Is Leibnizian calculus embeddable in first order logic?

Philosophy

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro-
          cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If,
          as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than
          modern Weierstrassian ones.

ar X iv :1 60 5. 03 50 1v 1 [ m at h. L O ] 1 1 M ay 2 01 6 IS LEIBNIZIAN CALCULUS EMBEDDABLE IN FIRST ORDER LOGIC? PIOTR B LASZCZYK, VLADIMIR KANOVEI, KARIN U. KATZ, MIKHAIL G. KATZ, TARAS KUDRYK, THOMAS MORMANN, AND DAVID SHERRY Abstract. To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on procedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. Keywords: First order logic; infinitesimal calculus; ontology; procedures; Leibniz; Weierstrass; Abraham Robinson Contents 1. Introduction 2 2. Examples from Leibniz 2 2.1. Series presentation of π/4 2 2.2. Leibniz convergence criterion for alternating series 3 2.3. Product rule 3 2.4. Law of continuity 4 2.5. Non-examples: EVT and IVT 5 3. What does "first-order" mean exactly? 5 4. Modern frameworks 7 5. Separating entities from procedures 10 6. A lid on ontology 11 7. Robinson, Cassirer, Nelson 12 7.1. Robinson on second-order logic 12 7.2. Ernst Cassirer 14 7.3. Edward Nelson 14 7.4. Not standalone 16 8. Conclusion 16 Acknowledgments 17 1 2 P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. References 17 1. Introduction Leibniz famously denied that infinite aggregates can be viewed as wholes, on the grounds that they would lead to a violation of the principle that the whole is greater than the part. Yet the infinitary idea is latent in Leibniz in the form of a distinction between assignable and inassignable quantities [Child 1920, p. 153], and explicit in his comments as to the violation of Definition V.4 of Euclid's Elements [Leibniz 1695b, p. 322]. This definition is closely related to what is known since [Stolz 1883] as the Archimedean property, and was translated by Barrow in 1660 as follows: Those numbers are said to have a ratio betwixt them, which being multiplied may exceed one the other [Euclid 1660]. Furthermore, Leibniz produced a number of results in infinitesimal calculus which, nowadays, are expressed most naturally by means of quantifiers that range over infinite aggregates. This tension leads us to examine a possible relationship between Leibnizian infinitesimal calculus and a modern logical system known as first order logic (FOL). The precise meaning of the term is clarified in Section 3. We first analyze several Leibnizian examples in Section 2. This text continues a program of re-evaluation of the history of infinitesimal mathematics initiated in [Katz & Katz 2012], [Bair et al. 2013] and elsewhere. 2. Examples from Leibniz Let us examine some typical examples from Leibniz's infinitesimal calculus so as to gauge their relationship to FOL. 2.1. Series presentation of π/4. In his De vera proportione (1682), Leibniz represented π 4 in terms of the infinite series 1 − 1 3 + 1 5 − 1 7 + . . . This is a remarkable result, but we wish to view it as a result concerning a specific real number, i.e., a single case, and in this sense involving no quantification, once we add a new function symbol for a black box procedure  called "evaluation of convergent series" (as well as a definition of π) (we will say a few words about the various implementations of  in modern frameworks in Section 4). LEIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 3 2.2. Leibniz convergence criterion for alternating series. This refers to an arbitrary alternating series defined by an alternating sequence with terms of decreasing absolute value tending to zero, such as the series of Subsection 2.1, or the series ∑ n (−1)n n determined by the alternating sequence (−1) n n . We will refer to such a sequence as a 'Leibniz sequence' for the purposes of this subsection. This criterion seems to be quantifying over sequences (and therefore sets), thus transcending the FOL framework, but in fact this can be handled easily by introducing a free variable that can be interpreted later according to the chosen domain of discourse. Thus, the criterion fits squarely within the parameters of FOL +  at level (3) (see Section 3). In more detail, we are not interested here in arbitrary 'Leibniz sequences' with possibly inassignable terms. Leibniz only dealt with sequences with ordinary (assignable) terms, as in the two examples given above. Each real sequence is handled in the framework R ⊂ ∗R by the transfer principle, which asserts the validity of each true relation when interpreted over ∗R. 2.3. Product rule. We have d(uv) dx = du dx v + dv dx u and it looks like we need quantification over pairs of functions (u, v). Here again we are only interested in natural extensions of real functions u, v, which are handled at level (3) as in the previous section. In [Leibniz 1684], the product rule is expressed in terms of differentials as d(uv) = udv + vdu. In Cum Prodiisset [Leibniz 1701c, p. 4647], Leibniz presents an alternative justification of the product rule (see [Bos 1974, p. 58]). Here he divides by dx and argues with differential quotients rather than differentials. Adjusting Leibniz's notation, we obtain an equivalent calculation d(uv) dx = (u + du)(v + dv) − uv dx = udv + vdu + du dv dx = udv + vdu dx + du dv dx = udv + vdu dx . Under suitable conditions the term du dv dx is infinitesimal, and therefore the last step udv + vdu dx + du dv dx = u dv dx + v du dx , (2.1) 4 P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. relying on a generalized notion of equality, is legitimized as an instance of Leibniz's transcendental law of homogeneity, which authorizes one to discard the higher-order terms in an expression containing infinitesimals of different orders. 2.4. Law of continuity. Leibniz proposed a heuristic principle known as the law of continuity to the effect that . . . et il se trouve que les règles du fini réussissent dans l'infini . . . ; et que vice versa les règles de l'infini réussissent dans le fini, . . . [Leibniz 1702a, p. 93-94], cited by [Knobloch 2002, p. 67], [Robinson 1966, p. 262], [Laugwitz 1992, p. 145], and other scholars. On the face of it, one can find numerous counterexamples to such a principle. Thus, finite ordinal number addition is commutative, whereas for infinite ordinal numbers, the addition is no longer commutative: 1 + ω = ω 6= ω + 1. Thus, the infinite realm of Cantor's ordinals differs significantly from the finite: in the finite realm, commutativity rules, whereas in the infinite, it does not not. Thus the transfer of properties between these two realms fails. Similarly, there are many infinitary frameworks where the law of continuity fails to hold. For example, consider the Conway–Alling surreal framework; see e.g., [Alling 1985]. Here one can't extend even such an elementary function as sin(x) from R to the surreals. Even more strikingly, √ 2 turns out to be (sur)rational; see [Conway 2001, chapter 4]. The surnaturals don't satisfy the Peano Arithmetic. Therefore transfer from the finite to the infinite domain fails also for the domain of the surreals. On the other hand, the combined insight of [Hewitt 1948], [ Loś 1955], and [Robinson 1961] was that there does exist an infinitary framework where the law of continuity can be interpreted in a meaningful fashion. This is the R ⊂ ∗R framework. While it is not much of a novelty that many infinitary systems don't obey a law of continuity/transfer, the novelty is that there is one that does, as shown by Hewitt, Loś, and Robinson, in the context of first-order logic. Throughout the 18th century, Euler and other mathematicians relied on a broad interpretation of the law of continuity or, as Cauchy will call it, the generality of algebra. This involved manipulation of infinite series as if they were finite sums, and in some cases it also involved ignoring the fact that the series diverges. The first serious challenge to this principle emerged from the study of Fourier series when new types of functions arose through the summation thereof. Specifically, Cauchy rejected the principle of the generality of algebra, LEIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 5 and held that a series is only meaningful in its radius of convergence. Cauchy's approach was revolutionary at the time and immediately attracted followers like Abel. Cauchy in 1821 was perhaps the first to challenge such a broad interpretation of the law of continuity, with a possible exception of Bolzano, whose work dates from only a few years earlier and did not become widely known until nearly half a century later. For additional details on Cauchy see [Katz & Katz 2011], [Katz & Katz 2012], [Borovik & Katz 2012], [Bascelli et al. 2014]. For Euler see [Kanovei, Katz & Sherry 2015] and [Bascelli et al. 2016]. 2.5. Non-examples: EVT and IVT. It may be useful to illustrate the scope of the relevant results by including a negative example. Concerning results such as the extreme value theorem (EVT) and the intermediate value theorem (IVT), one notices that the proofs involve procedures that are not easily encoded in first order logic. These 19th century results (due to suitable combinations of Bolzano, Cauchy, and Weierstrass) arguably fall outside the scope of Leibnizian calculus, as do infinitesimal foundations for differential geometry as developed in [Nowik & Katz 2015], [Kanovei, Katz & Nowik 2016]. There are axioms in FOL for a real closed field F (e.g., real algebraic numbers, real numbers, hyperreal numbers, Conway numbers). One of these axioms formalizes the fact that IVT holds for odd degree polynomials F [x]. In fact, one needs infinitely many axioms like (∀a, b, c)(∃x)[x3 + ax2 + bx + c = 0]. Meanwhile, IVT in its full form is equivalent to the continuity axiom for the real numbers [B laszczyk 2015]. 3. What does "first-order" mean exactly? The adjective 'first-order' as we use it entails limitations on quantification over sets (as opposed to elements). Now Leibniz really did not have much to say about properties of sets in general in the context of his infinitesimal calculus, and even declared on occasion that infinite totalities don't exist, as mentioned above. Note that Leibniz arguably did exploit second-order logic in areas outside infinitesimal calculus (see [Lenzen 1987], [Lenzen 2004]) but this will not be our concern here. Leibniz famously takes for granted second order logic in formulating his principle governing the identity of indiscernibles. While second order logic is possibly part of Leibniz's metaphysics it is not in any obvious way part of his infinitesimal calculus. Once we reach topics like Baire category, measure theory, Lebesgue integration, and modern functional analysis, quantification over sets 6 P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. becomes important, but these were not Leibnizian concerns in the kind of analysis he explored. In fact, the term "first order logic" has several meanings. We can distinguish three levels at which a number system could have firstorder properties compatible with those of the real numbers. Note that the real numbers satisfy the axioms of an ordered field as well as a completeness axiom. (1) An ordered field obeys those among the usual axioms of the real number system that can be stated in first-order logic (completeness is excluded). For example, the following commutativity axiom holds: (∀x, y) [x + y = y + x]. (2) A real closed ordered field has all the first-order properties of the real number system, regardless of whether such properties are usually taken as axiomatic, for statements which involve the basic ordered-field relations +,×, and ≤. This is a stronger condition than obeying the ordered-field axioms. More specifically, one includes additional first-order properties, such as extraction of roots (e.g., existence of a root for every odd-degree polynomial). For example, every number must have a cube root: (∀x)(∃y) [y3 = x], or every positive number have a square root: (∀x > 0)(∃y) [y2 = x]. (3) The system could have all the first-order properties of the real number system for statements involving arbitrary relations (regardless of whether those relations can be expressed using +,×, and ≤). For example, there would have to be a sine function that is well defined for infinitesimal and infinite inputs; the same is true for every real function. To do series, one needs a symbol for N, so as to define transcendental entities such as π or sine. We also introduce function symbols for whatever functions we are interested in working with; say all elementary functions occurring in Leibniz as well as their combinations via composition, differentiation, and integration. It follows from these examples that the first order qualification is connected with the intended domain of discourse, so that any quantifier related to objects outside the domain of discourse is qualified as not a first order one. It could be added however that all mathematical objects are, generally speaking, (represented by suitable) sets from the set theoretic standpoint, and hence all mathematical quantifiers are firstorder with respect to the background set universe (superstructure). The point with level (3) is that instead of quantifying over sequences or functions, we relate to each individual sequence or function, and LEIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 7 make sure that it has an analogue in the extended domain. Such an analogue of f is sometimes referred to as the natural extension of f . Then we can say something about the extension of every standard object in our system, e.g., function, without ever being able to assert anything about all functions. Thus, the product rule for differentiation is proved for the assortment of functions chosen in item (3) above. Remark 3.1. An alternative to the multitude of functional symbols would be to add a countable list of variables u, v, w, . . . meant to denote unspecified functions. The idea is to avoid quantifying over such variables, and use them as merely free variables. Then, for example, the product rule is the following statement: "if u, v are differentiable functions then the Leibniz rule holds for u and v", with u, v being free variables. Note that we use FOL in a different sense from that used in formalizing Zermelo–Fraenkel set theory (ZFC). When we seek hyperreal proxies, following the pioneering work of [Hewitt 1948] and [Robinson 1966], for Leibniz's procedural moves, the theory of real closed fields at level (2) is insufficient and we must rely upon level (3). Thus, Leibniz's series of Subsection 2.1 is expressible in FOL +  at level (3) but FOL level (2) does not suffice since π is not algebraic. Similarly, examples in Subsection 2.2 and Subsection 2.3 need symbols for unspecified functions which are not available at level (2). 4. Modern frameworks Based on the examples of Section 2, we would like to consider the following question: Which modern mathematical framework is the most appropriate for interpreting Leibnizian infinitesimal calculus? The frameworks we would like to consider are (A) a Weierstrassian (or "epsilontic") framework in the context of what has been called since [Stolz 1883] an Archimedean continuum, satisfying Euclid V.4 (see Section 2), namely the real numbers exclusively; and (B) a modern framework exploiting infinitesimals such as the hyperreals, which could be termed a Bernoullian continuum since Johann Bernoulli was the first to exploit infinitesimals (rather 8 P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. than "exhaustion" methods) systematically in developing the calculus.1 The series summation blackbox  (see Subsection 2.1) is handled differently in A and B. Framework A exploits a first-order "epsilontic" formulation that works in a complete Archimedean field. Thus, the convergence of a series ∑ i ui to L would be expressed as follows: (∀ǫ > 0)(∃n ∈ N)(∀m ∈ N) [ m ≥ n → ∣ ∣ ∣ ∣ ∣ m ∑ i=1 ui − L ∣ ∣ ∣ ∣ ∣ < ǫ ] . [Ishiguro 1990, Chapter 5] sought to interpret Leibnizian infinitesimal calculus by means of such quantified paraphrases, having apparently overlooked Leibniz's remarks to the effect that his infinitesimals violate Euclid V.4 [Leibniz 1695b, p. 322]. Meanwhile, framework B allows for an alternative interpretation in terms of the shadow (i.e., the standard part, closely related to Leibniz's generalized notion of equality) and hyperfinite partial sums as follows: for each infinite hypernatural H the partial sum ∑ H i=1 ui is infinitely close to L, i.e., (∀H) [ H infinite → H ∑ i=1 ui ≈ L ] . This is closer to the historical occurrences of the package  as found in Gregory, Leibniz, and Euler, as we argue below. In pursuing modern interpretations of Leibniz's work, a helpful distinction is that between ontological and procedural issues. More specifically, we seek to sidestep traditional questions concerning the ontology of mathematical entities such as numbers, and concentrate instead on the procedures, in line with Quine's comment to the effect that Arithmetic is, in this sense, all there is to number: there is no saying absolutely what the numbers are; there is only arithmetic. [Quine 1968, p. 198] 1The adjective non-Archimedean is used in modern mathematics to refer to certain modern theories of ordered number systems properly extending the real numbers, namely various successors of [Hahn 1907]. In modern mathematics, this adjective tends to evoke associations unrelated to 17th century mathematics. Furthermore, defining infinitesimal mathematics by a negation, i.e., as non-Archimedean, is a surrender to the Cantor–Dedekind–Weierstrass (CDW) view. Meanwhile, true infinitesimal calculus as practiced by Leibniz, Bernoulli, and others is the base of reference as far as 17th century mathematics is concerned. The CDW system could be referred to as non-Bernoullian, though the latter term has not yet gained currency. LEIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 9 Related comments can be found in [Benacerraf 1965]. If one could separate the "ontological questions" from the rest, then framework A would be more appropriate than framework B for interpreting the classical texts if and only if framework A provides better proxies for the procedural moves found in Leibnizian infinitesimal calculus than framework B does, and vice versa. The tempting evidence in favor of the appropriateness of a modern framework B for interpreting Leibnizian infinitesimal calculus is the presence of infinitesimals and infinite numbers in both, as well as the availability of hyperreal proxies for guiding principles in Leibniz's work such as the law of continuity as expressed in [Leibniz 1701c] and [Leibniz 1702a] as well as the transcendental law of homogeneity [Leibniz 1710b]; see [Katz & Sherry 2012], [Katz & Sherry 2013], [Sherry & Katz 2014], [Guillaume 2014]. To what extent Leibnizian infinitesimals can be implemented in differential geometry can be gauged from [Nowik & Katz 2015]. The question we seek to explore is whether the limitation of working with first order logic as discussed in Section 3 could potentially undermine a full implementation of a hyperreal scheme for Leibnizian infinitesimal calculus. With this in mind, let us consider Skolem's construction of nonstandard natural numbers [Skolem 1933], [Skolem 1934], [Skolem 1955]; see [Kanovei et al. 2013, section 3.2] for additional references. It turns out that one needs many, many nonstandard numbers in order to move from N to ∗N, e.g., in Henkin's countable model one has ∗N = N + (Z×Q). (4.1) Here we use Q to indicate that the galaxies are dense, so that between any pair of galaxies there is another galaxy (a galaxy is the set of numbers at finite distance from each other). Meanwhile Z indicates that each galaxy other than the original N itself is order-isomorphic to Z rather than to N, because for each infinite H the number H − 1 is in the same galaxy. Leibniz arguably did not have such a perspective. In other words, one needs to build up a considerable conceptual machinery to emulate Leibniz's probably rather modest arsenal of procedural moves. That is to say, we may be able to emulate all of Leibniz moves in a modern Bframework, such as Leibniz's infinite quantities, his distinction between assignable and inassignable quantities, and his transcendental law of homogeneity. However, the B-framework also enables us to carry out many additional moves unknown to Leibniz, for instance those related to the detailed structure of ∗N as in (4.1). Thus, the difference between 10 P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. the Leibnizian framework and a modern infinitesimal B-framework is large. On the other hand, a Weierstrassian A-framework may not cover all the moves Leibniz may make in his framework LEI, but one might argue that the difference between (A) and LEI is small. Thus, one may not necessarily have LEI ⊆ (A), but one might argue that the difference (A) − LEI is small. This may be taken as evidence that (A) and LEI are more similar to each other than (B) and LEI are. This could affect the assessment of appropriateness. Finally, could it be that neither the Weierstrassian nor the modern infinitesimal account is appropriate to cope with Leibnizian infinitesimal calculus? 5. Separating entities from procedures What would it mean exactly to separate ontological problems from procedural problems? A possible approach is to attempt to account for Leibniz's procedures in a framework limited to first order logic, with a small number of additional ingredients such as the relation of infinite proximity and the closely related shadow principle for passing from a finite inassignable quantity to an assignable one (or from a finite nonstandard number to a standard one), as in 2x + dx pq 2x. 2 As far as Skolem's nonstandard extension N ⊂ ∗N is concerned, anything involving the actual construction of the number system and the entities called numbers would go under the heading of the ontology of mathematical entities. Note that the first order theories of N and ∗N are identical, as shown by Skolem (for more details see Section 3). In this sense, not only is one not adding a lot, but in fact one is not adding anything at all at the level of the theory. What about the claim that Leibniz did not have this perspective? It is true that he did not have our perspective on the ontological issues involved in a modern construction of a suitable number system incorporating infinite numbers, but this needn't affect the procedural match. What about the claim that one has to build up a considerable conceptual machinery to emulate Leibniz's probably rather modest arsenal of procedural moves; that is to say, we may be able to emulate all of Leibniz's moves in the modern framework, but it also enables us to 2On occasion Leibniz used the notation " pq " for the relation of equality. Note that Leibniz also used our "=" and other signs for equality, and did not distinguish between "=" and " pq " in this regard. To emphasize the special meaning equality had for Leibniz, it may be helpful to use the symbol pq so as to distinguish Leibniz's equality in a generalized sense of "up to" from the modern notion of equality "on the nose." LEIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 11 carry out many moves that Leibniz would have never dreamt of ? As mentioned above, this is not the case at the level of first order logic. What about the claim that the difference between Leibniz framework and the infinitesimal framework is large? At the procedural level this is arguably not the case. What about the claim that the Weierstrass framework may not cover all of Leibniz's, but the difference, (A)− LEI, is small, indicating that (A) and LEI are more similar to each other than (B) and LEI are, affecting the assessment of appropriateness? What needs to be pointed out here is that actually the considerable distance in ontology between (A) and LEI is about the same is the distance between (B) and LEI. The Weierstrassian punctiform continuum where almost all real numbers are undefinable (so that no individual number of this sort can ever be specified, unlike π, e, etc.) is a far cry from anything one might have imagined in the 17th century. As far as the question Could it be that neither Weierstrass nor the infinitesimal account is appropriate to cope with Leibniz? this is of course possible in principle. However, we are interested here in the practical issue of modern commentators missing some compelling aspects of interpretation of Leibniz's work because of a self-imposed limitation to a Weierstrassian interpretive framework. 6. A lid on ontology It could be objected that one cannot escape so easily with the general argument along the lines of "Let's Ignore (ontological) Differences," or LID for short (putting a lid on ontology, so to speak). The LID proceeds as follows. We start with the 'real' L, i.e., the mathematician who lived, wrote, and argued in the 17th century. It seems plausible to assume that L based his reasoning on a mixture of first and second order arguments, without clearly differentiating between the two. In a reconstruction of L's arguments, one replaces the cognitive agent L by a substitute L1 who argues only in a first order framework. This entails, in particular, that L1 cannot distinguish between N and ∗N. However, it seems likely that L could distinguish the two structures, simply because he did not distinguish between the first and second order levels. In other words, the LID recommendation does not help because the distinction between first and second order does not only affect the ontology but also the epistemology of the historical agents involved. 12 P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. In sum, a modern infinitesimal reconstruction of L deals with a firstorder version of L, namely L1, and not with L. In line with his position on geometric algebra, [Unguru 1976] could point out that L1 is a modern artefact, different from the "real" L. Therefore additional arguments are needed in favor of the hypothesis that L and L1 are epistemologically sufficiently similar, but this seems difficult. In any case, a purely ontological assumption does not suffice. To respond to the L vs L1 distinction, note that the tools one needs are almost limited to first order logic, but not quite, since one needs the shadow principle and the relation of infinite proximity. Rather than arguing that L = L1, we are arguing that L = L1 + ǫ. Now the difference between calculus and analysis is that in calculus one deals mostly with first order phenomena (with the proviso as in Section 5), whereas in analysis one starts tackling phenomena that are essentially second order, such as the completeness property i.e., existence of the least upper bound for an arbitrary bounded set, etc. It seems reasonable to assume that what they were doing in the 17th century was calculus rather than analysis. As far as Unguru is concerned, he is unlikely to be impressed by interpretations of Leibnizian infinitesimals as quantified propositions or for that matter by reading Leibniz as if he had already read not only Weierstrass but also Russell à la Ishiguro, contrary to much textual evidence in Leibniz himself. We provide a rebuttal of the Ishiguro–Arthur logical fiction reading in [Bair et al. 2016] and [Bascelli et al. 2016]. 7. Robinson, Cassirer, Nelson 7.1. Robinson on second-order logic. The following quote is from Robinson's Non-standard Analysis : The axiomatic systems for many algebraic concepts such as groups or fields are formulated in a natural way within a first order language. . . , However, interesting parts of the theory of such a concept may well extend beyond the resources of a first order language. Thus, in the theory of groups statements regarding subgroups, or regarding the existence of subgroups of certain types will, in general, involve quantification with respect to sets of individuals. . . [Robinson 1966, section 2.6, p. 19] This appears to amount to a claim that the local ontology may indeed be often formulated in first-order terms, while the global ontology is deeply infected by second-order concepts. The latter may typically involve objects and arguments qualified as second or higher order with LEIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 13 respect to the former, which nevertheless are of the first-order type when considered as related to the background set universe. 14 P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. 7.2. Ernst Cassirer. Does the equation L = L1+ ǫ not amount to an underestimation of the historical Leibniz? Is it reasonable to assume that he only invented the calculus, and not analysis? According to Cassirer, the basic concepts of analysis were deeply soaked with philosophy, i.e., for Leibniz mathematical and philosophical concepts were intimately related:3 Leibniz himself asserted that the new analysis has sprung from the innermost source of philosophy, and he assigned to both regions [i.e., analysis and philosophy] the task to confirm and to elucidate each other.4 [Cassirer 1902, p. xi] If L = L1 + epsilon, i.e., if the historical Leibniz was mainly dealing with calculus, this may appear hardly compatible with Cassirer's perspective; see [Mormann & Katz 2013]. This impression would be, however, a misunderstanding. In order to forestall it, it merits being pointed out that developing the calculus was a great mathematical achievement of philosophical relevance. It is only today that the term calculus possesses a connotation of routine undergraduate mathematics, but not in the 19th century. As far as Cassirer is concerned, Leibniz was indeed doing analysis as l'Hôpital called it. It is not even sure Cassirer was aware of the more advanced analysis. Leibnizian calculus only seems "trivial" from the standpoint of properly 20th century mathematics. It is an advance in understanding when we make a distinction between Leibnizian calculus and analysis. We don't mean to diminish Leibniz's greatness by this distinction, nor do we suggest that Cassirer was wrong. He was merely using the term analysis in its 17–19th century sense rather than the sense in which we use it today. 7.3. Edward Nelson. As far as the passage from [Robinson 1966] is concerned, we find the following comment at the end of the paragraph: The following framework for higher order structures and higher order languages copes with these and similar cases. It is rather straightforward and suitable for our purposes. [Robinson 1966, section 2.6, p. 19] 3In support of this claim, Cassirer refers here in particular to Gottfried Wilhelm Leibniz, Die philosophischen Schriften, hrg. von Carl Immanuel Gerhardt, 7 Bde., Berlin 1875–1890, Bd. VII, S. 542. (Cassirer 1902, p. xi) 4In the original: "Leibniz selbst hat es ausgesprochen, dass die neue Analysis aus dem innersten Quell der Philosophie geflossen ist, und beiden Gebieten die Aufgabe zugewiesen, sich wechselseitig zu bestätigen und zu erhellen." LEIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 15 Robinson then proceeds to develop a solution, which roughly corresponds to level (3) as outlined in Section 3. The claim that we are dealing with first order logic plus standard part is in a sense a mathematical theorem, undermining the contention that "this seems hardly compatible, etc." Edward Nelson demonstrated that infinitesimals can be found within the ordinary real line itself in the following sense. Nelson finds infinitesimals in the real line by means of enriching the language through the introduction of a unary predicate standard and an axiom schemata (of Idealization), one of most immediate instances of which implies the existence of infinitely large integers and hence nonzero infinitesimals; see [Nelson 1977]. This is closely parallel to the dichotomy of assignable vs inassignable in Leibniz, whereas Carnot spoke of quantités désignées [Carnot 1797], [Barreau 1989, p. 46]. Thus, we obtain infinitesimals as soon as we assume that (1) there are assignable (or standard) reals, that obey the same rules as all the reals, and (2) there are reals that not assignable. In more detail, one considers the ordinary ZFC formulated in first order logic (here the term is used in a different sense from the rest of this article), adds to it the unary predicate and the axiom schemata, and obtains a framework where calculus and analysis can be done with infinitesimals. For further discussion see [Katz & Kutateladze 2015]. The passage from Robinson cited above does indicate that second order theory may often be interesting. However, in the case of the calculus/analysis as it was practiced in the 17th century, we are not aware of a single significant result that cannot be formulated in a system of type FOL +  (see Section 2 for examples of results that can). Arguably it was calculus (rather than analysis) that Leibniz invented, in the sense that there don't appear to be any essentially second order statements there. It may seem surprising that there could be a kind of pre-established harmony between a modern logical category, namely, first-order results, and a historical category, namely, results of 17th century calculus. This idea suggests further questions: does this only hold for the calculus, or is it always or often the case that a historically earlier realization of a theory covers only the first-order part of its successor. How do arithmetic and geometry behave in this respect? Would it really make sense to systematically distinguish between Leibniz and Leibniz1, Euclid and Euclid1, etc.? 16 P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. 7.4. Not standalone. Let us return to the comment In the context of Skolem's construction of nonstandard natural numbers and some related stuff, one is impressed by how many nonstandard numbers one needs to move from N to ∗N, e.g., in Henkin's model ∗N = N+ (Z×Q) and it goes without saying that Leibniz did not have this perspective, that was addressed briefly above. One could elaborate on the "impression" concerning "how many nonstandard numbers" one needs to define ∗N consistently and conveniently. An infinitely large number say H is not a standalone object, but rather lives in a community of numbers obeying certain laws which mathematicians anticipate as a goal of the construction of a nonstandard number system ∗N. Such a commitment to anticipated laws forces Skolem and others to add to N a suitable entourage of H along with H itself. What are the laws involved? Modern specialists in Nonstandard Analysis (NSA) stipulate that ∗N should satisfy the axioms of Peano Arithmetic and moreover, satisfy the same sentences of the language (not necessarily consequences of the axioms) that are true in N itself. This is called (the principle of) Transfer today. Mathematicians of the 17th (or even 19th) century had neither this perspective nor the tools consistently to define ∗N or ∗R. On the other hand, one can argue that there is no need for actually rigorously defining ∗N in order to make use of its benefits. One can argue that it is sufficient to have some idea of Transfer on top of an acceptance of infinitely large numbers per se (possibly as useful fictions, to borrow Leibniz's expression). We have argued that the Leibnizian Law of continuity is closely related to the Transfer principle; see [Katz & Sherry 2013]. Therefore the claim that Leibniz had not the slightest idea of this stuff (the "stuff" being the modern technique of building nonstandard models) is perhaps technically true, but it does not reflect all the aspects of the interrelations within the Leibniz/Weierstrass/NSA triangle. 8. Conclusion The vast oeuvre of Leibniz is still in the process of publication. In principle a lucky scholar might one day unearth a manuscript where Leibniz tackles a property equivalent to the completeness of the reals (after all the existence of the shadow is so equivalent), involving quantification over all sets of the number system and therefore second-order. However, this is unlikely in view of Leibniz's reluctance to deal with infinite collections, as mentioned above. If level (3) of first order logic is indeed suitable for developing modern proxies for the inferential moves LEIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 17 found in Leibnizian infinitesimal calculus, as we have argued, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. Acknowledgments M. Katz was partially funded by the Israel Science Foundation grant number 1517/12. References [Alling 1985] Alling, N. "Conway's field of surreal numbers." Transactions of the American Mathematical Society 287 (1985), no. 1, 365–386. [Bair et al. 2013] Bair, J.; B laszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz, K.; Katz, M.; Kutateladze, S.; McGaffey, T.; Schaps, D.; Sherry, D.; Shnider, S. "Is mathematical history written by the victors?" Notices of the American Mathematical Society 60, no. 7, 886-904. See http://www.ams.org/notices/201307/rnoti-p886.pdf and http://arxiv.org/abs/1306.5973 [Bair et al. 2016] Bair, J.; B laszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz, K.; Katz, M.; Kutateladze, S.; McGaffey, T.; Reeder, P.; Schaps, D.; Sherry, D.; Shnider, S. "Interpreting the infinitesimal mathematics of Leibniz and Euler." Journal for general philosophy of science, online first. See http://dx.doi.org/10.1007/s10838-016-9334-z and http://arxiv.org/abs/1605.00455 [Barreau 1989] Barreau, H. "Lazare Carnot et la conception leibnizienne de l'infini mathématique." In La mathématique non standard, 43–82, Fondem. Sci., CNRS, Paris. [Bascelli et al. 2014] Bascelli, T.; Bottazzi, E.; Herzberg, F.; Kanovei, V.; Katz, K.; Katz, M.; Nowik, T.; Sherry, D.; Shnider, S. "Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow." Notices of the American Mathematical Society 61, no. 8, 848-864. [Bascelli et al. 2016] Bascelli, T.; B laszczyk, P.; Kanovei, V.; Katz, K.; Katz, M.; Schaps, D.; Sherry, D. "Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania." HOPOS (Journal of the Internatonal Society for the History of Philosophy of Science) 6, no. 1, 117–147. See http://arxiv.org/abs/1603.07209, http://arxiv.org/abs/1603.07209 [Benacerraf 1965] Benacerraf, P. "What numbers could not be." Philosophical Review 74, 47–73. [B laszczyk 2015] B laszczyk, P. "A Purely Algebraic Proof of the Fundamental Theorem of Algebra." See http://arxiv.org/abs/1504.05609 [Borovik & Katz 2012] Borovik, A., Katz, M. "Who gave you the Cauchy–Weierstrass tale? The dual history of rigorous calculus." Foundations of Science 17, no. 3, 245-276. See http://dx.doi.org/10.1007/s10699-011-9235-x [Bos 1974] Bos, H. "Differentials, higher-order differentials and the derivative in the Leibnizian calculus." Archive for History of Exact Sciences 14, 1–90. [Carnot 1797] Carnot, L. Réflexions sur la métaphysique du calcul infinitésimal. Paris. 18 P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. [Cassirer 1902] Cassirer, E. Leibniz' System in seinen wissenschaftlichen Grundlagen. Gesammelte Werke, Hamburger Ausgabe, ECW 1, Hamburg, Felix Meiner Verlag, 1998. [Child 1920] Child, J. (ed.) The early mathematical manuscripts of Leibniz. Translated from the Latin texts published by Carl Immanuel Gerhardt with critical and historical notes by J. M. Child. The Open Court Publishing, ChicagoLondon. Reprinted by Dover in 2005. [Conway 2001] Conway, J. On numbers and games. Second edition. A K Peters, Natick, MA. [Euclid 1660] Euclid. Euclide's Elements; The whole Fifteen Books, compendiously Demonstrated. By Mr. Isaac Barrow Fellow of Trinity College in Cambridge. And Translated out of the Latin. London, 1660. [Gerhardt 1846] Gerhardt, C. (ed.) Historia et Origo calculi differentialis a G. G. Leibnitio conscripta. Hannover, Hahn. [Gerhardt 1850-1863] Gerhardt, C. (ed.) Leibnizens mathematische Schriften. Berlin and Halle: Eidmann. [Guillaume 2014] Guillaume, M. Review of "Katz, M.; Sherry, D. Leibniz's infinitesimals: their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond. Erkenntnis 78 (2013), no. 3, 571–625." Mathematical Reviews (2014). See http://www.ams.org/mathscinet-getitem?mr=3053644 [Hahn 1907] Hahn, H. "Über die nichtarchimedischen Grössensysteme." Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch Naturwissenschaftliche Klasse 116 (Abteilung IIa), pp. 601–655. [Hewitt 1948] Hewitt, E. "Rings of real-valued continuous functions. I." Transactions of the American Mathematical Society 64 (1948), 45–99. [Ishiguro 1990] Ishiguro, H. Leibniz's philosophy of logic and language. Second edition. Cambridge University Press, Cambridge. [Kanovei, Katz & Sherry 2015] Kanovei, V.; Katz, K.; Katz, M.; Sherry, D. "Euler's lute and Edwards' oud." The Mathematical Intelligencer 37, no. 4, 48–51. See http://dx.doi.org/10.1007/s00283-015-9565-6 and http://arxiv.org/abs/1506.02586 [Kanovei et al. 2013] Kanovei, V.; Katz, M.; Mormann, T. "Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics." Foundations of Science 18, no. 2, 259–296. See http://dx.doi.org/10.1007/s10699-012-9316-5 and http://arxiv.org/abs/1211.0244 [Kanovei, Katz & Nowik 2016] Kanovei, V.; Katz, K.; Katz, M.; Nowik, T. "Small oscillations of the pendulum, Euler's method, and adequality." Quantum Studies: Mathematics and Foundations, online first. See http://dx.doi.org/10.1007/s40509-016-0074-x and http://arxiv.org/abs/1604.06663 [Katz & Katz 2011] Katz, K.; Katz, M. "Cauchy's continuum." Perspectives on Science 19, no. 4, 426–452. See http://dx.doi.org/10.1162/POSC_a_00047 and http://arxiv.org/abs/1108.4201 LEIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 19 [Katz & Katz 2012] Katz, K.; Katz, M. "A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography." Foundations of Science 17, no. 1, 51–89. See http://dx.doi.org/10.1007/s10699-011-9223-1 and http://arxiv.org/abs/1104.0375 [Katz & Kutateladze 2015] Katz, M.; Kutateladze, S. "Edward Nelson (19322014)." The Review of Symbolic Logic 8 (2015), no. 3, 607-610. See http://dx.doi.org/10.1017/S1755020315000015 and http://arxiv.org/abs/1506.01570 [Katz & Sherry 2012] Katz, M.; Sherry, D. "Leibniz's laws of continuity and homogeneity." Notices of the American Mathematical Society 59, no. 11, 1550– 1558. See http://www.ams.org/notices/201211/rtx121101550p.pdf and http://arxiv.org/abs/1211.7188 [Katz & Sherry 2013] Katz, M.; Sherry, D. "Leibniz's infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond." Erkenntnis 78, no. 3, 571–625. See http://dx.doi.org/10.1007/s10670-012-9370-y and http://arxiv.org/abs/1205.0174 [Knobloch 2002] Knobloch, E. "Leibniz's rigorous foundation of infinitesimal geometry by means of Riemannian sums." Foundations of the formal sciences, 1 (Berlin, 1999). Synthese 133, no. 1-2, 59–73. [Laugwitz 1992] Laugwitz, D. "Leibniz' principle and omega calculus." [A] Le labyrinthe du continu, Colloq., Cerisy-la-Salle/Fr. 1990, 144-154 (1992). [Leibniz 1684] Leibniz, G. "Nova methodus pro maximis et minimis . . . " Acta Eruditorum, Oct. 1684. See [Gerhardt 1850-1863], V, pp. 220-226. English translation at http://17centurymaths.com/contents/Leibniz/nova1.pdf [Leibniz 1695b] Leibniz, G. "Responsio ad nonnullas difficultates a Dn. Bernardo Niewentiit circa methodum differentialem seu infinitesimalem motas." Acta Eruditorum Lipsiae (1695). In [Gerhardt 1850-1863], vol. V, p. 320-328. A French translation is in [Leibniz 1989, p. 316–334]. [Leibniz 1701c] Leibniz, G. "Cum Prodiisset. . . " mss "Cum prodiisset atque increbuisset Analysis mea infinitesimalis . . . " in [Gerhardt 1846, pp. 39–50]. Online at http://books.google.co.il/books?id=UOM3AAAAMAAJ&source=gbs_navlinks_s [Leibniz 1702a] Leibniz, G. To Varignon, 2 feb. 1702, in [Gerhardt 1850-1863], vol. IV, pp. 91–95. [Leibniz 1710b] Leibniz, G. "Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendentali." In [Gerhardt 1850-1863], vol. V, pp. 377-382. [Leibniz 1989] Leibniz, G. La naissance du calcul différentiel. 26 articles des Acta Eruditorum. Translated from the Latin and with an introduction and notes by Marc Parmentier. With a preface by Michel Serres. Mathesis. Librairie Philosophique J. Vrin, Paris. [Lenzen 1987] Lenzen, W. "Leibniz on How to Derive Set-Theory from Elementary Arithmetics." In: Proceedings of the 8th International Congress of Logic, Methodology, and Philosophy of Science, Moscow 3, 176-179. [Lenzen 2004] Lenzen, W. "Leibniz's logic." In The rise of modern logic: from Leibniz to Frege, 1-83, Handb. Hist. Log. 3, Elsevier/North-Holland, Amsterdam. 20 P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. [ Loś 1955] Loś, J. "Quelques remarques, théorèmes et problèmes sur les classes définissables d'algèbres." In Mathematical interpretation of formal systems, 98–113, North-Holland, Amsterdam. [Mormann & Katz 2013] Mormann, T., Katz, M. "Infinitesimals as an issue of neoKantian philosophy of science." HOPOS: The Journal of the International Society for the History of Philosophy of Science 3 (2013), no. 2, 236–280. See http://www.jstor.org/stable/10.1086/671348 and http://arxiv.org/abs/1304.1027 [Nelson 1977] Nelson, E. "Internal set theory: a new approach to nonstandard analysis." Bulletin of the American Mathematical Society 83, no. 6, 1165– 1198. [Nowik & Katz 2015] Nowik, T., Katz, M. "Differential geometry via infinitesimal displacements." Journal of Logic and Analysis 7:5 (2015), 1–44. See http://www.logicandanalysis.org/index.php/jla/article/view/237/106 [Quine 1968] Quine, W. "Ontological Relativity." The Journal of Philosophy 65, no. 7, 185-212. [Robinson 1961] Robinson, A. "Non-standard analysis." Nederl. Akad. Wetensch. Proc. Ser. A 64 = Indag. Math. 23 (1961), 432–440. Reprinted in Selected Works, see item [Robinson 1979], pp. 3-11. [Robinson 1966] Robinson, A. Non-standard analysis. North-Holland Publishing Co., Amsterdam. [Robinson 1979] Robinson, A. Selected papers of Abraham Robinson. Vol. II. Nonstandard analysis and philosophy. Edited and with introductions by W. A. J. Luxemburg and S. Körner. Yale University Press, New Haven, Conn. [Sherry & Katz 2014] Sherry, D., Katz, M. "Infinitesimals, imaginaries, ideals, and fictions." Studia Leibnitiana 44 (2012), no. 2, 166-192. (The article was published in 2014 even though the journal issue lists the year 2012.) See http://arxiv.org/abs/1304.2137 [Skolem 1933] Skolem, T. "Über die Unmöglichkeit einer vollständigen Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems." Norsk Mat. Forenings Skr., II. Ser. No. 1/12, 73-82. [Skolem 1934] Skolem, T. "Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen." Fundamenta Mathematicae 23, 150-161. [Skolem 1955] Skolem, T. "Peano's axioms and models of arithmetic." In Mathematical interpretation of formal systems, pp. 1–14. North-Holland Publishing, Amsterdam. [Stolz 1883] Stolz, O. "Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes." Mathematische Annalen 22 (4), 504–519. [Unguru 1976] Unguru, S. "Fermat revivified, explained, and regained." Francia 4, 774–789. Piotr B laszczyk is Professor at the Institute of Mathematics, Pedagogical University (Cracow, Poland). He obtained degrees in mathematics (1986) and philosophy (1994) from Jagiellonian University (Cracow, Poland), and a Ph.D. in ontology (2002) from Jagiellonian University. He authored Philosophical Analysis of Richard Dedekind's memoir LEIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 21 Stetigkeit und irrationale Zahlen (Cracow, 2008, Habilitationsschrift). He co-authored Euclid, Elements, Books V–VI. Translation and commentary (Cracow, 2013), and Descartes, Geometry. Translation and commentary (Cracow, 2015). His research interest is in the idea of continuum and continuity from Euclid to modern times. Vladimir Kanovei graduated in 1973 from Moscow State University, and obtained a Ph.D. in physics and mathematics from Moscow State University in 1976. In 1986, he became Doctor of Science in physics and mathematics at Moscow Steklov Mathematical Institute (MIAN). He is currently Principal Researcher at the Institute for Information Transmission Problems (IPPI), Moscow, Russia, and Professor at the Moscow State University of Railway Engineering (MIIT), Moscow, Russia. Among his publications is the book Borel equivalence relations. Structure and classification, University Lecture Series 44, American Mathematical Society, Providence, RI, 2008. Karin Katz (B.A. Bryn Mawr College, '82); Ph.D. Indiana University, '91) teaches mathematics at Bar Ilan University, Ramat Gan, Israel. Among her publications is the joint article "Proofs and retributions, or: why Sarah can't take limits" published in Foundations of Science. Mikhail G. Katz (B.A. Harvard University, '80; Ph.D. Columbia University, '84) is Professor of Mathematics at Bar Ilan University. Among his publications is the book Systolic geometry and topology, with an appendix by Jake P. Solomon, published by the American Mathematical Society; and the article (with T. Nowik) "Differential geometry via infinitesimal displacements" published in Journal of Logic and Analysis. Taras S. Kudryk (born 1961, Lviv, Ukraine) is a Ukrainian mathematician and associate professor of mathematics at Lviv National University. He graduated in 1983 from Lviv University and obtained a Ph.D. in physics and mathematics in 1989. His main interests are nonstandard analysis and its applications to functional analysis. He is the author of books about nonstandard analysis (in Ukrainian and English) and textbooks about functional analysis (in Ukrainian) coauthored with V. Lyantse. Kudryk has performed research in nonstandard analysis in collaboration with V. Lyantse and Vitor Neves. His publications appeared in Matematychni Studii, Siberian Journal of Mathematics, and Logica Universalis. Thomas Mormann is Professor at the Departamento de Lógica y Filosofıa de la Ciencia de la Universidad del Páıs Vasco UPV/EHU 22 P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. (Donostia-San Sebastián, Spain). He obtained a PhD in Mathematics from the University of Dortmund, and habilitated in Philosophy, Logic, and Philosophy of Science at the University of Munich. He published numerous papers in philosophy of science, history of philosophy of science, epistemology, and related areas. He is the editor of Carnap's "Anti-Metaphysical Writings" (in German). Presently, his main research interest is in the philosophy of Ernst Cassirer and, more generally, in the Marburg Neo-Kantianism. David Sherry is Professor of Philosophy at Northern Arizona University, Flagstaff, AZ. E-mail: firstname.lastname@example.com. He has research interests in philosophy of mathematics, especially applied mathematics and nonstandard analysis. P. B laszczyk, Institute of Mathematics, Pedagogical University of Cracow, Poland E-mail address : firstname.lastname@example.com V. Kanovei, IPPI, Moscow, and MIIT, Moscow, Russia E-mail address : email@example.com K. Katz, Department of Mathematics, Bar Ilan University, Ramat Gan 52900 Israel E-mail address : email@example.com M. Katz, Department of Mathematics, Bar Ilan University, Ramat Gan 52900 Israel E-mail address : email@example.com T. Kudryk, Department of Mathematics, Lviv National University, Lviv, Ukraine E-mail address : firstname.lastname@example.com T. Mormann, Department of Logic and Philosophy of Science, University of the Basque Country UPV/EHU, 20080 Donostia San Sebastian, Spain E-mail address : email@example.com D. Sherry, Department of Philosophy, Northern Arizona University, Flagstaff, AZ 86011, US E-mail address : email@example.com


Making Sense of Value

Philosophy

A book review of Elizabeth Anderson, Value in Ethics and Economics (Cambridge, Mass.: Harvard University Press, 1993). I will pass over her compelling critiques of cost-benefit analysis, rational desire theory, and "consequentialist" moral theories, among many topics she dispatches successfully, with fierce intelligence and wit. Instead I want to focus on the central justificatory strategy that underpins her defense of her pluralist, nonconsequentialist, rational attitude theory of value. Anderson states at the outset that she is not that interested in such metaethical questions as whether value judgments express beliefs or emotions or other attitudes (p. xiii). But I think her own answers to these questions might have implications for normative and pragmatic issues of concern to her. So I want to try tinkering with the metaethical arguments in ways that might streamline her treatment of the normative and pragmatic ones somewhat.

Making Sense of Value Author(s): Adrian M. S. Piper Source: Ethics, Vol. 106, No. 3 (Apr., 1996), pp. 525-537 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/2382269 . Accessed: 13/07/2013 09:50 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact email@example.com. . The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to Ethics. http://www.jstor.org This content downloaded from 184.108.40.206 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions Making Sense of Value Adrian M. S. Piper If I were to agree with everything Elizabeth Anderson says in this powerful and finely conceived work I would not be doing my job.1 So I will pass over her compelling critiques of cost-benefit analysis, rational desire theory, and "consequentialist"2 moral theories, among many topics she dispatches successfully, with fierce intelligence and wit. Instead I want to focus on the centraljustificatory strategy that underpins her defense of her pluralist, nonconsequentialist, rational attitude theory of value. Anderson states at the outset that she is not that interested in such metaethical questions as whether value judgments express beliefs or emotions or other attitudes (p. xiii). But I think her own answers to these questions might have implications for normative and pragmatic issues of concern to her. So I want to try tinkering with the metaethical arguments in ways that might streamline her treatment of the normative and pragmatic ones somewhat. I. EXPRESSIVE STATES To say that someone or something is valuable, on Anderson's account, is to say that it is rational, that is, makes sense for someone to value her or it. To value that person or thing intrinsically is to have a "complex of positive [or favorable] attitudes" toward her or it (pp. 2, 17, 124). Favorable attitudes toward people or things are diverse: they include being inspired, attracted, interested, pleased, awed, emotionally involved, attentive, or concerned (p. 2). Anderson does not define what she means by an attitude, but she describes it, first, as an expressive state, and second, as "partly constituted by norms that determine [its] proper objects" (p. 3). 1. Elizabeth Anderson, Value in Ethics and Economics (Cambridge, Mass.: Harvard University Press, 1993). All page references to this volume are parenthesized in the text. 2. I explain the scare quotes in "A Distinction without a Difference," in Midwest Studies in Philosophy, vol. 7, Social and Political Philosophy, ed. Peter A. French, Theodore E. Uehling, Jr., and Howard K. Wettstein (Minneapolis: University of Minnesota Press, 1982): 403-35. Ethics 106 (April 1996): 525-537 ? 1996 by The University of Chicago. All rights reserved. 0014-1704/96/0603-2002$01.00 525 This content downloaded from 188.8.131.52 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions 526 Ethics April 1996 I want to leave aside this second clause for the moment and concentrate on the notion that an attitude is an expressive state of the agent. What kind of state is an expressive state? A brain state? A dispositional state? An emotional state? Or perhaps all of the above? What does an expressive state express? Emotions? Thoughts? Desires? Impulses? Perceptions? Or all of the above? I will describe a yes answer to these questions as the inclusive conception of an attitude. On the inclusive conception, an attitude can express emotions, thoughts, desires, impulses, perceptions, or any of the other myriad internal states-brain, dispositional, emotional, and so on-that constitute our mental life. And to say that an attitude, on this conception, expresses any such state is to say that the bare presence of this state necessarily manifests its particular contents in overt physical change-as, for example, inner turmoil might manifest itself in rapid breathing, dilated pupils, increased heart rate, and agitated movement of the limbs. So the inclusive conception of an attitude implies that for any internal state constitutive of our mental life, there is necessarily some observable, physical manifestation of it. Now on Anderson's view there cannot be a necessary connection between any such state of the agent (whether brain state, dispositional state, or emotional state) and a particular physical manifestation of it. For that would conflict with her later attempt to distinguish appropriate from inappropriate expressions of an attitude (pp. 83, 129). If the connection between an internal state and a particular physical expression of it is a necessary one, there is no point in evaluating it as appropriate or inappropriate. That physical expression of it is not subject to reform. Similarly, Anderson's view cannot realistically imply that any such mental state-whether thought, desire, impulse, or perception-has some observable physical manifestation. For we learn to conceal and internalize our thoughts, desires, impulses, and perceptions in the process of socialization: to keep our thoughts to ourselves, to suppress our desires, to refrain from acting on our impulses, to register our perceptions without reacting to them. This skill of controlling and internalizing our reactions is, in essence, what the process of socialization teaches, and, as Nietzsche observed, it is the origin of the interiority of our mental lives. But any such state that, for reasons of socialization , individual constitution, or personal control, has no physically observable manifestation cannot be meaningfully described as an expressive state at all. So it cannot be true that, as the inclusive conception claims, all internal states constitutive of our mental lives are expressive states, nor, therefore, that all internal states constitutive of our mental lives conform to Anderson's conception of an attitude. However, for later purposes Anderson will want to insist that anything that is an attitude is necessarily expressive. She will also want This content downloaded from 18.104.22.168 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions Piper Making Sense of Value 527 to insist that an attitude can be expressed appropriately or inappropriately . Since the inclusive conception of an attitude implies the rejection of both of these features, Anderson should reject the inclusive conception of an attitude. By contrast, an exclusive conception of an attitude toward a person or thing might define it as a specifically emotional response, or disposition to so respond, to that person or thing, such that the emotion is caused in part by the agent's perceptions, thoughts, and beliefs about the object of valuation. On the exclusive conception, an attitude is an expressive state in that there is a necessary connection between the agent's inner emotional state and some overt physical manifestation. However, it excludes any necessary connection between any such emotional state of the agent and a particular physical manifestation, since Anderson needs to be able to distinguish between the attitude itself and the appropriateness with which it is expressed. For the same reason, the exclusive conception of an attitude excludes any necessary connection between the agent's inner emotional state and any intentional action, whether of execution or of omission. Thus, the exclusive conception of an attitude excludes certain thoughts, desires, impulses, and perceptions unless they bear the right kind of causal connection to emotions. And it leaves open whether a particular attitude is expressed only in the most subtle and minimal overt physical changes or in gross behavior or action of an appropriate or inappropriate kind. For these reasons, Anderson should accept the exclusive conception of an attitude. Let us assume, provisionally, that she does. Then to value intrinsically a person or thing, is, first of all, to respond with positive emotions to one's perceptions and beliefs about her or it. However, not just any favorable emotional response to a person or thing counts as valuing it. Positive valuations must be "governed by distinct standards for perception, emotion, deliberation, desire, and conduct" (pp. 2, 3). Parents who value-love-their children will feel not only proud when their children achieve, but alarmed when they are in danger, and disposed to rescue them. Parents who felt pride at their children's achievements but only indifference or regret rather than concern at their endangerment could not properly be said to value them at all. So to Anderson's account we can add that intrinsically valuing a person or thing requires not only a complex of favorable emotions and dispositions toward her or it, but also that this complex exhibit a certain internal consistency determined by our concept of valuation itself. Only a certain specific set of favorable responses, elicited under their appropriate conditions, counts as, for example, loving or respecting or admiring a person or thing. That is, our valuation concept provides both a criterion for identifying the constellation of favorable attitudes constitutive of it and also a standard of adequacy against which these responses can be measured. This content downloaded from 126.96.36.199 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions 528 Ethics April 1996 On Anderson's account, valuing a person or thing in a particular way requires that this constellation of favorable attitudes-perception, emotion, deliberation, desires, and conduct "express and thereby communicate one's regard for the object's importance" (p. 11). Here Anderson goes beyond the exclusive conception of an attitude I advocated above. That conception built in only the most minimal notion of expression, namely some overt physical manifestation of the agent's emotional state. The notion of expression Anderson invokes here builds in two further conditions: first, that an authentic valuation should "express [those] valuations in the world, . . . embody them in some social reality ... actually establish the relationship to the object of one's concern which is implicit in one's attitudes toward it" (p. 17). This seems right. An agent who is not moved to establish some such connection with the object she values either does not really or deeply feel what she claims to feel, or else her purported values have strictly armchair status. There is a natural causal link between emotion and behavior expressive of it in all cases, no less so in the case of that complex of emotions partly constitutive of valuation. If one really values something, one is disposed to act accordingly. However, we can grant this much without requiring, as Anderson does, that the connection between valuing and action be a necessary one. The connection may be merely sufficient, such that a failure or inability to express one's valuation in action would not imply that it was not an authentic valuation after all. II. EXPRESSIVE NORMS Anderson's second requirement, however, is that in genuinely expressing one's valuations in action, one thereby communicate one's regard for the object's importance to some possible observer or listener. To do this requires that others can identify our behavior as appropriate, that is, as meeting shared behavioral standards for expressing that valuation. This is the sense in which Anderson wants to claim that expressing valuations is governed by shared social norms relative to which others can recognize our behavior as expressing the valuations we intend to express by it. So in order to count as an authentic valuation , on Anderson's view, an agent must not only manifest overtly a positive valuational attitude. Indeed, she must not only also express that attitude in action that establishes a relationship to the object she values. In addition, the action must be intelligible to others as an expression of her regard for that object. If the action through which the agent expresses her connection and regard for the object valued does not (or could not) communicate that regard to other agents, she does not qualify as authentically valuing that object at all. This is a very strong claim. But there is no ambiguity in Anderson's formulation of it. She argues that "I am capable of valuing something This content downloaded from 126.96.36.199 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions Piper Making Sense of Value 529 in a particular way only in a social setting that upholds norms for that mode of valuation.... To care about something in a distinctive way, one must participate in a social practice of valuation governed by norms for its sensible expression" (p. 12). Earlier I suggested that intrinsically valuing a person or thing requires not only a complex of favorable emotions and dispositions toward her or it, but also that this complex exhibit a certain internal consistency determined by our concept of valuation itself. So, as we saw, loving one's children requires pride on some occasions, alarm and a disposition to rescue them on others. This suggestion was consistent with the exclusive conception of an attitude, since it did not require a necessary connection between these emotional and dispositional responses on the one hand and action on the other. By contrast, we can now see that Anderson's account does, and more. Her idea is that one must conform one's action to the shared behavioral norms prescribing appropriate expression of a particular mode of valuation in order to be said to value something in that way at all. The example she gives is that of honoring someone. Her claim is that if we do not physically do what counts socially as honoring her, for example, treating her deferentially, applauding her, or paying her obeisance under the appropriate circumstances, we cannot be said truly to honor her. This seems right. But these are all actions whose connections to valuational mental states of the agent is contingent at best. Honoring may usually include valuational attitudes such as respect , admiration, perhaps affection, or esteem. But the concept of honoring someone is not exhausted by the constellation of attitudes we take toward her, nor are any of those just named necessary conditions of it. It is much more closely linked to public performative rituals, and much more detached from any particular set of emotional states, in ways that other valuational attitudes are not. Honoring is in this way much more like promising or proclaiming than like appreciating or admiring. To claim that one does not really value something unless one conforms to social norms for expressing that valuation seems much too strong, for it makes impossible inarticulate or concealed valuation, being at a loss to express the depth or intensity or particular quality of one's valuation of a person or thing, or of doing so awkwardly, or of trying to express one's emotional response and failing. To say that there is a natural link between authentic valuation and action is not thereby to say that there is a necessary link between them, nor to say that the action expresses that valuation successfully, nor that it does so in accordance with shared social norms for expressing that valuation. A recent television commercial illustrates this point. A man takes a woman to a candlelight dinner at an expensive restaurant. The This content downloaded from 188.8.131.52 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions 530 Ethics April 1996 occasion is clearly a special one, through which he intends to express his feelings for her. Over drinks he gives her a small, wrapped gift, a jewel box seemingly intended for a ring. Her manner is expectant and loving. She open the box to find a ceramic pin with the cartoon face of a clown on it. She looks up at him in shock and astonishment. He smiles uncomfortably. "Next time, better call 1-800-FLOWERS," the voice-over intones. The implication is clear that he has expressed his affection for her inappropriately. Perhaps he has. But perhaps he intended to express something else. Perhaps he meant to portray himself as a clown and symbolically give himself to her. Or to communicate that they should lighten up in this relationship and not lose their sense of humor just because they were getting involved. Or to suggest that he loves her for the madcap clown she really is rather than the selfimportant facade she presents to the world, and to give her a means by which to advertise a more lighthearted self-image. The setting makes clear that he meant to show how much he values her, and also that he lacks the social resources for expressing in what exactly that valuation consists. Perhaps the culture contains no such resources, or perhaps he is just unschooled in the ways of love. Thus, as it stands, Anderson's analysis disses dorks, geeks, nerds, and dweebs. But even for the most highly socialized and sophisticated among us, sometimes there really are no words adequate to express our gratitude for another's support, nothing we can do to demonstrate the depth of our affection, no way to express our heartfelt appreciation -and simply saying this, or doing nothing, doesn't do the trick, either. This doesn't mean that we do not have those attitudes. If conforming to shared norms for expressing our valuations were a necessary condition of valuing something, it would be hard to see how new and original forms of social expression of these attitudes could arise, or what would motivate their creation, or how they could be recognized as original ways of expressing those attitudes. What of attitudes people in fact have that may not (yet) make sense to a society, such as the desire to rebel, detach, explore, or innovate? What of societies that do have norms needed to adequately express their members ' reflectively endorsed valuations, but none for valuations that are recognizably among the panoply of human valuations but not reflectively endorsed? And what of societies whose norms for expressing their members' valuations are themselves not reflectively endorsable ? Anderson agrees that "a social order can be criticized for failing to provide adequate normative vehicles for the expression of attitudes that have come to make sense to its members.... If a society lacks the social norms needed to adequately express its members' reflectively endorsed valuations, the rational thing to do is to invent and institute This content downloaded from 126.96.36.199 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions Piper Making Sense of Value 531 such norms" (p. 18). But if people can make reflectively endorsed valuations in the absence of adequate social norms for expressing them, then expressing those valuations in conformity with the norms cannot be a necessary condition for the existence of the reflectively endorsed valuations themselves. ILL. MAKING SENSE OF VALUE This brings us to Anderson's conception of a valuation's making sense, of its being reflectively endorsable. Just as Anderson equates having the constellation of favorable attitudes constitutive of a particular mode of valuation with expressing it in norm-governed action, she also equates rational valuation with expressing one's valuations through reflectively endorsable norm-governed action. Thus, Anderson wants to distinguish between valuing something and that thing's being valuable . She improves on Mill's formulation by stipulating that something is valuable if it makes sense for someone to value it (pp. 91-92, 102, 124) and if it meets standards it makes sense for someone to value (pp. 114-15). The concept of a thing's making sense is a central one for Anderson . Here she deploys two locutions. In some passages she speaks interpretively, of making sense of our attitudes. For example, she characterizes the quest for self-understanding as "an attempt to make sense of our own valuational responses to the world" (p. 3) and the coziness of a bedroom as making sense of "[one's] feeling snug when [one] retire[s] there" (p. 4). Later she suggests that "if either [of two very different and incommensurable ways of adequately expressing one's valuations of one's ends] makes adequate, but very different sense, of one's valuations, then reason permits the pursuit of either one" (p. 63), and that "one can make sense of one's own attitudes only by taking up a point of view from which others can also make sense of them" (p. 95), and that "to justify an evaluative claim is to appeal to reasons that make sense of particular attitudes toward the evaluated object" (p. 97), and finally that 'justification is concerned with making sense of our concerns and attitudes" (p. 111). In these passages Anderson treats some attitudes as a given. Making sense of them is then equivalent to finding the interpretation or explanation of given mental phenomena that makes them most comprehensible to oneself and others. On this reading, to justify one's valuations is just to explain them with reasons why. The interpretive locution has the advantage that it does not beg any questions about what valuations or attitudes any particular interlocutor might think one should have. It is a comparatively weak requirement on justification , in that it requires only that I understand your values, whether or not I share them. To complain that a valuation does not make sense This content downloaded from 22.214.171.124 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions 532 Ethics April 1996 is, on this reading, to complain not that it is personally unacceptable, but that it is unintelligible, that is, that it violates certain basic conditions of conceptual coherence and consistency. In other passages, however, she speaks prescriptively, of what it makes sensefor someone to do. Were two friends to become enemies, she argues, "it would make sense for [one] to stop cherishing" an ugly bracelet given her by the other (p. 19). Similarly, she says, it "makes sense for a person to value most [states of affairs] only because it makes sense for a person to care about the people, animals, communities, and things concerned with them" (p. 20), and "what it makes sense to do now essentially depends on what one has done in the past" (p. 34). Later she argues that "if goods are not commensurable, then it does not make sense to maximize their values" (p. 46). She defines a standard as "authentic if and only if ... it could make sense for a person to guide her responses by it" and as important to a person "if it makes sense for her to care about it" (p. 48). Similarly, she says that "which higher-order good it makes sense to use in justifying a person's choices depends on the context of decision" (p. 54), that "it makes sense to value different goods in different ways" (p. 72), and that "the conditions that make states of affairs valuable are not other states of affairs, but the people, animals, and things it makes sense to care directly about" (p. 85). In these passages, what it makes sense for someone to do is what there is reason for doing. To justify one's valuations is to demonstrate that the balance of reasons prescribes it. And to state that it makes sense for someone to do something is to state that the balance of reasons prescribes it. It is to advocate the doing of that thing. Thus, it presupposes and expresses a set of values with which one's listener is assumed to agree. The prescriptive locution in this sense imposes a much stronger condition on justification than the interpretive one. Anderson's explicit definition of making sense supports the prescriptive locution but not the interpretive one. To judge that one's valuations make sense, for Anderson, is "to judge that they would be endorsed" from a hypothetical, common point of view in which people can both achieve one another's valuations and also reflectively endorse them. The process of reaching this point of view is a process in which "people interpret and justify their valuations by exchanging reasons for them" (p. 3). Part of exchanging reasons for one's valuations is being "able to tell a story that makes sense of [an] ideal, that gives it some compelling point, that shows how the evaluative perspective it defines reveals defects, limitations, or insensitivities in the perspectives that reject these valuings" (p. 92). Moreover, this process of justification is objective, on Anderson's account, if the participants in this dialogue can reach agreement or make progress when they adhere to the following norms of rational discourse: This content downloaded from 22.214.171.124 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions Piper Making Sense of Value 533 A. They acknowledge the possibility of a permanent gap between their actual attitudes and rational ones B. They acknowledge the equal authority of others to offer criticisms and proposals C. No one competent to participate is excluded from the dialogue D. All apply reasons consistently to their own proposals and to others E. They aim for agreement or a common point of view F. They agree to work from mutually accepted reasons toward resolution of their differences G. The process contains methods for introducing new considerations as reasons and for criticizing what are currently taken to be reasons (p. 93) So one's valuations are valuable, that is, make sense, if they would be endorsed by others who, through adhering to norms A-G constitutive of the process of rational justification, comparison, and critique of one another's valuations, were to reach mutual agreement on their valuations. This process of justification is called for when people who have different values have some interest or need to reach agreement. It is possible, Anderson writes, when there is some overlap in the considerations each party accepts as counting for or against attitudes and judgments (p. 93). And it is required as a necessary condition of making sense of oneself and one's own values (pp. 94-95). This account of rational value does require that others share some values at the outset. It presupposes "a background of socially contingent and historically evolving social practices and conditions" (p. 102), as well as common ground, minimally, in "shared intuitions or in curiosity, trust, and a willingness to try alien practices" (p. 105). It also presupposes that all participants agree either on what counts as a reason for or against something (norms D and F) or on how to introduce new considerations as reasons for or against things (norm G). Together these presuppositions constitute a quite substantial area of shared values. Anderson's claim is that unless one's valuations conform to the hypothetical point of view delineated by these norms, one cannot make sense of oneself or them at all. IV. MAKING SENSE OF ONESELF In this discussion so far I have been concerned to make room in Anderson's metaethics for valuational attitudes people can have but not express, or express but not in socially familiar or acceptable ways. I have wanted to insist on the existence of such anomalous attitudes, as I will call them, even in the face of social ignorance or incomprehension of them. I will now want to insist on the rational value of some anomalous attitudes even though they do not meet some of the ratioThis content downloaded from 220.127.116.11 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions 534 Ethics April 1996 nality conditions Anderson requires. I will want to show that such anomalous attitudes can be valuable, that is, rational, even though they are not reflectively endorsable by other members of a social community governed by norms for their expression. Consider someone whose valuational attitudes are marginal with respect to all the social communities in which he moves. Suppose, for example, that the background of social practices and conditions in which he was raised is at odds with the two in which he now lives and commutes, that these two are at odds with each other, and that the two in which he now lives and commutes have alienated him from the one in which he was raised. Also suppose that because of his outsider status with respect to all three cultures, his intuitions, perceptions, and beliefs about the inhabitants of each are greatly at odds with the intuitions, perceptions, and beliefs the inhabitants of each culture have about themselves, and similarly at odds with the intuitions, perceptions , and beliefs each culture has about the others. Suppose further that this outsider status has virtually sated his curiosity and willingness to try alien practices by requiring him, as a condition of his own adaptation and survival, to study and gain extensive familiarity with the mores of each culture-to become more knowledgeable about each culture, in fact, than any single inhabitant of any of the three cultures is about her own. Moreover, suppose his outsider status has brought upon him repeated and consistent social ostracism, rejection, and punishment, so that his ability to trust any member of any of the three cultures is virtually nonexistent. It is not implausible that, as the result of his experience as an interloper in all three social communities, his conception of what counts as a reason in favor of certain basic matters might be equally at odds with others' conceptions, so that it would not be possible to apply certain reasons consistently both to his and to others' proposals. That certain lifestyles were socially isolating, for example, might count as a reason against them for members of any of the three communities, but as a consideration in their favor for him; that certain activities would bring one face to face with one's own mortality might count as a reason against them for others but as a reason for them for him; that certain kinds of relationships would fill his life with connection to others might count as a reason against them for him but a reason for them for others; and so forth. Similarly, it is not implausible that, given his experiences, he might not agree with members of any of the three social communities on how to introduce into the dialogue new considerations for or against things as reasons. By hypothesis he would be fully conversant with the practices members of all three cultures agreed on for doing this something analogous, let us suppose, to following Robert's Rules of Order. But he might justifiably think these practices inadequate for This content downloaded from 126.96.36.199 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions Piper Making Sense of Value 535 introducing considerations that were radically unlike the considerations members of these cultures were conditioned to recognize as reasons . He might think that precisely because of their social cohesion and conformity, there were certain sorts of quite important reasons that members of all three communities simply were not psychologically or socially equipped to consider-that they just wouldn't "get it." And he might think that only quite radical presentations of these considerations-in theatrical or otherwise dramatic symbolic form, perhaps, or in acts of self-immolation or antisocial destruction, might lead the light to dawn. Being unwilling or unable to perform such acts himself, he might conclude that there was no way for him to tell the story that made sense of his ideals, that would give it a compelling point or reveal the defects, limitations, or insensitivities in the perspectives of an audience of interlocutors whose experiences were so radically different from and limited relative to his own. Although he might fully understand their valuations, he might realistically conclude that there was no way for him to make his valuations intelligible to them. For all of these reasons, his valuations would not be rationally endorsable by other participants in the rational dialogue Anderson describes, nor might he think it worth his while even to participate in it. But this would not imply that he was unable to make sense of his own values. First, it would not imply this for the interpretive locution. He would be able to explain his attitudes and values in the same terms I have just described, offering reasons why he values and disvalues as he does that would enable us to understand his valuations even if we do not share them. Second, that his values were not rationally endorsable from Anderson's hypothetical common point of view would not imply that it would not make sense prescriptively for him to respond and act as he does. He could, by hypothesis, give realistic and wellgrounded reasonsfor valuing solitude, silence, and confrontation with mortality, such that we would be compelled to recognize the rational integrity of his perspective even if we did not share it. That these values would not make sense from the shared perspective of a community of participants engaged in rational dialogue of the kind Anderson describes does not imply that they do not make sense at all. Third and most important, that these anomalous values were not thus rationally endorsable would not imply that our marginalized agent could not make sense of himself. Anderson argues that "one can make sense of one's own attitudes only by taking up a point of view from which others can also make sense of them.... We can make sense of ourselves only by participating in practices of justification" (p. 95). By contrast, this agent would make sense of his own attitudes from a perspective to which no one else had access, namely, the perspective of having experienced the three disparate social communities in the unique and particular ways that he had. And he would be able This content downloaded from 18.104.22.168 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions 536 Ethics April 1996 to do this even though he had, by hypothesis, declined to participate in a social practice of justification of the kind Anderson advocates. Now Anderson argues that "making sense of ourselves is not a matter of theorizing about an object whose properties we cannot affect. We make ourselves intelligible to ourselves by cultivating attitudes that make sense to us, by determining to act in accord with ideals we accept that have survived critical scrutiny" (p. 91). She claims that part of the quest for self-understanding requires that when we recognize in ourselves attitudes that we cannot endorse from the hypothetical common point of view governed by shared social norms of discourse, we reform these attitudes "so that they make sense in the context of an enlarged self-understanding" (p. 96). Thus, self-understanding, on Anderson's view, requires active self-determination through the cultivation of attitudes that are reflectively endorsable from the hypothetical common point of view. Again I think this is too strong. It implies that we cannot make sense of what we cannot either endorse or improve, and so that those intractable and incorrigible parts of the self that are so necessary for bringing us face to face with our imperfections, our guilt, and our personal limitations must remain opaque or impenetrable to rational analysis. It also implies that we can easily improve what we cannot initially endorse, and I have yet to see an account of how this is supposed to work that does not degenerate into exhortations to bootstrap the triumph of the will over the flesh. Most people cannot even manage to stay on a low-cholesterol diet. But my main concern is what it implies for the possibility of social and cultural change. Earlier I asked how new and original expressions of valuational attitudes could arise, if conforming those expressions to shared social norms were a necessary condition of their existence. The same question can be asked about new and original valuational attitudes themselves. Demographically mobile societies such as this one are constantly creating marginalized agents of the sort just described. Through upward mobility we may move from our original class backgrounds to higher ones, and to different ethnic or cultural groups, through education and professional training; through downward mobility and economic contraction we may move in the reverse direction, and thereby into other new ethnic or lifestyle communities-perhaps even into homelessness or penal incarceration; through travel, new technologies, or contact with other cultures that create new possibilities for experience or lifestyle, we may find our most basic values or lifestyle preferences undergoing radical revision. Anderson conceives the relevant contrast along Marxist economic lines, as between individualistic and social conceptions of rational attitudes. But a society as marked by heterogeneous social values as this one owes its plethora of anomalous attitudes at least as much to its ethnic and class diversity and mobility as it does to its capitalist economic structure. This content downloaded from 18.104.22.168 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions Piper Making Sense of Value 537 Agents who undergo these social, economic, and cultural shifts are regularly confronted with disparities between their own anomalous attitudes and those that are socially endorsed by the community at hand. Under these circumstances, one always faces the choice of to which source authoritative weight should be ascribed. Either one may conclude that one's own values are inappropriate, and take steps to reform them in accordance with the norms of the community, or one may conclude that the norms of the community are inappropriate, and take steps to reform them in accordance with one's own values. Those who are strongly identified with the norms of a particular community will incline to choose the former alternative; marginalized agents by definition have a greater capacity to choose the latter. Without this capacity, it is hard to see how social and cultural value change could occur. When value change does occur, it does not require that one construct or even envision an alternative community that adopts and enacts the norms of rational dialogue Anderson describes, nor that one rely on such a hypothetical community to endorse and legitimate the anomalous attitudes one may know independently to be rational. A marginalized agent can recognize his anomalous values as rational if, to summarize briefly, (1) he can causally explain them by his experiences , (2) he can in turn cite these values as reasons for his behavior and attitudes, and (3) these values, and the experiences that form them, are internally coherent. Of course this does not imply that they are therefore morally acceptable to any actual or hypothetical community. Whether they are or not, what any actual or hypothetical community thinks about them is irrelevant to their rationality. Therefore, social and cultural value change does not require that "if our lives are to be meaningful, then we must adopt a perspective informed by the expressive theory as our global mode of deliberating about and justifying our actions, emotions, and attitudes" (p. 83). Since rationality is itself a value, it is unclear why Anderson, after successfully developing a fully pluralistic theory of value, thinks she needs to ground it with monistic and global criteria of rationality. Social and cultural value change requires only marginalized agents whose anomalous values are sufficiently secure, independent of the community's, and well grounded in their experience to furnish the distanced critical perspective from which the community's can be found to be lacking. That is the kind of agent for which I believe a genuinely pluralistic, rational attitude theory of value such as Anderson's can and must make room. This content downloaded from 22.214.171.124 on Sat, 13 Jul 2013 09:50:26 AM All use subject to JSTOR Terms and Conditions