In this essay, I provide an overview of the debate on infinite and essential idealizations in physics. I will first present two ostensible examples: phase transitions and the Aharonov– Bohm effect. Then, I will describe the literature on the topic as a debate between two positions: Essentialists claim that idealizations are essential or indispensable for scientific accounts of certain physical phenomena, while dispensabilists maintain that idealizations are dispensable from mature scientific theory. I will also identify some (...) attempts at finding a middle ground between the essentialists and dispensabilists camps. Finally, I will raise questions for future research on essential and infinite idealizations via the notion of exploration. (shrink)

We offer a framework for organizing the literature regarding the debates revolving around infinite idealizations in science, and a short summary of the contributions to this special issue.

The paper discusses the recent literature on abstraction/idealization in connection with the “paradox of infinite idealization.” We use the case of taking thermodynamics limit in dealing with the phenomena of phase transition and critical phenomena to broach the subject. We then argue that the method of infinite idealization is widely used in the practice of science, and not all uses of the method are the same. We then confront the compatibility problem of infinite idealization with scientific realism. (...) We propose and defend a contextualist position for realism and argue that the cases for infinite idealization appear to be fully compatible with contextual realism. (shrink)

1. Approximations of arbitrarily large but finite systems are often mistaken for infinite idealizations in statistical and thermal physics. The problem is illustrated by thermodynamically reversible processes. They are approximations of processes requiring arbitrarily long, but finite times to complete, not processes requiring an actual infinity of time.2. The present debate over whether phase transitions comprise a failure of reduction is confounded by a confusion of two senses of “level”: the molecular versus the thermodynamic level and the few (...) component versus the many component level. (shrink)

Williams and J. Fraser have recently argued that effective field theory methods enable scientific realists to make more reliable ontological commitments in quantum field theory than those commonly made. In this paper, I show that the interpretative relevance of these methods extends beyond the specific context of QFT by identifying common structural features shared by effective theories across physics. In particular, I argue that effective theories are best characterized by the fact that they contain intrinsic empirical limitations, and I extract (...) from their structure one central interpretative constraint for making more reliable ontological commitments in different subfields of physics. While this is in principle good news, this constraint still raises a challenge for scientific realists in some contexts, and I bring the point home by focusing on Williams’s and J. Fraser’s defense of selective realism in QFT. (shrink)

This paper argues against the view that the standard explanation of phase transitions in statistical mechanics may be considered a causal explanation, a distortion that can nevertheless successfully represent causal relations.

In this essay, I provide an overview of the debate on infinite and essential idealizations in physics. I will first present two ostensible examples: phase transitions and the Aharonov–Bohm effect. Then, I will describe the literature on the topic as a debate between two positions: Essentialists claim that idealizations are essential or indispensable for scientific accounts of certain physical phenomena, while dispensabilists maintain that idealizations are dispensable from mature scientific theory. I will also identify some attempts (...) at finding a middle ground between the essentialists and dispensabilists camps. Finally, I will raise questions for future research on essential and infinite idealizations via the notion of exploration. (shrink)

This contribution sheds light on the role of infinite idealization in structural analysis, by exploring how infinite elements and finite element methods are combined in civil engineering models. This combination, I claim, should be read in terms of a ‘complementarity function’ through which the representational ideal of completeness is reached in engineering model-building. Taking a cue from Weisberg’s definition of multiple-model idealization, I highlight how infinite idealizations are primarily meant to contribute to the prediction of structural (...) behavior in Multiphysics approaches. (shrink)

This contribution sheds light on the role of infinite idealization in structural analysis, by exploring how infinite elements and finite element methods are combined in civil engineering models. This combination, I claim, should be read in terms of a ‘complementarity function’ through which the representational ideal of completeness is reached in engineering model-building. Taking a cue from Weisberg’s definition of multiple-model idealization, I highlight how infinite idealizations are primarily meant to contribute to the prediction of structural (...) behavior in Multiphysics approaches. (shrink)

Thermodynamics and Statistical Mechanics are related to one another through the so-called "thermodynamic limit'' in which, roughly speaking the number of particles becomes infinite. At critical points (places of physical discontinuity) this limit fails to be regular. As a result, the "reduction'' of Thermodynamics to Statistical Mechanics fails to hold at such critical phases. This fact is key to understanding an argument due to Craig Callender to the effect that the thermodynamic limit leads to mistakes in Statistical Mechanics. I (...) discuss this argument and argue that the conclusion is misguided. In addition, I discuss an analogous example where a genuine physical discontinuity---the breaking of drops---requires the use of infinite idealizations. (shrink)

Infinite idealizations appear in our best scientific explanations of phase transitions. This is thought by some to be paradoxical. In this paper I connect the existing literature on the phase transition paradox to work on the concept of indispensability, which arises in discussions of realism and anti-realism within the philosophy of science and the philosophy of mathematics. I formulate a version of the phase transition paradox based on the idea that infinite idealizations are explanatorily indispensable to (...) our best science, and so ought to attract a realist attitude. I go on to offer a solution to the paradox by drawing a distinction between two types of indispensability: constructive and substantive indispensability. I argue that infinite idealizations are constructively indispensable to explanations of phase transitions, but not substantively indispensable. This helps to resolve the paradox, I maintain, since realist commitment tracks substantive, and not constructive, indispensability. (shrink)

The singularity arising from the violation of the Lipschitz condition in the simple Newtonian system proposed recently by Norton (2003) is so fragile as to be completely and irreparably destroyed by slightly relaxing certain (infinite) idealizations pertaining to elastic phenomena in this model. I demonstrate that this is also true for several other Lipschitz-indeterministic systems, which, unlike Norton's example, have no surface curvature singularities. As a result, indeterminism in these systems should rather be viewed as an artefact of (...) certain infinite idealizations essential for these models, depriving them of much of their intended metaphysical import. (shrink)

Amid a period of isolation and profound internal conflict, both in his life and in his thought, arises that which, according to Hölderlin, is “the general conflict in the human being”, namely the conflict between the “aspiration to limitation” and “the aspiration to the absolute”. The aim of this article is to analyze and, as much as possible, follow to its fullest extent, this fundamental thought: to see how it molds Hölderlin’s positions on existence and philosophy, how it meets the (...) most poignant philosophical concerns of the time and hence directly influences Hölderlin’s reading of Fichte, and how it leads the young poet to the notion of an infinite progression of philosophy, that is, to an infinite approximation to the ideal of knowledge. (shrink)

I analyze the role of infinite idealizations used in the renormalization group (RG hereafter) method in explaining universality across microscopically different physical systems in critical phenomena. I argue that despite the reference to infinite limit systems such as systems with infinite correlation lengths during the RG process, the key to explaining universality in critical phenomena need not involve infinite limit systems. I develop my argument by introducing what I regard as the explanatorily relevant property in (...) RG explanations: linearization* property; I then motivate and prove a proposition about the linearization* property in support of my view. As a result, infinite limit systems in RG explanations are dispensable. (shrink)

Idealization is ubiquitous in human cognition, and so is the inclination to be puzzled by it: what to make of ideal gas, infinitely large populations, homo economicus, perfectly just society, known to violate matters of fact? This is apparent in social science theorizing (from J. H. von Thünen, J. S. Mill, and Max Weber to Milton Friedman and Thomas Schelling), recent philosophy of science analyzing scientific modeling, and the debate over ideal and non-ideal theory in political philosophy (since John Rawls). (...) I will offer a set of concepts and principles to improve transparency about the precise contents of idealizations (in terms of negligibility, applicability, tractability, and early-step status) and their distinct functions (such as contributing to minimal modeling, benchmark modeling, and how-possibly modeling). (shrink)

Idealized scientific representations result from employing claims that we take to be false. It is not surprising, then, that idealizations are a prime example of allegedly inconsistent scientific representations. I argue that the claim that an idealization requires inconsistent beliefs is often incorrect and that it turns out that a more mathematical perspective allows us to understand how the idealization can be interpreted consistently. The main example discussed is the claim that models of ocean waves typically involve the false (...) assumption that the ocean is infinitely deep. While it is true that the variable associated with depth is often taken to infinity in the representation of ocean waves, I explain how this mathematical transformation of the original equations does not require the belief that the ocean being modeled is infinitely deep. More generally, as a mathematical representation is manipulated, some of its components are decoupled from their original physical interpretation. (shrink)

Vincent Ardourel discusses the eliminability of infinite limits in the explanations of phase transitions—an important point in the debate on the reducibility of thermodynamics to statistical mechanics. To this end, he examines alternative physical theories that deal with phase transitions in finite systems.

Classical accounts of intertheoretic reduction involve two pieces: first, the new terms of the higher-level theory must be definable from the terms of the lower-level theory, and second, the claims of the higher-level theory must be deducible from the lower-level theory along with these definitions. The status of each of these pieces becomes controversial when the alleged reduction involves an infinite limit, as in statistical mechanics. Can one define features of or deduce the behavior of an infinite idealized (...) system from a theory describing only finite systems? In this paper, I change the subject in order to consider the motivations behind the definability and deducibility requirements. The classical accounts of intertheoretic reduction are appealing because when the definability and deducibility requirements are satisfied there is a sense in which the reduced theory is forced upon us by the reducing theory and the reduced theory contains no more information or structure than the reducing theory. I will show that, likewise, there is a precise sense in which in statistical mechanics the properties of infinite limiting systems are forced upon us by the properties of finite systems, and the properties of infinite systems contain no information beyond the properties of finite systems. (shrink)

It has been suggested that the measuring apparatus used to measure quantum systems ought to be idealized as consisting of an infinite number of quantum systems. Let us call this the infinity assumption. The suggestion that we ought to make the infinity assumption has been made in connection with two closely related but distinct problems. One is the problem of determining the importance of the limitations on measurement incorporated into the Wigner-Araki-Yanase quantum theory of measurement. The other is the (...) measurement problem. These problems are not internal to the quantum mechanical formalism. They arise when we obtain particular formal results in applying the quantum mechanical formalism to measurement interactions. The formal results are problematic because they resist explanation. Various authors have claimed that by making the infinity assumption we can neglect the limitations on measurement. Bub (1988) and (1989) claims it allows us to solve the measurement problem. I will argue that both claims are unjustified. (shrink)

I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game‐theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and allowing transfinite game play. For each game, a secret codeword is hidden, which the codebreaker attempts to discover by making a series of guesses and receiving feedback as to their accuracy. In Wordle with words of any size from a finite alphabet of n letters, including (...) class='Hi'>infinite words or even uncountable words, the codebreaker can nevertheless always win in n steps. Meanwhile, the mastermind number, defined as the smallest winning set of guesses in infinite Mastermind for sequences of length ω over a countable set of colors without duplication, is uncountable, but the exact value turns out to be independent of, for it is provably equal to the eventually different number, which is the same as the covering number of the meager ideal. I thus place all the various mastermind numbers, defined for the natural variations of the game, into the hierarchy of cardinal characteristics of the continuum. (shrink)

It has been recently debated whether there exists a so-called “easy road” to nominalism. In this essay, I attempt to fill a lacuna in the debate by making a connection with the literature on infinite and infinitesimal idealization in science through an example from mathematical physics that has been largely ignored by philosophers. Specifically, by appealing to John Norton’s distinction between idealization and approximation, I argue that the phenomena of fractional quantum statistics bears negatively on Mary Leng’s proposed path (...) to easy road nominalism, thereby partially defending Mark Colyvan’s claim that there is no easy road to nominalism. (shrink)

When it comes to specifying the moral duties we bear towards future generations, most political philosophers position themselves on what could be regarded as a safe ground. A variant of the Lockean proviso is commonplace in the literature on intergenerational justice, taking the form of an obligation to bestow upon future people a minimum of goods necessary for reaching a certain threshold of well-being (Meyer, 2017). Furthermore, even this minimum is often frowned upon, given the non-identity problem and the challenges (...) this presents to the topic of justice between generations. Additional issues are raised at the level of non-ideal theory, the most significant being the problem of non-compliance (Gosseries and Meyer, 2009).In this paper I intend to probe the limits of “practical political possibility” (Rawls 1999), by inquiring whether embracing the sufficiency view (Frankfurt, 1987; Crisp, 2003; Benbaji, 2005) as a distributive pattern and capabilities as a metric can lead to more burdensome obligations for present generations. More specifically, I try to show that we have a duty to invest in research that aims at prolonging the lifespan of humans (the idea can already be found in the sufficientarian literature, for instance in Farrelly, 2007). Moreover, given the Earth’s limited resources, we ought to encourage the terraforming of other planets in order to make them inhabitable for (future) people.I argue that these two seemingly far-fetched projects are in fact worthwhile goals to pursue on the one hand, and moral obligations on the other hand. Nonetheless, they are not the only ones we ought to take on; for instance, we must simultaneously pursue them and try to improve the prospects of those who fall under a sufficiency threshold here and now. That is, specifying these (prima facie) duties towards future generations is connected with stronger obligations towards the current generation.Towards the end of the paper I engage in a discussion regarding the role of the feasibility constraint in a theory of justice, as rationales pertaining to feasibility are perhaps going to be the most recurrent criticisms raised against my proposal. To that end, I defend limitarian policies, which aim at setting an upper limit to how much money individuals are allowed to possess (Robeyns, 2017; Volacu and Dumitru, 2019). (shrink)

Elekes proved that any infinite-fold cover of a σ-finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I-almost everywhere infinite-fold covers of Polish spaces where I is a σ-ideal on the space and the set (...) of indices of the required subsequence should be in a fixed ideal ${{\mathcal{J}}}$ on ω. We introduce the notion of the ${{\mathcal{J}}}$ -covering property of a pair ${({\mathcal{A}}, I)}$ where ${{\mathcal{A}}}$ is a σ-algebra on a set X and ${{I \subseteq \mathcal{P}(X)}}$ is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal ${\mathcal{N}}$ and the meager ideal ${\mathcal{M}}$ . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals ${{\mathcal{J}}}$ on ω such that ${\mathcal{M}}$ has the ${{\mathcal{J}}}$ -covering property consists exactly of non weak Q-ideals. We also study the existence of smallest elements, with respect to Katětov–Blass order, in the family of those ideals ${\mathcal{J}}$ on ω such that ${\mathcal{N}}$ or ${\mathcal{M}}$ has the ${\mathcal{J}}$ -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails. (shrink)

The goal of this paper is to provide a detailed reading of John Venn's Logic of Chance as a work of logic or, more specifically, as a specific portion of the general system of so-called ‘material’ logic developed in his Principles of Empirical or Inductive Logic and to discuss it against the background of his Boolean-inspired views on the connection between logic and mathematics. It is by means of this situating of Venn 1866 [The Logic of Chance. An Essay on (...) the Foundations and Province of the Theory of Probability. With Especial Reference to Its Application to Moral and Social Science, London: Macmillan] within the entirety of his oeuvre that it becomes both possible to revisit and necessary to re-articulate its place in the history of the frequency interpretation of probability. For it is clear that if Venn's approach to logic not only allowed him to establish its foundations on the basis of a process of idealization and define it as consisting of so-called hypothetical infinite series, bu.. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:THE ACTUAL INFINITE IN ARISTOTLE Prolegomena: Philosophy and Theology Related HENEVER PHILOSOPHY is taken to be the handmaiden of theology, then the autonomy of reason is destroyed." Such a daim should be distinguished from a still 1stronger thesis. Compare: " A philosopher may not legitimately try to fortify an argument by bringing in new premises from another discipline which has a special aura of authority." Quite how Aristotle (...) would have reacted to the first generalization, if it had been suitably translated for his scrutiny, is a question on which commentators might agree to differ. But Aristotle would have been as!'eluctant as Aquinas, or Maimonides, or later Monotheists who treasured the Physics, to leave the second general claim unquestioned. Natural and revealed theology, like physics or mathematics, are domains where extremely subtle reasoner1s have studied and published widely for a very long time. So an elementary respect for induction suggests that philosophers should e:x;pect to benefit from,applying some premises and some ways of arguing, which are found in theology, to their own issues. I speak of premises and ways of arguing that philosophers qua philosophers have not provided before. Thinkers in all four disciplines could thus profit from each body of work. Even the law of averages is set pitilessly against the second general claim.1 1 The late Father F. C. Copleston, S.J., covers many important topics regarding theology and philosophy in Aquinets (Baltimore: Peng11in, 1955), pp. 12-14, 19, 54-63, 65-68, 74, 116, etc. But the point put forward here about human power to strengthen intuition in one field seems to be partly missed. Copleston also fails to dwell enough on infinity in various passages. See John King-Farlow, "The First ·way in Physical and Moral Space", The Thomist 427 428 JOHN KING-FARLOW By focussing on some arguments of Aristotle's against the admissibility of an,actual infinite, we shall find deeper, less obvims reasons for rejecting that second claim. Other philosophical points of intrinsic value should emerge as well foom the short investigation. But this one is especially interesting because of its curious pertinence to perennial disagreements among those who address the relations of theology, theism and philosophy. At any rate, the search for such deeper reasons will involve the application to Aristotle's texts of a somewhat modern way of using the term " intuition ". Introduotion: A Relevant Kind of Intuitive Thinking Diss,atisfaction with modern Platonism in higher mathematics may be added to disenchantment with Platonism's modern opponents. Such opposition may seem too technical for issues of intuition or, indeed, of common reason. And so one is led back to consider certain rightly famous arguments of Aristotle. But before turning to Aristotle and his grapplings with ideais and claims about an actual infinity, let us 1ook at a few very ordinary uses of the sometimes charismatic word "intuition." Webster's Dictionary (1978) offers: "immediate and instinctive perception of a truth; direct understanding without reasoning." In his classic study Evolution of JJ;Jathematieal Concepts, Wilder writes at least once in the spirit of Webster: "Counting with fingers (and toes) clearly involves intuitive recognition of (1-1)-correspontreme, excessive gene:msity, and, next, irion unwillingness to give ·anything to others, lies usually much closer to the first extreme. Between utter recklessness and self-paralyzing caution the Mean may oscillate in relative distance, according to the demands of the context. Near recklessness may offer the only means to win one kind of battle and a stubborn refusal to move may be needed for the nex>t victory. By pursuing such a nonmechanical and variably positioned ideal of a Mean, a really wise father usually oould show special favor to one specially deserving child in some ways and some contexts, yet generally show very satisfying love to a;ll his children. 11 Richard Bosley, " What Is a Mean? The Question Considered Compara· tively and Systematically." PhiloB. (shrink)

It is well known that Leibniz advocated the actual infinite, but that he did not admit infinite collections or infinite numbers. But his assimilation of this account to the scholastic notion of the syncategorematic infinite has given rise to controversy. A common interpretation is that in mathematics Leibniz’s syncategorematic infinite is identical with the Aristotelian potential infinite, so that it applies only to ideal entities, and is therefore distinct from the actual infinite that (...) applies to the actual world. Against this, I argue in this paper that Leibniz’s actual infinite, understood syncategorematically, applies to any entities that are actually infinite in multitude, whether numbers, actual parts of matter, or monads. It signifies that there are more of them than can be assigned a number, but that there is no infinite number or collection of them, which notion involves a contradiction. Similarly, to say that a magnitude is actually infinitely small in the syncategorematic sense is to say that no matter how small a magnitude one takes, there is a smaller, but there are no actual infinitesimals. In geometry one may calculate with expressions apparently denoting such entities, on the understanding that they are fictions, standing for variable magnitudes that can be made arbitrarily small, so as to produce demonstrations that there is no error in the resulting expressions. (shrink)

Let K be an infinite Galois extension of the rationals such that every finite subextension has odd degree over the rationals and its prime ideals dividing 2 are unramified. We show that its ring of integers is first-order definable in K. As an application we prove that equation image together with all its Galois subextensions are undecidable, where Δ is the set of all the prime integers which are congruent to −1 modulo 4.

We argue that C. Darwin and more recently W. Hennig worked at times under the simplifying assumption of an eternal biosphere. So motivated, we explicitly consider the consequences which follow mathematically from this assumption, and the infinite graphs it leads to. This assumption admits certain clusters of organisms which have some ideal theoretical properties of species, shining some light onto the species problem. We prove a dualization of a law of T.A. Knight and C. Darwin, and sketch a decomposition (...) result involving the internodons of D. Kornet, J. Metz and H. Schellinx. A further goal of this paper is to respond to B. Sturmfels’ question, “Can biology lead to new theorems?”. (shrink)

This compact booklet addresses informal logical aspects of infinite regress arguments. We know what infinite regress arguments are from such examples as Plato’s Third Man problem. It is presented here for tradition sake in its original formulation, where for convenience ‘man’ does duty for ‘human being’. Plato’s theory of abstract Ideas or Forms, in order to explain how it is that Phaedo and Meno are both men, posits their belonging to, participating in or falling under a higher ideal (...) abstract universal Man. The theory thereby takes the first irreversible step toward an infinite regress of ideal abstract universal men of increasingly higher order, beginning with the third MAN, for which there seems to be no legitimating explanatory rationale. A further example, to have several on hand, is the infinite regress that occurs on the assumption that every occurrent event has a temporally prior cause, or, simply, that every event has a cause that is also an event. The logical structure, class .. (shrink)

Two approaches to understanding the idealizations that arise in the Aharonov–Bohm effect are presented. It is argued that a common topological approach, which takes the non-simply connected electron configuration space to be an essential element in the explanation and understanding of the effect, is flawed. An alternative approach is outlined. Consequently, it is shown that the existence and uniqueness of self-adjoint extensions of symmetric operators in quantum mechanics have important implications for philosophical issues. Also, the alleged indispensable explanatory role (...) of said idealizations is examined via a minimal model explanatory scheme. Last, the idealizations involved in the AB effect are placed in a wider philosophical context via a short survey of part of the literature on infinite and essential idealizations. (shrink)

We introduce a notion of ideal type such that any two ideals with the same ideal type are isomorphic. From this we infer, under the axiom t = h, that each ideal which consists of all nowhere Ramsey sets contained in some family of infinite subsets of natural numbers is isomorphic with the ideal of all nowhere Ramsey sets.

The nature of of Infinite Number is discussed in a rigorous but easy-to-follow manner. Special attention is paid to Cantor's proof that any given set has more subsets than members, and it is discussed how this fact bears on the question: How many infinite numbers are there? This work is ideal for people with little or no background in set theory who would like an introduction to the mathematics of the infinite.

Preliminary words One important feature of Poincaré's conventionalism of geometry is linked to the relation between the abstract notion of space geometry and the representations of the free mobility of our bodies. In this sense «the group of rigid motions» identified by Helmholtz and Lie as the foundation of geometries of constant curvature is, according to Poincaré, an idealization of the primitive experience that acquaints us with the properties of space in the first place. 2 Furthermore, since Poincaré thinks that (...) the only adequate candidates for physical geometry – the geometries of constant curvature – are equivalent from a mathematical point of view (they are homeomorphic), we can choose any one of these. Actually, according to Poincaré, the idealization involved in passing from our own local motions to the group of motions makes use of temporal intuition: our notion of the large-space structure of space assumes that our displacements are infinitely iterables and this assumes temporal intuition. 3 Thus, it looks as Poincaré's notion of convention assumes some kind of idealization process. Indeed, the notion of convention involved in this cases seems to assume that in relation to a target system the choice between two idealizations is determined by some kind of equivalence relation, which in our case is displayed 1. (shrink)

Calhoun, W.C., Incomparable prime ideals of recursively enumerable degrees, Annals of Pure and Applied Logic 63 39–56. We show that there is a countably infinite antichain of prime ideals of recursively enumerable degrees. This solves a generalized form of Post's problem.

Interpreters have found it exceedingly difficult to understand how Hume could be right in claiming that his two definitions of ‘cause’ are essentially the same. As J. A. Robinson points out, the definitions do not even seem to be extensionally equivalent. Don Garrett offers an influential solution to this interpretative problem, one that attributes to Hume the reliance on an ideal observer. I argue that the theoretical need for an ideal observer stems from an idealized concept of definition, which many (...) interpreters, including Garrett, attribute to Hume. I argue that this idealized concept of definition indeed demands an unlimited or infinite ideal observer. But there is substantial textual evidence indicating that Hume disallows the employment of idealizations in general in the sciences. Thus Hume would reject the idealized concept of definition and its corresponding ideal observer. I then put forward an expert-relative reading of Hume’s definitions of ‘cause’, which also renders both definitions extensionally equivalent. On the expert-relative reading, the meaning of ‘cause’ changes with better observations and experiments, but it also allows Humean definitions to play important roles within our normative practices. Finally, I consider and reject Henry Allison’s argument that idealized definitions and their corresponding infinite minds are necessary for expert reflection on the limitations of current science. (shrink)

The notion of an ideal reasoner has several uses in epistemology. Often, ideal reasoners are used as a parameter of (maximum) rationality for finite reasoners (e.g. humans). However, the notion of an ideal reasoner is normally construed in such a high degree of idealization (e.g. infinite/unbounded memory) that this use is unadvised. In this dissertation, I investigate the conditions under which an ideal reasoner may be used as a parameter of rationality for finite reasoners. In addition, I present and (...) justify the research program of computational epistemology, which investigates the parameter of maximum rationality for finite reasoners using computer simulations. (shrink)

We describe the countably saturated models and prime models of the theory Thprin of Boolean algebras with a principal ideal, the theory Thmax of Boolean algebras with a maximal ideal, the theory Thac of atomic Boolean algebras with an ideal such that the supremum of the ideal exists, and the theory Thsa of atomless Boolean algebras with an ideal such that the supremum of the ideal exists. We prove that there are infinitely many completions of the theory of Boolean algebras (...) with a distinguished ideal that do not have a countably saturated model. Also, we give a sufficient condition for a model of the theory TX of Boolean algebras with distinguished ideals to be elementarily equivalent to a countably saturated model of TX. (shrink)

We show that if κ is an infinite successor cardinal, and λ > κ a cardinal of cofinality less than κ satisfying certain conditions, then no ideal on Pκ is weakly λ+-saturated. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

In the Transcendental Ideal Kant discusses the principle of complete determination: for every object and every predicate A, the object is either determinately A or not-A. He claims this principle is synthetic, but it appears to follow from the principle of excluded middle, which is analytic. He also makes a puzzling claim in support of its syntheticity: that it represents individual objects as deriving their possibility from the whole of possibility. This raises a puzzle about why Kant regarded it as (...) synthetic, and what his explanatory claim means. I argue that the principle of complete determination does not follow from the principle of excluded middle because the externally negated or ?negative? judgement ?Not (S is P)? does not entail the internally negated or ?infinite? judgement ?S is not-P.? Kant's puzzling explanatory claim means that empirical objects are determined by the content of the totality of experience. This entails that empirical objects are completely determinate if and only if the totality of experience has a completely determinate content. I argue that it is not a priori whether experience has such a completely determinate content and thus not analytic that objects obey the principle of complete determination. (shrink)

The topic of the ideal, that is, the topic of the possible or impossible human attainment of the absolute is ascribed divergent treatments throughout Kant’s work. Namely, it is either promptly accepted as possible by the critical Kant, and seen as something attainable by a means other than an infinite approximation (which would indeed imply a violation of autonomy, but denies the genuineness of the ideal), or it is rejected as impossible by the non-critical Kant, that is, it is (...) seen as something attainable only through an infinite approximation (which would involve an unconditional acceptance of heteronomy, but safeguards the authenticity of an aspiration to the ideal). Yet, the topic of the ideal receives a new, if not conciliatory, at least mutually explanatory approach in Kant’s Anthropology. Here – such is our proposition – Kant proposes a terminus medius between both conceptions of ideal, insofar as he is led to ponder on the mutual benefits of an autonomic possibility and an heteronomic impossibility of an infinite progression in thought; something which Kant proposes under the form of an almost-infinite, or an almost perennial, yet finite duration, to be endured until the attainment of an almost unreachable, yet indeed reachable practical ideal. A terminus medius which, we hope to prove, is none other than that at the root of Kant’s proposition of Pragmatic Anthropology as a mediating science in Kant’s fundamental scheme of human knowledges, and which therefore may be ultimately seen as the embodiment of Kant’s anthropo-cosmological, or indeed cosmopolitical dimension of thought, as expressed in Kant’s political and/or historical writings. (shrink)

By B2 we denote the σ-ideal of all subsets A of the Cantor set {0,1}ω such that for every infinite subset T of ω the restriction A∣{0,1}T is a proper subset of {0,1}T. In this paper we investigate set theoretical properties of this and similar ideals.

Using previous work of Baumgartner, Shelah and others, we describe, for each infinite cardinal θκ, the smallest κ-normal ideal J on Pκ such that.

Some time ago I wrote a paper about conceivability and knowledge. An anonymous referee rejected it on the grounds that the result had already been established in a short story by Jorge Luis Borges. Intrigued, I looked for the story but found no mention of it in Louis and Ziche’s extensive bibliography. I spent months consulting archives and electronic records to no avail. I had begun to doubt whether the story even existed when I had the curious good luck to (...) encounter Sir Thomas Browne of Pembroke College, Oxford, who assured me of the piece’s authenticity and introduced me to Brother Christian Rosenkreuz of Invisible College who in turn generously put me into correspondence with Borges himself. The literary defects in what follows reflect not a diminution of Borges’s undoubted power as a storyteller, but merely my limited ability as an amateur translator. Whatever the translation’s literary faults may be, I hope the story’s philosophical interest—i.e., an argument that ideal conceivers have higher-order knowledge of necessary truths—remains. [Trans.]. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 62.3 (2001) 483-503 [Access article in PDF] China and the Ideal of Order in John Webb's An Historical Essay.... Rachel Ramsey Scholars of seventeenth-century intellectual history have generally relegated John Webb to the footnotes of their work on universal language schemes, architectural history, and Sino-European relations. 1 In this essay I suggest that Webb's An Historical Essay Endeavoring a Probability that the Language (...) of the Empire of China is the Primitive Language (1669), which argues that for 5000 years China preserved the language spoken by Adam and Eve in the Garden of Eden, makes significant contributions to seventeenth-century intellectual history and to our understanding of early modern European perceptions of China. When Webb's Essay is placed within the context of the seventeenth-century debates about the "primitive" language of Eden, China's ancient history, and the idealization of that empire's harmonious and prosperous culture, its deeply political nature becomes apparent.The Jesuit accounts and those of other travelers to China were especially appealing to English readers living in the aftermath of the civil wars and the turmoil of the early years of the Restoration because they offered glimpses of a seemingly ideal state ruled by a stable monarchy and blessed with seemingly infinite resources and unimaginable wealth; however, these accounts included evidence that China had maintained an unbroken historical record that antedated the Flood, thus implicitly challenging the veracity of the Old Testament account of Noah and the Universal Deluge. Webb's Essay solves the problem posed by the Jesuit accounts of China's ancient history by placing them within a revised biblical narrative; this solution makes it possible for him to claim that China's [End Page 483] exemplary status is a result of its possession of the primitive language and its socio-political virtues are the consequence of its Noachian origins. The revised biblical narrative not only made Webb's encomium to that country more acceptable to a bible-reading public and protestant king but offered a politically-safe way for him to criticize the restored English monarchy by contrasting it to an idealized account of China's social and political state. More specifically, the circumstances of Webb's years of service in the Office of the King's Works help us further understand what attracted a seventeenth-century architect to the Jesuit accounts of China and what motivated him to write the first English treatise on the Chinese language. Webb's political career allows us to read AnHistoricalEssay as a politico-theological justification for a reasoned critique of the patronage system which Webb held accountable for his thwarted career ambitions. Webb's Career and the Politics of the Restoration John Webb lived through some of England's most turbulent history and witnessed firsthand the effects of monarchical instability and regicide, the uncertainties of the Commonwealth, and the shortcomings of the Restoration. Born to a Somerset family in 1611 and educated at the Merchant Taylors' school, Webb became Inigo Jones's pupil in 1628 and served as his "Clerk Engrosser" from 1633 to 1641 in the Office of the King's Works, where Jones was Surveyor. During the early years of the English revolution Webb served as his mentor's deputy in London after Jones fled north with Charles I, but he was dismissed from his post in 1643 following an "accusation from one Mr. Carter to the Committee of the Revenue, that the said Mr. Jones was at Oxford." 2 During his absence from the Office of the Works, Webb supposedly sent detailed plans of London's fortifications and smuggled jewelry to the king, for which he was briefly imprisoned. 3 After Charles I's execution in 1649, Webb worked on several country houses, including those of the Earl of Rutland at Belvoir, the Earl of Peterborough at Drayton, and Sir Justinian Isham at Lamport but he was firmly excluded from holding any political post as a former consort of the martyred King. 4 He held an appointment briefly... (shrink)

When McTaggart puts Spinoza on his short list of philosophers who considered time unreal, he is falling in line with a reading of Spinoza’s philosophy of time advanced by contemporaneous British Idealists and by Hegel. The idealists understood that there is much at stake concerning the ontological status of Spinozistic time. If time is essential to motion then temporal idealism entails that nearly everything—apart from God conceived sub specie aeternitatis—is imaginary. I argue that although time is indeed ‘imaginary’—in a sense (...) ‘no one doubts’ as Spinoza says—there is no good reason to infer that bodies, the infinite modes, and conatus are imaginary in the same sense. To avoid this conflation, we need to follow Spinoza in carefully distinguishing between tempus and duratio. Duration is not only real; it has all the structure needed to ground Spinozistic motion, bodies and conatus. (shrink)

Let ${{\mathcal J}\,(\mathbb M^2)}$ denote the σ-ideal associated with two-dimensional Miller forcing. We show that it is relatively consistent with ZFC that the additivity of ${{\mathcal J}\,(\mathbb M^2)}$ is bigger than the covering number of the ideal of the meager subsets of ω ω. We also show that Martin’s Axiom implies that the additivity of ${{\mathcal J}\,(\mathbb M^2)}$ is 2 ω .Finally we prove that there are no analytic infinite maximal antichains in any finite product of ${\mathfrak{P}{(\omega)}/{\rm fin}}$.

Pascal described human beings as ‘thinking reeds’, weak in flesh but magnificent in mind. While it is a poetic image, it is also an ambivalent one and may suggest an inappropriately dualist view of human nature. It is important to realise that not only are we thinking reeds but that we are thinking because we are reeds. In fact, rationality is reed-like itself, very much of a kind with the rest of human nature. It is now more than two and (...) half centuries since David Hume first pointed out the lack of an argument that would fully justify claims about matters of fact. Being neither made evident by our observations nor arising out of the mere considera-tion of relations of ideas, claims such as that the turkey will be fed dinner tomo-rrow – rather than being had for dinner have remained problematic ever since. Many attempts have been made to show that something of the beauty and certainty of reasoning about relations of ideas could be recaptured in our dealings with matters of fact, but all attempts have remained mere shadows of what we tried to grasp. Hume’s argument stands. An infinite being might watch countless sun-sets and yet should witness each new sun-rise with surprise, always withholding its judgement regarding what will follow. (shrink)

A family $$\mathcal {I}$$ I of subsets of a set X is an ideal on X if it is closed under taking subsets and finite unions of its elements. An ideal $$\mathcal {I}$$ I on X is below an ideal $$\mathcal {J}$$ J on Y in the Katětov order if there is a function $$f{: }Y\rightarrow X$$ f : Y → X such that $$f^{-1}[A]\in \mathcal {J}$$ f - 1 [ A ] ∈ J for every $$A\in \mathcal {I}$$ A (...) ∈ I. We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katětov order, where The Ramsey ideal consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph), The Hindman ideal consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory), The summable ideal consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent. (shrink)