Mathematical idealizations are scientific representations that result from assumptions that are believed to be false, and where mathematics plays a crucial role. I propose a two stage account of how to rank mathematical idealizations that is largely inspired by the semantic view of scientific theories. The paper concludes by considering how this approach to idealization allows for a limited form of scientific realism. ‡I would like to thank Robert Batterman, Gabriele Contessa, Eric Hiddleston, Nicholaos Jones, and Susan Vineberg (...) for helpful discussions and encouragement. †To contact the author, please write to: Department of Philosophy, Beering Hall, Purdue University, 100 N. University Street, West Lafayette, IN 47907-2098; e-mail: pincock@purdue.edu. (shrink)

This article examines the role of picturability in mathematical demonstration in the seventeenth and eighteenth centuries and draws attention to the general question of the role that picturability places in cognitive grasp. It suggests that mathematical demonstration is particularly applicable in cognitive grasp it allows the problematic to be identified with some precision. It also discusses infinitesimal analysis and the question of direct proof and evaluates the role of picturability in the analysis of human cognitive capacities.

Idealization in mathematics, by its very nature, generates a gap between the theoretical and the practical. This article constitutes an examination of two individual, yet similarly created, cases of mathematical idealization. Each involves using a theoretical extension beyond the finite limits which exist in practice regarding human activities, experiences, and perceptions. Scrutiny of details, however, brings out substantial differences between the two cases, not only in regard to the roles played by the idealized entities, but also in regard to (...) appropriate criteria for justifying the use of such entities. The background information supplied and the examples chosen for analysis in this paper were selected from the areas of measurement theory, probability theory, mathematical logic, and philosophy of mathematics. (shrink)

Husserl's contributions to the nature of mathematical knowledge are opposed to the naturalist, empiricist and pragmatist tendences that are nowadays dominant. It is claimed that mainstream tendences fail to distinguish the historical problem of the origin and evolution of mathematical knowledge from the epistemological problem of how is it that we have access to mathematical knowledge.

In The Explanatory Dispensability of Idealizations, Sam Baron suggests a possible strategy enabling the indispensability argument to break the symmetry between mathematical claims and idealization assumptions in scientific models. Baron’s distinction between mathematical and non-mathematical idealization, I claim, is in need of a more compelling criterion, because in scientific models idealization assumptions are expressed through mathematical claims. In this paper I argue that this mutual dependence of idealization and mathematics cannot be read in terms of symmetry (...) and that Baron’s non-causal notion of mathematical difference-making is not effective in justifying any symmetry-breaking between mathematics and idealization. The function of making a difference that Baron attributes to mathematics cannot be referred to physical facts, but to the features of quantities, such as step lengths or time intervals taken into account in the models. It appears, indeed, that it does not follow from Baron’s argument that idealizations do not help to carry the explanatory load at least for two reasons: mathematics is not independent of idealizations in modelling and idealizations help mathematics to carry the explanatory load of a model in different degrees. (shrink)

The notion of idealization has received considerable attention in contemporary philosophy of science but less in philosophy of mathematics. An exception was the ‘critical idealism’ of the neo-Kantian philosopher Ernst Cassirer. According to Cassirer the methodology of idealization plays a central role for mathematics and empirical science. In this paper it is argued that Cassirer's contributions in this area still deserve to be taken into account in the current debates in philosophy of mathematics. For extremely useful criticisms on earlier versions (...) I am grateful to B.P. Larvor and another anonymous journal referee. CiteULike Connotea Del.icio.us What's this? (shrink)

The theory of knowledge objectification, initially presented and developed by Luis Radford, has gained some traction in the field of mathematics education. As with any developing theory, its presentation contains statements that may contradict its stated intents; and these problems are exacerbated in its uptake into the work of other scholars. The purpose of this study is to articulate a Spinozist-Marxian approach, in which the objectification exists not in things—semiotic means that mediate interactions—but as real relation between people. As a (...) consequence, the concept of mediation is unnecessary and can be abandoned. A concrete classroom example from Radford’s own studies is used to exemplify and develop pertinent issues. In particular, the societal nature of the ideal—a synecdoche of relations between objects that reflect relations between people—should be added and the notion of mediation no longer is required. (shrink)

Domain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the model-theoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders’s proposed explanation does not suffice. I then develop and formalize a different approach to domain extension based on Dedekind’s Habilitationsrede, to which Manders’s account is compared. (...) I conclude with an examination of three possible stances towards extensions via ideal elements. (shrink)

In The Explanatory Dispensability of Idealizations, Sam Baron suggests a possible strategy enabling the indispensability argument to break the symmetry between mathematical claims and idealization assumptions in scientific models. Baron’s distinction between mathematical and non-mathematical idealization, I claim, is in need of a more compelling criterion, because in scientific models idealization assumptions are expressed through mathematical claims. In this paper I argue that this mutual dependence of idealization and mathematics cannot be read in terms of symmetry (...) and that Baron’s non-causal notion of mathematical difference-making is not effective in justifying any symmetry-breaking between mathematics and idealization. The function of making a difference that Baron attributes to mathematics cannot be referred to physical facts, but to the features of quantities, such as step lengths or time intervals taken into account in the models. It appears, indeed, that it does not follow from Baron’s argument that idealizations do not help to carry the explanatory load at least for two reasons: mathematics is not independent of idealizations in modelling and idealizations help mathematics to carry the explanatory load of a model in different degrees. (shrink)

Between 1942 and 1943 the editor of the journal «Domus» invited the most important Italian architects to design their ideal houses: fifteen projects designed by seventeen architects were published. They are most instructive to try to understand, firstly, what the philosophical notion of ideal means and, secondly, why mathematical and geometric tools are extensively used to work on ideality, namely, to design ideal houses. The first part of the article focuses on the philosophical foundations of ideality and, (...) after an overview of the fifteen projects, on the use of the golden ratio in two particularly meaningful cases. The second part of the article focuses on the cases in which there is a hidden use of the golden ratio, on the use of the modulus and on the use of the number 2. (shrink)

When Derrida translated and commented on Husserl’s manuscript The Origin of Geometry in 1962, he gave a central place to what Husserl called the Idea “in the Kantian sense”. This article reflects on the use and function of this Idea in Derrida’s reading of Husserl. It critically interrogates the relationship between the Idea in the Kantian sense and mathematical ideality, as well as the use of this Idea in the interpretation of the Thing (Ding) and the stream of (...) experience (Erlebnisstrom). In the centre of the discussion stands what Derrida calls “pure thinking”, on which he tries to ground the Idea in the Kantian sense. (shrink)

The problem of ‘applicability’ of mathematics to modern physical sciences has been labeled as an ‘unreasonably effective’ and unexplainable ‘miracle’ by prominent physicists such as Eugene Wigner a...

Review essay: Les mathématiques infinitésimales du IXe au XIe siècle. Volume 4: Ibn al-Hatham, méthodes géométriques, transformations ponctuelles, et philosophie des mathématiques (London: Al-Furq¸n Islamic Heritage Foundation, 2002), pp. xiii+1064+vi ¤ 106.71 ISBN 1 87399 260 2.

This study explored and compared the perspectives of Taiwanese in-service and pre-service high school mathematics teachers regarding ideal teaching behaviours; the perspectives of a nationwide sample of students were taken as the baseline. Fourteen factors contributing to ideal teaching behaviours were identified through exploratory factor analyses. Nine factors, including idea explanation and speedy lecture, were rooted in traditional Chinese culture; five factors, including concrete representation and student activities, were influenced by Western cultures. Three teacher profiles were identified through k-means clustering (...) analysis. The perspectives of in-service teachers were dominated by a painless meaning-emphasised profile; these teachers emphasised meaningful learning for students and avoided the fast pace and demanding requirements that can cause distress in students, whereas pre-service teachers were dominated by an all-round profile, revealing their openness to all factors. Compared with... (shrink)

We seek to elucidate the philosophical context in which the so-called revolution of rigor in inifinitesimal calculus and mathematical analysis took place. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind, and Weierstrass. The dominant current of philosophy in Germany at that time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our main thesis is that Marburg Neo-Kantian philosophy formulated (...) a sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the "great triumvirate" of Cantor, Dedekind, and Weierstrass. Expressed in terms of modern mathematics, the Marburg philosophers saw the introduction of both infinitesimals and limits as completions whose prototype was Dedekind's of the rational number system resulting in the real numbers. At least partially,, this idea of "completions" can be captured in terms of a category-theoretical description of the conceptual development of modern mathematics. The feasibility of such a modern reformuation may be taken as evidence that the philosophical resources of Marburg neo-Kantianism may be of interest even for contemporary philosophy of mathematics. (shrink)

This paper examines the role of mathematical idealization in describing and explaining various features of the world. It examines two cases: first, briefly, the modeling of shock formation using the idealization of the continuum. Second, and in more detail, the breaking of droplets from the points of view of both analytic fluid mechanics and molecular dynamical simulations at the nano-level. It argues that the continuum idealizations are explanatorily ineliminable and that a full understanding of certain physical phenomena cannot be (...) obtained through completely detailed, non-idealized representations. (shrink)

This book tells a single story, in many voices, about a serious and sustained set of changes in mathematics teaching practice in a high school and how those efforts influenced and were influenced by a local university. It includes the writings and perspectives of high school students, high school teachers, preservice teacher candidates, doctoral students in mathematics education and other fields, mathematics teacher educators, and other education faculty. As a whole, this case study provides an opportunity to reflect on reform (...) visions of mathematics for all students and the challenges inherent in the implementation of these visions in US schools. It challenges us to rethink boundaries between theory and practice and the relative roles of teachers and university faculty in educational endeavors. (shrink)

The reconstruction of Leibniz’s metaphysics that Deleuze undertakes in The Fold provides a systematic account of the structure of Leibniz’s metaphysics in terms of its mathematical foundations. However, in doing so, Deleuze draws not only upon the mathematics developed by Leibniz—including the law of continuity as reflected in the calculus of infinite series and the infinitesimal calculus—but also upon developments in mathematics made by a number of Leibniz’s contemporaries—including Newton’s method of fluxions. He also draws upon a number of (...) subsequent developments in mathematics, the rudiments of which can be more or less located in Leibniz’s own work—including the theory of functions and singularities, the Weierstrassian theory of analytic continuity, and Poincaré’s theory of automorphic functions. Deleuze then retrospectively maps these developments back onto the structure of Leibniz’s metaphysics. While the Weierstrassian theory of analytic continuity serves to clarify Leibniz’s work, Poincaré’s theory of automorphic functions offers a solution to overcome and extend the limits that Deleuze identifies in Leibniz’s metaphysics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. This essay is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

In empirical modeling, mathematics has an important utility in transforming descriptive representations of target system into calculation devices, thus creating useful scientific models. The transformation may be considered as the action of tools. In this paper, I assume that model idealizations could be such tools. I then examine whether these idealizations have characteristic properties of tools, i.e., whether they are being adapted to the objects to which they are applied, and whether they are to some extent generic.

For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could mathematics be knowledge (...) about? (2) How do we distinguish significant from insignificant mathematics? This is a fundamental philosophical problem concerning the nature of mathematics. But it is also a practical problem concerning mathematics itself. In the absence of the solution to the problem, there is the danger that genuinely significant mathematics will be lost among the unchecked growth of a mass of insignificant mathematics. This second problem cannot, it would seem, be solved granted knowledge-inquiry. For, in order to solve the problem, mathematics needs to be related to values, but this is, it seems, prohibited by knowledge-inquiry because it could only lead to the subversion of mathematical rigour. Both problems are solved, however, when mathematics is viewed from the perspective of wisdom-inquiry. (1) Mathematics is not a branch of knowledge. It is a body of systematized, unified and inter-connected problem-solving methods, a body of problematic possibilities. (2) A piece of mathematics is significant if (a) it links up to the interconnected body of existing mathematics, ideally in such a way that some problems difficult to solve in other branches become much easier to solve when translated into the piece of mathematics in question; (b) it has fruitful applications for (other) worthwhile human endeavours. If ever the revolution from knowledge to wisdom occurs, I would hope wisdom mathematics would flourish, the nature of mathematics would become much more transparent, more pupils and students would come to appreciate the fascination of mathematics, and it would be easier to discern what is genuinely significant in mathematics (something that baffled even Einstein). As a result of clarifying what should count as significant, the pursuit of wisdom mathematics might even lead to the development of significant new mathematics. (shrink)

In a recent paper as an alternative to models based on the notion of ideal mathematical point, characterized by a property of separatedness, we considered a viewpoint based on the notion of continuous change, making use of elements of a non-classical logic, in particular the fuzzy sets theory, with events represented as spatiotemporally blurred blobs. Here we point out and discuss a number of aspects of this imperfect symbolic description that might potentially be misleading. Besides that, we analyze its (...) relation to various concepts used commonly to model physical systems, denoted by terms like: point, set, continuous, discrete, infinite, or local, clarifying further how our viewpoint is different and asking whether, in light of our main postulate, any of these notions, or their opposites, if exist, are in their usual meanings suitable to accurately describe the natural phenomena. (shrink)

When is a law too idealized to be usefully applied to a specific situation? To answer this question, this essay considers a law in hydrogeology called Darcy''s Law, both as it is used in what is called the symmetric-cone model, and as it is used in equations to determine a well''s groundwater velocity and hydraulic conductivity. After discussing Darcy''s law and its applications, the essay concludes that this idealized law, as well as associated models and equations in hydrogeology, are not (...) realistic in the sense required by the D-N account. They exhibit what McMullin calls mathematical idealization, construct idealization, empirical-causal idealization, and subjunctive-causal idealization. Yet this lack of realism in hydrogeology is problematic for reasons unrelated to the status of the D-N account. These idealizations are also problematic in applied situations. Their problems require developing two supplemental criteria, necessary for their productive application. (shrink)

Idealization is a necessity. Stripping away levels of complexity makes questions tractable, focuses our attention, and lets us develop comprehensible, testable models. Applying such models, however, requires care and attention to how the idealizations incorporated into their development affect their predictions. In epistemology, we tend to focus on idealizations concerning individual agents' capacities, such as memory, mathematical ability, and so on, when addressing this concern. By contrast, this dissertation focuses on social idealizations, particularly those pertaining to salient social categories (...) like race, sex, and gender. In Chapter II, Privilege and Superiority, we begin with standpoint epistemology, one of the earliest efforts to grapple with the ways that social structures affect our epistemic lives. I argue that, if we interpret standpoint epistemologists' claims as hypotheses about the ways that our social positions affect access to evidence, we can fruitfully employ recent developments in evidence logic to study the consequences. I lay the groundwork for this project, developing a model based on neighborhood semantics for modal logic. Adapting this framework to standpoint epistemology helps to clarify the meaning of terms like "epistemic privilege" and "superior knowledge" and to elucidate the differences between various accounts of standpoint epistemology. I also address a longstanding criticism of these views: Longino's (1990) bias paradox, which suggests that there is no objective position from which to judge the goodness of a particular standpoint. Chapter III, Evidence in a Non-Ideal World, turns to the broader social context, looking at how ideology affects the availability of evidence. Throughout the chapter, I take the formation, justification, and maintenance of racist, sexist, and otherwise oppressive beliefs as a central case. I argue that these beliefs are, at least sometimes, formed as a result of evidential distortion, a structural feature of our epistemic contexts that skews readily available evidence in favor of dominant ideologies. Because they are formed this way, such beliefs will appear justified on prominent accounts of justification, both internalist and externalist. As a result, epistemic norms that fail to account for such non-ideal conditions will deliver verdicts that are not only counter-intuitive, but also morally unpalatable. This, I argue, reveals a kind of structural epistemic injustice, especially where oppressive ideology is involved and suggests the need for epistemic norms that are sensitive to agents' social contexts. Much of the discussion in Chapters II and III depends on social categories like race and gender, arguing that they have a distinctive influence on our epistemic lives. In Chapter IV, I Know You Are, But What Am I?, my co-author and I focus on social categories themselves, distinguishing between self-identity, social identity, and social role. We self-identify as gay or straight, men or women, couch-potatoes or gym rats. Sometimes, these identities affect our social roles---the way we are perceived and treated by others---and sometimes they do not. This relationship between our internal identities and our preferred public perceptions begs for explanation. On our account, this relationship is captured by what we refer to as 'social identity'---roughly, internal identities made available to others. We argue that this account of social identity plays an illuminating role in structural explanation of discrimination and individual behavior, dissolves puzzles surrounding the phenomenon of 'passing', and explains certain moral and political obligations toward individuals. (shrink)

The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative (...) accounts, it appears as a necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality, generality, and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain. (shrink)

Idealizations are ubiquitous in physics. They are distortions or falsities that enter into theories, laws, models, and scientific representations. Various questions suggest themselves: What are idealizations? Why do we appeal to idealizations and how do we justify them? Are idealizations essential to physics and, if so, in what sense and for which purpose? How can idealizations provide genuine understanding? If our motivation for believing in the existence of unobservable entities like electrons and quarks is that they are indispensable to our (...) best theories, should we also believe in the existence of indispensable idealizations? This Element will tackle such questions and offer an opinionated and selective introduction to philosophical issues concerning idealizations in physics. Topics to be covered include the concept of and reasons for introducing idealization, abstraction, and approximation, possible taxonomy and justification, and application to issues of mathematical Platonism, scientific realism, and scientific understanding. (shrink)

The problem of the origin of geometry is crucial for understanding the formation and development of Derrida’s early conception of historicity. Mathematical idealities offer the most powerful example of meanings that are fully transmissible through history. Against Husserl’s explanation of the particular, Derrida considers that the logic and progression of mathematical idealities can only be explained if they are referred to non-intentional and pre-subjective movements of production and development of significations: language itself, which is structured as non-phonetic writing. (...) Historicity is, to Derrida’s eyes, the non-intentional and non-present structure at work in the formation and transmission of meaning. Therefore, pure transmissibility of meaning is an essentially equivocal and creative process. However, Derrida’s analyses fail at understanding the logic of mathematical progression. He explains it by generalizing consciousness’ inner temporality to what he describes as being the “dialecticity” of non-present temporal modes. But the dialecticity of mathematical concepts is not reducible to the dialecticity of temporal modes of experience. We cannot characterize the pre-intentional conditions of historicity if we put into brackets the concrete field in which history becomes factual, i.e. in which meaning and appearing actually historialize: the effective progression of objects. (shrink)

Idealization is commonly understood as distortion: representing things differently than how they actually are. In this paper, we outline an alternative artifactual approach that does not make misrepresentation central for the analysis of idealization. We examine the contrast between the Hodgkin-Huxley (1952a, b, c) and the Heimburg-Jackson (2005, 2006) models of the nerve impulse from the artifactual perspective, and argue that, since the two models draw upon different epistemic resources and research programs, it is often difficult to tell which features (...) of a system the central assumptions involved are supposed to distort. Many idealizations are holistic in nature. They cannot be locally undone without dismantling the model, as they occupy a central position in the entire research program. Nor is their holistic character mainly related to the use of mathematical and statistical modeling techniques as portrayed by Rice (2018, 2019). We suggest that holistic idealizations are implicit theoretical and representational assumptions that can only be understood in relation to the conceptual and representational tools exploited in modeling and experimental practices. Such holistic idealizations play a pivotal role not just in individual models, but also in defining research programs. -/- . (shrink)

Idealized scientific representations result from employing claims that we take to be false. It is not surprising, then, that idealizations are a prime example of allegedly inconsistent scientific representations. I argue that the claim that an idealization requires inconsistent beliefs is often incorrect and that it turns out that a more mathematical perspective allows us to understand how the idealization can be interpreted consistently. The main example discussed is the claim that models of ocean waves typically involve the false (...) assumption that the ocean is infinitely deep. While it is true that the variable associated with depth is often taken to infinity in the representation of ocean waves, I explain how this mathematical transformation of the original equations does not require the belief that the ocean being modeled is infinitely deep. More generally, as a mathematical representation is manipulated, some of its components are decoupled from their original physical interpretation. (shrink)

Ecologists attempt to understand the diversity of life with mathematical models. Often, mathematical models contain simplifying idealizations designed to cope with the blooming, buzzing confusion of the natural world. This strategy frequently issues in models whose predictions are inaccurate. Critics of theoretical ecology argue that only predictively accurate models are successful and contribute to the applied work of conservation biologists. Hence, they think that much of the mathematical work of ecologists is poor science. Against this view, I (...) argue that model building is successful even when models are predictively inaccurate for at least three reasons: models allow scientists to explore the possible behaviors of ecological systems; models give scientists simplified means by which they can investigate more complex systems by determining how the more complex system deviates from the simpler model; and models give scientists conceptual frameworks through which they can conduct experiments and fieldwork. Critics often mistake the purposes of model building, and once we recognize this, we can see their complaints are unjustified. Even though models in ecology are not always accurate in their assumptions and predictions, they still contribute to successful science. (shrink)

Graduate-level historical study is ideal for students intending to specialize in the topic, as well as those who only need a general treatment. Part I discusses traditional and symbolic logic. Part II explores the foundations of mathematics, emphasizing Hilbert’s metamathematics. Part III focuses on the philosophy of mathematics. Each chapter has extensive supplementary notes; a detailed appendix charts modern developments.

The ArgumentThis paper is a study of the role of language in scientific activity. It recommends that language be viewed as a community's means of patterning its affairs. Language represents where the boundaries of the community are and who is entitled to speak within it, and it displays the structures of authority in the community. Moreover, language precipitates the community's view of what the world is like, such that linguistic usages can be taken as referring to that world. Thus, language (...) connects on the one side to a community's practical life, and, on the other, to the reality on which it “reports.”The example treated is Robert Boyle's view of the language proper for experimental practice, and his arguments against the value and legitimacy of mathematical representations within the experimental community. Boyle argued that the use of mathematics was inappropriate because (i) it would restrict the number of potential participants; (ii) it would give rise to unwarranted expectations of the certainty and accuracy to be expected of real physical inquiries; (iii) it would suggest views of natural law and God's relations with the natural world which were incorrect and possibly dangerous; and because (iv) the world upon which the experimentalist operated was not such as was represented by mathematical idealizations. I try to establish that decision about the appropriate language for the experimental community was a moral choice. (shrink)

Mathematical models of biological patterns are central to contemporary biology. This paper aims to consider what these models contribute to biology through the detailed consideration of an important case: Hamilton’s selfish herd. While highly abstract and idealized, Hamilton’s models have generated an extensive amount of research and have arguably led to an accurate understanding of an important factor in the evolution of gregarious behaviors like herding and flocking. I propose an account of what these models are able to achieve (...) and how they can support a successful scientific research program. I argue that the way these models are interpreted is central to the success of such programs. (shrink)

This paper explores some of the constructive dimensions and specifics of human theoretic cognition, combining perspectives from (Husserlian) genetic phenomenology and distributed cognition approaches. I further consult recent psychological research concerning spatial and numerical cognition. The focus is on the nexus between the theoretic development of abstract, idealized geometrical and mathematical notions of space and the development and effective use of environmental cognitive support systems. In my discussion, I show that the evolution of the theoretic cognition of space apparently (...) follows two opposing, but in truth, intrinsically aligned trajectories. On the epistemic plane, which is the main focus of Husserl’s genetic phenomenological investigations, theoretic conceptions of space are progressively constituted by way of an idealizing emancipation of spatial cognition from the concrete, embodied intentionality underlying the human organism’s perception of space. As a result of this emancipation, it ultimately becomes possible for the human mind to theoretically conceive of and posit space as an ideal entity that is universally geometrical and mathematical. At the same time, by synthesizing a range of literature on spatial and mathematical cognition, I illustrate that for the theoretic mind to undertake precisely this emancipating process successfully, and further, for an ideal and objective notion of geometrical and mathematical space to first of all become fully scientifically operative, the cognitive support provided by a range of specific symbolic technologies is central. These include lettered diagrams, notation systems, and more generally, the technique of formalization and require for their functioning various cognitively efficacious types of embodiment. Ultimately, this paper endeavors to understand the specific symbolic-technological dimensions that have been instrumental to major shifts in the development of idealized, scientific conceptions of space. The epistemic characteristics of these shifts have been previously discussed in genetic phenomenology, but without devoting sufficient attention to the constructive role of symbolic technologies. At the same time, this paper identifies some of the irreducible phenomenological and epistemic dimensions that characterize the functioning of the historically situated, embodied and distributed theoretic mind. (shrink)

Purity has been recognized as an ideal of proof. In this paper, I consider whether purity continues to have value in contemporary mathematics. The topics (e.g., algebraic topology, algebraic geometry, category theory) and methods of contemporary mathematics often favor unification and generality, values that are more often associated with impurity rather than purity. I will demonstrate this by discussing several examples of methods and proofs that highlight the epistemic significance of unification and generality. First, I discuss the examples of algebraic (...) invariants and of considering a mathematical object from several different perspectives to illustrate that the methods used in contemporary mathematics favor impurity. Then I consider an example from category theory which demonstrates how unification and generality are related to impurity and that impure solutions can be explanatory. In light of this discussion, we see that purity only has marginal value within contemporary mathematics which instead prioritizes the epistemic values associated with impurity. (shrink)

At the beginning of the present century, a series of paradoxes were discovered within mathematics which suggested a fundamental unclarity in traditional mathemati­cal methods. These methods rested on the assumption of a realm of mathematical idealities existing independently of our thinking activity, and in order to arrive at a firmly grounded mathematics different attempts were made to formulate a conception of mathematical objects as purely human constructions. It was, however, realised that such formulations necessarily result in a mathematics (...) which lacks the richness and power of the old ‘platonistic’ methods, and the latter are still defended, in various modified forms, as embodying truths about self-existent mathematical entities. Thus there is an idealism-realism dispute in the philosophy of mathematics in some respects parallel to the controversy over the existence of the experiential world to the settle­ment of which lngarden devoted his life. The present paper is an attempt to apply Ingarden’s methods to the sphere of mathematical existence. This exercise will reveal new modes of being applicable to non-real objects, and we shall put forward arguments to suggest that these modes of being have an importance outside mathematics, especially in the areas of value theory and the ontology of art. (shrink)

_Philosophy of Mathematics_ is an excellent introductory text. This student friendly book discusses the great philosophers and the importance of mathematics to their thought. It includes the following topics: * the mathematical image * platonism * picture-proofs * applied mathematics * Hilbert and Godel * knots and nations * definitions * picture-proofs and Wittgenstein * computation, proof and conjecture. The book is ideal for courses on philosophy of mathematics and logic.

Examining how, and to what effect, the phrase “truly international” became central to the rhetoric and organization of the American-hosted 1950 International Congress of Mathematicians, this essay traces the negotiation of a “truly international” discipline from mathematicians’ first international congresses around the turn of the century across two world wars and their divisive interlude. Two failed attempts to host international congresses of mathematicians in the United States, for 1924 and 1940, defined the stakes for those who became the principal organizers (...) for 1950. Combining American organizational records with contexts and sources that extend across and beyond traditional mathematical centers in Europe and North America, the essay shows how a small cohort of American mathematicians marshaled an emphatic but ambiguous “international” rhetoric to guide policies and command cooperation and support while responding to persistent challenges. Their adaptations and compromises left a lasting mark on the terms and achievements of international inclusion, cooperation, and hegemony in mathematics. (shrink)