Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof.

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and (...) proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim. (shrink)

Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case on which (...) to focus in analyzing mathematical depth. After introducing the theorem, four accounts of mathematical depth will be considered. (shrink)

This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.

Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, he used (...) several keywords such as "active intuition" and "self-reflection" from Nishida's philosophy. In this paper, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, after reviewing Takeuti's proof-theoretic results briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy and explain the future goals of our project. (shrink)

Over the years many mathematicians have voiced a preference for proofs that stay “close” to the statements being proved, avoiding “foreign”, “extraneous”, or “remote” considerations. Such proofs have come to be known as “pure”. Purity issues have arisen repeatedly in the practice of arithmetic; a famous instance is the question of complex-analytic considerations in the proof of the prime number theorem. This article surveys several such issues, and discusses ways in which logical considerations shed light on these issues.

Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy in Kyoto's (...) school. For the proving the consistency of such systems, it is crucial to prove the well-foundedness of ordinals called "ordinal diagrams" developed for it. Takeuti presented such arguments several times in order to show that they are admitted in his stand point. As a starting point of investigating his finitist stand point, we formulate the system of ordinal notations up to ε0 and reconstruct the well-foundedness arguments of them. (shrink)

Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...) naïf, il semblerait que la première soit pure et la seconde impure. Des objections à cette vue naïve sont ici considérées et réfutées. Concernant la preuve euclidienne, la question relève de la logique, notamment de la définissabilité arithmétique de l’addition en termes de successeur et de divisibilité telle que démontrée par Julia Robinson, tandis qu’en ce qui concerne la preuve topologique, la question relève de la sémantique, notamment pour tout ce qui touche au problème de savoir ce qui est « inclus » dans le contenu d’énoncés particuliers.A proof is pure, roughly, if it draws only on what is « close » or « intrinsic » to the statement being proved. The infinitude of prime numbers, a classical theorem of arithmetic, is a rich case study for philosophical investigation of purity. Two different proofs of this result are considered, namely the classical Euclidean proof and a more recent « topological » proof by Furstenberg. Naively the former would seem to be pure and the latter to be impure. Objections to these naive views are considered and met. In the case of the former the issue rests on logical matters, specifically the arithmetic definability of addition in terms of successor and divisibility shown by Julia Robinson, while in the case of the latter the issue rests on semantic matters, specifically with respect to what is « contained » in the content of particular statements. (shrink)

In this paper I begin by extending two results of Solovay; the first characterizes the possible Turing degrees of models of True Arithmetic (TA), the complete first-order theory of the standard model of PA, while the second characterizes the possible Turing degrees of arbitrary completions of P. I extend these two results to characterize the possible Turing degrees of m-diagrams of models of TA and of arbitrary complete extensions of PA. I next give a construction showing that the conditions Solovay (...) identified for his characterization of degrees of models of arbitrary completions of PA cannot be dropped (I showed that these conditions cannot be simplified in the paper. (shrink)

Review of Dowek, Gilles, Computation, Proof, Machine, Cambridge University Press, Cambridge, 2015. Translation of Les Métamorphoses du calcul, Le Pommier, Paris, 2007. Translation from the French by Pierre Guillot and Marion Roman.

When mathematicians think of the philosophy of mathematics, they probably think of endless debates about what numbers are and whether they exist. Since plenty of mathematical progress continues to be made without taking a stance on either of these questions, mathematicians feel confident they can work without much regard for philosophical reflections. In his sharp–toned, sprawling book, David Corfield acknowledges the irrelevance of much contemporary philosophy of mathematics to current mathematical practice, and proposes reforming the subject accordingly.

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late nineteenth century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis (in the sense of the infinitesimal calculus) received much attention in the nineteenth century. They helped instigate what Hans Hahn called a “crisis of intuition”, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this “crisis” as (...) follows: Mathematicians had for a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof... (p. 67) Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called “logicization”, “when the discipline requires nothing but purely logical fundamental concepts and propositions for its development.” On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals to visual representations. This would be a radical reformation of past practice, necessary, according to its advocates, for avoiding “unclarities and mistakes” like the one exposed by Peano. (shrink)

This collection of essays explores what makes modern mathematics ‘modern’, where ‘modern mathematics’ is understood as the mathematics done in the West from roughly 1800 to 1970. This is not the trivial matter of exploring what makes recent mathematics recent. The term ‘modern’ (or ‘modernism’) is used widely in the humanities to describe the era since about 1900, exemplified by Picasso or Kandinsky in the visual arts, Rilke or Pound in poetry, or Le Corbusier or Loos in architecture (a building (...) by the latter graces the cover of this book’s dust jacket). (shrink)

We review Solomon Feferman's 1998 essay collection In The Light of Logic (Oxford University Press).

This paper examines the idea of objective phenomenology, or a way of understanding the phenomenal character of conscious experiences that doesn’t require one to have had the kinds of experiences under consideration. My central thesis is that structural facts about experience—facts that characterize purely how conscious experiences are structured—are objective phenomenal facts. I begin by precisifying the idea of objective phenomenology and diagnosing what makes any given phenomenal fact subjective. Then I defend the view that structural facts about experience are (...) objective. I also argue that structural facts about experience, despite being objective, nevertheless still give rise to an explanatory gap. (shrink)

This is a discussion of the state of discussion on pragmatic reasons for belief.

While many aspects of cognition have been shown to be shared between humans and non-human animals, there remains controversy regarding whether the capacity to mentally time travel is a uniquely human one. In this paper, we argue that there are four ways of representing when some event happened: four kinds of temporal representation. Distinguishing these four kinds of temporal representation has five benefits. First, it puts us in a position to determine the particular benefits these distinct temporal representations afford an (...) organism. Second, it provides the conceptual resources to foster a discussion about which of these representations is necessary for an organism to count as having the capacity to mentally time travel. Third, it enables us to distinguish stricter from more liberal views of mental time travel that differ regarding which kind(s) of temporal representation is taken to be necessary for mental time travel. Fourth, it allows us to determine the benefits of taking a stricter or more liberal view of mental time travel. Finally, it ensures that disagreement about whether some species can mentally time travel is not merely the product of unrecognised disagreement about which temporal representation is necessary for mental time travel. We argue for a more liberal view, on the grounds that it allows us to view mental time travel as an evolutionarily continuous phenomenon, and to recognise that differences in the ways that organisms mentally time travel might reflect different temporal representations, or combinations thereof, that they employ. Our ultimate aim, however, is to create the conceptual framework for further discussion regarding what sorts of temporal representations are required for mental time travel. (shrink)

Perception is riddled with cases of reliable misperception. These are cases in which a perceptual state is tokened inaccurately any time it is tokened under normal conditions. On the face of it, this fact causes trouble for theories that provide an analysis of perceptual content in non-semantic, non-intentional, and non-phenomenal terms, such as those found in Millikan (1984), Fodor (1990), Neander (2017), and Schellenberg (2018). I show how such theories can be extended so that they cover such cases without giving (...) up their core commitments. (shrink)

The fallacy fallacy is either the misdiagnosis of fallacy or the supposition that the conclusion of a fallacy must be a falsehood. This paper explores the relevance of these and related errors of reasoning for the appraisal of arguments, especially within virtue theories of argumentation. In particular, the fallacy fallacy exemplifies the Owl of Minerva problem, whereby tools devised to understand a norm make possible new ways of violating the norm. Fallacies are such tools and so are vices. Hence a (...) similar problem arises with argumentative vices. Fortunately, both instances of the problem have a common remedy. (shrink)

This chapter addresses an ambiguity in some of the literature on rational peer disagreement about the use of the term 'rational'. In the literature 'rational' is used to describe a variety of normative statuses related to reasons, justification, and reasoning. This chapter focuses most closely on the upshot of peer disagreement for what is rationally required of parties to a peer disagreement. This follows recent work in theoretical reason which treats rationality as a system of requirements among an agent's mental (...) states. It is argued that peer disagreement has either no, or a very limited, affect on what rationality requires of an agent in a given circumstance. This is in part because of difficulties generated by a novel example of evidence of evidence of p being evidence against p. This example calls into question the mechanisms whereby peer disagreement might affect what is rationally required of an agent. The chapter also reevaluates the importance of actual peer disagreement against the backdrop of prior expectations about whether disagreement is believed to be likely, arguing that peer disagreement is most likely to change what is rationally required of an agent when it is believed to be unlikely. (shrink)

To be conscious—according to a common metaphor—is for the “lights to be on inside.” Is this a good metaphor? I argue that the metaphor elicits useful intuitions while staying neutral on controversial philosophical questions. But I also argue that there are two ways of interpreting the metaphor. Is consciousness the inner light itself? Or is consciousness the illuminated room? Call the first sense subjectivity (where ‘consciousness’ =def what makes an entity feel some way at all), and the second sense phenomenal (...) character (where ‘consciousness’ =def what it feels like to be an entity). I use this distinction—as well as the metaphor of the inner light—to clarify some philosophical questions about whether consciousness comes in degrees, whether consciousness is multidimensional, and whether there are borderline cases of consciousness. (shrink)

Many say that ontological disputes are defective because they are unimportant or without substance. In this paper, we defend ontological disputes from the charge, with a special focus on disputes over the existence of composite objects. Disputes over the existence of composite objects, we argue, have a number of substantive implications across a variety of topics in metaphysics, science, philosophical theology, philosophy of mind, and ethics. Since the disputes over the existence of composite objects have these substantive implications, they are (...) themselves substantive. (shrink)

The aim of the Analytic of Concepts is to derive and deduce a set of pure concepts of the understanding, the categories, which play a central role in Kant’s explanation of the possibility of synthetic a priori cognition and judgment. This chapter is structured around two questions. First, what is a pure concept of the understanding? Second, what is involved in a deduction of a pure concept of the understanding? In answering the first, we focus on how the categories differ (...) from the pure forms of sensibility and examine whether they are known only to be the pure forms of human thinking or rather the forms of discursive cognition as such. In answering the second, we draw a distinction between the application and the exemplification of the categories and use it to identify different ways of understanding Kant’s project in the Transcendental Deduction of the Categories. These questions connect because the answer to the first determines how we should understand the kind of entitlement which is to be established in the Deduction. (shrink)

For Kant, knowledge involves certainty. If “certainty” requires that the grounds for a given propositional attitude guarantee its truth, then this is an infallibilist view of epistemic justification. Such a view says you can’t have epistemic justification for an attitude unless the attitude is also true. Here I want to defend an alternative fallibilist interpretation. Even if a subject has grounds that would be sufficient for knowledge if the proposition were true, the proposition might not be true. And so there (...) is sometimes still rational room for doubt. The goal of this paper is to present four different models of what “certainty” amounts to, for Kant, each of which is compatible with fallibilism. (shrink)

The norm of monogamy is pervasive, having remained widespread, in most Western cultures at least, in spite of increasing tolerance toward more diverse relationship types. It is also puzzling. People willingly, and often with gusto, adhere to it, yet it is also, prima facie at least, highly restrictive. Being in a monogamous relationship means agreeing to give up certain sorts of valuable interactions and relationships with other people and to severely restrict one’s opportunities for sex and love. It is this (...) restrictiveness which has led for calls among some philosophers for the justification of the norm, and some have argued that it cannot be justified. John McMurtry, for example, writing in 1972, compared monogamy to private property and argued that it serves the capitalist order (McMurtry 1972, 596–97). More recently, Harry Chalmers (2019) has argued that monogamy is morally impermissible. In this chapter we argue that rather than justify the norm, we ought to explain and understand its pervasiveness. Not everyone who follows the norm is doing so blindly, irrationally, or immorally. Indeed, many people are monogamous because being monogamous supports their needs and best enables them to flourish. We suggest that the human need for trust and attachment could underlie many people’s desires for monogamy and, therefore, explain monogamy’s pervasiveness. Focusing on the human need for trust and attachment security can also help us to understand why sexual infidelity can feel like such a betrayal. It can make sense to mutually agree to exclusivity in relationships, in spite of the apparent restrictions this brings. In a sense, not all restrictions are restrictive, because holding ourselves to limits on our behavior can in fact help us to invest in and experience the things we value most. (shrink)

Matteo Ricci (1552-1610) was an Italian Jesuit and one of the first Christian missionaries to China in the modern era. He was a genuine polymath—a translator, cartographer, mathematician, astronomer, and musician. Above all, Ricci was a missionary for the gospel. As we briefly examine his 1603 seminal work, The True Meaning of the Lord of Heaven, our hope is that we, as evangelical educators, will perceive some of the deeper principles necessary for our own missionary work among unreached people.

This entry discusses the notion of a unit of normativity. This notion may be understood in two distinct ways. One way to understand a unit of normativity is as some particular type of assignment of normative status, e.g., a requirement, an ought, a reason, or a permission. A second way to understand a unit of normativity is as a measure of a quantity of normativity, perhaps associated with the numerical assignment given to the strength of reasons. This entry outlines some (...) basic differences among units of normativity in the first sense, noting that they vary slightly depending on whether one is talking about normativity in a more general or more robust sense. This entry also discusses in more detail the question of whether there might be a unit of normativity in the second sense. It discusses the relevant metaphysical questions. It also provides an explanation of why reasons can be assigned numerical strengths, even if there are no units of normativity in the sense of measurements of quantities of normativity. (shrink)

This paper outlines a new pluralistic theory of normative reasons for belief, welfarist pluralism, which aims to explain how there can be basic alethic/epistemic reasons for belief and basic pragmatic/non-alethic reasons for belief that can combine to determine what one ought to believe. The paper shows how this non-derivative first-order pluralism arises from a purely welfarist account of the foundations of theoretical normativity, thereby combining foundational pragmatism with non-derivative pluralism about normative reasons for belief. In addition, this paper outlines how (...) welfarist pluralism provides a new and potentialy attractive account of the value of truth (or its indicators), namely that the value of truth (or its indicators) is its direct contribution to individuals' personal good or wellbeing. (shrink)

Let a smooth experience be an experience with perfectly gradual changes in phenomenal character. Consider, as examples, your visual experience of a blue sky or your auditory experience of a rising pitch. Do the phenomenal characters of smooth experiences have continuous or discrete structures? If we appeal merely to introspection, then it may seem that we should think that smooth experiences are continuous. This paper (1) uses formal tools to clarify what it means to say that an experience is continuous (...) or discrete, and (2) develops a discrete model of the phenomenal characters of smooth experiences. As a result, I'll argue that introspection leaves open whether smooth experiences are continuous or discrete. Yet I’ll also argue—perhaps surprisingly—that the discrete theory may better fit our introspective evidence. (shrink)

I argue that the following theses are both popular among evidentialists but also jointly inconsistent with evidentialism: 1) Time-Slice Mentalism: one’s justificational properties at t are grounded only by one’s mental properties at t; 2) Experience Ultimacy: all ultimate evidence is experiential; and 3) Sleep Justification: we have justified beliefs while we have dreamless, nonexperiential sleep. Although I intend for this paper to be a polemic against evidentialists, it can also be viewed as an opportunity for them to clarify their (...) views. Furthermore, the paper is not only relevant to evidentialists. For example, the arguments of this paper could give Time-Slice Mentalists a reason to deny evidentialism. (shrink)

The classical Confucian philosopher Xunzi proposed a naturalistic virtue ethics account of ritual: rituals are practices that channel human emotion and desire so that one develops virtues. In this paper I show that William of Auxerre’s Summa de Officiis Ecclesiasticis can be understood as presenting a similar account of ritual. William places great emphasis on the emotional power of the liturgy, which makes participants like the blessed in heaven by developing virtue. In other words, he has a virtue ethics of (...) ritual closely aligned with that of Xunzi. Xunzi’s writings on ritual illuminate and enrich one’s reading of the Summa de Officiis. But unlike Xunzi, William is not a naturalist with regard to ritual: although much of William’s language about the causal power of liturgy can be explained in Xunzian terms, Christian liturgy has an irreducible supernatural element. (shrink)

How is it that a speaker S can at once make it obvious to an audience A that she intends to communicate some proposition p, and yet at the same time retain plausible deniability with respect to this intention? The answer is that S can bring it about that A has a high justified credence that ‘S intended p’ without putting A in a position to know that ‘S intended p’. In order to achieve this S has to exploit a (...) sense in which communication can be lottery-like. After defending this view of deniability I argue that it compares favorably to a rival account recently developed by Dinges and Zakkou (2023). (shrink)

In recent years, the science and the philosophy of consciousness has seen growing interest in structural questions about consciousness. This is the Editorial Introduction for a special volume for Philosophy and the Mind Sciences on “Structuralism in Consciousness Studies.”.

This overview of the philosophy of certainty will distinguish two types of certainty, specify controversial theses about certainty from recent literature, and explain some of the arguments for and against those theses.

Our aim in this chapter is to shed light on Kant’s account of the pure forms of sensibility by focusing on a somewhat neglected issue: Kant’s restriction of his claims about space and time to the case of human sensibility. Kant argues that space and time are the pure forms of sensibility for human cognizers. But he also says that we cannot know whether space and time are likewise the pure forms of sensibility for all discursive cognizers. A great deal (...) of attention has focused on the first of these claims, both on how Kant argues for it and how it relates to transcendental idealism. But a satisfactory interpretation must also account for the second claim, and it must account for the fact that Kant endorses both of them. What we need is an explanation of why Kant thinks our knowledge that space and time are the pure forms of sensibility extends to all human beings but no further. (shrink)

The official definitions that appear at the beginning of four of the five parts of the "Ethics" do not include an account of "existence." However Spinoza does provide a definition of “existence” in the scholium to proposition 45 of Part 2. This is an odd place for such an important doctrine, and all the more so given that the account there differs from anything resembling commonsense. In this paper I show that, for Spinoza, to exist is to be eternal. Existence (...) in the strict sense is eternal existence. (shrink)

This is a critical, but sympathetic, examination of the manifesto for naturalized metaphysics that forms the first chapter of James Ladyman and Don Ross's 2006 book, Every Thing Must Go, but it has wider implications than this description suggests.

This entry summarizes arguments for and against the thesis that knowledge entails belief.

The fact that physical laws often admit certain kinds of space-time symmetries is often thought to be problematic for substantivalism --- the view that space-time is as real as the objects it contains. The most prominent alternative, relationism, avoids these problems but at the cost of giving abstract objects (rather than space-time points) a pivotal role in the fundamental metaphysics. This incurs related problems concerning the relation of the physical to the mathematical. In this paper I will present a version (...) of substantivalism that respects Leibnizian theses about space-time symmetries, and argue that it is superior to both relationism and the more orthodox form of substantivalism. (shrink)

This chapter articulates two concerns that Karl Jaspers raised (with Hannah Arendt) about the common practice of viewing moral evil as unintelligible. The first is that this involves exoticizing the act and/or perpetrator in such a way that moral condemnation becomes difficult. The second is that it can lead us to treat the perpetrator, place, or victim as tainted or stained by a force whose motives we cannot grasp; this in turn can lead to magical thinking about evil as somehow (...) contagious or contaminating. After distinguishing some of the main categories of evil discussed in the western tradition, I examine ways in which moral evil, in particular, has been characterized as unintelligible, and try to discern which of them raises these Jaspersian concerns. I argue that there are at least two conceptions of “radical evil”—not the Kantian one, but the ones articulated by Hannah Arendt—that do so. (shrink)

I defend a neo-Kantian view wherein we are capable of being completely autonomous and impartial and argue that this ability can ground normativity. As this view includes an existentialist conception of the self, I defend radical choice, a primary component of that conception, against arguments many take to be definitive. I call the ability to use radical choice “existentialist voluntarism” and bring it into a current debate in normative philosophy, arguing that it allows that we can be distanced from all (...) ends at once so as to be completely impartial. Finally, I indicate how this can be the source of normativity as it provides a purely impartial reason for being rational. (shrink)

This paper argues against the correctness of the more popular -first views (fittingness first, reasons first, and value first). The aim of this paper is to argue that there are at least two categories of normative properties that are basic and that one of those categories is value. Although the emphasis of this paper is on reductionist -first accounts, most of the arguments work equally well against weaker views that are only committed to different categories of normative properties' being linked (...) by a necessary bi-conditional. -/- . (shrink)

In this chapter, we argue that a distinct concept of “aesthetic hope” emerges from the way Kierkegaard’s Aesthete treats hope [Haab] and its relationship to recollection [Erindring] in “The Unhappiest One” and “Rotation of Crops.” We first show that aesthetic hope is distinct from the two other kinds of hope discussed by Kierkegaard: temporal hope and eternal hope. We then consider the suggestion that aesthetic hope is also an expression of despair – an inverse hope against hope, which seeks to (...) avoid disappointment by hoping for things that are in some sense certain. The aesthete’s recommendation that we hope in such a way illuminates Kierkegaard’s view of the “dialectic” of temporal hope and eternal hope. Finally, we explore the treatment of hope in Either/Or as essentially involving a controlled, attentional element that anticipates some contemporary trends in the philosophy of hope. (shrink)