The philosophical bases underlying methodological and decision-making processes for environmental issues are rarely questioned, and yet have important consequences. What commonly results is that first order solutions are technical ways of addressing problems which limit human relation to nature. Martin Heidegger’s phenomenology makes a distinction between “thatness” and “whatness.”“What a thing is” is depicted by modern science with “being as continual presence.” “That a thing is” refers to nature’s capacity for disclosure and withdrawal, that being is both “presence and absence.” (...) This essay evaluates thepragmatic prospects of heightening an approach on the “thatness” of nature. (shrink)
At least four different frameworks — psychiatric, cognitive, functional and decision-making — are used in the evaluation of competence, all of which remain more or less unrelated in the literature. In the first section of this paper we consider various meanings of competence, in order to arrive at a definition of the term relevant to the medical and legal setting. Patient or client competence, we conclude, refers to the practical abilities that individuals employ in pursuing their own autonomous goals in (...) life. We then show how a systematic categorization of these practical abilities — which we call a taxonomy of practical judgment — allows us to show when the traditional frameworks for the evaluation of competence may or may not be useful in the evaluation of a particular competence.In the final section we explore some of the normative considerations underlying the taxonomy. For instance, competence is not only related to intrinsic abilities but to resources available in the community. Here we touch on questions related to the fair distribution of community resources. (shrink)
Abstract The Suttas indicate physical conditions for success in meditation, and also acceptance of a not?Self life?principle (primarily viññana) which is (usually) dependent on the mortal physical body. In the Abhidhamma and commentaries, the physical acts on the mental through the senses and through the ?basis? for mind?organ and mind?consciousness, which came to be seen as the ?heart?basis?. Mind acts on the body through two ?intimations?: fleeting modulations in the primary physical elements. Various forms of r?pa are also said to (...) originate dependent on citta and other types of r?pa. Meditation makes possible the development of a ?mind?made body? and control over physical elements through psychic powers. The formless rebirths and the state of cessation are anomalous states of mind?without?body, or body?without?mind, with the latter presenting the problem of how mental phenomena can arise after being completely absent. Does this twin?category process pluralism avoid the problems of substance?dualism? (shrink)
The two books discussed here make important contributions to our understanding of the role of spacetime concepts in physical theories and how that understanding has changed during the evolution of physics. Both emphasize what can be called a ‘dynamical’ account, according to which geometric structures should be understood in terms of their roles in the laws governing matter and force. I explore how the books contribute to such a project; while generally sympathetic, I offer criticisms of some historical claims concerning (...) Newton, and argue that the dynamical account does not undercut ontological issues as the books claim. *Received January 2009; revised March 2009. †To contact the author, please write to: Department of Philosophy, 1423 University Hall MC 267, University of Illinois at Chicago, 601 S. Morgan Street, Chicago, IL 60607; e‐mail: email@example.com. (shrink)
It is argued that Minkowski space-time cannot serve as the deep structure within a ``constructive'' version of the special theory of relativity, contrary to widespread opinion in the philosophical community.
The implications for the substantivalist–relationalist controversy of Barbour and Bertotti's successful implementation of a Machian approach to dynamics are investigated. It is argued that in the context of Newtonian mechanics, the Machian framework provides a genuinely relational interpretation of dynamics and that it is more explanatory than the conventional, substantival interpretation. In a companion paper (Pooley [2002a]), the viability of the Machian framework as an interpretation of relativistic physics is explored. 1 Introduction 2 Newton versus Leibniz 3 Absolute space versus (...) an affine connection 4 Anti-relationalist arguments 5 Rehabilitating relationalism 6 Dynamics on the relative configuration space 7 Intrinsic particle dynamics 8 Conclusion. (shrink)
The existence of a definite tangent space structure (metric with Lorentzian signature) in the general theory of relativity is the consequence of a fundamental assumption concerning the local validity of special relativity. There is then at the heart of Einstein's theory of gravity an absolute element which depends essentially on a common feature of all the non-gravitational interactions in the world, and which has nothing to do with space-time curvature. Tentative implications of this point for the significance of the vacuum (...) solutions in general relativity, and for the issue of quantising gravity, are briefly examined. (shrink)
The purpose of this paper is to evaluate the `Lorentzian Pedagogy' defended by J.S. Bell in his essay ``How to teach special relativity'', and to explore its consistency with Einstein's thinking from 1905 to 1952. Some remarks are also made in this context on Weyl's philosophy of relativity and his 1918 gauge theory. Finally, it is argued that the Lorentzian pedagogy---which stresses the important connection between kinematics and dynamics---clarifies the role of rods and clocks in general relativity.
In this new edition, Arthur Fine looks at Einstein's philosophy of science and develops his own views on realism. A new Afterword discusses the reaction to Fine's own theory. "What really led Einstein . . . to renounce the new quantum order? For those interested in this question, this book is compulsory reading."--Harvey R. Brown, American Journal of Physics "Fine has successfully combined a historical account of Einstein's philosophical views on quantum mechanics and a discussion of some of the (...) philosophical problems associated with the interpretation of quantum theory with a discussion of some of the contemporary questions concerning realism and antirealism. . . . Clear, thoughtful, [and] well-written."--Allan Franklin, Annals of Science "Attempts, from Einstein's published works and unpublished correspondence, to piece together a coherent picture of 'Einstein realism.' Especially illuminating are the letters between Einstein and fellow realist Schrodinger, as the latter was composing his famous 'Schrodinger-Cat' paper."--Nick Herbert, New Scientist "Beautifully clear. . . . Fine's analysis is penetrating, his own results original and important. . . . The book is a splendid combination of new ways to think about quantum mechanics, about realism, and about Einstein's views of both."--Nancy Cartwright, Isis. (shrink)
The quantum theory of de Broglie and Bohm solves the measurement problem, but the hypothetical corpuscles play no role in the argument. The solution finds a more natural home in the Everett interpretation.
In a comparison of the principles of special relativity and of quantum mechanics, the former theory is marked by its relative economy and apparent explanatory simplicity. A number of theorists have thus been led to search for a small number of postulates - essentially information theoretic in nature - that would play the role in quantum mechanics that the relativity principle and the light postulate jointly play in Einstein's 1905 special relativity theory. The purpose of the present paper is to (...) resist this idea, at least in so far as it is supposed to reveal the fundamental form of the theory. It is argued that the methodology of Einstein's 1905 theory represents a victory of pragmatism over explanatory depth; and that its adoption only made sense in the context of the chaotic state state of physics at the start of the 20th century - as Einstein well knew. (shrink)
Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now some of my fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, and there (...) are quite a few highly technical journals in logic, such as The Journal of Sym-. (shrink)
It is still perhaps not widely appreciated that in 1905 Einstein used his postulate concerning the ‘constancy’ of the light-speed in the ‘resting’ frame, in conjunction with the principle of relativity, to derive numerical light-speed invariance. Now a ‘weak’ version of the relativity principle (or, alternatively, appeal to the Michelson—Morley experiment) leads from Einstein's light postulate to a condition that we call universal light-speed constancy. which is weaker than light-speed invariance. It follows from earlier independent investigations (Robertson ; Steigler ; (...) Tzanakis and Kyritsis ) that this condition is none the less sufficient to derive the Lorentz transformations up to a scale factor, given the well-known kinematic principle of ‘reciprocity’. In this paper, we follow Robertson and explore the kinematics consistent with universal light-speed constancy without imposing reciprocity, and we recover the Lorentz transformations by further appeal only to the weak relativity principle and spatial isotropy. (shrink)
In a recent paper in the British Journal for the Philosophy of Science, Kosso discussed the observational status of continuous symmetries of physics. While we are in broad agreement with his approach, we disagree with his analysis. In the discussion of the status of gauge symmetry, a set of examples offered by ’t Hooft has influenced several philosophers, including Kosso; in all cases the interpretation of the examples is mistaken. In this paper we present our preferred approach to the empirical (...) significance of symmetries, re-analysing the cases of gauge symmetry and general covariance. (shrink)
It is argued that awareness of the distinction between dynamical and variational symmetries is crucial to understanding the significance of Noether's 1918 work. Specific attention is paid, by way of a number of striking examples, to Noether's first theorem, which establishes a correlation between dynamical symmetries and conservation principles.
Abstract The historical evolution of the principle of relativity from Galileo to Einstein is briefly traced, and purported difficulties with Einstein's formulation of the principle are examined and dismissed. This formulation is then compared to a precise version formulated recently in the geometrical language of spacetime theories. We claim that the recent version is both logically puzzling and fails to capture a crucial physical insight contained in the earlier formulations. The implications of this claim for the modern treatment of general (...) dynamical symmetries are discussed. (shrink)
I discuss a rarely mentioned correspondence between Einstein and Swann on the constructive approach to the special theory of relativity, in which Einstein points out that the attempts to construct a dynamical explanation of relativistic kinematical effects require postulating a fundamental length scale in the level of the dynamics. I use this correspondence to shed light on several issues under dispute in current philosophy of spacetime that were highlighted recently in Harvey Brown’s monograph Physical Relativity, namely, Einstein’s view on (...) the distinction between principle and constructive theories, and the consequences of pursuing the constructive approach in the context of spacetime theories. r 2008 Elsevier Ltd. All rights reserved. (shrink)
The vacuum is fast emerging as the central structure of modern physics. This collection brings together philosophically-minded specialists who engage these issues in the context of classical gravity, quantum electrodynamics, and the grand unification program. The vacuum emerges as the synthesis of concepts of space, time, and matter; in the context of relativity and the quantum this new synthesis represents a structure of the most intricate and novel complexity. This book is a work in modern metaphysics, in which the concepts (...) of substance and space interweave in the most intangible of forms, the background and context of our physical experience: vacuum, void, or nothingness. (shrink)
A comparison is made of the traditional Loschmidt (reversibility) and Zermelo (recurrence) objections to Boltzmann's H-theorem, and its simplified variant in the Ehrenfests' 1912 wind-tree model. The little-cited 1896 (pre-recurrence) objection of Zermelo (similar to an 1889 argument due to Poincare) is also analysed. Significant differences between the objections are highlighted, and several old and modern misconceptions concerning both them and the H-theorem are clarified. We give (...) particular emphasis to the radical nature of Poincare's and Zermelo's attack, and the importance of the shift in Boltzmann's thinking in response to the objections as a whole. (shrink)
Quantum field theory, one of the most rapidly developing areas of contemporary physics, is full of problems of great theoretical and philosophical interest. This collection of essays is the first systematic exploration of the nature and implications of quantum field theory. The contributors discuss quantum field theory from a wide variety of standpoints, exploring in detail its mathematical structure and metaphysical and methodological implications.
Internal global symmetries exist for the free non-relativistic Schrodinger particle, whose associated Noether charges---the space integrals of the wavefunction and the wavefunction multiplied by the spatial coordinate---are exhibited. Analogous symmetries in classical electromagnetism are also demonstrated.
It is argued that an unheralded moment marking the beginnings of relativity theory occurred in 1889, when G. F. FitzGerald, no doubt with the puzzling 1887 Michelson-Morley experiment fresh in mind, wrote to Heaviside about the possible effects of motion on inter-molecular forces in bodies. Emphasis is placed on the difference between FitzGerald's and Lorentz's independent justifications of the shape distortion effect involved. Finally, the importance of the their `constructive' approach to kinematics---stripped of any commitment to the physicality of the (...) ether--- will be defended, in the spirit of Pauli, Swann and Bell. (shrink)
The aim of this paper is to illustrate four properties of the non-relativistic limits of relativistic theories: (a) that a massless relativistic field may have a meaningful non-relativistic limit, (b) that a relativistic field may have more than one non-relativistic limit, (c) that coupled relativistic systems may be ''more relativistic'' than their uncoupled counterparts, and (d) that the properties of the non-relativistic limit of a dynamical equation may differ from those obtained when the limiting equation is based directly on exact (...) Galilean kinematics. These properties are demonstrated through an examination of the non-relativistic limit of the familiar equations of first-quantized QED, i.e., the Dirac and Maxwell equations. The conditions under which each set of equations admits non-relativistic limits are given, particular attention being given to a gauge-invariant formulation of the limiting process especially as it applies to the electromagnetic potentials. The difference between the properties of a limiting theory and an exactly Galilean covariant theory based on the same dynamical equation is demonstrated by examination of the Pauli equation. (shrink)
An important contribution to the foundations of probability theory, statistics and statistical physics has been made by E. T. Jaynes. The recent publication of his collected works provides an appropriate opportunity to attempt an assessment of this contribution. * Review of E. T. JAYNES (1983): Papers on Probability, Statistics and Statistical Physics. Edited by R. D. Rosenkrantz. D. Reidel Publishing Company. US $49.50. Pp. xxiv + 434. We are grateful to Harvey Brown, Kenneth Denbigh, Udi Makov and Oliver Penrose (...) for their valuable criticisms of this paper. (shrink)
The purpose of the paper is to explore different aspects of the covariance of (mostly) non-relativistic quantum mechanics. First, doubts are expressed concerning the claim that gauge fields can be 'generated' by way of imposition of (local) gauge covariance of the single-particle wave equation. Then a brief review is given of Galilean covariance in the general case of external fields, and the connection between Galilean boosts and gauge transformations. Under time-dependent translations (and hence non-instantaneous boosts) the geometric phase associated with (...) Schrödinger evolution is non-invariant, and the significance of this result is briefly analysed. The covariance properties of Schrödinger dynamics are then brought to bear on certain versions of the modal interpretation of quantum mechanics. The conclusion that it is only relational properties that can be considered coordinate- or gauge-independent elements of reality is reinforced by appeal to the theory of quantum reference frames due to Aharonov and Kauffher. (This paper appeared in "From Physics to Philosophy", J. Butterfield and C. Pagonis (eds.), Cambridge University Press (1999); pp. 45-70.). (shrink)
The kinematical principle of Equal Passage Times (EPT) was introduced by Winnie in his 1970 derivation of the relativistic coordinate transformations compatible with arbitrary synchrony conventions in one-dimensional space. In this paper, the claim by Winnie and later Giannoni that EPT is a direct consequence of the relativity principle is questioned. It is shown that EPT, given Einstein's 1905 postulates, is equivalent to the relativistic (synchrony independent) clock retardation principle, and that for standard synchrony it reduces to an isotropy condition (...) for contraction (and dilation) effects. (shrink)
One of the widespread confusions concerning the history of the 1887 Michelson-Morley experiment has to do with the initial explanation of this celebrated null result due independently to FitzGerald and Lorentz. In neither case was a strict, longitudinal length contraction hypothesis invoked, as is commonly supposed. Lorentz postulated, particularly in 1895, any one of a certain family of possible deformation effects for rigid bodies in motion, including purely transverse alteration, and expansion as well as contraction; FitzGerald may well have had (...) the same family in mind. A careful analysis of the Michelson-Morley experiment (which reveals a number of serious inadequacies in many text-book treatments) indeed shows that strict contraction is not required. (shrink)
Machine generated contents note: -- Acknowledgements -- Preface; A.McRobbie -- Notes on Contributors -- Introduction; C.Scharff & R.Gill -- PART I: SEXUAL SUBJECTIVITY AND THE MAKEOVER PARADIGM -- Pregnant Beauty: Maternal Femininities under Neoliberalism; I.Tyler -- The Right to Be Beautiful: Postfeminist Identity and Consumer Beauty Advertising; M.M.Lazar -- Spicing It Up: Sexual Entrepreneurs and The Sex Inspectors; L.Harvey & R.Gill -- '(M)Other-in-Chief: Michelle Obama and the Ideal of Republican Womanhood'; L.Guerrero -- Scourging the Abject Body: Ten Years Younger (...) and Fragmented Femininity under Neoliberalism; E.Tincknell -- PART II: NEGOTIATING POSTFEMINIST MEDIA CULTURE -- Are You Sexy, Flirty, Or A Slut? Exploring 'Sexualisation' and How Teen Girls Perform/Negotiate Digital Sexual Identity on Social Networking Sites; J.Ringrose -- 'Feminism? That's So Seventies': Girls and Young Women Discuss Femininity and Feminism in America's Next Top Model; A.L.Press -- Media 'Sluts': 'Tween' Girls' Negotiations of Postfeminist Sexual Subjectivities in Popular Culture; S.Jackson & T.Vares -- Is 'the Missy' a New Femininity?; J.Kim -- PART III: TEXTUAL COMPLICATIONS -- Of Displaced Desires: Interrogating 'New' Sexualities abd 'New' Spaces in Indian Diasporic Cinema; B.Bose -- Notes on Some Scandals: The Politics of Shame in Vers le Sud; S.Wearing -- The Limits of Cross-Cultural Analogy: Muslim Veiling and 'Western' Fashion and Beauty Practices; C.Pedwell -- PART IV: NEW FEMININITIES: AGENCY AND/AS MAKING DO -- Through the Looking Glass? Sexual Agency and Subjectification Online; F.Attwood -- Reckoning with Prostitutes: Performing Thai Femininity; J.Haritaworn -- Migrant Women Challenging Stereotypical Views on Femininities and Family; U.Erel -- Negotiating Sexual Citizenship: Lesbians and Reproductive Health Care; R.Ryan-Flood -- PART V: NEW FEMINISMS, NEW CHALLENGES -- The New German Feminisms: Of Wetlands and Alpha-Girls; C.Scharff -- The Contradictions of Successful Femininity: Third-Wave Feminism, Postfeminism and 'New' Femininities; S.Budgeon -- Skater Girlhood: Resignifying Femininity, Resignifying Feminism; D.H.Currie, D.M.Kelly & S.Pomerantz -- Will These Emergencies Never End? Some First Thoughts about the Impact of Economic and Security Crises on Everyday Life; G.Bhattacharyya -- Index. (shrink)
Considerable work within the modern 'space-time theory' approach to relativity physics has been devoted to clarifying the role and meaning of the principle of relativity. Two recent discussions of the principle within this approach, due to Arntzenius (1990) and Friedman (1983), are found to contain difficulties.
Demonstration and self-evidence, by E.D. Simmons.--The significance of the universal ut nune, by J.A. Oesterle.--William Harvey, M.D.: modern or ancient scientist? by H. Ratner.--Medicine and philosophy in the eleventh and twelfth centuries: the problem of elements, by R.P. McKeon.--The origins of the problem of the unity of form, by D.A. Callus.--The celestial movers in medieval physics, by J.A. Weisheipl.--Gravitational motion according to Theodoric of Freiberg, by W.A. Wallace.--"Mining all within," Clarke's notes to Rohault's Traité de physique, by M.A. Hoskin.--Darwin's (...) dilemma, by C. DeKoninck.--[phi]úsló: the meaning of nature in the Aristotelian philosophy of nature, by S. O'F. Brennan.--Order in the philosophy of nature, by M. Glutz.--Motionless motions, by R.A. Kocourek.--Time, the measure of movement, by Sister M. Jocelyn.--Evolution and entropy, by V.E. Smith. (shrink)
The space CS(R) has a unique “Borel structure” in the following sense. Note that there is a natural mapping from R¥ onto CS(R}; namely, taking ranges. We can combine this with any Borel bijection from R onto R¥ in order to get a “preferred” surjection F:R ® CS(R).
Analysis of Emmy Noether’s 1918 theorems provides an illuminating method for testing the consequences of “coordinate generality”, and for exploring what else must be added to this requirement in order to give general covariance its far-reaching physical significance. The discussion takes us through Noether’s first and second theorems, and then a third related theorem due originally to F. Klein. Contact will also be made with the contributions of, principally, J.L. Anderson, A. Trautman, P.A.M. Dirac, R. Torretti and the father of (...) the whole business, A. Einstein (an apparent shift in Einstein’s thinking on the significance of general covariance between 1916 and 1918 is highlighted). (shrink)
S. Adams, W. Ambrose, A. Andretta, H. Becker, R. Camerlo, C. Champetier, J.P.R. Christensen, D.E. Cohen, A. Connes. C. Dellacherie, R. Dougherty, R.H. Farrell, F. Feldman, A. Furman, D. Gaboriau, S. Gao, V. Ya. Golodets, P. Hahn, P. de la Harpe, G. Hjorth, S. Jackson, S. Kahane, A.S. Kechris, A. Louveau,, R. Lyons, P.-A. Meyer, C.C. Moore, M.G. Nadkarni, C. Nebbia, A.L.T. Patterson, U. Krengel, A.J. Kuntz, J.-P. Serre, S.D. Sinel'shchikov, T. Slaman, Solecki, R. Spatzier, J. Steel, D. Sullivan, S. (...) Thomas, A. Valette, V.S. Varadarajan, B. Velickovic, B. Weiss, J.D.M. Wright, R.J. Zimmer. (shrink)
The purpose of this study was to empirically investigate the role of pluralistic ignorance in perceptions of unethical behavior. Buckley, Harvey, and Beu (2000) suggested that pluralistic ignorance plays a role such that individuals mistakenly believe that others are more unethical than they actually are. In two studies, we confirmed that pluralistic ignorance influences perceptions of ethics in a manner consistent with what Buckley et al. suggested. The implications of pluralistic ignorance in perceptions of ethics are discussed with suggestions (...) for how pluralistic ignorance might be reduced and how research in this area may be extended. (shrink)
Based on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable in R#. (...) The reason is interesting: if PA is slightly weakened to a subtheory P+, it admits the complex ring C as a model; thus QRF is chosen to be a theorem of PA but false in C. Inasmuch as all strictly positive theorems of R# are already theorems of P+, this nonconservativity result shows that QRF is also a nontheorem of R#. As a consequence, Ackermann's rule γ is inadmissible in R#. Accordingly, an extension of R# which retains its good features is desired. The system R##, got by adding an omega-rule, is such an extension. Central question: is there an effectively axiomatizable system intermediate between R# and R##, which does formalize arithmetic on relevant principles, but also admits γ in a natural way? (shrink)
Recent studies have characterized R. Hayyim of Volozhin (1749-1821) as actively preparing for the messianic redemption. Contrary to this, I maintain that R. Hayyim was concerned with the immediate potential of the world in the context of historical time (divre ha-yamim), rather than with a world-to-come, in the context of messianic time (aharit ha-yamim). To R. Hayyim, the existence of the cosmos and the stability of a dynamic life-giving universe are fraught with contingency because of the interdependent organic naturality shared (...) by man and world. This contingency is constant, placing man's conduct as the central determinant of the world's future. R. Hayyim's theory and practice attest to a concern for the immanent and transcendent aspects of this world and for the meanings of our everyday experienced relationship to its variegated pragmata. (shrink)
The subtle, almost ineffable, and ineffable cardinals were introduced in an unpublished 1971 manuscript of R. Jensen and K. Kunen, and a number of basic facts were proved there. These concepts were extended to that of k-subtle, k-almost ineffable, and k-ineffable cardinals in [Ba75], where a highly developed theory is presented.
Let k be a positive integer. There is a longest finite sequence x1,...,xn in k letters in which no consecutive block xi,...,x2i is a subsequence of any other consecutive block xj,...,x2j. Let n(k) be this longest length. We prove that n(1) = 3, n(2) = 11, and n(3) is incomprehensibly large. We give a lower bound for n(3) in terms of the familiar Ackerman hierarchy. We also give asymptotic upper and lower bounds for n(k). We view n(3) as a particularly (...) elemental description of an incomprehensibly large integer. Related problems involving binary sequences (two letters) are also addressed. We also report on some recent computer explorations of R. Dougherty which we use to raise the lower bound for n(3). (shrink)
We say that E is R-sparse if f(Ek) has no interior, for each k 2 N and f : Rk ! R de nable in R. (Throughout, \de nable" means \de nable without parameters".) In this note, we consider the extent to which basic metric and topological properties of subsets of R de nable in (R;E)# are determined by the corresponding properties of subsets of R de nable in (R;E), when R is an o-minimal expansion of (R;<;+;0;1) and E is (...) R-sparse. The precise statement of the main result is a bit complicated, but we can state some special cases now. (shrink)
An ideal in a commutative ring R with unit is a nonempty I Õ R such that for all x,y Œ I, z Œ R, we have x+y and xz Œ I. A set of generators for I is a subset of I such that I is the least ideal containing that subset.
Let R be an o-minimal expansion of (R. <. +) and (ϕk)k∈N be a sequence of positive real numbers such that limk→+∝ ƒ(ϕk)/ϕk+1 = 0 for every ƒ: R → R definable in R. (Such sequences always exist under some reasonable extra assumptions on R, in particular, if R is exponentially bounded or if the language is countable.) Then (R. (S)) is d-minimal, where S ranges over all subsets of cartesian powers of the range of ϕ.