We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New V yields a system (...) equivalent to full Zermelo-Fraenkel set theory with a principle of global choice. This advances Boolos's longstanding interest in the foundations of set theory. (shrink)

Idealization and abstraction are central concepts in the philosophy of science and in science itself. My goal in this paper is suggest an account of these concepts, building on and refining an existing view due to Jones Idealization XII: correcting the model. Idealization and abstraction in the sciences, vol 86. Rodopi, Amsterdam, pp 173–217, 2005) and Godfrey-Smith Mapping the future of biology: evolving concepts and theories. Springer, Berlin, 2009). On this line of thought, abstraction—which I call, for (...) reasons to be explained, abstractness—involves the omission of detail, whereas idealization consists in a deliberate mismatch between a description and the world. I will suggest that while the core idea underlying these authors’ view is correct, they make several assumptions and stipulations that are best avoided. For one thing, they tie abstractness too close to truth. For another, they do not allow sufficient room to the difference between idealization and error. Taking these points into account leads to a refined account of the distinction, in which abstractness is seen in terms of relative richness of detail, and idealization is seen as closely connected with the knowledge and intentions of idealizers. I lay out these accounts in turn, and then discuss the relationship between the two concepts, and several other upshots of the present way of construing the distinction. (shrink)

INTRODUCTION! In almost every area of contemporary philosophy the impact of Franz Brentano or his pupils can be detected. ...

A co-authored article with Roy T. Cook forthcoming in a special edition on the Caesar Problem of the journal Dialectica. We argue against the appeal to equivalence classes in resolving the Caesar Problem.

The use of “levels of abstraction” in philosophical analysis (levelism) has recently come under attack. In this paper, I argue that a refined version of epistemological levelism should be retained as a fundamental method, called the method of levels of abstraction. After a brief introduction, in section “Some Definitions and Preliminary Examples” the nature and applicability of the epistemological method of levels of abstraction is clarified. In section “A Classic Application of the Method ofion”, the philosophical fruitfulness (...) of the new method is shown by using Kant’s classic discussion of the “antinomies of pure reason” as an example. In section “The Philosophy of the Method of Abstraction”, the method is further specified and supported by distinguishing it from three other forms of “levelism”: (i) levels of organisation; (ii) levels of explanation and (iii) conceptual schemes. In that context, the problems of relativism and antirealism are also briefly addressed. The conclusion discusses some of the work that lies ahead, two potential limitations of the method and some results that have already been obtained by applying the method to some long-standing philosophical problems. (shrink)

The logical status of abstraction principles, and especially Hume’s Principle, has been long debated, but the best currently availeble tool for explicating a notion’s logical character—permutation invariance—has not received a lot of attention in this debate. This paper aims to fill this gap. After characterizing abstraction principles as particular mappings from the subsets of a domain into that domain and exploring some of their properties, the paper introduces several distinct notions of permutation invariance for such principles, assessing the (...) philosophical significance of each. (shrink)

On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek (...) to explain one way in which it may be extended to encompass the theory of real numbers, introducing the reals, by means of suitable further abstraction principles, as ratios of quantities. (shrink)

ion is arguably one of the most important methods in modern science in analysing and understanding complex phenomena. In his book The Philosophy of Information, Floridi (The philosophy of information. Oxford University Press, Oxford, 2011) presents the method of levels of abstraction as the main method of the Philosophy of Information. His discussion of abstraction as a method seems inspired by the formal methods and frameworks of computer science, in which abstraction is operationalised extensively in programming languages (...) and design methodologies. Is it really clear what we should understand by levels of abstraction? How should they be specified? We will argue that levels of abstraction should be augmented with annotations, in order to express semantic information for them and reconcile the method of level of abstraction (LoA’s) with other approaches. We discuss the extended method when applied e.g. to the analysis of abstract machines. This will lead to an example in which the number of LoA’s is unbounded. (shrink)

Focusing especially on Science and the Modern World, this article explores Whitehead's understanding of the social contexts and repercussions of mathematical and scientific abstraction. It investigates his remarks on the need to offset pernicious practices of abstraction in the context of a renewed concern with the link between conceptuality and materiality in social theory. Whitehead's inquiry into the problematic legacy of Galileo and scientific materialism is then contrasted with a different diagnosis of the abstractive maladies of modern society, (...) the one put forward under the Marxist rubric of `real abstraction'. While both stances allow us to explore the social repercussions of abstractive practices, it is argued that an understanding of the practically abstract character of capitalism permits us to identify the limits of Whitehead's pedagogical wish to reform our culture of abstraction. (shrink)

One of the main problems plaguing neo-logicism is the Bad Company challenge: the need for a well-motivated account of which abstraction principles provide legitimate definitions of mathematical concepts. In this article a solution to the Bad Company challenge is provided, based on the idea that definitions ought to be conservative. Although the standard formulation of conservativeness is not sufficient for acceptability, since there are conservative but pairwise incompatible abstraction principles, a stronger conservativeness condition is sufficient: that the class (...) of acceptable abstraction principles be strictly logically symmetrically class conservative . The article concludes with an examination of which classes of abstraction principles meet this criteria. (shrink)

One of the largest ongoing debates in scholarship on Avicenna concerns his epistemology of the first acquisition of intelligible forms or concepts. “Emanationists” hold that intelligibles are emanated by the separate Active Intellect directly into human minds. “ionists” hold that intelligibles are abstracted by the human intellect from sensory images. Neither of these positions has a satisfactory grip on Avicenna’s philosophy. I propose that the two positions can be reconciled because Avicenna states in many texts that what the AI emanates (...) is a power, and not the various intelligible forms. I argue that this can only be the power of abstraction itself. This new interpretation does greater justice to Avicenna’s system and reveals his unique place in the history of epistemology. (shrink)

While Frege’s own attempt to provide a purely logical foundation for arithmetic failed, Hume’s principle suffices as a foundation for elementary arithmetic. It is known that the resulting system is consistent—or at least if second-order arithmetic is. Some philosophers deny that HP can be regarded as either a truth of logic or as analytic in any reasonable sense. Others—like Crispin Wright and I—take the opposed view. Rather than defend our claim that HP is a conceptual truth about numbers, I explain (...) one way it may be possible to extend our view beyond elementary arithmetic to encompass the theory of real numbers. My approach has affinities to the leading ideas of Frege’s own treatment of the reals, although differing in one fundamental way. I attempt, like the HP approach to elementary arithmetic, to obtain the reals very directly by means of abstraction principles without any essential reliance on a theory of sets. This is the most natural way of extending the neo-Fregean position to the reals. (shrink)

The international debate on the ethics and legality of autonomous weapon systems (AWS), along with the call for a ban, primarily focus on the nebulous concept of fully autonomous AWS. These are AWS capable of target selection and engagement absent human supervision or control. This paper argues that such a conception of autonomy is divorced from both military planning and decision-making operations; it also ignores the design requirements that govern AWS engineering and the subsequent tracking and tracing of moral responsibility. (...) To show how military operations can be coupled with design ethics, this paper marries two different kinds of meaningful human control (MHC) termed levels of abstraction. Under this two-tiered understanding of MHC, the contentious notion of ‘full’ autonomy becomes unproblematic. (shrink)

We argue against theory-of-mind interpretation of recent false-belief experiments with young infants and explore two other interpretations: enactive and behavioral abstraction approaches. We then discuss the differences between these alternatives.

The concept of the environment appears to be indispensably involved in adaptive explanation. Quite what its role is, however, is a matter of some dispute. The environment is customarily viewed as the dual of the organism; a wholly external, discrete, autonomous cause of evolution. On this view, the external environment is the principal cause of the adaptedness of form, and the determinant of what it is to be an adaptation. I argue that this conception of the environment neither adequately explains (...) nor individuates evolutionary adaptations. Instead, adaptation, properly construed, is an evolutionary response to affordances. The environment, traditionally construed, underdetermines an organism’s affordances. Instead, I argue that the environment takes its place in evolutionary models not as a discrete causal entity, but as an abstraction. (shrink)

Aquinas’s process of abstraction of the particular thing into a universal concept is of pivotal importance for grounding his philosophy and theology in a natural framework. Much has been said and written regarding Aquinas’s doctrine of abstraction, yet recent studies still consider it to be ‘nothing more than a kind of magic.’ This problematic claim is not without foundation, for in trying to understand exactly how this process works, we are constantly faced with an unbridgeable abyss and the (...) repeated vague explanations made by Aquinas. The plain truth is that Aquinas explains what abstraction ought to do and yet, most of the time, he does not explain how it is to be done. This paper intends to show that although Aquinas does not present us with a mechanism for the theory of abstraction, we are nevertheless able to construct a viable mechanism which accords with Aquinas’s guidelines. The aim of constructing a working mechanism that corresponds to what Aquinas demands of the process of abstraction is twofold: First, it attempts to extricate Aquinas’s doctrine of abstraction from the claims of being a quasi-natural doctrine. Second, a viable process of abstraction which is derived from Aquinian-Aristotelian sources can approach contemporary cognitive problems from a fresh point of view. (shrink)

We may define words or concepts, and we may also, as Aristotle and others have thought, define the things for which words stand and of which concepts are concepts. Definitions of words or concepts may be explicit or implicit, and may seek to report preexisting synonymies, as Quine put it, but they may instead be wholly or partly stipulative. Definition by abstraction, of which Hume’s principle is a much discussed example, seek to define a term-forming operator, such as the (...) number operator, by fixing the truth-conditions of identity-statements featuring terms formed by means of that operator. Such definitions are a species of implicit definition. They are typically at least partly stipulative. Definitions of things, or real definitions, are, by contrast, typically conceived as true or false statements about the nature or essence of their definienda, and so not stipulative. There thus appears to be an obvious and head-on clash between taking Hume’s principle as an implicit and at least partly stipultative definition of the number operator and taking it as a real definition, stating the nature or essence of cardinal numbers. This paper argues that this apparent tension can be resolved, and that resolving it sheds light on part of the epistemology or essence and necessity, showing how some of our knowledge of essence and necessity can be a priori. (shrink)

This is a technical supplement to "Abstraction and Grounding", forthcoming in /Philosophy and Public Affairs/.

I consider three explanatory strategies from recent systems biology that are driven by mathematics as much as mechanistic detail. Analysis of differential equations drives the first strategy; topological analysis of network motifs drives the second; mathematical theorems from control engineering drive the third. I also distinguish three abstraction types: aggregations, which simplify by condensing information; generalizations, which simplify by generalizing information; and structurations, which simplify by contextualizing information. Using a common explanandum as reference point—namely, the robust perfect adaptation of (...) chemotaxis in Escherichia coli—I argue that each strategy invokes a different combination of abstraction types and that each targets its abstractions to different mechanistic details. (shrink)

Kit Fine’s book is a study of abstraction in a quite precise sense which derives from Frege. In his Grundlagen, Frege contemplates defining the concept of number by means of what has come to be called Hume’s principle—the principle that the number of Fs is the same as the number of Gs just in case there is a one-to-one correspondence between the Fs and the Gs. Frege’s discussion is largely conducted in terms of another, similar but in some respects (...) simpler, proposal—to define the concept of direction by laying down the principle that the direction of a line a is the same as that of line b if and only if lines a and b are parallel. Such principles have come to be known as abstraction principles. More generally, abstraction principles are ones of the shape: ∀α∀β ↔ α ≈ β), where ≈ is an equivalence relation on entities of the type over which α and β vary and, if the principle is acceptable, § is a function from entities of that type to objects. Since ‘the direction of’ is intended to stand for a function from objects of a certain sort to other objects, the Direction Equivalence is a first-order, or in Fine’s terminology objectual, abstraction; since ‘the number of’ is intended to stand for a function from concepts to objects, Hume’s principle is a higher-order, or as Fine says conceptual, abstraction. As is well-known, Frege himself abandoned the idea of defining number implicitly or contextually by means of Hume’s principle—adopting instead an explicit definition of the number of Fs as the extension of the concept concept equinumerous to the concept F— because he thought that an adequate definition should settle the question whether Julius Caesar, for example, is or is not the number of some concept, but that the proposed implicit definition cannot do so. But interest in abstraction principles has revived in the last couple of decades, largely as a result of Crispin Wright’s attempt to show that in spite of the many difficulties confronting it—including the notorious Julius Caesar problem— Hume’s principle can after all serve as a foundation for arithmetic. (shrink)

The nature of light is a focus of Thomas Hobbes’s natural philosophical project. Hobbes’s explanation of the light of lucid bodies differs across his works, from dilation and contraction in Elements of Law to simple circular motions in De corpore. However, Hobbes consistently explains perceived light by positing that bodily resistance generates the phantasm of light. In Letters I.XIX–XX of Philosophical Letters, fellow materialist Margaret Cavendish attacks the Hobbesian understanding of both lux and lumen by claiming that Hobbes has illicitly (...) made abstractions from matter. In this paper, I argue that Cavendish’s criticisms rely on an incorrect understanding of the nature of Hobbesian geometry and the role it plays in Hobbes’s natural philosophy. Rather than understanding geometry as wholly abstract, Hobbes attempts to ground geometry in different ways of considering bodies and their motions. Furthermore, Hobbes’s own criticisms of abstraction suggest that he would share many of the worries she raises but deny that he falls prey to them. (shrink)

I present a general theory of abstraction operators which treats them as variable-binding term- forming operators, and provides a reasonably uniform treatment for definite descriptions, set abstracts, natural number abstraction, and real number abstraction. This minimizing, extensional and relational theory reveals a striking similarity between definite descriptions and set abstracts, and provides a clear rationale for the claim that there is a logic of sets (which is ontologically non- committal). The theory also treats both natural and real (...) numbers as answering to a two-fold process of abstraction. The first step, of conceptual abstraction, yields the object occupying a particular position within an ordering of a certain kind. The second step, of objectual abstraction, yields the number sui generis, as the position itself within any ordering of the kind in question. (shrink)

Needs matter; but not all needs do. Needs range from the urgent and dire to the insignificant and even trivial. Given this, a central task for the needs theorist is to give some account of which ne...

I offer a new interpretation of Feyerabend’s ‘conquest of abundance’ narrative. I consider and reject both the ontological reading as implausible and the ‘historical’ reading as uncompelling My own proposal is that the ‘conquest of abundance’ be understood in terms of an impoverishment of the richness of human experience. For Feyerabend, such abundance is ‘conquered’ when individuals internalize distorting epistemic prejudices including those integral to the theoretical conceptions associated with the sciences. I describe several ways, identified by Feyerabend, in which (...) individuals can be led to occlude the richness of their experience in ways that are existentially impoverishing. (shrink)

Mauricio Suárez and Agnes Bolinska apply the tools of communication theory to scientific modeling in order to characterize the informational content of a scientific model. They argue that when represented as a communication channel, a model source conveys information about its target, and that such representations are therefore appropriate whenever modeling is employed for informational gain. They then extract two consequences. First, the introduction of idealizations is akin in informational terms to the introduction of noise in a signal; for in (...) an idealization we introduce ‘extraneous’ elements into the model that have no correlate in the target. Second, abstraction in a model is informationally equivalent to equivocation in the signal; for in an abstraction we “neglect” in the model certain features that obtain in the target. They conclude that it becomes possible in principle to quantify idealization and abstraction in informative models, although precise absolute quantification will be difficult to achieve in practice. (shrink)

The paper is a study of the logic of existence, negation, and order in the Neoplatonic tradition. The central idea is that Neoplatonists assume a logic in which the existence predicate is a comparative adjective and in which monadic predicates function as scalar adjectives that nest the background order. Various scalar predicate negations are then identifiable with various Neoplatonic negations, including a privative negation appropriate for the lower orders of reality and a hyper-negation appropriate for the higher. Reversion to the (...) One can then be explained as the logical inference of hyper-negations from mundane knowledge. Part I develops the relevant linguistic and logical theory, and Part II defends Wolfson and the scalar interpretation against the more traditional Aristotelian understanding of Whittaker and others of reversion as intensional abstraction. (shrink)

We argue that two powerful error-theoretic concepts provide a general framework that satisfactorily accounts for key aspects of the explanation of physical patterns. This method gives an objective criterion to determine which mathematical models in a class of neighboring models are just as good as the exact one. The method also emphasizes that abstraction is essential for explanation and provides a precise conceptual framework that determines whether a given abstraction is explanatorily relevant and justified. Hence, it increases our (...) epistemological understanding of how one should go about reconstructing scientific practices by making clear that, at a fundamental level, a key aspect of mathematical modeling consists in exactly solving nearby problems. (shrink)

Recall, if you will, the standard objections to the traditional doctrines. While the most subtle of the competing doctrines is, in my opinion, the Aristotelian and scholastic account of abstraction, the objection to this doctrine is that it requires a realism which is too immediate, so that the forms of one's present state of knowledge are allowed to pass as the forms of nature. And although, as I understand it, Aristotelian mathematics is gained by abstraction from an already (...) fairly abstract matter, one naturally expects in this context one true geometry, rather than alternate geometries. At the same time the strength of this view lies in its confidence that our abstractions must refer to the real world. The defect of the standard Lockian doctrine is that it contains an inner inconsistency, the nature of which we shall notice in due course. And upon application the doctrine explodes at once. On the other hand, the strength of the Lockian doctrine, as well as the doctrine of Berkeley and Hume, lies in its reluctance to allow needless entities entrance into reality. The weakness of the doctrine of Berkeley and Hume is that in moving from the impossible image of Locke's doctrine, the emphasis on particularity requires the agency of custom or habit to join the particulars of sense into classes. But the edges of such classes remain always indefinite. Habit, or custom, would seem not to allow the clean separation of abstractions from each other which the results of intellection give us reason to demand in any adequate doctrine. The weakness of the Kantian doctrine lies in its inability to explicate an antecedent reality; that is, in affirming at last the incognizable thing in itself; while the strength of this doctrine derives from its ability to allow the construction of precise abstractions in a somewhat autonomous manner. (shrink)

In this article I argue that structure in chemistry is a creature of abstraction: attending selectively to structural similarities, we neglect differences. There are different ways to abstract, so abstraction is interest dependent. So is structure. First, there are two different and mutually irreducible notions of structure in chemistry: bond structure and geometrical structure. Second, structure is relative to scale : the same substance has different structures at different scales, and relationships of structural sameness and difference vary across (...) the scales. However, these facts have no tendency to undermine structure’s claim to reality, or its metaphysical seriousness. (shrink)

We investigate the class of strongly distributive pregroups, a common abstraction of MV-algebras and Abelian l-groups which was introduced by E.Casari. The main result of the paper is a representation theorem which yields both Chang's representation of MV-algebras and Clifford's representation of Abelian l-groups as immediate corollaries.

Charles Mills has argued against ideal theory in political philosophy on the basis that it contains idealizations. He calls for political philosophers to do more nonideal theory, namely political theory that pays more attention to the most visible oppressions in society, such as those based on race, gender, and class. Mills's argument relies on a distinction between idealization and abstraction. Idealizations involve adding false assumptions to one's model, which is unacceptable, whereas abstractions merely leave out details without undermining descriptive (...) power. By studying formal models of injustice, I argue that the idealization/abstraction distinction is unhelpful. Either the distinction exists only relative to one's modeling purposes, or all models in political theory contain idealizations. Either way, the distinction does not help Mills's cause. Furthermore, there are arguments from philosophy of science for the epistemic benefits of idealizations. However, Mills's call for greater emphasis on the most visible mechanisms of oppression can be supported without relying on an idealization/abstraction distinction. I provide three alternative reasons for why we should prefer political theories that place more emphasis on race-, class-, and gender-based oppression. (shrink)

As Nelson Goodman highlighted, there are two main senses of “abstract” that can be found in discussions about abstract art. On the one hand, a representation is abstract if it leaves out certain features of its target. On the other hand, something can be abstract to the extent that it does not represent a concrete subject. The first sense of “abstract” is well-known in philosophy of science. For example, philosophers discuss mathematical models of physical, biological, and economic systems as being (...) abstract in this sense. However, it is the second sense that dominates discussions of abstract art in aesthetics. For example, abstract art was (and is) considered revolutionary precisely for being non-figurative. Through an analysis of artists including Kandinsky, Malevich, and Mondrian, we develop a reading of this second sense, which we call “generative abstraction,” as opposed to “subtractive” abstraction. Generative abstraction is a process in which a new artifact is created which does not represent the initial concrete target system that inspired it (if there was one), where the artifact’s features are explored for their own sake, and where the “language” of the new artifact is in some way more “universal.” Focusing on this sense of abstraction is helpful in revealing the complexity of the process of crafting an abstract artifact, in problematizing the notion that abstraction can always be un-done (or concretized), as well as revealing new ways for abstractions to be epistemically (un)successful. (shrink)

In the philosophy of science, abstraction has usually been analyzed in terms of the interface between our experience and the design of our concepts. The often implicit assumption here is that such interface has a definite identifiable and universalizable structure, determining the epistemic correctness of any abstraction. Our claim is that, on the contrary, the epistemic grounding of abstraction should not be reduced to the structural norms of such interface but is also related to the constraints on (...) the cognitive processes of specific abstractions. This suggests that we should understand abstraction as embodied in different kinds of abstraction practices. (shrink)

The first part of this paper surveys the current literature in the history of nineteenth-century mathematics in order to show that the question “Did the increasing abstraction of mathematics lead to a sense of anxiety?” is a new and valid question. I argue that the mathematics of the nineteenth century is marked by a growing appreciation of error leading to a note of anxiety, hesitant at first but persistent by 1900. This mounting disquiet about so many aspects of mathematics (...) after 1850 is seldom discussed. The second part explores the issue of anxiety in mathematical life through an interesting account of an address made by a mathematician in 1911, Oscar Perron. The third and final part ventures some conclusions about the value of anxiety as a question for historians of mathematics to pursue. (shrink)

This article offers some hypotheses to explain Later Stone Age lithic miniaturization: the systematic creation of small stone flakes on the finest-grained materials. Fundamentally, this phenomenon appears to represent the prioritization of stone tool sharpness over longevity, and a disposable mode of using stone tools. Ethnographic evidence from Australasia, the Andaman Islands, and Africa is used to suggest some specific functions for miniaturized lithics, as well as their relationship to other aspects of Later Stone Age material culture, including ochre crayons, (...) shell beads, and notched bones. Miniaturized lithic functions are hypothesized to have a common basis in the cognitive capacity for abstraction: having ideas about ideas. The technological and social affordances of abstraction may have given later Homo sapiens significant adaptive advantages over other members of our genus. (shrink)

Résumé Cet article propose de revenir sur le programme d’histoire concrète de l’abstraction proposé par Jean-Claude Perrot dans les années 1990, pour montrer quels en sont les apports pour l’histoire des mathématiques. En mettant l’accent sur les dynamiques sociales, culturelles et matérielles dans lesquelles s’élaborent les connaissances, ce programme fournit en effet des outils pour analyser non seulement les modalités de circulation des mathématiques, mais surtout les effets concrets des pratiques symboliques, souvent laissées de côtés dans les travaux historiens (...) sur les sciences. Plus encore, le programme d’histoire concrète de l’abstraction constitue en objet d’histoire le processus qui permet aux traces matérielles des activités savantes passées et aux modalités de raisonnements qu’elles véhiculent de parvenir jusqu’à nous tout en transcendant leurs conditions initiales de production. Il offre ainsi un cadre analytique pour étudier le rapport des mathématiques au temps historique. (shrink)

Recently, Horsman et al. have proposed a new framework, Abstraction/Representation theory, for understanding and evaluating claims about unconventional or non-standard computation. Among its attractive features, the theory in particular implies a novel account of what is means to be a computer. After expounding on this account, I compare it with other accounts of concrete computation, finding that it does not quite fit in the standard categorization: while it is most similar to some semantic accounts, it is not itself a (...) semantic account. Then I evaluate it according to the six desiderata for accounts of concrete computation proposed by Piccinini. Finding that it does not clearly satisfy some of them, I propose a modification, which I call Agential AR theory, that does, yielding an account that could be a serious competitor to other leading account of concrete computation. (shrink)