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)

If numbers were identified with any of their standard set-theoretic realizations, then they would have various non-arithmetical properties that mathematicians are reluctant to ascribe to them. Dedekind and later structuralists conclude that we should refrain from ascribing to numbers such ‘foreign’ properties. We first rehearse why it is hard to provide an acceptable formulation of this conclusion. Then we investigate some forms of abstraction meant to purge mathematical objects of all ‘foreign’ properties. One form is inspired by Frege; the (...) other by Dedekind. We argue that both face problems. (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)

ion turns equivalence into identity, but there are two ways to do it. Given the equivalence relation of parallelness on lines, the #1 way to turn equivalence into identity by abstraction is to consider equivalence classes of parallel lines. The #2 way is to consider the abstract notion of the direction of parallel lines. This paper developments simple mathematical models of both types of abstraction and shows, for instance, how finite probability theory can be interpreted using #2 abstracts (...) as “superposition events” in addition to the ordinary events. The goal is to use the second notion of abstraction to shed some light on the notion of an indefinite superposition in quantum mechanics. (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)

Linguistics of Saying is to be analyzed in the speech act conceived as an act of knowing. The speaking, saying and knowing subject, based on contexts and the principles of congruency and trust in the speech of other speakers, will create meanings and interpret the sense of utterances supplying the deficiencies of language by means of the intellective operations mentally executed in the act of speech. In the intellective operations you can see three steps or processes: first the starting point, (...) intuition or aísthesis; second, the process of abstraction; and third, the inverse: the process of determination or fixing the construct created. (shrink)

Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, (...) we show that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to stability-like acceptability criteria on abstraction principles, the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli and Fine, and supervaluational ideas coming out of Hodes' work. (shrink)

Berkeley and Hume object to Locke's account of abstraction. Abstraction is separating in the mind what cannot be separated in reality. Their objection is that if a is inseparable in reality from b, then the idea of a is inseparable from the idea of b. The former inseparability is the reason for the latter. In most interpretations, however, commentators leave the former unexplained in explaining the latter. This article assumes that Berkeley and Hume present a unified front against (...) Locke. Hume supplements Berkeley's argument just where there are gaps. In particular, Hume makes explicit something Berkeley leaves implicit: The argument against Locke depends on the principle that things are inseparable if and only if they are identical. Abstraction is thinking of one of an inseparable pair while not thinking of the other. But doing so entails thinking of something while not thinking of it. This is the fundamental objection. (shrink)

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)

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)

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.

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)

IN THIS PAPER I intend to suggest that the tantalizing notion of dialectics as found, for example, in Hegel might be approached by reflecting upon some aspects of the notion of abstraction as found, for example, in Aquinas. It is more or less by accident that the two thinkers I wish to discuss here are Aquinas and Hegel rather than Aristotle and Marx, or even Meinong and Theodor Adorno. It certainly is possible to compare two metaphysicians of the stature (...) of Aquinas and Hegel, particularly if one restricts oneself to a well-defined issue. However, in this paper I do not intend to advance any such comparison in the strict sense of the term. Rather, I would like to show that certain basic features of the Hegelian and post-Hegelian notion of dialectics can be understood as an attempt at avoiding certain conclusions that almost inevitably have to be drawn if one accepts the classical conception of abstraction. (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)

Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven (...) to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic. (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)

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)

According to a simple similarity theory of representation, x represents y because x and y share some properties. In Meaning and Mental Representation, Robert Cummins rejects this account for representations that play a role in cognition because, among other things, a similarity theory of representation precludes a satisfactory account of an essential cognitive task, namely abstraction. Intelligent beings have representations of classes and properties, and we need an account for such representations. Cummins argues that a causal covariance theory of (...) representation, even a very simple version of it, is superior to a similarity theory because causal covariance at least makes abstraction possible. I want to argue that this is not so. Covariance theories of representation do not achieve abstraction. (shrink)

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.

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)

Charles S. Peirce’s second trichotomy, which introduces the concepts of iconicity, indexicality, and symbolicity, is probably the only piece of his semiotic that is familiar to visual artists and designers. Although the concepts have found their way into the academy, their utility in the field has been reduced for a couple of reasons. First, as with all of Peirce’s philosophy, his second trichotomy is a concept that is subtle, fluid, and difficult to fully grasp in a sound bite. Second, there (...) has simply been no bridge concept that would form a working connection between that philosophy in its logical guise and the studio practice in the visual arts. The purpose of this article is to remedy that situation by investigating the subtle ways the second trichotomy functions within the visual sphere, and to then suggest a model that can serve to bridge the divide between pure theory and practice. The article makes four main points: first, using examples from visual identity and the graphic arts, it demonstrates how the modes of icon, index, and symbol tend to be blended; second, examples from fine art are used to illustrate how the concept of abstraction, as used in the art world, can only be partially accounted for within the second trichotomy, but can be modeled by supplying a syntactical supplement; third, it expands on and elaborates a previously sketched model, the visual gamut, which makes it possible to classify visual entities according to their position within a map of semantic and syntactic space; finally, it concludes by suggesting ways this enhanced version of the visual gamut model might be used in the analysis of, or creation of, art and design, presenting suggestions for further study. (shrink)

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)

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)

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)

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.

Cantor's abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor's proposal based upon the set theoretic framework of Bourbaki - called BK - which is a First-order set theory extended with Hilbert's ε-operator. Moreover, it is argued that the BK system and the ε-operator provide a faithful reconstruction of Cantor's insights on cardinal numbers. (...) I will introduce first the axiomatic setting of BK and the definition of cardinal numbers by means of the ε-operator. Then, after presenting Cantor's abstractionist theory, I will point out two assumptions concerning the definition of cardinal numbers that are deeply rooted in Cantor’s work. I will claim that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to those of Zermelo-von Neumann and Frege-Russell. On the basis of these similarities, I will make use of the BK framework in meeting Frege's objections to Cantor's proposal. A key ingredient in the defence of Cantorian abstraction will be played by the role of representative sets, which are arbitrarily denoted by the ε-operator in the BK definition of cardinal numbers. (shrink)

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)

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)

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.

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)