As before, logic tells us less about the meaning of a successful prediction. Quine believes, however, that if we are to talk about empirical support, we must talk about it accruing to larger bodies of theory involved in the implication of empirical consequences. Importantly, we have also noticed that mathematical truths are revisable along with the others. That mathematics is involved in theory which receives empirical support is a major part of his ultimate acceptance of mathematical realism. address before moving on, however, is exactly what the scope of this co nfirmation might be. While Quine remained committed to holism throughout his career, his view on this matter did change. Early on, he offered a rather strong view, claiming that all of scientific theory was to be thought of as involved in every empirical t From a Logical Point of View 42). Later on, however, he sort of way, but it diverts attention from what is more to the point: clusters of sentences just
11 inclusive enough to have critical semantic mass. By this I mean a cluster sufficient to imply an Quine combines this holism with a thorough rejection of the a priori status of any kind same token, no statement is FLPV 43) All of scientific theory, under this picture, is a human creation designed to predict experience. Yet, product of human artifice as it is, we should still believe scientific theory is true when it performs the function for which it was designed ( Ways 251). As we have already seen, empirical evidence significantly underdetermines scientific practice as an additio nal consideration to guide the acceptance of theory. Science offers several such considerations, each of which a single piece of theory might exemplify to a certain degree. The more it does so for each, the more strongly does scientific practice recommend we accept Pursuit 20) If we were to think of adherence with empirical evidence as a virtue as well, then we might even think of the entire process of scientific investigation as the applicatio n of the virtues to hypotheses and, in the case of competing theories, the discovery of which of them best exemplifies the most virtues considering their relative importance. Quine attempted several similar lists of these virtues but never quite seemed to settle on one. Satisfaction of empirical evidence remained the most important consideration. Second in importance tended to be theoretical simplicity ( Pursuit 15). Beyond this, there was greater variation. Perhaps the most developed list is to be found in his Pursuit of Truth where they are,
12 (20). Earlier, in Word and Object 21) in addition to empirical success. Even the addition of these virtues leaves scientific theory underdetermined. They are, however, a part of science nonetheless. We have yet to make much sense of the relation of theory to experience, however. Traditional accounts of observation often involve the reception of information about external objects and phenomena through our sensory receptors. This is not quite how Quine conceives of the relation of experience to scientific knowl edge in general. In keeping with his naturalized epistemology, Quine accepts that objects, existing in four dimensions, cause the firing of our sensory receptors by various means. He accepts that the firing of these receptors is the beginning of the neurol ogical process whose result is our knowledge of these very objects and more ( Ways 228). Quine even offers us the psychologico Pursuit 19). Ind eed, it is these moments of sensory receptors firings that Quine believes are the subject of the theoretical prediction we recently discussed ( Pursuit 2). The question Quine must address is how this sensory stimulation can relate to theory overall. He resp onds by isolating a class of sentences he calls observation sentences as more closely keyed to sensory stimulation than others. Attempting to be true to Quine is somewhat tricky here as his precise understanding of how we are to think of, and identify, the se observation sentences went through a few revisions
13 given observat ion sentence is to bring little or no more to the scientific process in its involvement than the implication of the occurrence of one or more of the sensory events with community in question. His overall philosophy leads Quine to find a criterion of identity for these sentences in the linguistic behavior of the members of the commun ity in question. From here, Quine looks for an absence of, or very limited willingness to, assent or dissent from these sentences based on anything other than certain firings of sensory receptors on the part of relevant community members. The prescribed formula for narrowing in on which sentences were observation sentences for a community changed slightly from time to time. A late formulation was that observation sentences were those sentences which all fluent speakers of the language would assent or diss firings (Quine, From Stimulus to Science 22). It is these sentences in virtue of which theory in general can be tested. These will be occasion sentences, some of which will be cast in categoric observation sentences ( Pursuit 10). We then test a theory by setting up a situation where the antecedent holds and checking to see if the consequent holds as well. If it does, then the theory
14 has successfully predicted experience, if it does not, then the prediction is a failure and re visions are in order. We had identified Quine as taking empirical science as authoritative and had proceeded to fill in his account of the practice and conclusions of science in trying to arrive at his mathematical realism. So far, we have talked almost exclusively about theory, its empirical not accidental. For Quine, bodies of theory with empirical consequences are the primary bearers of confirmation and meaning. Individual sentences receive confirmation and meaning derivatively by way of their inclusion in these larger, empirically meaningful bodies. All along we have been talking about theory. Just to clarify, this is simply declarative verbiage. Quine uniformly regards scientific content and conclusions in terms of theory. We might be wondering theory by means of talking about the sentences to which those beliefs wo uld lead us to assent ( Theories and Things ). But mathematical realism is a doctrine about objects; to make sense of it, we must first ask how our knowledge of entities fits into this broader conception of science. Within our larger world theory, we can m ake sense of individual terms appearing in these sentences as referring of assuming objects becomes a question of verbal reference to objects. To ask what the assuming of an object consists in is to ask what referring Theories and Things 2). Objects receive scientific support and acceptance only as the portions of theory that reference them receive scientific support and acceptance.
15 Objects, in this picture, become a derivative feature of scientific discovery. Quine Theories 2). Quin e denies, however, that this is meant as skeptical and declares that he fully accepts the existence of the objects of science ( Ways 251). The rationale, of course, is that helping theory predict experience in this way is the only real evidence there ever w as for such objects and that to draw anti realist conclusions from the fact is questionably coherent ( Ways 251). Naturalized ontology, then, becomes a matter of distilling from the final theoretical account produced by science overall, its ontological cont all conceptual scheme which is to accommodate science in the broadest s ense; and the considerations which determine a reasonable construction of any part of that conceptual scheme, for example, the biological or the physical part, are not different in kind from the considerations which determine a reasonable construction of t FLPV 17) Given his philosophy so far, these conclusions inform the reasoning he can employ in making ontological conclusions. In terms of his actual account, this is an essentially two step process. First, as accepting objects is a matter of a ccepting theory that references them, he must settle on which locutions are genuinely referential ones. Second, he must determine, to some degree, how these locutions play into the final scientific account of the world. Perhaps most of us are used to thin king of names as the most truly referential locutions,
16 referential locutions are the bound variables of quantification of symbolic logic, as well as all their appropriate natural and artificial linguistic analogues. The most common symbols for are referential only inasmuch as they relate, appropriately, to these ones we have identified (Quine, Quiddities 180). To believe an obj ect exists is one and the same, says Quine, as counting that object among those over which these quantifiers range ( FLPV 13). With this idea in place, theory can not be true unless there is such an entity in the range of its quantified variables ( FLPV 13 14). Accordingly, an individual who accepts the truth of these theories must accept the existence of the entities to which it is committed. With this criterion of commitment in hand, Quine has an important tool for drawing naturalized ontological conclusions. Making such conclusions, however, will involve the additional step of critically examining the conclusions of science to some degree. This approach might appe ar an initially disheartening one, however. The natural place to look in attempting this project is the actual language in which scientist cast their discoveries. Upon an examination in ordinary (or scientific) language. The idea of a boundary between being and nonbeing is a philosophical one, an idea of technical science in a broad sense. Scientists and philosophers seek a comprehensive system of Theories 9). In response, Quine proposes the recasting of scientific language in terms of a regimented notation. This notation would be specified at the outset. Quine proposes
17 Theo ries 9). These are essentially the resources of symbolic logic with first order quantification. In recasting the scientific theory using these resources, the philosopher would have to preserve the empirical support of the original form of the statements of science. The ontological assertion of science, however, would be made very definite by the nature of the resources for reinterpretation specified at the outset. Those things we would ultimately determine exist would simply be those over which we ultimat ely determined the quantifiers of the newly recast theory should range. There is more to this project of recasting science than simply trying to recapitulate the thoughts of the original scientists in new, more precise terms. This recasting of science give s the philosopher the chance to contribute to the actual substance of scientific conclusions. Quine, as we saw, does not even think that a highly precise ontology is implied by scientific theory in its original form. In seeing the various ways that a scien tific conclusion cast in its original form might be recast in the favored terms, the philosopher proceeds by choosing the interpretation which best satisfies the scientific virtues. In other words, the philosopher is One way Quine sees that philosophers can contribute in this latter respect is by Stimulus 18). He seems to see this justified primarily by the scientific virtue of simplicity. He tells us that tives that impel scientists to seek ever simpler and clearer theories adequate to the subject matter of their special sciences are motives for the simplification and clarification of the broader framework shared by all the sciences. Here the objective is c ( W&O 161).
18 Quine admits that simplicity is a vague notion in regard to theory. In regard to ontology, however, he understands it as demanding economy, and uniformity. His very choice of the resources of the f inal, regimented notation was a step in this direction ( W&O 161). The devices of this notation for making ontological assertions were the first order quantifiers. In allowing only these, Quine anticipates and demands that science ultimately accept an ontol ogy consisting of objects to the exclusion of other kinds of existences. This will means that task falls to Quine of reinterpreting talk implying different kinds of existences, like mass nouns, in objectual terms. There is more to the project of trimming of science, however. We might draw an unproblematically fuzzy line between physical objects and abstract objects. Physical objects should be familiar enough to us. The category includes material objects as well as some more exotic types which physics has u ltimately come to countenance like fields of force. More marginal cases might include things like distinctly mental entities. Common characterizations of abstract objects include their nonspacial and nontemporal existence as well as an a causal nature. Exa mples include properties, platonic forms, and, above all, mathematical objects. Mathematical objects are simply those objects primarily dealt with in pure mathematics like numbers, functions, geometric forms and sets. Perhaps Quine has nothing against th e idea of abstract objects per se. However, these represent two distinct kinds of objects and a uniform ontology is to be preferred, on grounds of simplicity, to one that involves both ( Stimulus 56). Further, physical objects are preferred, above all other W&O 238). Physics, as it is now, doing without efforts at excision. Modern empirical scientific theory is, of course, th oroughly quantitative.
19 Physical measurements are taken and presented, statistical regularities are assessed and patterns are charted and graphed. This involves all manner of mentions of mathematical, and thus abstract, objects in scientific theory. Terms a pparently referring to mathematical objects, then, appear all over scientific theory which is most definitely involved in the derivation of science into the f inal terms he proposes, the mentions will be reinterpreted as ontologically noncommittal ones and empirical evidence will not ultimately demand we accept the existence of these objects. The position that abstract entities do not exist and, rather, only con crete Quine provides, it must be at this stage of the reinterpretation of scientific language into the regimented notation. stic that mathematical entities might be eliminated from the final ontology of science and, therefore, that naturalized mathematical nominalism was a possibility. Ultimately, however, he concluded it could not be done. Mathematical entities, he came to bel ieve, were indispensible for empirical science. Why exactly did he conclude this? The scientific theory he seemed to view as most important to the failure of s in this generous sense constitute a fairly lavish universe, but more is wanted notably numbers. Measurement (is needed for) the formulation of quantitative laws. These are the mainstays of scientific theory, and they call upon the full resources of the r constants; we must quantify over numbers. Admitting numbers as the values of variables means Theories 14) There is just no proper way to express these laws, and thus fully
20 capture the content of scientific theory, without quantifying over numbers. These laws are, of course, heavily involved in the derivation of the overall empirical theory and, therefore, receive the best kinds of scientific support. Both mathematical abstract and physical objects are, there fore, a part of our ultimate scientific ontology we should accept from our naturalistic perspective. At this point, we have all the essentials of a Quinean naturalized argument for mathematical realism. Some comments are in order concerning this position, however. If it is not already clear, nothing about the practice of pure mathematics alone is responsible for this Quine discusses as ultimately authorit ative. Rather, a Quinean might understand it as the deductive discovery of logical relations among mathematical assertions as well as the development and anticipation of new such assertions. Many of these assertions are existential. Had Quine succeeded in excising mathematicalia from the ultimate ontology of empirical science, this whole process would come to be understood as a meaningless formalism; At least, it would inasmuch as it was put in terms of mathematical existences ( Stimulus 56). Accepting math ematical assertions as truths, however, he still does not accept them as necessary truths as many realist views might. Quine understands those mathematical assertions he does ultimately accept as revisable in light of failed empirical prediction like all o ther appearance of their necessity, and the fact that revision is very unlikely to actually happen to mathematical assertions, as resulting from the scientific pri nciple of minimum mutilation we mentioned earlier ( Pursuit 15). All manner of otherwise independent branches of science rely on
21 a relatively limited number of mathematical truths to express their discoveries. This means that even a small revision of mathem atical theory would require us to undertake revisions all over science to accommodate it making it, always, a very undesirable move. Even though he has ultimately accepted the existence of mathematical objects, this is ntological simplicity and parsimony in regard to mathematical abstract objects. Set theory is a distinct area of research within mathematics itself. It is a well known fact, however, that nearly all mathematical objects can be identified with certain sets. Once other necessary terms have been defined in their appropriate set theoretical terms, the truths concerning that object in its original setting can be successfully deduced from the axioms of set theory. All the mathematical demands of science can be me t by set theory in this way. Quine sees a mathematical ontology of sets as preferable to the potentially diverse one offered by face value applied mathematics. He argues, then, that we should accept the existence of the full cumulative hierarchy of sets an d understands the mathematical objects of science as being among them by the means I identified above ( Stimulus 40). The full range of mathematical objects, e.g. natural, rational, irrational, and complex numbers all simply become different kinds of sets. Even then, he tells us: in questioning their meaningfulness, we do keep their sentences as meaningful, but only because they are built of the same lexicon and grammatical constructions that are needed in applicable mathematics. It would be an intolerably pedantic tour de force to gerrymander our
22 grammar in such a way as to account the inapplicable flights ungrammatical while preserving TDR 58). There is much more to recommend set theory however. By accepting sets, and by understanding there to exist a set with all and only the members of any arbitrary collection of objects as elements, we can meet almost all of the nonmathematical abstr act needs of science as well ( Stimulus 40). Properties, for example, can be identified with the set of the elements of which they are true. Similarly, we can identify things like species, aggregates, and other scientific totalities as the sets containing t he objects categorized accordingly. So, then, Quine is a somewhat reluctant mathematical realist who is led to accept the existence of mathematical objects by, above all, their necessity for ultimately confirmed empirical scientific theory. He was led to t his conclusion, in turn, by his somewhat idiosyncratic picture of the nature of science and scientific confirmation. He arrived at this account by pursuing his program of naturalized epistemology, a program he prescribed in response to his understanding of naturalism as the appropriate course for philosophy overall. By reasoning from the scientific virtues in deciding how to cast science into its final form, the abstract, mathematical ontology he ultimately accepts consists in, precisely, the full cumulativ e hierarchy of sets.
23 As we saw, Quine fully accepts that we should believe scientific theory, that the external world science describes and assumes is fully real, and that scientific methods are up to the task of producing increasingly accurate descriptions of this world. So far in our study, the term wider, philosophical setting, however, this term signifies thos e approaches to scientific theory that is, the way of full belief in scientific theory and methods and the network of views that are its articulations. But this is not the only option. Many thinkers, for many different reasons, harbor do ubts about some aspect of the scientific process which leads them to doubt the full legitimacy of the scientific enterprise as a truth seeking one. Given the success of science, full repudiation of scientific theory is an implausible way to realize these d oubts. A position called instrumentalism, however, allows for a plausibly limited attitude to the abilities of the scientific endeavor. Instrumentalism is an attitude that can be held in regard to a piece of theory which combines the eschewing of full beli ef in the portion of theory itself with the belief that the world behaved as though it were true. In other words, a scientific instrumentalist regards scientific theory as useful, but not true in the full sense. As many harbor various misgivings about the ability of science to truly discern reality the way of instrumentalism as an attitude towards science remains a live and acceptable one. The cluster of views which express these doubts about the scientific process and prescribe attitudes like instrumentali sm toward scientific theory realism debate. It
24 is this very debate that is the primary concern of the next philosopher we will be examining, Ian Hacking. While it has been underrepresented so far in this study, anti philosophy overall is significant. Hacking himself shows some rather strong anti realist tendencies when he weighs in on the debate in its usual form. Our primary focus in this chapter Representing and Intervening. While realism in the broad of this traditional form of the debate is not. Rather, he hopes to provide a more complex picture of how to think of the debate and to offer some conclusions based on the reevaluation that follows. For Hacking, this reevaluation primarily results from the recognition of the hitherto underappreciated philosophical importance of scientific experiment and engineering. The primary concer n of the debate, as I have discussed it, is the proper attitude we should hold toward scientific theory. The notion of theory has had an important role in our discussion so far. For Quine, theory is simply declarative verbiage; statements that bear on the no less precedent in going philosophy of science. To understand his usage of the term, we might only mediately knowable constitution and nature of reality. the term, is one half of a dichotomy that separates scientific information on epistemological grounds. The traditional way of drawing this distinction is by contrasting theory with observation. Observation
25 is information manifest in the sensory experience of the observer. Theory, in contrast, is probabilistic arguments to support it once it has been suggested. Until it receives a great deal of support, it is rather te nuously held. Of course, the distinction need not be this precise to be meaningfully applied. In a reasonably naturalistic epistemological framework that separates one body of scientific knowledge from another as being categorically less well confirmed, t hat accepting the legitimacy of the notion of observation and theory as we have described them, does not see the two categories as exhaustively categorizing t he kinds of knowledge we can hope for. Indeed, we should already be rather familiar with the notion of deep theory from our discussion of Quine. There, we had discussed theory that received confirmation by no other D and application of the virtues. Among the methods of Hacking emphas izes the epistemic importance of observation over other, less direct, means of acquiring information to a greater degree than does Quine. To Hacking, then, our knowledge of the existence and behavior of mundane, macroscopic objects is fairly stable while d eep theory is dubious. Quine, however, had no problem accepting both kinds of information as truth. Hand in rtmanteau word for all that ragbag of stuff
26 these entities as decisively confirmed in the case of useful theory, Hacking believes their purely theoretical status makes them a legitimate object of an anti realist attitude. theory/observation distinction we are better able to tackle his treatment of realism. I have already mentioned realism and ant i realism and that the debate concerning them is the primary philosophical positions that cluster around the possibilities of full belief and instrumentalism in reg ard to scientific theory respectively. A bit more discussion of the actual nature of the views might be in order. Realism tends to hold that the theory with which science deals is either true or false and scientific practice should be regarded as both seek ing truth as well as approaching it. Further, for most realists, if an assertion is true then it accurately describes the nature of the world and reality. Science, then, can reveal and has revealed much about reality as it truly is. Anti realism on the oth er hand questions the ability of science to describe the world or, perhaps, even to produce true conclusions. Perhaps the most obvious option for the anti realist is to think of scientific theory as either true or false but being beyond the possibility of confirmation or proper refutation. On the other hand, the anti realist may take the next step and declare scientific theory not even to be properly thought of as bearing truth values (Hacking 21). Notice that these positions concern only the status of theo term. In keeping with the epistemological role he assigns observation, Hacking seems unwilling to consider positions which do not recognize at least some basic scientific notions and phenomena. If anyone was going to doubt the truth of observed information then it would be
27 the anti realists but, in using the popular example of a theoretical entity, the electron, Hacking of electricit y and of inheritance but we construct theories about tiny states and processes and (21). Also, the anti the kinds of things that that these macroscopic, observed phenomena and features of our world are known to us in a way which eliminates the possibility of antirealism in regard to them. This, I believe, is the primary cash value of his seeming acceptance of observation as a superior means of acquiring information. realism d ebate. A full description of this idea, however, must take account of his belief in the accuracy of something like the Kuhnian conception of the movement of scientific theory. The realism debate becomes one about the ultimate behavior of this movement. He tells us that he according to Hacking, realists tend to believe that the scientific progress of the selection of bodies of theory will terminate in a final account. At least they will think that there is a decisive progress towards such a thing. This notion is very much like the one C.S. Peirce had posited in his definitio n of truth. The anti realist on the other hand, does not believe in any such final scientific account of the world. The difference between these two positions becomes the difference between faith and skepticism as Hacking tells us, in order to be a realist like this one
28 realism, then, becomes one of the disbelief in this ultimate, final account. m/anti realism debate as he sees it actually existing in the larger philosophical landscape. Of his contributions to the debate he offers in the course of his book, he takes his most significant to be those concerning the nature of the debate itself rather than those concerning the superiority of any one side of it. Above all, he tells us that, because of the fact that the debate concerns only theory and the activities of theorizing, it is fundamentally inconclusive (31). We saw hints as to why this might b e in our discussion above; the difference between realism and anti realism final account to be reached by scientific inquiry. That the debate takes place in ter ms of theory, says Hacking, is symptomatic of a more general philosophical preoccupation with scientific theory, a preoccupation which has lead to the neglect and misunderstanding of the philosophical importance of other scientific activities like experime nt and engineering (150). should greater attention be paid to it for the sake of better understanding this distinct activity, Hacking believes that a closer anal ysis of experimental practice will help overcome the complete stalemate he has diagnosed in the realism debate (273 4). Here then, we see his more positive contribution. Hacking reminds us that we might be realists in regard to theoretical entities without also being realists toward the theory in which they are mentioned, a position he calls entity realism (27). He believes, further, that a realism, just like this, is supported, and supported fairly decisively, by experimental practice (38).
29 But what is i t about experimental practice that makes so much difference? Simply put, Hacking believes that it is the successful manipulation of theoretical entities which allows them to emerge from their theoretical status. There are truly theoretical entities to Hack but once we begin to manipulate them regularly, they shed this status and become something more stable and assured. Hacking tells us, in regard to the once theoretical entity, the electron, theoretical entities Hacking has in mind for this argument is precisely the kind that is carried out all the time in laboratory science. The entities that have emerged from theoretical status by these means are cal realism enters because, while we have a sort of functional knowledge of these entities which gains the relevant degree of certainty along with our knowledge of their existence, no o ne, full blown theory that describes them is able to achieve this status along with them by the same process. We will spend much of the rest of this chapter filling in the details of this suggestion. At the outset, however, I should note an ambiguity I d clear enough in his support of the suggestion at the level of generality that I have described, his understanding of the more precise nature of it seems somewhat unclear. Specifically, he seems to be submitting two distinct doctrines as the content of his position. Confusingly, he nowhere explicitly acknowledges this. This could simply be because he believes them to dovetail so neatly as to not require separate presentation and argument. While he may be right, fo r the purposes of this study, we will treat them separately. Roughly, one doctrine submits manipulation as a distinctly effective means of accumulating evidence for an entity. The other construes it as
30 either a sort of criterion of commitment to an entity in the spirit of Quine, or a strong inducer of belief on the part of the experimenter who uses the entity. these two suggestions are mutually supportive, even if th ey are not so to the degree Hacking might think. Indeed, before we discuss them in detail, it would be best for us to examine the features of these positions shared in common before we examine them individually. Most import is the very notion of manipulati on itself. The fact that we are dealing with theoretical entities means that manipulation will have to be a somewhat complex process. It proceeds, we shall see, by way of the construction of machines to employ causal networks to interact with the entity. I will offer more on this later. For now, we should ask how manipulation might even be possible given the nature of the entities we are dealing with. Hacking believes it is done by means of properties of the entity se properties which allow for the various physical effects the entity might have in other parts of the world. Accordingly, it is precisely our knowledge of some subset of the total collection of these causal properties of various entities that allows us to manipulate them. manipulation a bit here. For Hacking, manipulation means using an entity to generate effects in the causal powers of electrons, the more we can build devices that achieve well understood effects in other parts of nat ure. By the time that we can use the electron to manipulate other
31 gene rate these effects. They enter both sides of an act of manipulation in this conception by both bringing the effect about directly and allowing for the entity to be controlled so as to make this possible. While they may achieve experimental status eventual ly, these entities begin as mere features of theory. As such, they enter scientific consideration with a fairly robust description of their nature and abilities. This being the case, our ability to manipulate the hypothetical entity must initially proceed by the exploitation of these similarly theoretical properties. As the entity separates itself from the theory in which it entered consideration, what of this original content does it take with it? If knowledge of these causal properties is compatible with a theory anti realism of the kind Hacking describes, what is the nature and appearance of our knowledge of these properties? incompatible accounts of it, all of which agree in describing various causal powers which we are the knowledge of these properties we can glean from this statement. First is that the knowledge of them h as an intertheoretical meaning, something of which Quine had questioned the kn owledge of the causal properties of experimental entities. The knowledge of the causal properties, once they have been discovered, remains fixed and proven for good. Theories, on
32 the other hand, come and go according to the broadly Kuhnian picture. A theor y can contribute by suggesting an entity and, in its own terms, describing some of its causal properties; the entity so suggested may go on to be successful as a manipulated, experimental entity. Subsequent theories must account for this kind of knowledge in their own terms or, if not, so much the worse for them. We see the scientific superiority of these entities. As we have seen, then, these causal properties are intertheoretical in the sense that knowledge of them remains while theories that express them shift. They are also intertheoretical, however, in the sense that there may not be much in the way of a strong, unified, formulable expression of their nature and kind. This means they are something basically different from the kind of theory we have disc ussed so far. Knowledge of an entity is, above all, a skill, namely, a skill at manipulati on and not necessarily and exhaustively articulable one. Hacking conveys this in his own terms by saying that, in knowing these properties, we have numerous different theories or models about which an experimenter can be rather agnost this knowledge is more a functional one of the kind that eludes complete formulation in terms of language anyway. In talking about the removal of the ent ity from a purely theoretical status by virtue of its role in experiment, Hacking seems inclined to talk about this progress as occurring in stages. In spray
33 begin as a feature of a larger piece of theory proper, acting to help it do its work of explaining and predicting. In the next stage we would start to measure it an d discern the causal properties successfully manipulated suggests a de finite, final stage that would follow our suggested that we can use the electron to manipulate other parts of nature in a systematic way, the electron has ceased to be something hypothetical, something inferred. It has ceased to be More than this, he tells us that, in the case of electrons, a final stage of knowledge is tandard emitters with which we can spray positrons and electrons standard emitters here suggests that the thoroughness or regularity of our understanding of the ti me to see what sense there is in talking about these stages for each suggestion. Now, however, we are in a position to see how the manipulation of entities qua construction of machines might go. Manipulation proceeds by the construction and operation of m achines that employ the entity and its causal properties. For various purposes, Hacking believes that experiment consist, largely, in the creation of functional machines that reveal certain features of nature to us (167
34 us that this is how manipulation must be carried out given the nature of the entities themselves. We might discern several phases to the process of the construction of these machines. The whole process begins with our specifying an end we want to achieve. In an already mentioned example, the alteration of the charge on a ball of niobium was the desired end. machine that will be used. (263) this is a general, probably mostly qualitative, idea of an arrangement of other experimental entities and the relevant exertion of their causal properties she must bring about. Because she has imbibed the functional nature of so many entities, she is rea dily able to formulate one possible network of such entities which, together, could conceivably generate this effect. Then the machine is built, surely after the idea is honed and carefully planned. The job, however, is not over once the design is constru process is also highly demonstrative of our functional understanding of the entities in question. using our causal knowledge of these entities that makes this possible. Further, we alter the
35 These hunches, again, arise from our causal sense of these entities. Really, all the entities that were involved in this way were manipulated in a certain sense. Further, all that were counted on in the design of the machine were experimental entities. Notice the role they played in the design of the experiment and our knowledge of them that allowed this. We will see, better, how we gain this causal understanding, how this process might confer confirmation and how the involvement in this process affects the attitude of the experimenter in regard to the entities suggestions for the significance for all this in detail. manipulation as evidential. The importance of manipulat ion, according to this suggestion lies in the confirmation it is able to confer to the entity manipulated. While we might identify this suggestion as the one which receives the least explicit acknowledgement by Hacking in his text, there is undoubtedly an reg (262). The precise formulation I offer for this proposal is as follows manifestly successful manipulation of an entity confers a distinctly significant degree of confirm ation to the suggestion (Hacking 255) The formulation is my own, but I think it captures reasonably well the best interpretation of what Hacking is really suggesting here. As can
36 sometimes be the case with Hacking, this assertio n is painted in with a broad brush and the details of the argument can be more suggestive than precise. What seems clear enough, however, is that the rationale for this argument has to do with the fact that successful manipulation demonstrates to us our un derstanding of the entity itself. Specifically, it demonstrates our understanding of the causal properties of the entity. That this is what he has in mind for this approach to understanding manipulation seems clear enough from an account he offers us of h electrons at a suspended ball of niobium to reduce its charge hoping to detect quarks. He tells us now there are standard emitt ers with which we can spray positrons and electrons and that is precisely what we do with them. We understand the effects, we understand the causes, and we There is, of course, a lot going on here, what we shoul d note most of all is that an demonstrated our knowledge of the entities that were being used. Hacking formulates this suggestion most explicitly as follows: kinds of evidence for a postulated or inferred entity is (sic) that we can begin to measure it or otherwise begin to understand its causal powers. The best evidence, in turn, that we have this kind of understanding is that we can set out, from scratch, to build machines
37 that will work fairly reliably, taking advantage of this or that causal nexus. Hence engineering, Much of what we have already discussed is mentioned in this passage. I t should also help hint at some of the details of the argument of concern before we commence our closer analysis of it. We see reference to building machines that employ causal powers, a process we had already associated with the manipulation of an entity. Here also we see the role of this manipulation as confirming our understanding, we will return to this passage but first we should get a better feel for how this actual process of confirmation accumulation actually proceeds by examining an example in deta il. For this example, we move back a bit farther in the history of the scientific understanding and interaction with the electron. In 1908, Thomas Millikin set about, most explicitly, to examine the charge of the electron. To do this, he let tiny oil dro plets pass between want to get into details but suffice it to say that, from a number of such velocities, Millikin was able to discern I.) whether or not there was a minimum unit of electrical charge and II.) what that charge was. The electron had been hypothesized as a tiny entity which was the bearer of this smallest unit of electrical charge. Indeed, the electron was very hypothetical at the time. I.) was no t a foregone conclusion. The support to be had from manipulation was very much wanting for the entity. So what exactly should we think of this experiment as achieving given our attempt to understand ppears to be a measurement of the electron, that is, it is a determination of the charge. Hacking does seem to consider the charge
38 adumbrated earlier, given the nat ure of the activity and the residual hypothetical nature of the electron itself, we would place it in the measurement stage. However, notice that manifestly successful manipulation, in a significant sense, was actually involved in this procedure. More sp exhibited the phenomenon that was the result which, ultimately convinced Millikan of the increased the cer tainty of the existence of the entity itself as well. Of course, continuing to follow through the rationale of our above passage, we should suppose that not much support is conferred by an experiment like this because of the simplicity of the experiment it self. That is, we do little in the way of demonstrating our understanding of the entity by engaging in an experiment that demands limited knowledge of the electron to be built successfully. The more prominent result of this experiment, however, is the pr oduction of a measurement, specifically, the exact charge of the electron, result II.). While Millikan surely had a range of expected charges for the minimum charge, provided there was one of course, he did not know what it would be. The measurement gleane d from this particular experiment, while it information that we can use in subsequent experiment to manipulate the entity. It is by engaging in such an experiment, and demonstrating our understanding of the measurement we took as the measurement of entity by redeploying it, that we actually do achieve this kind of confirmation for the entity.
39 Likely as not, every significant moment in the progress of the acquisiti on of this confirmation involves some combination like this. It seems, then, that even as we confirm an entity, we gather the material that makes this confirmation possible. This conclusion does not square well with talk of stages as we discussed above. If we have to talk of such stages, it cannot be in a way which is deeply meaningful for the process of confirmation. A hypothetical stage would be one in which an entity will not have been regarded with any particular experimental scrutiny and whose scientif ic role would simply be to act as a placeholder in an explanation. A measurement stage is one in which research has kicked off regarding the entity but is directed primarily at gathering information about it. The beginning of the measurement stage would s ignal the beginning of the acquisition of the strong confirmation for the entity which would continue and increase throughout the research regarding it. Hacking had discussed a final stage. Indeed, it was this stage which had really convinced him. The mos t significance I can see for this stage while we discuss this notion of confirmation is as an indicator to the lay person that a certain level of confirmation for the entity has been achieved. Using an entity to learn about others will not involve much in the way of confirmation for the entity itself given the uncertainty regarding the entity measured. Along with the easy assumption that the relevant scientists are competent, it indicates that a deep and broad understanding of the causal properties of the e ntity, at least, an understanding deep and broad enough for the given purpose, had been achieved. If the experimenters are competent, we can be sure that they would not use an entity they did not understand to glean information about another entity they di d not understand. The information so gathered could not be trusted. It is precisely this second order knowledge, knowing they understand the entity that is, that Hacking
40 had identified as essential and it was this knowledge that we gained from manipulating the entity successfully. Aside from this, however, the confirmation proceeds gradually, with the knowledge of the causal properties proceeding abreast of, or at least, leapfrogging with, the confirmation for the entity itself. We have to believe Hacking believes this to be the case when, knowing what we do about his thoughts on the nature of the manipulation of experimental entities, he tells us build devices that ach Having discussed the general idea of how the accumulation of the confirmation for the entity is supposed to proceed according to Hacking, we can examine, in greater detail, how the actual just ification for this picture of confirmation should be understood. Perhaps it is not properties) is that we can build machines from scratch that will work fairly understand or measure it or otherwise begin to u Why would this be the case exactly? It is certainly important for the overall argument. Just what do I mean by that? Many deeper explanations of the reality we take ourselves to be observing offer themselves, al l of which are quite good. Some may be better at solving some problems and others at others but there may be some places where problems are multiplied by theory shifts and other times where ability to solve these problems underdetermines the problems thems elves. Many hypothesized pictures of invisible reality
41 begin as metaphors for macroscopic phenomena such as waves and fluids. Because a metaphor is helpful in understanding the visible consequences of an invisible feature of reality, there are still many q uestion we could ask about how well these metaphorical pictures properly describe that feature of reality. The question of the mere existence of an entity, even a theoretical entity, seems a relatively straightforward affair in comparison. Existence, we tend to think, is the same thing for entities of all kinds, perhaps abstract entities excepted. Further, enti ties have certain behaviors and traits characteristics of them. Among the most important are that entities bear properties, occupy space, perform actions, are identical to themselves, persist through time and so on. Not all entities bear all these properti es. But if we see evidence of the effects of these properties, we can be quite confident an entity is responsible. The support conferred by manipulation is little To fully understand how the argument might go, we return to the construction of the experimental machines. In our earlier discussion of these machines, all parts had been rk as we status. Here, this means that we are dealing with an entity which has yet to receive significant confirmation by means of its manipulation by means of t he machines. The process by which this occurs will be familiar but slightly different. that their causal powers are exerted, at least when the machine is activated, in a certain specific way. This creates a definite, well understood matrix of causal forces. I emphasize, again, that
42 there is an understanding of the nature, and applicability, of causal powers and entities definite and consistent enough so as to allow for a respectably hard distinction between causal and theoretical content. I mention this because, in the clearest and most epistemically relevant cases of this kind of manipulation, the specific arrangement of the machine is intended to cause an entity of a cer tain description to bring about a certain effect and the description itself is involved in the design of this machine only insofar as it conveys causal/experimental content. This entity, by the way, is the uncertain one, the one we take to be in need of co nfirmation as the other entities are not but will have such confirmation conferred to it by the experiment. The various features of the causal matrix each exploit different causal features of the described entity to bring about this effect. We might, then, be able to distinguish between the properties of the entity in virtue of which the other entities in the apparatus interact with it, and those most directly responsible for the observable, ultimate effects. Given how well understood the machine itself is we have very good reason to believe that if there is an entity that fits the causal description employed to construct the machine, then the machine will, ultimately, produce the effect. The arrangement I described above was a two step process in keeping w ith our earlier description. The initial step is the actual planning and construction of the machine. The second step is the debugging stage. Both of these proceed similarly according to our thorough understanding of the entities involved and the second st ep is necessary more because of the nature of our knowledge of the entities involved than because of a lack of it. When we have gotten the machine to work regularly, we have very good reason to believe that it is because all the parts functioned togethe r. We have assumed that our
43 knowledge of the other entities was, effectively, beyond dispute. It was the described entity used to bring about the effect that was in danger of not existing. If the machine ultimately works, then the final conclusion that the entity that fits the description as it was used to construct the machine is, in naturalistic terms, virtually incontrovertible. The contrary assertion, that the entity does not exist in light of a successful experiment of this kind, would make of the mach We can gather more information from this insight about the nature of the kind of actual experiment. For example, the more complex the array of causal agents involved, the more unlikely it would be that the machine would function without the existence of the entity itself. Further, as time goes by, we may come to be justified in asserti ng of a single entity that it has many and diverse causal properties which allow it to be deployed using many very different machines to achieve all kinds of effects. While this may be, it seems that the use of one cluster of causal properties that allow o ne or two machines to regularly function as we have described will provide more confirmation then subsequent experiments. This is because such an experiment would already be more or less conclusive, there is only so much others could add. They could, it mu st be mentioned, contribute significantly if they employed causal agents that were known with greater certainty, a possibility we had not explicitly mentioned earlier. We more or less assumed that the other experimental entities employed in the constructio n of machines were known to exist and function in a certain way. An infinite regress threatens if we insist on casting the argument this way. After all, how were these others proven
44 to exist? I believe Hacking would have us understand this process to have begun with the In any case, I have offered only one facet of a potentially rather complex argument for seemed to offer us a second order structure of justification when he told us: to understand or measure it or otherwise begin to unde rstand its causal powers. The best evidence, in turn, that we have this kind of understanding is that we can set out, from scratch, to build machines that will work fairly reliably, taking advantage of this or that causal nexus. Hence engineering, not theo I believe this resonates with the understanding I have offered so far. We identify an entity as a single bearer of a number of causal and theoretical properties as part of a larger theory. We perform several experiments that would seem to yield further information about the entity so described. This further information would seem to be knowledge of the entity itself. This, in turn, would contribute to the knowledge of the existence of the entity as it meant the entity continually yielded information when subjected to a battery of measurements as an entity ought to. A problem arises because our understanding of the theoretical world is such that such measurements may mean many things other than that we a re actually measuring an entity. Hacking makes this point, in terms of an example with which we are already familiar, when he tells us:
45 experiment in which they are manipu lated). There were ever so many more things for the skeptic to find out: there was that nagging worry about inter molecular forces acting on the oil charges, but no It is all too easy to construe such a measurement as so many other things than a measurement of the entity in which we are interested. This kind of worry is not a new one. In Dialogues Concerning Natural Religion the skeptical Phi lo, while offering a spree of incompatible explanations of the designed appearance of creation, offers the suggestion that it could have just as easily been two or more intelligences that were responsible. Given the pervasiveness and generality of the noti on of an entity, this seems to be indicative of what the most general kind of worry might be that Hacking here expresses. Could the causal properties responsible for the experimental measurements and detections we gather not be those of many entities rathe r than one? To refute Philo on his own terms, we would have to find, completed, a phenomenon that would only be present if there was one, creating intelligence. The clearest be made to function if a single entity bearing certain causal properties actually exists. We can even choose which properties we want to show the entity possesses by carefully choosing the causal forces in the design of the machine so as not to bring about the desired result unless the entity in question possesses the properties we have singled out. I must take credit for at least the form of the forgoing argument. Nowhere does it appear in anything like the explicit terms in which I have offered it above in the work of
46 some of his suggestions. There are other things to be said in favor of this evidential suggestion. Other arguments might be offered with less or ev en no reliance on a substantial account of observation like the preceding needed. Hacking seems to suggest that manipulation is a notion closely tied to our notion of existence. If we succeed in an activity we interpret as manipulation time and time again, the step to the existence is a straightforward one. Less abstractly than that, however, this process of manipulation, as Hacking describes it, is an intensely empirical one. By this, I mean that the way he describes the rebuilding and tinkering needed to get the machines working implies a quick and complex responsiveness to the knowledge conditioning behavior of nature. The knowledge resulting from it all is complex in a way which eludes neat formulation. All this seems to recommend it as stronger, episte mically, than theory as we have been discussing it. We will have more to say, about this matter. For now, however, that should be enough on the topic of the suggestion of manipulation as a distinctly effective means of gathering confirmation for an entity position. Recall that the second version concerned the attitude of scientists to the entities with which they dealt. We shift our discussion to it now. I mentioned at the outset the further a criterion of commitment, in the spirit of Quine, or an inducer of belief in the entity for the researcher who manipulates it. Again, this ambiguity itself is not acknowledged in any explicit
47 some ent ities, namely the ones they use I contend they cannot help being so (262). While it might be argued these assertions are the same, it is far from obvious. I will treat them as being different. Perhaps this seems like quibbling, but it makes a difference. The central feature of this insight is the recognition that engaging in an activity a researcher regards as the manipulation of an entity moves her toward a realism regarding that entity, in one way or another, in a way and to a degree it does not in the case of the theorist who employs an entity as a part of a broader portion of theory. Here again, we see the significance of the distinct activity of experiment and its contribution to the realism discussion as mere theorists do not engage in actual manipu lation. Hacking emphasizes that this insight is not so much a matter of the experimenter becoming convinced by the evidence for the entity per se. Already then, we have something of a separation of this insight from the earlier evidential one. We recogni ze, first, a distinction between an entity discussed merely because it is a feature of a successful theory which helps that theory explain and predict phenomena and an entity discussed, specifically, because it can words, they are entities which we can use (Hacking 263). Needless to say, we will continue to talk about the latter kind of entity even when the original theory in which it was introduced, or, indeed, an y theory that explains other stable features of our understanding, is discarded. In the case of the purely theoretical entity, as long as it fails to be useful in the relevant way, is likely to stick around only as long as the theory describing it does. and the knowledge of their causal properties, to accumulate in a way that theory might not.
48 More importantly than the persistence of the scientific relevance of a manipula ted entity over theory, however, is the relation of the experimenter to this knowledge. We note that just about all conceivable uses of theory and the entities mentioned in it, that is, explanation, prediction, and so on, are compatible with an anti realis m in regard to that theory on the part of the experimenter and the theorist. Indeed, a theorist might quite deliberately suggest an entity simply because it makes a theory a bit more useful even while she regards it as false. On the other hand, no clear po sition of anti realism is possible for an entity one takes oneself to actually be using, that is, manipulating. Hacking expresses the contrast between these two kinds of entities as one between s noted, we might also note the persistence of the ambiguity between criterion of commitment and belief. The difference is one manipulation to involve her belief in the entity thought of as being manipulated or not. If we think of belief as an utterly free activity, then we might possibly assert that, even while someone thinks of an activity she is engaged in in terms of the manipulation of an entity, she might still remain an instrumentalist in regard to the entity. This sounds like a strange thing to do, certainly, but the possi bility should be mentioned. I f it is possible, then the criterion of commitment would remain, presumably, that one must accept the entity one takes oneself to be manipulating on pain of the above kind of strange contradiction. Of course, the contradiction is not a genuine logical one, simply a very unappealing one, so the criterion is not even completely hard and fast. What is sure is that even if the suggestion that manipulation implies belief fails, the criterion of commitment always remains. I have allowed for this rather extreme
49 possibility that the scientist does not believe in the entity but, to be fair, use in the relevant sense seems ver y good evidence indeed that the scientist does, in fact, believe in the entity. But surely all manipulation does not involve this distinct kind of realism on the part of rued all kinds of activities as manipulation, properly speaking, that would certainly not seem to involve anything like belief in, or commitment to, the entity on the part of the relevant researcher. Here our earlier, more straightforward, talk of stages o sense and can play a more important role in the suggestion itself. It seems the stage where this realism is really in place for an experimenter is that in which she is willing to use it to experiment on someth ing else. The significance in the evidential case had simply been that an properties has been reached. Here, however, it is the willingness to use the entity in this way w hich indicates the complete regard of the experimenter of the entity as a dependable tool rather than a questionable theoretical posit. The actual fact of the use of the entity to explore others on the part of the experimenter is still more an indicator o f the final stage than the inducer of it. While we could certainly say that the willingness of an experimenter to use an entity to explore another certainly seems to is not even that we use electrons to experiment on something else that makes it impossible to doubt discussed where it applies. What this willingness indicat es is the confidence in the usefulness of the entity for any of several ends.
50 Here too a distinct measurement stage can also be discussed. It is at this stage where we are learning the causal properties, the measurements are being collected and the entit y has yet to prove its mettle. While hard distinctions drawn along these lines had not been readily forthcoming in the suggestion of manipulation as confirmation, here they are to a greater degree. The measurement stage involves a progression of combinatio ns of an attitude of willingness to reject as fiction, that entity, the ratio between the strength of which increases with each successful manipulation/measurement in this stage. A number of features of the final attitude separate it discretely from that of the theoretical stage of an entity: purely hypothetical entities are simply used to account for other phenomena while the experimental entity is something that m ust be accounted for and might not be explained. Also, there is the more obvious willingness to use the experimental entity for a given purpose if its properties indicate it could be as opposed to unwillingness. The measurement stage, for an individual, co nsists in all the possible combinations of attitudes and propensities between these extremes.
51 Chapter 3: Quine, Hacking, and the Scientific Confirmation of Entities With our groundwork behind us, we can now directly engage the problem of interest to us the degree of scientific support for mathematical realism. Hinging on the ultimate answer to this question is the necessity of adopting a position of mathematical realism upon the adoption of a broadly Quinean naturalism. Indeed, in addressing this que stion, we will remain strictly within the confines set by naturalism, taking science as authoritative and employing the resources science provides. Further, our dealing with the question will consist primarily in our We will ultimately conclude that Hacking provides an account of scientific confirmation that allows us to avoid ontological commitment to mathematical entities within a broadly naturalized framework contrar might be mathematical nominalists even while adopting a naturalistic approach to philosophy. This option of a naturalistic nominalism follows the broader conclusion we will reach by followin g Hacking that mathematical entities are, epistemically, categorically weaker than other scientific entities. In order for the approach we have outlined to produce the conclusion we have ralistic and it must allow for the possibility of nominalism given the Quinean, naturalized understanding of science and its naturalism on its sleeve. Indeed, we
52 being motivated and supported by the work and attitudes of actual scientists, however. All the basic notions he employed, theory, observation, experiment, manipulation and so on, were all heavily informed by their meanings and use in the practice of empirical science itself. Hacking even brings some interesting variety to the approach of naturalism by raising th e issue of different kinds of experts within one field. I mean, of course, the issue of the authority of the experimenter or engineer as opposed to the theorist. However, if there is a well developed and coherent naturalistic outlook lying behind s work, I am not so much interested in it for the sake of this project. Quine already provides us with a naturalism sufficient to provide the parameters by which we might judge the stronger of two purportedly naturalized arguments. We need only concern our selves with the to perform in this way. Indeed, in presenting ma naturalized framework. The ultimate way to decide between a Quinean and Hackingian perspective in these terms, and where the two diverge, will be to determine which of them rich and developed one, but these considerations lay at the heart of his, and, perhaps, any other possible naturalism. With the terms of the debate b etween the two set, we notice that a disagreement between them that we cannot immediately resolve may require some form of empirical research for its decision. Here we run up against a limitation; due to the limited scope of this
53 project, we cannot actuall y engage in any such research and must leave this work to others interested in the question with the resources to do it. In such cases, then, we must simply do our best to resolve the issue and, where we cannot, make conclusions taking our limitations into consideration This brings us to the issue at hand; our goal is to argue for the possibility of naturalized nominalism from Hacking style considerations. But what is this alleged nominalistic conclusion r the importance Hacking lays on manipulation and observation above other methods for the confirmation of entities. Simply put, mathematical entities are in no even remotely obvious way either observable or manipulable. Quine had argued that theory that fe atured mathematical entities had sensory consequences. consequences to call their own of a kind that would satisfy a substantial theory of observation. According to Hacking then, these are theoretical entities. In addition, the abstractness of mathematicalia has one nearly unavoidable consequence. As certain as we might be that they are not observable, we can be at least as certain that mathematical entities lack the causal properties Hacking identifies as enabling the possibility of their manipulation and, therefore, the possibility of their removal from this theoretical status. Mathematical entities seem stuck firmly in the realm of theory. Theory, of course, was less conf irmed than other bodies of knowledge in a way which, Hacking told us, left open the option of maintaining an attitude of instrumentalism in regard to it. It seems we could maintain instrumentalism.
54 confirmation of these entities to allow us to single out mathematical entities as less well confirmed in their existence than other scientific en tities, thereby making sense of nominalistic simple task. The thought of Quine and Hacking both boas ted respectable internal coherence but, looking back, they appear to have been on different wavelengths. Responsible for this science was shaped, in large part, by conceptual tools and philosophical assumptions developed in, and closely tied to, the empiricism of the day. The philosophical apparatus I have in mind include his emphasis on theory, his doctrine of reification and his dealing with epistemology primari ly on the level of sensory intake. Hacking explicitly rejects many of these as unhelpful for effect. In this respect, we might see in Hacking an even stronger nat uralistic spirit than in Quine. employed by scientists. Whatever the case, the differences between these positions make comparison and the kind of assessment we a re hoping to undertake difficult. This problem might be ameliorated by for realism had, at its core, a certain conception of the nature of scientific confirmation Hacking is very concerned with this as well and the nominalist conclusion we sketched arose out of his view of confirmation. Indeed, in Hacking we see a thorough account of scientific confirmation incompatible with that of Quine. If we are to remain with
55 assessment of the fate of our anticipated nominalism conclusion should take place on the level of the comparison of these two theories of confirmation. Here, however, we come up against our anticipated limitations. Since our debate has become a matter of determining the nature of scientific confirmation, that is, an epistemological program of naturalized epistemology. This means ou r question becomes which of these accounts more accurately describes the psychological processes by which we arrived at our scientific account of the world. This would seem to imply that the question should be dealt with exclusively by the empirical scienc e of psychology of the kind we might see practiced in university departments and research institutions. While it is true that this science provides vital and, ultimately, authoritative considerations on the matter, it does not have exclusive access to the Indeed, he had offered many arguments for his own conception of the nature of scientific confirmation that would not normally be raised in the course of carrying ou t empirical psychology and were more appropriate for the philosopher even if they bore on the same problems. In lieu of a decisive account from empirical psychology, we can do the philosophical work in anticipation of such an account and even do our best to bring to bear the social scientific material relevant to that matter we have at our disposal. Indeed, Hacking offers us more than of it with a significant d egree of research into the actual practice of research scientists. Again, there are different degrees to which we can realize the account of one or the other in the final
56 account. How we do so will have significant bearing on the degree to which different scientific conclusions are supported, particularly our nominalistic conclusions. We will cover a number of such possible realizations, beginning with an account of confirmation which is thoroughly are the least controversial. We will then move in stages, each closer than the last, to what I believe to be a more thoroughly Hacking style picture of the same issue. At each of these stages, I will examine anew the fate of the nominalistic conclusion we discussed. Empirical psychology and other relevant sciences must ultimately decide which of these accounts is the most accurate one. We begin, then, by working firmly in the Quinean framework. Above all, at this point, we do not question the centrality an d power of the hypothetico deductive method as Quine sees it where the confirmation accumulated thereby is allocated. With these assumptions in place at thi s point, there seems little of the philosophy of Hacking we have come to know that can be brought to bear. We must narrow in on the insights on whose significance and interpretation Hacking and Quine can agree. Does Hacking raise any issues of this kind th at significantl y disrupt o science. We should look in detail at how this might go. Hacking had called out philosophy for its failure to attend to the actual practice of scientific experiment and for focusing too closely on theorizing. In more closely examining the actual practice of experiment, he noted the centrality of the manipulation of theoretical entities
57 by machines. What exactly does Quine make of the process of experimental manipulation? Quine had said that prediction was the means of the confirmation of theory. What does the manipulation of entities and the construction and tweaking of machines have to do with that? Is Quine just wrong about experimental practice? Quine even says that manipulation of nature is a goal of scienc e, but not a means of accruing confirmation for theory ( Pursuit of Truth 2). Putting manipulation as a special case of the account of the significance of exp eriment that Quine offers. We just have to shift our understanding of what is actually going on to see it. For Quine, we recall, a theory was tested by means of the observation categoricals that could be derived from it. An actual test would occur when th e antecedent held. Whether the theory passed the test depended on whether or not the consequent held as well. In the case of, say, the theory of the electron, Hacking says the experimentalists concerned with exploring the electron would build machines that employed its causal properties. Quine would understand the observation sentence whose assent would be prompted by the ordered set of the firing of the sensory receptors which would lead the experimenter to believe the machine had functioned as the consequ ent of the relevant observation categorical. The antecedent would be the observation sentence, or, more likely, the conjunction of observation sentences, announcing that the machine had been constructed, and should be functioning correctly. Holism enters i n a big way here of course there is the whole litany of pathological worries and features of the theory of the electron that Quine tells us will be involved by way of holism. There is also, however, the theory of all the other entities substantially involv ed in the construction of the
58 apparatus. This large portion of theory, of course, would receive confirmation when the machine functioned and would be altered, that is, rejected in its full form, when it failed to. modate the activity of manipulation as a means of gathering confirmation relatively easily. But there are many other features of of science. Some of them are a insights. Most obvious is the fact that Hacking emphasizes the degree to which the be captured by the ory but which, yet, still plays significantly into getting the experimental the linguistic expressions they would lead the believer to assent to was a powerful one. Here, however, the knowledge of the experimenter seems to strain the possibility that all beliefs will be adequately captured by dealing with the sentences related to them in this way. We will discuss this in significant detail later in our analysis. There is a cogent response for Quine here however. If inexpressible beliefs are involved, and they may not be, then we can still simply talk about confirmation accruing to them along with the other expressible ones. The proper conclusions of science, then, will be of a mixed kind. Some will be theory in a straightforward form while others will be left, uninterpreted, as the content of the beliefs of the scientist. Quine wou ld certainly consider this a regress, he had
59 account, however, is still intact; these uninterpreted beliefs would be involved in the holistic confirmation as would the other expressible ones. talk of beliefs has significantly compli cated the matter of ontological commitment. Quine had assuming of an object consists in is to ask what referring to an Theories and Things 2) upon the shift to talk about the sentences to which that belief would lead the individual in question to assent. If we cannot shift to talk of sentences when talking about a belief, no remotely obvious way of discerning the commitments of that belief presents itself. This does not mean that these beliefs are not commi tted to objects, it only means that we have no easy way of knowing which objects they are committed to. Further, this does not free us from commitment to mathematical objects at this stage. Engaging in our we enforce a thorough holism and see the large body of physical theory that describes the experimental entities in question as confirmed by the battery of experiments in which the entity in question is involved. This theory, of course, goes far beyond kn owledge that might qualify as inexpressible in the manner above. This large body of physical theory will, of course, include the kind of quantified sentences that are truly committed to mathematical objects and, thus, we see mathematical entities as confir med along with them. We will, however, return to this issue. experimental work Hacking describes, we run up against another problem that Hacking raises for an account of confir
60 n their more properly theoretical aspect. Often, many of these would be actually involved in an experiment; what sense does it make to talk about confirmation in this case? This worry can be overcome without too much trouble, however. The fact that there a re so many theories involved in experimental confirmation simply means that the confirmation that does accrue is distributed between them. Scientists would hold out for an experiment or series of experiments that definitively refuted one and left the other Until then, a strong theoretical realist would have to remain agnostic about which was true. doubts about the possibility of the standards of science terminating in a final account. He is convinced that sensory information far underdetermines theory and that even the virtues and any additional selective forces built into scientific procedure are not enough to narrow in on a final account. His realism is that of bel ieving the current body of scientific theory that is best confirmed and in use, something Hacking had identified as entirely optional. Possibly, the uncontroversial insight Hacking offers that most effectively challenges the Quinean picture of scientific confirmation concerns the sheer amount of tweaking and debugging that goes on after a machine fails to work before a theory is actually rejected. We had fit the process of machine building and operation into the Quinean model of experiment. According to t his model, however, if the consequent of the relevant observation categorical fails to hold when the antecedent does hold then the theory is simply rejected. In the case Hacking describes, however, when the consequent fails to be realized after the first a ttempt, this is only the beginning of a complex process of debugging. We can see allowing for some of this kind of
61 activity, but it does basically weigh against the understanding of testing that Quine submits. The test of theory was supposed to be, precise ly, the prediction of future sensory stimulation. If a yet, the theory is not rejected (instead held as true while the technicians tweak the machine) then it wou ld appear that prediction is not the central concern in testing. Quine had expressly allowed for some tweaking before theory is rejected. He had also explicitly said tha t the amount of tweaking that could be allowed must be very limited ( Ways of Paradox 247). The kind of revision that Hacking had discussed was anything but limited. He said that many different prototypes needed to be built based on the same idea before one could be made to work. Often, a keen eye was needed to see just what had to be done in order to get the allowances to account for. While all this does strain ways out even if none of them are terribly satisfying. We might say that prediction of the conclusion is a probabilistic one. The theory would predict that a number of different, but very similar devices would perform the same function, to each of which the same theory would attach a certain probability of working. The debugging of the experimenter would be altering their first machine to be one of these. The probability would enter by way of a pre mise expressing an anticipation of otherwise unexplained interference. We might also see some of construction. Perhaps, also, we could see the experimenter as actua lly rejecting the theory in
62 the original form and replacing it by very subtly different versions. Once a machine fails to work, the theory which had guided its construction could simply be replaced by a slightly different theory which differed only in its low level articulation of the precise nature of the entities involved, leaving the theory proper, more or less intact. This is a more damaging insight than the others we have covered until now. We have d not come out entirely unscathed. We will see later that this insight does weigh in favor of what we might now identify as a more thoroughly Hacking style account. While we had seen interesting breaches in the Quinean framework while being very conservat mathematical realist conclusion, in more or less the form it exists in his thought, remains un derstanding of the nature of science, even while we remain essentially within the framework Quine sets. We had noted that bodies of theory were involved in the derivation of the observation categoricals that described the machines that manipulated the theo retical entities perspective to more thoroughly inform our understanding of the nature of scientific confirmation. We do this by scaling back the scope of the scientif ic information involved by means of holism in the actual empirical confirmation conferred by experiments of the kind Hacking describes. ntain that empirical confirmation would always accrue to
63 our entire body of scientific theory. It would not make sense, then, to talk about certain kinds of saw, however, Quine abandoned this strong holism, instead proposing that smaller portions of theory more directly involved in the actual derivation of the observation categoricals were those we should view as the relevant bearers of confirmation and disconfirma tion. Given this suggestion, perhaps, we should think of the bodies of causal properties for a given entity, and not those which Hacking identifies as theory, being precisely those which are involved via holism. If this were so, then only the body of causa l properties, properly identified, was that which actually received confirmation from experiment. concerned with such side assertions as the assumption that all our i nstruments are working properly and delivering to us the correct information, that we ourselves are perceiving and understanding the information as it should be understood, and any number of other assumptions necessary to construe what we are seeing as a s uccessful manipulation. This need not make much of a difference however. We are only hoping to say that manipulation involves, and proves, our knowledge of a limited range of properties that might properly be called in addition to these, assumptions eliminating addition of these facts negating such worries will not be a significant impediment. Already, the burden of proof has fallen on Hacking. The assumption in the immediately preceding discussion is that the causal properties are the only ones substantially involved in the derivation of observational consequences of employing a theoretical way in a certain way.
64 Indeed, t consists in the construction of machines for the purpose of manipulation as he describes it, then a decisively greater degree of confirmation will have accrued already for the causal properties of argument for the scientific support of mathematical realism. Quine, w e recall, had told us that a object in order to be true. Further, he told us that scientifically confirmed theory quantified over mathematical objects and did so ev en after attempts to eliminate them in the course of casting scientific theory in the regimented notation. This was, primarily, because it was necessary to quantify over them in the formulation of broad physical laws, although, this was not the only reason These physical laws fall pretty squarely in the realm of deep theory. They are, by no means, clear examples of causal properties of entities in the relevant sense. The lesson seems clear enough then; If Hacking is right about the points we have already mentioned, and we maintain a strongly Quinean framework as we have been doing, the theories in which mathematical entities are irreducibly referenced are, systematically, weakly supported given the prevalence of manipulation for empirical scientific work. Notice the qualification in the above assertion that it was only the theory in which mathematical entities appeared irreducibly that seemed to be in a position of receiving less empirical support. To be sure, mathematical entities will continue to be ment ioned in the theory used to describe the causal properties of experimental entities. Measurement of the properties
65 of the entities in question is, after all, an important part of the learning phase, inasmuch as we can talk about one, in our interaction wit h a theoretical entity. Therefore, if nothing else, we can expect names of mathematical entities to appear in the description of the causal properties reporting these measurements. We have on our hands, then, a slew of mentions of mathematical entities in the theory that will be receiving the bulk of the empirical confirmation. If we want to take advantage of these new lessons taught by Hacking to lessen the degree of empirical support for the mathematical entities allegedly so named, we will have to go thr ough the work of reconstruing the mere mentions of the numbers and classes so as to avoid ontological commitment. Perhaps we might have not bothered to do this when we simply accepted the existence of mathematical entities, viewing scientific experimental procedure as Quine did. Once one accepts classes, then many notions can be cast in rather simple terms as we saw We might object that the mere fact that scientists do focus most of the resources set aside for research on this specific kind of theory is a bad reason to suppose the remaining portion is less supported in any fundamental way. This is, after all, what seems to be the core scientists ought to spread their degree? In other words, if we think that manipulation represents a kind of experiment that only ma nipulation and do what we can to test theory as well? This is problematic. Hacking had even admitted that theory might be quite useful for prediction and so on. If this is the case, then there may not be so much of a discrepancy
66 between the confirmation, under our current Quinean viewpoint, for the understanding of the causal nature of experimental entities and that of theory proper. The latter would simply be receiving its confirmation in a different setting than the proper experimental setting. The most ready answer to this concern may simply be that the causal properties of experimental entities do actually successfully fill explanatory and predictive roles that deeper theory would otherwise be needed for. In other words, theory only tends to succeed at prediction inasmuch as it expresses the causal properties of experimental entities. Perhaps this is a bit of a shaky response. This problem highlights the important difference between Hacking and Quine, namely, that manipulation ought to be able to confer a distinctly great degree of support to the result of the greater amount of successful experiment in the relevant area. If we are to better realize the meat What we want, then, is for the bodies of theory relevantly involved in the manipulation of an entity to receive distinctly significant degrees of support. It is one thing to simpl y say they do, but for this suggestion to have much content we will need to actually identify these assertions and discern the degree to which our remaining Quinean assumptions affect the outcome. How do we do this? It is difficult to say really. I had tal ked about only manifestly successful acts of manipulation being able to confer the kind of support that Hacking had talked about. This meant that, perhaps, if there is an accurate narrative of causal interaction from the entity to the effect in question, t hen the theory receives a greater degree of confirmation. This entity in question. How would we include this in our suggestion? Perhaps, included in this causal
67 narrative, would have to be reference to a manipulator. That is, there would have to be deliberate human action intending to generate the effect as part of the narrative. a specific kind of body of theory as that which should receive more confirmation than anything else? Perhaps it is that its confirmation by manipulation advances in discrete degrees that simply approach certainty with greater rapidity than confirmation con ferred by any other means. 1 To harden the separation even more, we might suggest that the rate of confirmation attainable by non manipulation means is such that it does not approach certainty while the former does or at least, some greater measure of certa inty between 0 and 1, with repeated experiments 2 Formulating the above suggestions in exact terms would mean involving a rather precise inductive principle. The kind of discrepancies in positive support my preceding discussion represents simply seem the epistemic superiority of manipulation given the current parameters of our discussion. I do not intend to suggest or defend such a principle. This can be a project for another day. Talking about different degrees and rates of confirmation in this way would involve a be based solely on the ability of a piece of theory to predict future sensory input but, also, its being of a cert ain form. Making this kind of distinction involves elevating, to the level of prediction of empirical content, 1 Perhaps confirmation accrues, for such theory, by something like the rate of increase expressed by the series ( 1 / 2 ) = 1 while the rate for other kinds of confirmation might involve a series with a slower rate of accumulation. 2 Perhaps, for example, we could model the accumulation of the causal theory with the formula ( 1 / 2 ) = 1 and other kinds with ( 1 / 3 ) = 1
68 considerations internal to our world theory itself. Perhaps this is a bit strong however; perhaps what we are doing is merely singling out certai n portions of theory as more responsive to empirical support. We would base this move on the various arguments we have already offered for the distinct significance of manipulation. We remain in the Quinean framework in that we are focusing on the theory t hat is actually employed, by the scientist, in the course of engaging in the experiment. That is to say, we are talking about the actual theory that the experimenters use to guide their experiments Our supposition so far is that the theory in which we s hould suppose mathematics to be irreducibly contained is part of that portion of theory that does not tend to become involved in the experimental manipulation of entities. Whether we adopt the former or the latter strategy determines just how big the diffe rence between experimental and proper theory will be. But what is the payoff of all this? We are still accepting mathematics in a straightforwardly Quinean Nothing about our preceding discussion motivated us to change our assertion that the acceptance of deep theory demanded the acceptance of mathematical objects. In other words, our Quinean perspective continued to lead us to see quantification as truly co mmittal. Further, since we are remaining in a Quinean framework, even though we were hesitant to directly take up the issue of the precise formulation of empirical support, we should not believe that the theory marked as less receptive to empirical suppor t is scientifically un acceptable. Our strongest conclusion was simply that it was categorically weaker than experimental knowledge. Perhaps this conclusion is already enough to satisfy some with doubts about mathematical entities. The nominalist need not s ettle for this alone at this stage,
69 however. The possibility of a nominalistic conclusion comes with the possibility of instrumentalism. The approach I have in mind here would involve acceptance of all scientific theory above a certain level of confirmatio n and rejection of all theory below it as being of, at best, instrumental value. The nominalist could specifically choose a level that would exclude deep theory and admit experimental information as per the discrepancy in confirmation we suggested above. If our earlier discussion is correct, experimental knowledge would not involve commitment to mathematical entities. Hereby, the mathematical nominalist could remain both a nominalist and a naturalist. We might wonder at the degree to which instrumentalism of this kind is properly naturalistic. Hacking, of course, had argued that instrumentalism in regard to deep scientific theory was, indeed, a legitimate option and we might think of this specific approach to instrumentalism as precisely the realization of this suggestion within our current Quinean parameters. That a naturalized nominalist might choose to exclude mathematical entities by these means would simply be a result of her conviction as to the nonexistence of mathematical entities. The strength of h er conviction in this nominalism, of course, would have to be greater than that of her belief in the full truth of the theory she would be regarding as merely instrumental in order to follow through on it in this situation. Here, then, we have our nominali stic option. There is a bit of potential seesawing here depending on how exactly things go on the experimental side as it concerns the involvement of theory committed to mathematical entities. If confirmation ultimately demands that we do accept some math ematical entities, we might be no better off in regard to our need to accept mathematical entities than we were when we
70 started. How is this exactly? Recall Quine had proposed we simply accept sets instead of numbers or, indeed, any other distinctly mathem atical objects as they were all that would be needed to do all the work we would otherwise need these objects to do. If, by whatever means, we are forced into accepting numbers as we would have to if some of the deep theory worked its way into the well con firmed portions we have been discussing, we would probably be better off simply accepting sets instead. Even if we only had to accept the real numbers, sets would probably be the objects to go with as they have proven so useful and clear as foundational ob jects (although we would probably want to stop short of the full cumulative hierarchy). In any case, if we did accept these objects, we might think we could still see Hacking as having shown mathematical entities to be systematically less well supported t han other objects referenced in science. But if we end up accepting mathematical objects and placing them in these higher reaches, the reasoning that had demanded we reinterpret those portions of theory to avoid an overly complex ontology might give way to the counteracting forces exerted by those principles which demand we avoid excessive complexity in our theories. The reinterpretations we would have to do in order to make this asymmetry in confirmation come off would indeed be somewhat cumbersome, perhap s it would be best to simply accept the measurements and so on to be what they initial appear to be abstract numerical objects. Exactly how this would actually go would depend on how strong a consolation the limited degree of support these entities would r eceive by maintaining the reinterpretation was (that is, how strongly it was motivated by the virtues which had previously encouraged mathematical nominalism).
71 I have been wont to characterize the above discussion as a heavily Quinean one. While we had m scientific practice that Quine endorses. This meant characterizing scientif ic method, uniformly, as the accumulation of empirical confirmation by way of the holistic application of H D and the other virtues as well as an understanding of all confirmation accruing, in proper science, to assertions cast in the regimented language Q uine had suggested. We had only moderate luck trying to realize our anticipated nominalist conclusion and, even then, there were a number of problems. What went wrong? Our strict maintenance of the overall Quinean framework is to blame for these results. For one thing, this goes to show just how strongly the combination of the Quinean understanding of science with the actual content of scientific theory implies the conclusion of mathematical realism. For another, it shows that if we want the conclusion ex pressed above to follow, then we will have to move away from this science. This will involve moving manipulation to a more significant place in our understanding of scientifi c confirmation. To do this properly, and in a way which will allow for continued comparability with Quine, we should take a closer look at manipulation itself. As the etymology of the term would seem to suggest, the common sense meaning of is likely to be heavily keyed to that vague cluster of activities that we do, in the normal course of things, to and with other things with our hands. Whether or not this is the case, there is a perfectly respectable general meaning of this term well suite d to the kinds of
72 activities scientists engage in. We had assumed a reasonable familiarity and understanding of this general notion in our discussion of Hacking. We may say that an entity is manipulated when a change pertaining to it has been brought abo ut by interactions with it involving the excitation of its causal powers by actions deliberately intended for this end This definition is probably not completely adequate, but it is good enough for our purposes. Most importantly, this definition lays bare the assumptions needs to be a world of objects with causal powers to manipulate of course. If we are to further make sense of states that these entities might have, we are likely going to have to make some sense of many other notions such as space, the finer points of the nature of matter, charge, mass and so on. This is a fairly robust ontology, especially when we add to it the whole network of well understood macroscopic objects Hacking seems to see as delivered to us by direct observation. Noting that the ontology is robust, of course, is not to say that the fact is surprising. Indeed, this is probably the minimum of ontological resources we could expect to n eed for science itself and the fulfillment of its ends. The robustness of an ontology, of course, is not even likely to concern Hacking very much. To focus attention on it still shows Quinean inclinations. Robustness aside, the issue of the resources need ed to make sense of manipulation is very important because, if we are right to see Hacking as telling us that this is the best way to accumulate confirmation for a theoretical entity, these resources will be baked into the very foundations of theoretical s cience in a way that the content of theories proper will not. If not this, perhaps, then at least they are more firmly held in place than are the resources needed for
73 the theories that come and go. What this means is that the ontological framework for a si gnificant portion for science is already in place and, assuming the basic method of science remains the same in the future, actual science will simply involve the filling in of this framework according to the conditioning parameters provided by nature. Th ese parameters are precisely those provided by the twin epistemic tools of observation, and manipulation as Hacking sees them. The best confirmed visible world, then, is enomena the world that is best confirmed for us is one composed of minimally characterized entities, albeit with potentially inscrutable deeper natures, that causally interact with each other in well understood ways, in addition to other mysterious forces which interfere with experimental machines. The ontology of any future science, then, is bound to include the one described above. It need not be limited to this of course. If we decide to be realists in regard to theory, then our ontology may end up being a more complex one. However, even if we do decide to be theoretical realists, Hacking seems to say, the world our future science will deliver to us will be one that we can be sure will contain the ontology already described. If H D is a weaker means of confirmation than observation and manipulation then, in the event that a theory that does contradict the information conveyed by these two begins to enjoy H D success to such a degree that it edges out all other candidates, we will have no choice but to be anti realists, that is instrumentalists, in r egard to it. This is because it will be disproven by the better confirmed fact of the ontology needed for manipulation and observation. In such a situation, perhaps those
74 with realist inclinations would simply accept the best confirmed theory on the releva nt matter that did not contradict manipulation and observation. We have begun to observe the consequences of our move away from the Quinean 3 his way of rebuildin g the boat produces an end product quite different than that of Quine in the same position. of mathematical entities. A system of scientific proof that definitively fav ors manipulated entities would have little support to offer mathematical entities. Perhaps now with the reorientation of our perspective away from Quine, we can more fully realize this intuition. A significant obstacle to this successful realization still presents itself. What about the causal over mathematical entities for their expression. If this is so, and they really are strongly confirmed along with the knowl edge of the existence of manipulated entities in the process of manipulation, we may have to accept mathematical existences even with our new Hackingian perspective. Have we made any progress? To best answer this question, we might take a closer look than we had before at the actual process of building machines that Hacking regards as so important. The only knowledge that will be confirmed hereby will be that actually involved, or close to that involved, in the process of manipulation. Perhaps an analysis of the process will give us a needed better feel for 3 The Positivist philosopher Otto Neurath offers an illustrative parable of naturalism often quoted by Quine. In this parable, a sailor must rebuild a ship even as he stays afloat in it. The sailor represents the naturalized philosopher, and the ship, her a ccount of science and the world.
75 where we stand in the way of the confirmation of mathematical entities. Hacking had emphasized the process of tinkering and tooling with the machine once it had already been constructed. This process was an important feature of the process of confirmation by manipulation. Hacking pointed out that, often enough, the machines we construct do not initially work as we might expect. It may be necessary to make minor adjustments here and there in the machine in order to produce the expected and needed result. interference per se, but, rath er, applying our knowledge of the causal properties of the entities concerned to achieve the desired results (Hacking 265). In support of this understanding of the nature of the process is the observation that the relevant tinkering need not even arise as an actual reaction to evident problems with, or failures in, the machines themselves. Instead, the experimenter may tweak the machine, or the conditions under which it operates, in response to conceived and anticipated problems. The anticipation of these problems results from, again, an understanding of the powers of the entity manipulated as part of the experiment. Hacking gives the example of an apparatus designed to emit electrons for the purpose of experimentally determining the properties of a hypoth etical natural force. A certain experimenter worried that dust on the laser being used to dislodge the electrons from their source might lead to problems with the final results of the experiment and, from then on, cleaned the emitter regularly (Hacking 69 70). Hacking means us to understand these activities as minimally theoretical and as proceeding from our diffuse understanding of the nature of the entity. In the preceding example, there may be a modest clump of theory involved in the
76 anticipation of prob lems and the formulation of a response, but it would be nothing particularly deep, systematic, or continuous with heavily quantitative, broader understanding (Hacking 271). We had already discussed the nature of the knowledge of these entities which seeme d to shy away from definite linguistic formulation. This seems to be borne out by our closer analysis of this facet of the process of manipulation. No matter what, however, we can count on a thin layer of theory accompanying the process of adjustment of th e kind described above. This is the theory expressing the effectiveness of the acts of adjustment taken for the desired end. We can think of this layer of theory as completely expressible in terms of what I will call engineering directives. These are stat If we assume, rather optimistically, that this sentence form characterizes all theory properly involved in the process of adjustment, we can safely conclude that this feature of the process of manipulation is in no danger of confirming mathematical entities along with it. There are, of course no mathematical expressions involved in the general form of the expressions and include irreducible references to mathematical entities. They will, more than likely, be closer to expressions of observable features of the machine or phenomena. Mathematics is only likely to appear in the form of individual names, perhaps expressing precise degrees to which adjustments must be made. We have not challenged the Quinean conception of ontological commitment and, therefore, these locutions are not committal.
77 While the activity of adjustment may be important, it is only one feature of the overall activity of manipulation. The other is the actual design and constru ction of the machines. Perhaps we could argue that the construction of the machine is not properly to be considered part of the manipulation itself. This seems dishonest, however, as this construction proceeds almost exclusively according to our understand ing of the causal nature of the entities themselves. Further, the cogency of thinking of using the machine as manipulating the entity itself ought to be closely tied to the basic understanding of the machine as a functioning one that exploits the entity in then our prospects for avoiding mathematical entities become far dimmer. Forgoing broad and respectably precise expressions of the nature of the entity itself as an active and full blo wn feature of the process of putting together experimental devices would seem to make of this process an intolerably haphazard one. Hacking tells us that these machines rarely work as we would hope on the first try and one machine may not even ever work, n eeding to be abandoned as a failed prototype. But much of this failure is likely to result from a limited understanding of outside and ambient forces and factors rather than a general and fundamental vagueness in the nature of machine itself. Even still, we might hope to avoid mathematical commitment by finding purely qualitative theory involved in the construction of these machines. This is a bit more promising an angle as Hacking sees many such assertions at work in his close analysis of the experiment o f the electron emitter I mentioned above. In this experiment, a substance is provided from which electrons will be removed. In order to facilitate this removal, the substance is painted with Gallium Arsenide and kept in a near vacuum for the extent of the experiment. To remove the
78 electrons, it is struck with a laser composed of linearly polarized photons. The electrons emitted from the substance will be polarized in a predictable ratio of directions, as hoped. These electrons are then moved for a distance prominent description of the manipulation of an experimental entity. We do not need to go into the actual purpose of the experiment to appreciate this (Hacking 268 270). The laws that were involved in t he construction of this machines were, often enough, this would involve naming charges). Hacking tends to encourage us to regard the deeper quantitative understanding of these laws as unimportant to their role in the experiment itself. Even still, however, it seems unlikely to expect that no quantitative law s will be involved in the process itself. The configuration of the charge of the magnets and the accompanying use of precise electric charge will surely assume the truth of some laws relating and expressing the o get myself in trouble by talking in too much detail about something I am not fully qualified to discuss. However, Hacking, in a different context, Franz law which states that the ratio of the thermal to the electric co nductivity is equal to LT, where T is the absolute temperature and L is a constant x y ((Txz&Eyz) s(Pxys& r constant).
79 If our understanding of the causal properties really is quantitative in this sense, are we really forced to assume that mathematical entities exist, confirmed to the same degree as the manipulated experimental entities? If so, this would be quite an upset for Hac philosophy and would therefore, be an interesting conclusion in itself. Mathematical entities, it would seem, are the very model of a theoretical entity. Not only have we not manipulated them, we have no reason to assu me we ever can or will be able to. It would be a strange result indeed which attributed to mathematical entities anything like causal properties in the relevant sense. That they could be confirmed to the same degree would mean that manipulation need not be performed on the actual entity to confer to them the strong confirmation Hacking had discussed. act that it was so readily replaced and altered by the normal process of scientific work. This is certainly not the case with thought though there may be, the Hackingian need not accept mathematical entities. Indeed, something very much like our anticipated nominalist conclusion is possible once we fully exploit the shift in our view on scientific confirmation. The question is w hether mathematical entities are supported in their existence by the process of confirmation Hacking describes. If we hope to avoid mathematical entities, the way to attack this question is to produce a more thorough analysis of exactly what it is that is supported by this process of confirmation. Indeed, we have talked vaguely about the possibility of separating off a component of theory describing an entity that, specifically, conveys its causal
80 nature from the more properly theoretical component. How mi ght this actually go and what scientists. There are, of course, plent y of seemingly suitable linguistic expressions of these components of scientific knowledge that actually appear in the practice of experimental science. different t entities, of which there were often several incompatible examples for a single entity that could be used in different situations (Hacking 217). Then there was the vague, taci t knowledge from which the hunches Hacking had mentioned seemed to arise. This classification is, surely, not exhaustive. Given the diversity of what we are seeing already, we might think the hope of identifying, by some neat criterion, theory proper and removing it from the expression of the causal properties seems like a distant one. This has a lot to do with the sheer vagueness of the notion of theory itself. We had formulated a sort of functional understanding of what theory, as opposed to more stable kinds of knowledge, consisted in. Nothing we have talked about so far, however, would be suitable for the purposes of purging the verbal expression of our experimental understanding of an entity of this kind of content. While it might be a disappointment to have to abandon the hope of precisely describing theory, we may yet hope for a precise description of causal content by shifting our attention to the nature of this content itself. This approach looks much more promising already. We had adduced specifi
81 and the nature of manipulation itself, as a guide to precisely describing experimental content. We need simply determine the nature of the information proven by the process of ma nipulation. What was proven, no matter how we looked at it, was that experimental entities possessed certain properties whose cash value, if nothing else, was the changes of physical state it could bring about for an entity with some other given causal pro perty. Precisely describing the information proven by manipulation would seem to at least involve a full, detailed account of causal properties and how they causal entities to interact. While a full understanding of the nature of this topic would certainly need to elaborate in detail of the description of the causal natures of these entities while avoiding direct reliance on even this unexplained notion. What I have in mind here is a schematic formulation of the kind of and involve centrally in their reasoning, as they engage in the manipulation of an entity. The formulation is as follows In situation X, entity A, because of its possessing property Y, causes entity B in virtue of its possessing property Z, to produce effect W. Again, I emphasize that this is a schematic form of the sentence; the capital letters a re simply place holders for other the appropriate physica l attributes. We should understand this schematic sentence as a tool for querying experimentalists, as they engage in their experimental work, on the reasoning they actually use as it involves this pure experimental content. That is, rather than simply as king what generalizations are being
82 used in an experiment, an approach that would invite all content, theoretical or not, we ask for causes, entities, and their interaction on this level (i.e. experimental content) by demanding conformity to the prescribed form in their response. The work of discerning the basic nature of a cause is avoided and passed on to the scientist in very small part. The sentence form, then, is seen as playing the role of constraining the response of the scientist so they will provid e the information we are looking for. What we are currently asking of the scientist, then, is a complete narration of the reasoning process behind the construction of a machine in terms of sentences adhering to the form we have described which ought to del Hacking discusses. This said, we would have to be liberal with the statements that we would accept as conforming, in the important respects, to our prescribed form. It would be best, also, to allow that some of the blan ks need not be filled in, and that not all cases of causes and effects of theoretical entities will fit comfortably in this form. There are more important worries, however, concerning the fact that theory may yet enter into the responses scientists give. I f Hacking is right about the difference between pure theory and experimental knowledge, then scientists ought to have some kind of awareness of which scientific conclusions are which. However, experimenters could still, and probably would, be realists in r egard to theory. Demanding they be cast entirely in the sentences of the form we described may not be enough to ensure that experimenters do not include theoretical content in the accounts of their experimental work. In other words, merely asking for any n arrative of the construction and operations of experimental machines the relevant experimenters might offer may well not be enough to narrow in on the
83 scientists t hat they should refrain from including theory in their responses to the queries given the vague nature of the notion. While this is a step in the right direction, we will need more assurance we will not be catching theoretical content by our strategy These worries will require us to shift our strategy a bit. My strategy of talking about the causal reasoning concerning these experimental entities is clearly modeled on that of Quine for procuring workable formulations of beliefs. To reiterate, Quine had tende d to talk about Hacking, more than anyone else covered here, gives us reasons to doubt the full legitimacy of this strategy, we will address the problems tha later. For now, I will, along with Quine, make this move to discussing disposition to assent as engage in an act of experimental manipulation of an entity will allow us the full latitude we will need to avoid the worries we had raised. as being so complex as to very possibl y elude formulations like these. I am only dealing with reasoning concerning the manipulated entity inasmuch as it can be expressed. Our driving assumption, based, of course, on what Hacking has told us, is that all such reasoning can be expressed in the f orm I have offered already. But there is more than what can actually be expressed about the overall process of manipulation than what can be expressed about the reasoning concerning cause and effect and the physical interaction of entities per se. Perhaps the sentences of the form we framed earlier might remind us of the
84 composed entirely of sentences of this form, if nothing else, could be counted on to accomp any and narrate the construction and alteration of an experimental machine. We now have this additional sentence form which, we might believe, can fully express the deeper reasoning lete characterization, possessing minimal theoretical content, of the reasoning involved in the actual construction of the machine characterized by sentences beginning with an engineering directive in a conjunction of the causal sentences we recently described. There may need to be some bare explanation appended to some of these sentences to close any residual gap in explanation between the two components of the different forms. With these language forms framed, we have what we need to narrow in on the minimal content we have been hoping for. We had talked about shifting focus to the sentences of the preceding form to which the experimenter would assent in the course of developing a machine to achie ve a certain end rather than those they would voluntarily offer in response to some predetermined query about this process. We could identify this minimal content for a certain machine as the narrative to which the relevant scientist would assent expressin g the reasoning behind the machine which the same scientist would also indentify as the most conservative/least substantial of all possible such narratives. We would have to demand, in addition to this, that the narrative be cast entirely in terms of the s entence form we have already described. This latter demand is not to impose on scientists constraints that they would not otherwise think legitimate. Since we believe that scientists accept manipulation as a superior means of the confirmation of entities a
85 of constraints is simply framed in order to narrow in precisely on the content so proven as these scientists ought to recognize it. While we should not assume that scientists can explicitly point out properly experimental information, as opposed to theory, when asked, we should assume that they know what information is best confirmed by certain acts of manipulation. Our newly framed criteria ought to be up to the task of identifying at least the minimal experi mental information scientists involved in the manipulation of an entity would see as best confirmed by this process. The expressions we have identified above, if Hacking is right, are guaranteed to be supported by the success by manipulation. We might thi nk ourselves limited here by the fact that we are only dealing with experiments that have actually happened. There is a possibility of expanding our scope however we might talk about all possible uses to which an entity might be put and the machines that w ould be constructed for them. This again, would involve recourse to dispositions of the experimenters. In other words, we are talking about the minimal discourse of explanation and reasoning a scientist has a disposition to offer for all possible experimen tal goals. This includes the goal of getting machines they had already built to work. This collection of narratives we just talked about is quite an unwieldy thing. What exactly is the significance of it? The totality of the information in this collection represents the barest expression of the (expressible) knowledge, of the experimenter queried, of the causal nature of all the experimental entities with which she is familiar. The overall content of all the expressions so collected is the minimal, causal c ontent we had hoped for. There will be more we are justified in calling experimental knowledge as per Hacking. This minimal knowledge is that we can be sure is confirmed by the manipulation of an entity.
86 the way she does, that is, to achieve these experimental ends, specifically identifies it as the knowledge of the causal properties we had identified in our discussion of Hacking. That is, this is the knowledge which, if nothing else, we know we know to th e degree of strength that regular manipulation confers. This is little more than the material of induction for stronger generalizations all of which, I believe will be less confirmed than this information itself. We have, then, the minimal content we were looking for. This inquiry into the expressible content minimally demonstrated by the manipulation of entities has been, perhaps, more involved than expected. Indeed, Hacking, being anti theoretical as he is, probably did not have conclusions like those I now find myself reaching from his position in mind when he offered it. Whether or not this is the case I do believe it is a justified conclusion and one we needed to make in order to deal with the question that had faced us. Indeed, this was a rather simp le question for such an involved aside. It was simply this: might we find ourselves committed to mathematical entities by means, merely, of the expression of our knowledge of the causal properties? Needless to say, I do not believe that such an expression is to be found among this minimal material. Theory may be vague, but there is nothing vague about the theoretical status of Observation and, to a lesser degree, manipulation are also rather vague. Representing as they do complementary portions of our knowledge, satisfactorily defining one would mean defining the other. It is only the marginal cases on which definite formulations might founder and mathematical entities are no marginal case, being definitely neith er manipulable nor
87 observable. This being the case, we can be quite sure that a portion of scientific belief which irreducibly refers to mathematical entities is also theory. It seemed unavoidable that expressing the causal nature of entities would involv e stating quantitative laws that would involve precisely the kind of commitment to abstract entities we are talking about. Theory that ought to receive a distinctly lower degree of we see this situation for what it really is. No such quantitative law is the proper expression that receives support from the manipulation of entities. They are too strong, that is, inasmuch as they assert the existence of abstract entities they are theor y and undersupported. Conversely, they only need be supported to the degree that they assert the causal content we had discussed. With the preceding, we see a way the linguistic expression of such content might be gathered in a general setting. So how doe s theory relate to the causal consequences it might have? Quine specifically addresses himself to an anticipated instrumentalism of a kind we will find enlightening in this Ways 246). In this essay, he describes the entities, as conceptual devices included in theory to act as tools to help that theory fulfill the goal of organizing sensory experience. In d oing so, he acknowledges their use as instrumental in entity in question, this instrumental device, as being no more than that and accept that the observational con sequences of the theory will come to pass. In response to this instrumentalism, he pursues a line which, by now, certainly should not surprise us. He responds
88 that the success in anticipation of sensory information just is the evidence for an entity, obser ved and unobserved alike (Quine, Ways 251). eliminating statements that would appear to commit us to abstract entities. In the case of an cept the experimental consequences, that is, we accept the consequences of the assertion of the experimental form that expresses minimal content in the way we already described. It is these that are most supported by the evidence. As for the law itself, th e fact of its commitment to mathematical existences means it remains on the level of theory. It might be true, but it is not supported as well as the experimental content which can be derived from it is. Therefore, when any such law is in place, we can be sure that it remains on the level of all theory by virtue of its commitment to mathematical entities. We may yet regard it as a mere instrument for organizing and deploying our understanding of the experimental entities. This is not the same as the possib ility of instrumentalism that Quine had examined. instrumentalism he addressed? They do not hold because of our earlier shift in our understanding of the nature of scienti fic confirmation. Evidence for theory is no longer simply its ability to predict. Really strong evidence is playing a key role in the manipulation we had described. That is, strong support is limited to the description of the physical interaction of entiti es. Quantitative laws also assert the existence of mathematical entities. Therefore, they are not supported to this degree.
89 of understanding how it actually works. T his meant examining how scientists and the like actually did form the beliefs that they did instead of trying to find a grounding for science. We in the experimen have all the justification we need. If it really is true that scientists come to be lieve in the entities they manipulate to a greater degree than other, theoretical entities, then Quine would have to recognize it as an important part of confirmation as Hacking did. Indeed, perhaps even the arguments I had offered earlier was unnecessary and a mere elaboration or rationalization of the way scientists actually arrive at their beliefs. This further determination is, of course, more a matter for empirical researchers into the psychology and sociology of science. To Quine, the general suspici on that mathematical objects were removed from the normal ken of what can be known because of their a causal nature, and that, therefore, mathematical realism was doomed to go undersupported by science, was a complete mistake. Not only this, it was a mista ke which we would have to amend if we adopted a naturalistic existence of the mathematical objects. With Hacking, an attention to the nature of experiment gave us a richer picture of the way confirmation and scientific work went. Most importantly, we learned that manipulation of entities with causal powers may well be a stronger means of acquiring confirmation for an entity than the means Quine had suggested. The su spicion about
90 naturalistic perspective. After all, no one would try to claim that mathematical entities had causal powers. The actual application turned out to be more complicated than it seemed. Ultimately, mathematical entities as basically undersupported. The definitive answer to the question of whether a more Hackingian or a more Quinean approach to understanding science and scientific confirmation is more accurate is an issue for another day. This is because the real way to decide chara cterizes the matter. From Quine, we learned that mathematical entities were not fundamentally different from other theoretical entities, that the role they play in scientific theory does involve the asse rtion of their existence and that there are sensory consequences of the substantial mathematical assumptions we make. From Hacking, however, we learned that science did not necessarily say that, for these reasons, mathematical entities were supported every bit as much as any other scientific entity. The manipulated ones are more assured. Actual nominalism would involve a hefty instrumentalism, albeit one which is not too sweeping and which Hacking seems to suggest we could maintain along with naturalism. Ev en moderate scientific realists would still find themselves accepting mathematical entities. What we have learned here is more important than how well supported is nominalism or mathematical realism. We have found that some of our common sense suspicions a bout mathematical entities are not completely groundless. Even if we accept entities we can still recognize a hierarchy of support among the entities we do recognize. Mathematical entities are stuck at a low level of confirmation because of their
91 abstract nature. We are, then, able to understand the choice of the acceptance of their existence as more a practical one than a necessary conclusion of science in a way we could not with some other entities. The mathematical realism we discussed is just the resul t of the legitimate, but disputable, belief that there is more reason to think that science produces true theory than that mat hematical entities are not real
Naturalized Mathematical Realism Having examined, and a rgued against, Quine in detail, we proceed to offer a brief sketch of a defense of our thesis of the possibility of a Hackingian naturalized nominalism against the naturalized mathematical realism of John P. Burgess and Gideon Rosen. The mathematical reali sm of this writing duo is related to, but distinct from, that of Quine. While clearly drawing inspiration from Quine, their argument, and the accompanying naturalized approach to philosophical problems, differs from his considerably. We had argued, against Quine, that Hacking offers an understanding of science that allows for naturalized, mathematical anti realist for our thesis. While length constraints mean I cannot argue against this argument in any great detail, I can offer a brief sketch of a response. development of a coherent and general naturalism or, even, a positive conception of n aturalized mathematical realism. Rather, they are mostly interested in difusing the relatively prevalent project of mathematical nominalism as we find it in the philosophy of the last 20 or 30 years. I am not concerned with nominal ism here but, rather, only with the naturalistic support for mathematical entities. We should note this about Burgess and Rosen at the outset because it explains both why some features of their philosophy might appear fuzzy or underdeveloped in
this genera l setting and why their realism has a somewhat negative quality reflected in the of generality, involves a heavy empha sis on the final authority of scientific experts on matters relating to their area of expertise. Accordingly, then, their naturalism demands we defer to the word and assessment of the relevant experts while engaging in our philosophical inquiries where thi s is possible. Of course, it is not entirely obvious to whom exactly we are to defer. We Developed institutions of training, discussion, and acceptance of c onclusions surround many areas of specialization. B&R call those institutions to whose certified experts we should A Subject with no Object 217). While they offer no explicit criterion for qualification as a science, we are probably alrea dy familiar with what should qualify, for the deference they demand of us is of a very mundane kind. If we had a very complex question about, say, physics, most reasonable laypeople in our milieu would think the most appropriate way for them to resolve it would be to consult a reputable physics textbook or specialist. Similarly, in the face of a conclusion provided by sound physics, this same person would think the best response would be to accept it or, at least, not to reject it by her own reasoning disag reement on such matters being the place of other experts. the various forms of enquiry we call science do not speak to every question. There is no such thing as political science, for instance; yet no professed naturalist ma intains that we should abstain from having pol ASWNO 65). Of course, ompelled to defer
to it on the matters on which it deals. Unifying features of the disciplines that fit this description, then, include a set of standards which is able to produce effectively unanimous agreement among specialists on certain conclusions; de ference would make little sense otherwise. I would imagine there also would have to be some kind of demonstrated usefulness, veracity, or acceptability on the part of the science, to justify the attitude of deference of the reasonable lay people (B&R, ASWN O 3 5) been discussing hold even when dealing with questions that are raised in a philosophical context. Philosophy, while it concerns itself with questions of very wide scope, is not a science according to our description and thus is not an authority according to B&R. If a philosopher wants to make an assertion concerning a field dealt with by a science, she will not be taking a naturalized approach, say B&R, unless she b elieves that the experts in the science would accept her reasoning ( ASWNO 205 210). As in the case with Quine, the best way to test if they do, say B&R, would be to send a paper explaining the reasoning into a relevant professional journal and judge the ar gument based on the response ( ASWNO 210). Otherwise, where philosophy might weigh on a scientific question, as in physics, mathematics or so on, it must accept the conclusions of these other disciplines on the matter first before it can contribute. B&R ca ll ASWNO 205 210). We can envision a justification of this kind of naturalism following from some form of the naturalized epistemologi cal approach. Nothing about their naturalism, so far, is considerations he viewed as scientific in making philosophical conclusions. Burgess and Rosen
are wary of the pos sibility of nonscientists applying scientific methods of reasoning but would practicing scientists in the relevant fields. Testing whether they were satisfactory in this way would involve something like the journal test offered above. B&R differ decisively from Quine, however, in recognizing pure mathematics as a must be of a restricted variety making invidious distinctions, marginalizing some sciences (the mathematical) and privileging others ASWNO 211) surprising given our discussion so far. Mathematics meets all the criteria for sciencehood we had offered above. Hesitant to characterize scientific methods though they are, B&R recognize that mathematical methods differ from those o f other sciences in not being empirical. Burgess tells us owing to its distinctive meth odology of deductive proof, a special case the only truly scientific method is the empirical H D one. More importantly, it means their naturalism demands that all assertions issuing from pure mathematics are to be believed. existence of mathematical entities. This alone, then, is enough to lead B&R to accept mathematical 517). There is more to their argument than the preceding, however. We recall their goal had been that of difusing nominalism, naturalized and otherwise. They develop many aspects of their position in defending and bolstering it from potential detractors of this kind. One of these features of their philosophy is particularly relevant to the present discussion. In our forgoing discussion of H acking and Quine, we had relied on the possibility of a general anti realism or
might attempt to purge the ontology of science of mathematical entities by claim ing that mathematical existence theorems and claims are of merely instrumental value. In order for this themselves withhold full belief in the existence asser tions of mathematics. This would involve mathematicians holding some attitude weaker than belief toward these assertions B&R call this As naturalized realists, B&R do maintain that mathematicians fully believe the existence ilent reservations, and that they rely on them in both theoretical and practical contexts. They use them as premises in demonstrations intended to convince other experts of novel claims, and together with other assumptions, as premises in arguments intende d to persuade others to they take themselves to have refut ed attitude hermeneutic nominalism. The anti instrumentalism it had allowed. With our goal being the discernment of the possibility of a Hackingian anti phy becomes our primary area of interest. I do not assert that attitude hermeneutic nominalism is correct. However, once we conduct a deeper analysis of the nature of the expert belief on the
matter, I think we will find room to argue for our conclusion of the epistemic inferiority of mathematical objects a la I move, then, to an analysis of the actual beliefs of mathematicians. My resources for conducting such an analysis are irregular and limited This being the case, my conclusions must be tentative and suggestive and I will leave the work of producing conclusive results to later research. I turn to the writer Reuben Hersch for material on the realism of mathematicians. Hersch is a trained mathem atician, so his word carries authority on the subject according to conscious platonism is nearly What is Mathematics Really 11). So far, this is no real contradic tion of Burgess and Rosen although it should strike us that he describes the platonism mathematicians outright with behaviors they displayed in the scientific con texts we had mentioned. There is hardly anything inarticulate and half propensity to behave in these ways or about the place of the existence theorems in the mathematics employed in these contexts. By describing the realis m of mathematicians this way, Hersch seems to imply there is something lacking about it. We get more than his word alone to this effect when he tells us that the realism that hearted and shame general, then, according to Hersch, regard realism as a limited and problematic view ( Mathematics 11). weaker kind than expected. However, looking further int mathematicians may actually tend to defer to philosophers on the issue. He tells us that
Mathemat ics usually discuss philosophical issues. We think someone else has taken that over Mathematics 41). This certainly sounds like deference to philosophers in something like the way B&R h ad imagined. unable to produce consensus among its practitioners as to the acceptability of a philosophical be understood as the belief that philosophical arguments are among the best to be had for the determination of the question of realism and, since philosophers are the most familiar with these, it is they who should be responsible for determining the solut realism as problematic as their belief that realism does not hold up to properly philosophical scrutiny as they understand it. It falls back on philosophers, then, to answer the question of realism by the best arguments that present themselves. This preceding portion of my response amounts to an undersupported sociological assertion. To substantiate it, we would need to engage in some deeper research guided by our goals and conclusions. Before we can sketch how this might go, we must address a difficulty. Burgess and Rosen had noted that mathematicians confidently affirm mathematical existence theorems and assert them in those contexts we had noted earlier. Hersch and I, however, have produced evidence to the effect that mathematicians might only hesitantly affirm realism, if they do so at all, when queried. Indeed, Hersch and others have even noted professional mathematicians actually making claims with a decidedly anti realist ring to them. Hersch rel ays Mathematics
of the famous mathematicians who have expressed themselves on the question have in one alism. Having done so, it becomes a problem for me that mathematicians appear to be contradicting themselves with these divergent responses on realism. This naturalism insists, after all, that the expert word is authoritative. I respond to this dilemma by hypothesizing that the response of the mathematician on the issue of realism is context dependent and by suggesting that the contexts differ in the scope of relevant reasons considered in giving the response. We might call the context that B&R focus on the mathematical context, and the other the philosophical context. It is in this philosophical context that mathematicians express reservations about mathematical realism, suggest the authority of philosophical reasoning and, if their sympathies turn this way express adherence to anti realism of whatever form. I suggest, further, that it is in this latter context that mathematicians consider all relevant arguments including those offered in the purely mathematical context. If I am right, then the ultimately a uthoritative word of the experts is the one they offer in this context and, accordingly, we have reason to believe that we should turn to philosophy for our answer to the question. I had addressed this issue in order to envision more conclusive research o n it and substantiate our latter conclusion. The best way I can see to do such research would be to administer a survey to a group of professional mathematicians and ask their views on the issue. Our earlier considerations showed that we must keep in mind what context we are asking the question in. If we are confident about my suggestion on the authority of the philosophical context, then our goal must be to somehow induce this context in the survey. To be true to
o, we must take another factor into consideration. views on the question of realism. They imply that cases of mathematicians dissenting from anti realism are the Here, then, are two considerations that we must take into account in designing our question. Much debate and thought can, and would need to, go into this design. For now, I suggest something like the following oo vague we easiest way of inducing the philosophical context. When math ematical standards alone are considered, this question is almost entirely vacuous. With the impression that the survey is a sincere, well informed and well designed one, the substance of the question should be clear. Further, this seems to fare well on the score of not grilling mathematicians. The question seems, overall, to be both nonabusive and not loaded. I ask as directly as possible after the information I wish to know without trying to force or suggest a correct answer. Much more would need to be t aken into consideration before the question would be a satisfactory one. Perhaps we would need to think of some way to test my theory of the two contexts with the survey. Perhaps the question could be even less leading. The list goes on. I leave my suggest ion as is and move on assuming we were right to conclude that mathematicians defer to philosophy as our best evidence suggests they do. Having done so, we find ourselves dealing directly with philosophy again. Perhaps this would seem to make the possibilit y of
remaining true to B&R impossible. However, only one feature of their thought has fallen apart here, we can still offer good naturalistic answers to the question within philosophy. One way would be to formulate a naturalized theory of confirmation and apply it to the question of mathematical realism. A naturalized theory of confirmation would be a general formula for determining the scientific support for an assertion. Above all, it would be judged by the degree to which scientists could agree that it produced the same confirmation levels as going, applicable, more local scientific standards on matters to which it applied. The support conferred to the more general theory by so corroborating a more local standard or belief would be proportional in streng th to how strongly that standard or belief was felt to be true by the relevant scientist. In this case, we would use it to determine the scientific support for mathematical entities. There would be more to consider than simply corroboration of going scient ific beliefs. We would also consider beliefs more directly concerning the nature of such a general theory of confirmation, sundry philosophical arguments pertinent to the matter and, of course, the facts of the actual experience of doing mathematics that m athematicians could provide. Indeed, B&R had addressed positions that argued for nominalism along these lines. They had responded that if a theory of confirmation produced nominalist conclusions, then it had not been calibrated to corroborate the scienti fic conclusions of mathematics, specifically, the should corroborate This raises an issue I should touch on, if only briefly. What if the other sciences we have calibrated this theory on have separate scientific and philosophical contexts as well? I believe that there are, actually such different contexts in other sciences. Indeed, the
history of physics yields some rather famous anti realist physicists including Pierre Duhem and Er nst Mach ( ASWNO 61). However, I suspect that, when issued a study similar the one we had framed for mathematicians, most physicists would not dee m philosophers the authority on issues of realism regarding their subject. I think that mathematicians would be willing to see the issue of mathematical realisms a philosophical issue as other scientists would not in the case of their respective sciences. W e would have to design another survey for this as well. B&R had objected to such an approach, inasmuch as it was an approach to nominalism, saying that if a theory of confirmation produced nominalist conclusions then it failed the naturalistic test of cor roborating scientific standards of their local conclusions, in this case, mathematical standards and the existence theorems. I hope that, at this point, this worry no longer lingers; the question of mathematical realism is, in the mind of mathematicians, a n undecided issue the likes of which such a theory should decide. realism within the naturalized framework B&R provide. At this poi regularly manipulated entities are decisively confirmed while entities yet to b e manipulated, may legitimately be regarded with an attitude of anti realism with a scale of different degrees of theory scores rather well in this regard. He had argued for his theory on very naturalistic grounds, presenting it as directly reflecting the views of many scientists, although had focused primarily on experimental physicists. He had gone further than this, however, offering brief considerations of his v iew as it fared in microbiology, medicine, and even the social sciences.
must leave for another day. Abstract objects are, of course, permanently theoretical as their a to the argument we have summarized here would show that abstract entities are epistemically l objects. It would do so, first, in the sense that the manipulation of the latter has left them better confirmed. Second, it would show abstract, mathematical objects to be epistemically inferior to even causally active theoretical entities in their inabi lity to ever attain the superior level of confirmation that manipulation confers. With this we have the point we had hoped for.
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