We discuss abstraction principles in the context of modal and temporal logic. It is argued that abstractionism conflicts with both serious presentism and serious actualism.

In this paper, I present two crucial problems for Wolff’s metaphysics of quantities: 1) The structural identification problem and 2) the Pythagorean problem. The former is the problem of uniquely defining a general algebraic structure for all quantities; the latter is the problem of distinguishing physical quantitative structure from mathematical quantities. While Wolff seems to have a consistent and elegant solution to the first problem, the second problem may put in jeopardy his metaphysical view on quantities as spaces. After drawing (...) a parallelism between Wolff’s treatment of quantitative structures and Frege’s conception of quantitative domain, I propose a solution to the Pythagorean problem based on the idea that mathematical structures are the result of applying an abstraction principle on physical quantitative structures. In particular, I propose the view that abstraction may be seen as the operation of structure determination which transforms concrete physical quantities (i.e. undetermined structures) into abstract mathematical quantities (fully determined structures of thin and shallow objects). (shrink)

The idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume’s Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one-to-one: (HP) #xx=#yy iff xx≈yy. The principal aim of this article is to use the notion of (...) grounding to develop this sort of abstractionism. The appeal to grounding enables a unified response to the two main challenges that confront abstractionism. First, we must explicate the metaphor of meta- physical “cheapness.” Second, we must rebut the “bad company” objection, which rejects abstraction principles like (HP) as tarnished by their similarity to inconsistent principles like Frege’s Basic Law V. By enforcing a simple requirement that all abstraction be properly grounded, we propose a unified solution to these two hard, and prima facie unrelated, problems. On our view, grounded abstraction simultaneously ensures “cheap” abstracta and permissible abstraction. (shrink)

In his "Two concepts of possible worlds", Peter Van Inwagen explores two kinds of views about the nature of possible worlds : abstractionism and concretism. The latter is the view defended by David Lewis who claims that possible worlds are concrete spatio-temporal universes, very much like our own, causally and spatio-temporally disconnected from each other. The former is the view of the majority who claims that possible worlds are some kind of abstract objects – such as propositions, properties, states (...) of affairs, or sets of numbers. In this paper, I will develop this view in an 'extreme abstractionist' way, appealing to a 'modal bundle theory', and I will try to show that it is preferable to the standard abstractionist ones. Finally, I will compare this kind of abstractionism to concretism, only to find that the difference between the two is minimal. (shrink)

One prominent criticism of the abstractionist program is the so-called Bad Company objection. The complaint is that abstraction principles cannot in general be a legitimate way to introduce mathematical theories, since some of them are inconsistent. The most notorious example, of course, is Frege’s Basic Law V. A common response to the objection suggests that an abstraction principle can be used to legitimately introduce a mathematical theory precisely when it is stable: when it can be made true on all sufficiently (...) large domains. In this paper, we raise a worry for this response to the Bad Company objection. We argue, perhaps surprisingly, that it requires very strong assumptions about the range of the second-order quantifiers; assumptions that the abstractionist should reject. (shrink)

"The main aim of the paper is to compare two types of abstractionistic accounts of fictional objects, and to analyze their consequences for interpretation of existential quantification. According to a proponent of general abstractionistic theory, fictional objects have abstract nature in a way similar to contracts, marriages, and the likes. This view is an alternative to strongly realistic accounts of fictional objects, defended by Terence Parsons or David Lewis. Within abstractionistic theories, as in all philosophical areas, one can find divergences (...) of opinions. The main differences between two of them – Peter van Inwagen's and Edward Zalta's – are connected with the interpretation of existential quantification. According to van Inwagen, “being” is the same as “existence” and its sense is captured by the existential quantifier. Edward Zalta's theory is much closer to the Meinongian Theory of Objects. He argues for the need of distinguishing between “being” and “existence” and for invoking nonexistent objects. Because of that he suggests an alternative interpretation of quantification. Admittedly, there is one abstractionistic source for both theories, but their ontological consequences are different - van Inwagen is a staunch opponent of nonexistent objects, and Zalta describes his own theory as "Meinongian".". (shrink)

Abstractionism in the philosophy of mathematics aims at deriving large fragments of mathematics by combining abstraction principles (i.e. the abstract objects $\S e_1, \S e_2$, are identical if, and only if, an equivalence relation $Eq_\S$ holds between the entities $e_1, e_2$) with logic. Still, as highlighted in work on the semantics for relevant logics, there are different ways theories might be combined. In exactly what ways must logic and abstraction be combined in order to get interesting mathematics? In this (...) paper, we investigate the matter by deriving the axioms of second-order Peano Arithmetic from Frege's Basic Law V (the extension of $F$ is identical with the extension of $G$ if, and only if, $F$ and $G$ are extensionally equivalent) in the presence of a relevant higher-order logic. The results are interesting. Not only must we take on logic as true, and not only must we apply our logic to abstraction principles, but also we have to apply our theory of abstraction back to the logic in order to arrive at arithmetic. Thus, what Abstractionism gives us is not simply what we get from abstraction via logic, but also what we get from logic via abstraction. (shrink)

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

A brief review of Øystein Linnebo's Thin Objects. The review ends with a brief discussion of cardinal number and metaphysical ground.

We note that a plural version of logicism about arithmetic is suggested by the standard reading of Hume's Principle in terms of `the number of Fs/Gs'. We lay out the resources needed to prove a version of Frege's principle in plural, rather than second-order, logic. We sketch a proof of the theorem and comment philosophically on the result, which sits well with a metaphysics of natural numbers as plural properties.

We can introduce singular terms for ordered pairs by means of an abstraction principle. Doing so proves useful for a number of projects in the philosophy of mathematics. However there is a question whether we can appeal to the abstraction principle in good faith, since a version of the Caesar Problem can be generated, posing the worry that abstraction fails to introduce expressions which refer determinately to the requisite sort of object. In this note I will pose the difficulty, and (...) then propose a solution. Since my solution appeals to a plausible constraint on the introduction of new expressions to a language, it is of interest independently of the particular case of terms for pairs. Since these provide the occasion for discussion we should nonetheless review the use of abstraction for pairs before the argumentative business of the paper commences. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter 2 argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter 3 provides an abstraction principle for epistemic (hyper-)intensions. Chapter 4 advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter 5 applies the fixed points of the modal μ-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal μ-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μ-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter 6 advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter 7 provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters 2 and 4 are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters 8-12 provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter 8 examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of Ω-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter 9 examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ω-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter 11 provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter 12 avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter 13 examines modal responses to the alethic paradoxes. Chapter 14 examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

Neo-Fregeans such as Bob Hale and Crispin Wright seek a foundation of mathematics based on abstraction principles. These are sentences involving a relation called the abstraction relation. It is usually assumed that abstraction relations must be equivalence relations, so reflexive, symmetric and transitive. In this article I argue that abstraction relations need not be reflexive. I furthermore give an application of non-reflexive abstraction relations to restricted abstraction principles.

Reification is to abstraction as disease is to health. Whereas abstraction is singling out, symbolizing, and systematizing, reification is neglecting abstractive context, especially functional, historical, and analytical-level context. William James and John Dewey provide similar and nuanced arguments regarding the perils and promises of abstraction. They share an abstraction-reification account. The stages of abstraction and the concepts of “vicious abstractionism,” “/the/ psychologist’s fallacy,” and “the philosophic fallacy” in the works of these pragmatists are here analyzed in detail. For instance, (...) in 1896 Dewey exposes various fallacies associated with reifying dualistic reflex arc theory. The conclusion prescribes treatments (pluralism and assumption archaeology) for de-reifying ill models (i.e., universalized, narrowed, and ontologized models) in contemporary scientific fields such as cognitive science and biology. (shrink)

This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by (...) Hale and Wright and examined in Hale (2013), and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest that observational type theory can be applied to first-order abstraction principles in order to make abstraction principles recursively enumerable. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics. (shrink)

This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open versus closed intervals (...) is discussed. Finally, it is suggested that one reason there is a common structure between Aristotle's account of the continuum and that found in Cantor's definition of the real number continuum is that our intuitions about the continuum have their source in the experience of the real spatiotemporal world. A plea is made to consider Aristotle's abstractionist philosophy of mathematics anew. (shrink)

How do we form concepts like those of three, bicycle and red? According to Kant, we form them by carrying out acts of comparison, reflection and abstraction on information provided by the senses. Kant's answer raised numerous objections from philosophers and psychologists alike. "Kant e la formazione dei concetti" argues that Kant is able to rebut those objections. The book shows that, for Kant, it is possible to perceive objects without employing concepts; it explains how, given those perceptions, we can (...) form categories and empirical concepts; and it argues that theories like Kant's - abstractionist theories of concept formation - are more plausible than is often assumed. (shrink)

The Gestalt principle of isomorphism reveals the primacy of subjective experience as a valid source of evidence for the information encoded neurophysiologically. This theory invalidates the abstractionist view that the neurophysiological representation can be of lower dimensionality than the percept to which it gives rise.

How to think a unique and determinative turn in analytic philosophy of mind? To answer this question this article first presents an attempt to render clear that analytic phenomenology, by contrast with conceptions of phenomenology of the XXth century, beneficially dispenses with several methodological and conceptual assumptions that were assumed to be compulsory, as phenomenological reduction, a notion of synthesis, and a philosophical notion of the a priori. It then presents some eventual difficulties to the achievement of a phenomenological turn (...) within analytic philosophy, which are, the neglect of historicity, abstractionism, the acknowledgement of the place of language in our lives, and solipsism. It finally presents several demands that concern the felicity of contemporary analytic phenomenologies, namely, anti-abstractionism, fallibilism, attention to polyadic relations, and the integration of ecological and decolonial concerns of our cultures. (shrink)

The tendency in semiotics toward unnecessary abstractionism and antinaturalism is criticized. More broadly, a transformation is proposed from abstractionism, with its fetishism of signs, to an animism of signs in which the imagination and the signs it gives birth to not only reconnect with the biocultural heritage, but also animate an idea of culture as involving living purpose, not simply inert code. See the revised version of this chapter in my book, Meaning and Modernity.

The Caesar problem arises for abstractionist views, which seek to secure reference for terms such as ‘the number of Xs’ or #X by stipulating the content of ‘unmixed’ identity contexts like ‘#X = #Y’. Frege objects that this stipulation says nothing about ‘mixed’ contexts such as ‘# X = Julius Caesar’. This article defends a neglected response to the Caesar problem: the content of mixed contexts is just as open to stipulation as that of unmixed contexts.

When painter Fritz Janschka arrived from Vienna to teach at Byrn Mawr College in October, 1949, he entered a culture seemingly as alien to his art as one can imagine. Janschka is one of the co­founders of the Vienna School of Fantastic Realism, a group of painters who studied at the Academy of Fine Arts in Vienna shortly after World War Two. The fantastic realists cultivated a precisely controlled craft informed by traditional methods and modernist sensibilities, incorporating collectively the entire (...) range of Western painting, past and present, in their attempts to come to grips with a shattered world. New York Abstract Expressionism and Viennese Fantastic Realism, two mid-twentieth century art movements, represent diametrically opposed ways of painting, yet both schools sprang out of surrealism. The Fantastic Realists were widely known in Europe and hardly known in America. Given the dominant abstractionist tendencies of post-World War Two art this is understandable. For these reasons Janschka’s openness to the techniques of the New York School in the early 1950s forms an unlikely meeting point of seemingly irreconcilable positions, one suggestive of possibilities for the meeting of abstraction and figuration, and more broadly, for addressing contemporary dehumanizing abstractionism, which I discuss in this essay. (shrink)

In the essay I intend to revalue the descriptive principle in the contemporaryAmazonian geography, as presented to us by the geographer from Pará EidorfeMoreira (1960). Laterally, I call the attention of Amazonian geographers to thesensitivity of his work, which is not present in the bibliography of the training coursesin Geography in Pará. The methodological strategy is descriptive-interpretative with aphenomenological tone. I conclude: the refusal of the description is installed by aprejudiced effect of our current formation in relation to the procedures (...) consideredtraditional; the production of knowledge, eager for explanation/analysis, can produceethical violence; the abstraction - which flirts with abstractionism - imposed by thehasty generalization of certain ideas about the Amazon, from geographical studies,installs an artificial cut between the symbolic and the emotional in the act of makinggeography in our region. (shrink)

This paper argues in the first place that nominalists are right in insisting against ontological realists that semantic universality does not require commitment to universal entities. However, Ockham, in his zeal to get rid of Scotus’s universal entities, swept under the carpet the issue of universal representational content of genuinely universal symbols, conflating it with the mere indifference of the information content of non-distinctive singular representations. Buridan did come up with an abstractionist theory of the formation of genuinely universal representational (...) content to solve the resulting issues, however, the paper argues further, his solution is committed to attributing the sort of “aspectuality” to universal absolute concepts that his Ockhamist semantics denies to them. The conclusion of the paper suggests how Aquinas’s “moderate realism” can provide a consistent solution without the ontological extravaganza of ontological realists, without conflating the mere indifference of singular representation with genuine universality, and without having to deny aspectuality to our quidditative universal concepts formed by abstraction. (shrink)

In recent metaphysics, the questions of whether fictional entities exist, what their nature is, and how to explain truths of statements such as “Sherlock Holmes lives at 221B Baker Street” and “Holmes was created by Arthur Conan Doyle” have been subject to much debate. The main aim of my thesis is to wrestle with key proponents of the abstractionist view that fictional entities are abstract objects that exist (van Inwagen 1977, 2018, Thomasson 1999 and Salmon 1998) as well as Walton’s (...) (1990) pretense view, which denies the existence of such entities. In the process, I propose modifications to these views to deal with problems they face and show how the modifications better account for the philosophical data. -/- Key abstractionists (van Inwagen 1977, Thomasson 1999) make a strict distinction between discourse within fiction, in which statements about literary characters cannot be literally true, and discourse about fiction, as it occurs in literary criticism, where statements about fictional characters can be literally true. Fictional objects are postulated to account for the truth of the latter. This runs into trouble because statements thought to be literally true are not literal. (Yagisawa 2001, Friend 2002) I provide a uniform analysis to account for the truth of statements involving fictional characters by appealing to a presupposition involving a metaphor in both contexts. The presupposition is that there is an x such that x is fictional; x is likened to a real person; and x is and ought to be treated/counted as a real person for all relevant intents and purposes. -/- More generally, I adopt Everett and Schroeder’s (2015) realist view that fictional characters are ideas constituted by mental representations. This, to me, better accounts for how fictional characters are created within the world’s causal nexus (unlike non-spatiotemporal entities in abstractionism), among other things. One key challenge they face is to explain how ideas can possess properties such as being a detective. I present a fine-grained version of their view, according to which the mental representations constituting fictional entities encode mind-dependent properties. Moreover, I explain how reference to such representations is possible, using Bencivenga’s (1983) Neo-Kantian view of reference and Karttunen’s (1976) view on discourse referents. Finally, I suggest that the identity of fictional characters is interest-relative. The constant, and sometimes radical, change of properties that, fictional characters can undergo is taken to be a consequence of the fact that unified mental representations are bundles of simpler mental representations. As change occurs, simpler representations are replaced by others. -/- A key theme that runs through the thesis is that neither fictionality nor pretense is relevant to the semantics of fictional sentences—a claim bolstered by Matravers’ (2014) arguments. Whether or not my account works, this claim, as well as the new philosophical data I bring up, are some of the challenges I pose to the heart of established views. (shrink)

We can introduce singular terms for ordered pairs by means of an abstraction principle. Doing so proves useful for a number of projects in the philosophy of mathematics. However there is a question whether we can appeal to the abstraction principle in good faith, since a version of the Caesar Problem can be generated, posing the worry that abstraction fails to introduce expressions which refer determinately to the requisite sort of object. In this short paper I will pose the difficulty, (...) and then propose a solution. (shrink)

Abstract:Certain advocates of the so-called “neo-logicist” movement in the philosophy of mathematics identify themselves as “neo-Fregeans” (e.g., Hale and Wright), presenting an updated and revised version of Frege’s form of logicism. Russell’s form of logicism is scarcely discussed in this literature and, when it is, often dismissed as not really logicism at all (in light of its assumption of axioms of infinity, reducibility and so on). In this paper I have three aims: firstly, to identify more clearly the primary meta-ontological (...) and methodological differences between Russell’s logicism and the more recent forms; secondly, to argue that Russell’s form of logicism offers more elegant and satisfactory solutions to a variety of problems that continue to plague the neo-logicist movement (the bad company objection, the embarrassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that neo-Russellian forms of logicism remain viable positions for current philosophers of mathematics. (shrink)

The question of how we come to accept new theories is a central area of inquiry in scientonomic discourse. However, there has yet to be a formal discussion of the subjective reasons an agent may have for accepting theories. This paper explores these epistemic reasons and constructs a historically sensitive definition of reason. This formulation takes an abstractionist stance towards the ontology of reasons and makes use of a composite basing relation. The descriptive and normative components of reasons are fully (...) formulated in scientonomic terms through the application of the newly introduced notion of implication, and its separation from the notion of inference. In addition, the paper provides scientonomic definitions for sufficient reason, support, and normative inference. The fruitfulness of this formulation of reasons is illustrated by a few examples. (shrink)

I discuss two arguments against the view that reasons are propositions. I consider responses to each argument, including recent responses due to Mark Schroeder, and suggest further responses of my own. In each case, the discussion proceeds by comparing reasons to answers and goals.

Neo-Fregean logicists claim that Hume's Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A longstanding problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck's Two-sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn't. (...) In fact, 2FA is not conservative over $n$-th order logic, for all $n \geq 2$. It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic. (shrink)

is happyness productive? See book by Roger Scruton, "On Pessimism".

The question whether numbers are objects is a central question in the philosophy of mathematics. Frege made use of a syntactic criterion for objethood: numbers are objects because there are singular terms that stand for them, and not just singular terms in some formal language, but in natural language in particular. In particular, Frege (1884) thought that both noun phrases like the number of planets and simple numerals like eight as in (1) are singular terms referring to numbers as abstract (...) objects. (shrink)