Abstraction and idealization are the two notions that are most often discussed in the context of assumptions employed in the process of model building. These notions are also routinely used in philosophical debates such as that on the mechanistic account of explanation. Indeed, an objection to the mechanistic account has recently been formulated precisely on these grounds: mechanists cannot account for the common practice of idealizing difference-making factors in models in molecular biology. In this paper I revisit the debate and (...) I argue that the objection does not stand up to scrutiny. This is because it is riddled with a number of conceptual inconsistencies. By attempting to resolve the tensions, I also draw several general lessons regarding the difficulties of applying abstraction and idealization in scientific practice. Finally, I argue that more care is needed only when speaking of abstraction and idealization in a context in which these concepts play an important role in an argument, such as that on mechanistic explanation. (shrink)

Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the core (...) of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism. (shrink)

Proponents of mechanistic explanation all acknowledge the importance of organization. But they have also tended to emphasize specificity with respect to parts and operations in mechanisms. We argue that in understanding one important mode of organization—patterns of causal connectivity—a successful explanatory strategy abstracts from the specifics of the mechanism and invokes tools such as those of graph theory to explain how mechanisms with a particular mode of connectivity will behave. We discuss the connection between organization, abstraction, and mechanistic explanation and (...) illustrate our claims by looking at an example from recent research on so-called network motifs. (shrink)

This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.

In this paper we describe and interpret the formal machinery of abstraction processes in which the domain of abstracta is a subset of the domain of objects from which is abstracted.

Kant’s subject as source of universality and necessity in human understanding is Modern Philosophy's solution to the old problem of the universals, a solution which appeared to supersede once and for all the Aristotelian theory of abstraction. The present paper intends to show how Aquinas's Aristotelian doctrine on abstraction may stand the Kantian challenge and resolve the old problem when three principles are brought into play: 1) the same perfection can subsist in two different modes of being, and thus the (...) universal content of knowledge can be present in the particular reality (Summa, I, 84, 1); 2) things which are really distinct between themselves can be known in their distinction, even when they cannot exist separately (metaphysical distinction justifies gnoseological separation); 3) as Cornelio Fabro suggests, it is artificial to separate in the object of experience the object’s different levels of objectivity (quality, configuration and meaning) and, as a result, human experience could be considered the source not only of sensible content but also of intelligible content. This “defense” of Thomistic abstraction intends to offer an account of knowledge which reunites again epistemology with common sense experience, so that philosophy may no longer be a Platonic abstraction but a meaningful explanation of life. (shrink)

This intervention concerns the different statute of abstraction in Marx's work. By means of a critical confrontation with Chris Arthur's work, Finelli presents his thesis of the presence of a double theory and fuction of abstraction in Marx's work. In the early Marx, until the German Ideology, abstraction is, in accordance with the traditional meaning of this term, a product of the mind, an unreal spectre. More exactly, it consists in negating the common essence belonging to labouring humanity and projecting (...) it, as alienated universal, into the idea of philosophy, into the state of politics and into the money of the market. In the later Marx, the nature of abstraction is, rather than mental, practical. It is directly related to the quantity without quality of capitalist labour, and it is the product of the systemic connection of machines to labour-power. In contrast to Arthur, Finelli maintains that practical abstraction in the Marx of Capital is not located in the zone of exchange and the market, where there is the mediation of money. On the contrary, it is located in the zone of production, which, for Marx, is a social ensemble not mediated by money but by relations of technological domination. (shrink)

What is the best way to analyse abstraction in scientific modelling? I propose to focus on abstracting as an epistemic activity, which is achieved in different ways and for different purposes depending on the actual circumstances of modelling and the features of the models in question. This is in contrast to a more conventional use of the term ‘abstract’ as an attribute of models, which I characterise as black-boxing the ways in which abstraction is performed and to which epistemological advantage. (...) I exemplify my claims through a detailed reconstruction of the practices involved in creating two types of models of the flowering plant Arabidopsisthaliana, currently the best-known model organism in plant biology. This leads me to distinguish between two types of abstraction processes: the ‘material abstracting’ required in the production of Arabidopsis specimens and the ‘intellectual abstracting’ characterising the elaboration of visual models of Arabidopsis genomics. Reflecting on the differences between these types of abstracting helps to pin down the epistemic skills and research commitments used by researchers to produce each model, thus clarifying how models are handled by researchers and with which epistemological implications. (shrink)

Which abstraction principles are acceptable? A variety of criteria have been proposed, in particular irenicity, stability, conservativeness, and unboundedness. This note charts their logical relations. This answers some open questions and corrects some old answers.

Abstract: Laws of computer science are prescriptive in nature but can have descriptive analogs in the physical sciences. Here, we describe a law of conservation of information in network programming, and various laws of computational motion (invariants) for programming in general, along with their pedagogical utility. Invariants specify constraints on objects in abstract computational worlds, so we describe language and data abstraction employed by software developers and compare them to Floridi's concept of levels of abstraction. We also consider Floridi's structural (...) account of reality and its fit for describing abstract computational worlds. Being abstract, such worlds are products of programmers' creative imaginations, so any "laws" in these worlds are easily broken. The worlds of computational objects need laws in the form of self-prescribed invariants, but the suspension of these laws might be creative acts. Bending the rules of abstract reality facilitates algorithm design, as we demonstrate through the example of search trees. (shrink)

Abstraction is seen as an active process which both enlightens and obscures. Abstractions are not true or false but relatively enlightening or obscuring according to the problem under study; different abstractions may grasp different aspects of a problem. Abstractions may be useless if they can answer questions only about themselves. A theoretical enterprise explores reality through acluster of abstractions that use different perspectives, temporal and horizontal scales, and assumes different givens.

The way we understand the environment is analogous to the way we draw a map. Drawing insights from this analogy, this paper shows how the abstraction that occurs in ecological explanation can lead to damaging distortion. It is mistaken, therefore, to assume that by abstraction we can easily determine the correct variables for controlling a given ecosystem as if it were ideally closed. Recent work shows that the environment is a global composite with a very high degree of internal dependence (...) between its parts, a kind of dependence that makes it more holistic than a machine and perhaps even as holistic as an organism. This paper was presented at the 6th European Meeting of Jesuits in Science on “Science and Culture”, Frankfurt a. M., Germany (8th - 12th September 1999). (shrink)

While the fundamental laws of physics are time-reversal invariant, most macroscopic processes are irreversible. Given that the fundamental laws are taken to underpin all other processes, how can the fundamental time-symmetry be reconciled with the asymmetry manifest elsewhere? In statistical mechanics, progress can be made with this question. What I dub the ‘Zwanzig–Zeh–Wallace framework’ can be used to construct the irreversible equations of SM from the underlying microdynamics. Yet this framework uses coarse-graining, a procedure that has faced much criticism. I (...) focus on two objections in the literature: claims that coarse-graining makes time-asymmetry ‘illusory’ and ‘anthropocentric’. I argue that these objections arise from an unsatisfactory justification of coarse-graining prevalent in the literature, rather than from coarse-graining itself. This justification relies on the idea of measurement imprecision. By considering the role that abstraction and autonomy play, I provide an alternative justification and offer replies to the illusory and anthropocentric objections. Finally, I consider the broader consequences of this alternative justification: the connection to debates about inter-theoretic reduction and the implication that the time-asymmetry in SM is weakly emergent. 1Introduction 1.1Prospectus2The Zwanzig–Zeh–Wallace Framework3Why Does This Method Work? 3.1The special conditions account3.2When is a density forwards-compatible?4Anthropocentrism and Illusion: Two Objections 4.1The two objections in more detail4.2Against the justification by measurement imprecision5An Alternative Justification 5.1Abstraction and autonomy5.2An illustration: the Game of Life6Reply to Illusory7Reply to Anthropocentric8The Wider Landscape: Concluding Remarks 8.1Inter-theoretic relations8.2The nature of irreversibility. (shrink)

Several modern accounts of explanation acknowledge the importance of abstraction and idealization for our explanatory practice. However, once we allow a role for abstraction, questions remain. I ask whether the relation between explanations at different theoretical levels should be thought of wholly in terms of abstraction, and argue that changes of the quantities in terms of which we describe a system can lead to novel explanations that are not merely abstractions of some more detailed picture. I use the example of (...) phase transitions as described by statistical mechanics and thermodynamics to illustrate this, and to demonstrate some details of the relationship between abstraction, idealization, and novel explanation. (shrink)

Neologicists have sought to ground mathematical knowledge in abstraction. One especially obstinate problem for this account is the bad company problem. The leading neologicist strategy for resolving this problem is to attempt to sift the good abstraction principles from the bad. This response faces a dilemma: the system of ‘good’ abstraction principles either falls foul of the Scylla of inconsistency or the Charybdis of being unable to recover a modest portion of Zermelo–Fraenkel set theory with its intended generality. This article (...) argues that the bad company problem is due to the ‘static’ character of abstraction on the neologicist's account and develops a ‘dynamic’ account of abstraction that avoids both horns of the dilemma. 1 Introduction2ion Reconstructed3 From Bad to Worse4 Abstraction Reconceived5 Bad Company Made GoodA AppendixA.1SemanticsA.2LogicA.3ConsistencyA.4Strength. (shrink)

For various reasons several authors have enriched classical first order syntax by adding a predicate abstraction operator. “Conservatives” have done so without disturbing the syntax of the formal quantifiers but “revisionists” have argued that predicate abstraction motivates the universal quantifier’s re-classification from an expression that combines with a variable to yield a sentence from a sentence, to an expression that combines with a one-place predicate to yield a sentence. My main aim is to advance the cause of predicate abstraction while (...) cautioning against revisionism. In so doing, however, I shall pursue a secondary aim by conveying mixed blessings to those who hold the view that in the logical sense of “existence” some existing object is such as to exist contingently. Advocates of this view must concede Williamson’s recent contention that the domain of unrestricted objectual quantification could not have been narrower than it is actually, but predicate abstraction affords them some hope of accommodating this concession. (shrink)

Wright claims that “the epistemology of good abstraction principles should be assimilated to that of basic principles of logical inference”. In this paper I follow Wright’s recommendation, but I consider a different epistemology of logic, namely anti-exceptionalism. Anti-exceptionalism’s main contention is that logic is not a priori, and that the choice between rival logics should be based on abductive criteria such as simplicity, adequacy to the data, strength, fruitfulness, and consistency. This paper’s goal is to lay down the foundations for (...) an application of the anti-exceptionalist methodology to abstraction principles. In Sect. 1 I outline my strategy. I show that anti-exceptionalism has a bearing on the so-called Bad Company problem (Sect. 2), and explore some versions of anti-exceptionalism about abstraction (Sect. 3). I then consider new criteria for choosing between rival abstraction principles, and, in particular, between consistent revisions of Frege’s Basic Law V (Sect. 4). I finally consider how the abstractionist can compare between competing criteria (Sect. 5), and argue that anti-exceptionalists should be pluralists about abstraction (Sect. 6). (shrink)

The method of “abstraction” had been the centerpiece of earlier attempts at founding a Marxist philosophy of science – from Engels to Soviet Marxism. This paper confronts Marx’s writings on abstraction with contemporary views of the method, stemming mostly from the analytic and (post-)empiricist traditions. In Section 2, I reconstruct the roles that abstraction was to play, according to Marx, in the construction of a scientific theory, focusing exclusively on his own writings. The analysis reveals certain rules, left mostly implicit (...) in Marx, for the correct application of the method of abstraction. These are discussed in Section 3. The first rule states that concepts of the historically specific aspects of target systems (e.g., the capitalist economy) cannot be defined simply by means of transhistorical concepts. The second rule prohibits abstraction from the explanatorily relevant aspects which pertain, in Marx’s vocabulary, to the “essence” of the target system. In Section 3, I confront Marx’s notion of “abstraction” with contemporary statements on the method. It is shown that it covers both abstraction and idealization as understood in some modern accounts (notably, that of M. R. Jones). Under this approach, abstraction involves the tacit omission of properties, which are simply left unspecified. In contrast, idealization consists in the explicit counterfactual ascription of properties (and values of magnitudes). Finally, the representational goals pertaining to Marx’s “abstraction” are discussed, using distinctions due to M. Weisberg. It is shown that Marx was a proponent of “minimalist idealization”, focusing on the identification of causally relevant mechanisms that characterize all capitalist societies. I conclude with a suggestion for further research. (shrink)

Abstraction and generalization are two processes of reasoning that have a special role in the construction of scientific theories and models. They have been important parts of the scientific method ever since the nineteenth century. A philosophical and historical analysis of scientific practices shows how abstraction and generalization found their way into the theory of the logic of science of the nineteenth-century philosopher Charles S. Peirce. Our case studies include the scientific practices of Francis Galton and John Herschel, who introduced (...) composite photographs and graphical methods, respectively, as technologies of generalization and thereby influenced Peirce’s logic of abstraction. Herschel’s account of generalization is further supported by William Whewell, who was very influential on Peirce. By connecting Herschel’s scientific technology of abstraction to Peirce’s logical technology of abstraction—namely, diagrams—we highlight the role of judgments in scientific observation by hypostatic abstractions. We also relate Herschel’s discovery-driven logic of science and Peirce’s open-ended diagrammatic logic to the use of models in science. Ultimately, Peirce’s theory of abstraction is a case of showing how logic applies to reality. (shrink)

The main thesis of this book is that abstraction, far from being confined to higher forms of cognition, language and logical reasoning, has actually been a major driving force throughout the evolution of creatures with brains. It is manifest in emotive as well as rational thought. Wending its way through the various facets of abstraction, the book attempts to clarify - and relate - the often confusing meanings of the word 'abstract' that one may encounter even within the same discipline. (...) The unusual synoptic approach, which draws upon research in psychology, neural network theory, child language acquisition, philosophy and consciousness studies, as well as a variety of linguistic disciplines, cannot be compared directly to other books on the market that touch upon just one particular aspect of abstraction. It is aimed at a wide readership - anyone interested in the nature of abstraction and the cognitive processing and purpose behind it. (shrink)

We provide an explicit taxonomy of legitimate kinds of abstraction within constitutive explanation. We argue that abstraction is an inherent aspect of adequate mechanistic explanation. Mechanistic explanations—even ideally complete ones—typically involve many kinds of abstraction and therefore do not require maximal detail. Some kinds of abstraction play the ontic role of identifying the specific complex components, subsets of causal powers, and organizational relations that produce a suitably general phenomenon. Therefore, abstract constitutive explanations are both legitimate and mechanistic.

Fine and Antonelli introduce two generalizations of permutation invariance — internal invariance and simple/double invariance respectively. After sketching reasons why a solution to the Bad Company problem might require that abstraction principles be invariant in one or both senses, I identify the most fine-grained abstraction principle that is invariant in each sense. Hume’s Principle is the most fine-grained abstraction principle invariant in both senses. I conclude by suggesting that this partially explains the success of Hume’s Principle, and the comparative lack (...) of success in reconstructing areas of mathematics other than arithmetic based on non-invariant abstraction principles. (shrink)

We develop a functional abstraction principle for the type-free algorithmic logic introduced in our earlier work. Our approach is based on the standard combinators but is supplemented by the novel use of evaluation trees. Then we show that the abstraction principle leads to a Curry fixed point, a statement C that asserts C ⇒ A where A is any given statement. When A is false, such a C yields a paradoxical situation. As discussed in our earlier work, this situation leaves (...) one no choice but to restrict the use of a certain class of implicational rules including modus ponens. (shrink)

We characterize abstraction in computer science by first comparing the fundamental nature of computer science with that of its cousin mathematics. We consider their primary products, use of formalism, and abstraction objectives, and find that the two disciplines are sharply distinguished. Mathematics, being primarily concerned with developing inference structures, has information neglect as its abstraction objective. Computer science, being primarily concerned with developing interaction patterns, has information hiding as its abstraction objective. We show that abstraction through information hiding is a (...) primary factor in computer science progress and success through an examination of the ubiquitous role of information hiding in programming languages, operating systems, network architecture, and design patterns. (shrink)

Capacities for abstract thinking and problem‐solving are central to human cognition. Processes of abstraction allow the transfer of experiences and knowledge between contexts helping us make informed decisions in new or changing contexts. While we are often inclined to relate such reasoning capacities to individual minds and brains, they may in fact be contingent on human‐specific modes of collaboration, dialogue, and shared attention. In an experimental study, we test the hypothesis that social interaction enhances cognitive processes of rule‐induction, which in (...) turn improves problem‐solving performance. Through three sessions of increasing complexity, individuals and groups were presented with a problem‐solving task requiring them to categorize a set of visual stimuli. To assess the character of participants’ problem representations, after each training session, they were presented with a transfer task involving stimuli that differed in appearance, but shared relations among features with the training set. Besides, we compared participants’ categorization behaviors to simulated agents relying on exemplar learning. We found that groups performed superior to individuals and agents in the training sessions and were more likely to correctly generalize their observations in the transfer phase, especially in the high complexity session, suggesting that groups more effectively induced underlying categorization rules from the stimuli than individuals and agents. Crucially, variation in performance among groups was predicted by semantic diversity in members’ dialogical contributions, suggesting a link between social interaction, cognitive diversity, and abstraction. (shrink)

The neo-Fregean program in the philosophy of mathematics seeks a foundation for a substantial part of mathematics in abstraction principles—for example, Hume’s Principle: The number of Fs D the number of Gs iff the Fs and Gs correspond one-one—which can be regarded as implicitly definitional of fundamental mathematical concepts—for example, cardinal number. This paper considers what kind of abstraction principle might serve as the basis for a neo- Fregean set theory. Following a brief review of the main difficulties confronting the (...) most widely discussed proposal to date—replacing Frege’s inconsistent Basic Law V by Boolos’s New V which restricts concepts whose extensions obey the principle of extensionality to those which are small in the sense of being smaller than the universe—the paper canvasses an alternative way of implementing the limitation of size idea and explores the kind of restrictions which would be required for it to avoid collapse. (shrink)

Philosophers have often claimed that general ideas or representations have their origin in abstraction, but it remains unclear exactly what abstraction as a psychological process consists in. We argue that the Lockean aspiration of using abstraction to explain the origins of all general representations cannot work and that at least some general representations have to be innate. We then offer an explicit framework for understanding abstraction, one that treats abstraction as a computational process that operates over an innate quality space (...) of fine-grained general representations. We argue that this framework has important philosophical implications for the nativism-empiricism dispute, for questions about the acquisition of unstructured representations, and for questions about the relation between human and animal minds. (shrink)

Feminists, critical race scholars, and other social‐justice theorists sometimes object to “abstraction” in liberal normative theory. Arguing that oppression affects individual agents in powerful yet subtle ways, they contend that allegedly abstract theories often reinforce oppressive power structures. Here I critically examine and ultimately reject Onora O'Neill's “abstraction without idealization” as a solution to this problem. Because O'Neill defines abstraction as simply the “bracketing of certain predicates,” her methodology fails to guide decisions about what to bracket and what to include (...) in the theory. Moreover, it may not be possible to abstract without also relying on some particular ideals. While abstraction is unavoidable, I conclude that it must be employed with greater attention to sociopolitical hierarchies and by using ideals that do not collude with structures of oppression. I discuss the work of Susan Babbitt and Elizabeth Anderson as examples of how nonoppressive ideals might be incorporated into normative theory. (shrink)

According to Charles Travis, Frege’s principle “always to sharply separate the psychological from the logical, the subjective from the objective” involves a move called “the fundamental abstraction.” I try to explain what this abstraction is and why it is interesting. I then raise a problem for it, and describe what I think is a better way to understand Frege’s principle.

The aim of this essay is to emphasize a number of important points that will provide a better understanding of the history of philosophical thought concerning scientific knowledge. The main points made are: (a) that the principal way of viewing abstraction which has dominated the history of thought and epistemology up to the present is influenced by the original Aristotelian position; (b) that with the birth of modern science a new way of conceiving abstraction came into being which is better (...) characterized by the term idealization, the name that was later, in fact, to be used by scientists to describe their scientific activity; (c) that, however, on account of the influence of empirical and inductive philosophy, scientists have often not had sufficient methodological awareness of this new way of viewing abstraction; (d) that this new concept of abstraction has frequently been expressed in the framework of philosophies that lie outside the mainstream of contemporary epistemology or even exhibit marked anti-scientific tendencies; (e) that the theme of idealization has been taken up again in the last few decades and a great contribution in this direction has been made by the so-called Pozna school of methodology. (shrink)

Talkers are recognized more accurately if they are speaking the listeners’ native language rather than an unfamiliar language. This “language familiarity effect” has been shown not to depend upon comprehension and must instead involve language sound patterns. We further examine the level of sound-pattern processing involved, by comparing talker recognition in foreign languages versus two varieties of English, by English speakers of one variety, English speakers of the other variety, and non-native listeners. All listener groups performed better with native than (...) foreign speech, but no effect of language variety appeared: Native listeners discriminated talkers equally well in each, with the native variety never outdoing the other variety, and non-native listeners discriminated talkers equally poorly in each, irrespective of the variety's familiarity. The results suggest that this talker recognition effect rests not on simple familiarity, but on an abstract level of phonological processing. (shrink)

This paper investigates the roles that abstraction and representation have in activities associated with language. Activities such as associative learning and counting require both the abilities to abstract from and accurately represent the environment. These activities are successfully carried out among vocal learners aside from humans, thereby suggesting that nonhuman animals share something like our capacity for abstraction and representation. The identification of these capabilities in other species provides additional insights into the development of language.

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)

The dual-code proposal of number representation put forward by Cohen Kadosh & Walsh (CK&W) accounts for only a fraction of the many modes of numerical abstraction. Contrary to their proposal, robust data from human infants and nonhuman animals indicate that abstract numerical representations are psychologically primitive. Additionally, much of the behavioral and neural data cited to support CK&W's proposal is, in fact, neutral on the issue of numerical abstraction.

We focus on the theory of abstraction proposed by the target article. We suggest that abstraction varies at different levels of learning, cognitive development, or cognitive ability. We argue that this theory does not specify how abstraction is done at each of these levels. Because of these weaknesses, the theory cannot explicate how individuals differ in mental time travel at different phases of life or different levels of cognitive ability.

Gilead et al. offer a thoughtful and much-needed treatment of abstraction. However, it fails to build on an extensive literature on abstraction, representational diversity, neurocognition, and psychopathology that provides important constraints and alternative evidence-based conceptions. We draw on conceptions in software engineering, socio-technical systems engineering, and a neurocognitive theory with abstract representations of gist at its core, fuzzy-trace theory.

A perceptual learning experiment provides evidence that the mental lexicon cannot consist solely of detailed acoustic traces of recognition episodes. In a training lexical decision phase, listeners heard an ambiguous [f–s] fricative sound, replacing either [f] or [s] in words. In a test phase, listeners then made lexical decisions to visual targets following auditory primes. Critical materials were minimal pairs that could be a word with either [f] or [s] (cf. English knife–nice), none of which had been heard in training. (...) Listeners interpreted the minimal pair words differently in the second phase according to the training received in the first phase. Therefore, lexically mediated retuning of phoneme perception not only influences categorical decisions about fricatives (Norris, McQueen, & Cutler, 2003), but also benefits recognition of words outside the training set. The observed generalization across words suggests that this retuning occurs prelexically. Therefore, lexical processing involves sublexical phonological abstraction, not only accumulation of acoustic episodes. (shrink)

Formal abstraction is a key instrument Aquinas employs to secure the possibility of mathematics conceived as a theoretical Aristotelian science. In this concept, mathematics investigates quantitative beings, which are grasped by means of formal abstraction in their necessary, universal, and changeless properties. Based on this, the paper divides into two main parts. In the first part (section II) I explicate Aquinas’s conception of (formal) abstraction against the background of the Aristotelian theory of science and mathematics. In the second part (section (...) III) I present and critically assess the problems associated with formal abstraction in mathematics. With all due respect to Aquinas’s genius, I find his conception of formal abstraction (as well as mathematics) unsatisfactory and list the main reasons for its failure in the conclusion. (shrink)

Non-reductive physicalists have long held that the special sciences offer explanations of some phenomena that are objectively superior to physical explanations. This explanatory “autonomy” has largely been based on the multiple realizability argument. Recently, in the face of the local reduction and disjunctive property responses to multiple realizability, some defenders of non-reductive physicalism have suggested that autonomy can be grounded merely in human cognitive limitations. In this paper, I argue that this is mistaken. By distinguishing between two kinds of abstraction (...) I show that the greater explanatory relevance of some special science predicates (to certain explananda) is both non-anthropocentric and not solely based on considerations of multiple realizability. This shows that the explanatory autonomy of the special sciences is safe from the local reduction and disjunctive property strategies, even if they are successful responses to the multiple realizability argument. (shrink)

This article improves two existing theorems of interest to neologicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation E is defined without nonlogical vocabulary, then the bicardinal slice of any equivalence class—those equinumerous elements of the equivalence class with equinumerous complements—can have one of only three profiles. The improvements to Fine’s theorem allow for an analysis of (...) the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan’s relative categoricity theorem. (shrink)

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.

In Die Grundlagen der Arithmetik, Frege attempted to introduce cardinalnumbers as logical objects by means of a second-order abstraction principlewhich is now widely known as ``Hume's Principle'' (HP): The number of Fsis identical with the number of Gs if and only if F and G are equinumerous.The attempt miscarried, because in its role as a contextual definition HP fails tofix uniquely the reference of the cardinality operator ``the number of Fs''. Thisproblem of referential indeterminacy is usually called ``the Julius Caesar (...) problem''.In this paper, Frege's treatment of the problem in Grundlagen is critically assessed. In particular, I try to shed new light on it by paying special attention to the framework of his logicism in which it appears embedded. I argue, among other things, that the Caesar problem, which is supposed to stem from Frege's tentative inductive definition of the natural numbers, is only spurious, not genuine; that the genuine Caesar problem deriving from HP is a purely semantic one and that the prospects of removing it by explicitly defining cardinal numbers as objects which are not classes are presumably poor for Frege. I conclude by rejecting two closely connected theses concerning Caesar put forward by Richard Heck: (i) that Frege could not abandon Axiom V because he could not solve the Julius Caesar problem without it; (ii) that (by his own lights) his logicist programme in Grundgesetze der Arithmetik failed because he could not overcome that problem. (shrink)

Cases where analogy has played a significant role in the formation of a new scientific concept are well-documented. Yet, how is it that genuinely new representations can be constructed from existing representations? It is argued that the process of ‘generic modeling’ enables abstraction of features common to both the domain of the source of the analogy and of the target phenomena. The analysis focuses on James Clerk Maxwell's construction of the electromagnetic field concept. The mathematical representation Maxwell constructed turned out (...) to be a system of abstract laws that when applied to electromagnetic systems yield laws of a dynamical system that will not map back onto the mechanicals domains used in their construction. (shrink)

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)

Fitch's basic logic is an untyped illative combinatory logic with unrestricted principles of abstraction effecting a type collapse between properties (or concepts) and individual elements of an abstract syntax. Fitch does not work axiomatically and the abstraction operation is not a primitive feature of the inductive clauses defining the logic. Fitch's proof that basic logic has unlimited abstraction is not clear and his proof contains a number of errors that have so far gone undetected. This paper corrects these errors and (...) presents a reasonably intuitive proof that Fitch's system K supports an implicit abstraction operation. Some general remarks on the philosophical significance of basic logic, especially with respect to neo-logicism, are offered, and the paper concludes that basic logic models a highly intensional form of logicism. (shrink)