It is proved in this paper that the positive abstraction scheme is consistent with extensionality only if one drops equality out of the language. The theory obtained is then compared with GPK, a wellknown set theory based on an extended positive comprehension scheme.

Neo-Fregean logicism attempts to base mathematics on abstraction principles. Since not all abstraction principles are acceptable, the neo-Fregeans need an account of which ones are. One of the most promising accounts is in terms of the notion of stability; roughly, that an abstraction principle is acceptable just in case it is satisfiable in all domains of sufficiently large cardinality. We present two counterexamples to stability as a sufficient condition for acceptability and argue that these counterexamples can be (...) avoided only by major departures from the existing neo-Fregean programme. (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)

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.

Peano's axiomatizations of geometry are abstract and non-intuitive in character, whereas Peano stresses his appeal to concrete spatial intuition in the choice of the axioms. This poses the problem of understanding the interrelationship between abstraction and intuition in his geometrical works. In this article I argue that axiomatization is, for Peano, a methodology to restructure geometry and isolate its organizing principles. The restructuring produces a more abstract presentation of geometry, which does not contradict its intuitive content but only puts (...) it into a particular form. (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)

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.

Ethicists of care have objected to traditional moral philosophy's reliance upon abstract universal principles. They claim that the use of abstraction renders traditional theories incapable of capturing morally relevant, particular features of situations. I argue that this objection sometimes conflates two different levels of moral thinking: the level of justification and the level of deliberation. Specifically, I claim that abstraction or attention to context at the level of justification does not entail, as some critics seem to think, a (...) commitment to abstraction or attention to context at the level of deliberation. It follows that critics who reject a theory's use of abstraction at the level of justification have not shown that the theory recommends abstraction at the level of deliberation and that it, therefore, compels the deliberating agent to overlook morally salient details. (shrink)

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)

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)

Frege's views regarding analysis and synomymy have long been the subject of critical discussion. Some commentators, led by Dummett, have argued that Frege was committed to the view that each thought admits of a unique ultimate analysis. However, this interpretation is in apparent conflict with Frege's criterion of synonymy, according to which two sentence express the same thought if one cannot understand them without regarding them as having the same truth–value. In a recent article in this journal, Drai attempts to (...) reconcile Frege's criterion of synonymy with unique ultimate analysis by holding that, for Frege, if two sentences satisfy the criterion without being intensionally isomorphic, at most one of them is a privileged representation of the thought expressed. I argue that this proposal fails, because it conflicts not only with Frege's views of abstraction principles but also with slingshot arguments (including one presented by Drai herself) that accurately reflect Frege's commitment to the view that sentences alike in truth–value have the same Bedeutung. While Drai helpfully connects Frege's views of abstraction principles with such slingshot arguments, this connection cannot become fully clear until we recognise that Frege rejects unique ultimate analysis. (shrink)

Through a comparison of everyday practices surrounding dairy production and breastfeeding in Croatia, this book explores the way that humans work to transform milk into human or animal milks.

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)

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)

This paper contributes to the philosophical accounts of generalisation in formal modelling by introducing a conceptual framework that allows for recognising generalisations that are epistemically beneficial in the sense of contributing to the truth of a model result or component. The framework is useful for modellers themselves because it is shown how to recognise different kinds of generalisation on the basis of changes in model descriptions. Since epistemically beneficial generalisations usually de-idealise the model, the paper proposes a reformulation of the (...) well-known distinction between abstraction and idealisation. A reformulated notion of abstraction is needed because the extant accounts yield wrong judgments when model-modifications introduce implicit assumptions. (shrink)

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)

The first amounts, roughly, to "It is necessarily the case that any President of the U.S. is a citizen of the U.S." But the second says, "the person who in fact is the President of the U.S, has the property of necessarily being a citizen of the U.S," Thus, while (2) is clearly true, it would be reasonable to consider (3) false.

Explains why Bob Hale's proposed notion of weak sense cannot explain the analyticity of Hume's principle as he claims. Argues that no other notion of the sort Hale wants could do the job either.

Abstraction principles and grounding can be combined in a natural way Modality: metaphysics, logic, and epistemology, Oxford University Press, Oxford, pp 109–136, 2010; Schwartzkopff in Grazer philosophische studien 82:353–373, 2011). However, some ground-theoretic abstraction principles entail that there are circles of partial ground :775–801, 2017). I call this problem auto-abstraction. In this paper I sketch a solution. Sections 1 and 2 are introductory. In Sect. 3 I start comparing different solutions to the problem. In Sect. 4 I (...) contend that the thesis that the right-hand side of an abstraction principle is prior to its left-hand side motivates an independence constraint, and that this constraint leads to predicative restrictions on the acceptable instances of ground-theoretic abstraction principles. In Sect. 5 I argue that auto-abstraction is acceptable unless the left-hand side is essentially grounded by the right-hand side. In Sect. 6 I highlight several parallelisms between auto-abstraction and the puzzles of ground. I finally compare my solution with the strategies listed in Sect. 3. (shrink)

This essay has three general themes. The first involves the claim that nudging threatens human agency. My basic response is that human agency is fully retained (because nudges do not compromise freedom of choice) and that agency is always exercised in the context of some kind of choice architecture. The second theme involves the importance of having a sufficiently capacious sense of the category of nudges, and a full appreciation of the differences among them. Some nudges either enlist or combat (...) behavioral biases but others do not, and even among those that do enlist or combat such biases, there are significant differences. The third general theme is the need to bring various concerns (including ethical ones) in close contact with particular examples. A legitimate point about default rules may not apply to warnings or reminders. An ethical objection to the use of social norms may not apply to information disclosure. Here as elsewhere, abstraction can be a trap. We continue to learn about the relevant ethical issues, about likely public reactions to nudging, and about differences across cultures and nations. Future progress will depend on a high level of concreteness, perhaps especially in dealing with the vexing problem of time-inconsistency. (shrink)

The paper contends that abstraction lies at the core of the philosophical and methodological rupture that occurred between Husserl and his mentor Franz Brentano. To accomplish this, it explores the notion of abstraction at work in these two thinkers’ methodological discussions through their respective claims regarding the structure of consciousness, and shows that how Husserl and Brentano analyze the structure of consciousness conditions and strictly delineates the nature and reach of their methods of inquiry. The paper pays close (...) attention to intentionality, founding (Fundierung), and qualitative modification (qualitative Modifikation) understood as principles of consciousness. While intentionality has been the topic of numerous discussions surrounding these two thinkers’ work, founding and qualitative modification have slipped under the radar despite the fact that they hold—as I intend to show—the key to shedding new light on their respective methods of inquiry. More specifically, the paper explicates the ways in which Brentano’s notion of founding (understood as a relationship between whole acts) and his failure to identify radical-qualitative modification as a principle of consciousness preclude him from offering a successful model for philosophical universalizing thought. It is my hope to show that Husserl’s rethinking of founding as a relationship between certain structural moments of acts along with his novel notion of qualitative modification ground his attempt to carve a new method of philosophical inquiry—albeit jejune and ambiguous at the time of the Investigations, yet nevertheless able to successfully negotiate the problems faced by his predecessor. (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)

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)

I argue that we cannot adequately characterize idealization and abstraction and the distinction between the two on the grounds that they have distinct semantic properties. By doing so, on the one hand, we focus on the conceptual products of the two processes in making the distinction and we overlook the importance of the nature of the thought processes that underlie model-simplifying assumptions. On the other hand, we implicitly rely on a sense of abstraction as subtraction, which is unsuitable (...) for explicating scientific model construction. Instead, I argue that a sense of abstraction as extraction is more suitable. Finally, I suggest a different way to distinguish the two processes that avoids these problems. Namely, that both idealization and abstraction could be understood as particular modes of application of the same cognitive process: selective attention. (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)

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 laterHomo sapienssignificant adaptive advantages over other members of our genus. (shrink)

The realism grounding St. Thomas Aquinas’s pre-modern natural science defends the reception of similitudes of the forms of things known by abstraction. Modern natural science challenges this abstractio- nist account by recasting «form» in the leading role of principle of intelligibility—instead of forms, modern science discovers laws. Thomistic realism is prima facie incompatible with this account. Following Charles De Koninck, this essay outlines a rapprochement between the epistemology of pre-modern, Thomistic natural science and its modern successor. I argue that (...) natural forms are noetic limits towards which physical laws tend, and our grasp of this tendency uses a mode of knowledge comparable to what St. Thomas termed universal in repraesentando. (shrink)

From the advent of general purpose, Turing-complete machines, the relation between operators, programmers and users with computers can be observed as interconnected informational organisms (inforgs), henceforth analysed with the method of levels of abstraction (LoAs), risen within the philosophy of information (PI). In this paper, the epistemological levellism proposed by L. Floridi in the PI to deal with LoAs will be formalised in constructive terms using category theory, so that information itself is treated as structure-preserving functions instead of Cartesian (...) products. The milestones in the history of modern computing are then analysed through constructive levellism to show how the growth of system complexity lead to more and more information hiding. (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)

Linnebo and Pettigrew have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They recognize that this version of structuralism is vulnerable to the well-known problem of non-rigid structures. This paper offers a solution to the problem for this version of structuralism. The solution involves expanding the languages used to describe mathematical structures. We then argue that this solution is philosophically acceptable to those who endorse mathematical structuralism based on Fregean abstraction principles.

An informal theory is set forth of relations between abstract entities, includingcolors, physical quantities, times, andplaces in space, and the concrete things thathave them, or areat orin them, based on the assumption that there are close analogies between these relations and relations between abstractsets and the concrete things that aremembers of them. It is suggested that even standard scientific usage of these abstractions presupposes principles that are analogous to postulates of abstraction, identity, and other fundamental principles of set theory. (...) Also discussed is the significance of important disanalogies between sets and physical abstractions, including especiallymodal andtemporal aspects of physical abstractions, which is related to the problem of the characterizingconstancy, of colors, physical attributes, and locations in space. (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)

We propose that abstraction is an interpersonal process and serves a social function. Research on shared reality shows that in communication, people raise their level of abstraction in order to create a common understanding with their communication partner, which can subsequently distort their mental representation of the object of communication. This work demonstrates that, beyond building accurate models, abstraction also functions to build socially shared models – to create a shared reality.

Play behaviour is notorious for constituting a much debated, yet little clarified field of research. In this article, attempts are made to reach conclusions on the relation between human play and the play of other animals (especially cat play), as well as on the very character of play. The concept of Umwelt is reviewed, as are definitions of animal play, categorization of animal play and the role of meta-communication in playful behaviour. For some, play is a symbol of everythingthat is (...) good. The author of the current article does not deny that social morality may have originated from play behaviour, but stresses the existence of cruelty play, which leads to additional assumptions. Another notion that is treated in some detail is perceptual play, which proves to demonstrate complex semiotic play that is related first of all to signification. At the end of the article an alternative categorization of animal play is suggested, in which the fundamental role of mind games is emphasized. Throughout the text, examples of play behaviour are offered by the two domestic cats Muki and Maluca. (shrink)