Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of (...) mathematical theorems can cover at most one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

Mathematical pluralism can take one of three forms: (1) every consistent mathematical theory consists of truths about its own domain of individuals and relations; (2) every mathematical theory, consistent or inconsistent, consists of truths about its own (possibly uninteresting) domain of individuals and relations; and (3) the principal philosophies of mathematics are each based upon an insight or truth about the nature of mathematics that can be validated. (1) includes the multiverse approach to set theory. (2) (...) helps us to understand the significance of the distinguished non‐logical individual and relation terms of even inconsistent theories. (3) is a metaphilosophical form of mathematical pluralism and hasn't been discussed in the literature. In what follows, I show how the analysis of theoretical mathematics in object theory exhibits all three forms of mathematical pluralism. (shrink)

A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis ('SIA'), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis ('CA') without resort to the method of limits. Formally, however, unlike Robinsonian 'nonstandard analysis', SIA conflicts with CA, deriving, e.g., 'not every quantity is either = 0 or not = 0.' Internally, consistency is maintained by using intuitionistic logic (without the law of excluded middle). This paper examines problems of interpretation resulting from this 'change of logic', (...) arguing that standard arguments based on 'smoothness' requirements are question-begging. Instead, it is suggested that recent philosophical work on the logic of vagueness is relevant, especially in the context of a Hilbertian structuralist view of mathematical axioms (as implicitly defining structures of interest). The relevance of both topos models for SIA and modal-structuralism as appled to this theory is clarified, sustaining this remarkable instance of mathematical pluralism. (shrink)

IntroductionScientific pluralism is generally understood in the backdrop of scientific monism. So is mathematical pluralism. Though there are many culture-dependent mathematical practices, mathematical concepts and theories are generally taken to be culture invariant. We would like to explore in this paper whether mathematical pluralism is admissible or not.Materials and methodsMathematical pluralism may be approached at least from five different perspectives. 1. Foundational: The view would claim that different issues within mathematics need support (...) of different foundations, apparently incompatible with one another. 2. Ontological: The world itself is dappled—the mathematical counterpart of which can be traced to the admission of non-Euclidean spaces and also in simultaneous acceptance of set-theoretic and category theoretic entities in the ontology of mathematics. 3. The third route is epistemological. It follows from the view that the nature of reality is so complex that different aspects of it require alternative modes of representation and explanation sometimes severally, sometimes simultaneously, e.g., classical, constructivist, computer-aided and different finitist mathematics can be used depending on the knowledge situation. 4. The fourth route is semantic. Fuzzy mathematics, we know, gives up bivalence in theory of truth, and ascribes truth to mathematical propositions in degrees. 5. The last one is Programmatic: for some, mathematical pluralism is a stance, a programme. It provides a framework which strikes at the root of cultural hegemony and nourishes a climate of intellectual humility and tolerance. It is not my intention to hold the brief of any of these versions. I just want to present to the readers a more or less complete picture of the scenario, though not an exhaustive one. ConclusionUnlike monism, mathematical pluralism does not rule out any possibility—even the possibility of having a unitary over-arching framework of interpretation remains as an open option. This paper highlights the fact that motivations for upholding pluralism in mathematics have been many and one can be a pluralist just by subscribing to any of the views listed above. (shrink)

PurposeThis paper aims to establish that a certain sort of mathematical pluralism is true. MethodsThe paper proceeds by arguing that that the best versions of mathematical Platonism and anti-Platonism both entail the relevant sort of mathematical pluralism. Result and ConclusionThis argument gives us the result that mathematical pluralism is true, and it also gives us the perhaps surprising result that mathematical Platonism and mathematical pluralism are perfectly compatible with one another.

Mathematical pluralism is the view that there is an irreducible plurality of pure mathematical structures, each with their own internal logics; and that qua pure mathematical structures they are all equally legitimate. Mathematical pluralism is a relatively new position on the philosophical landscape. This Element provides an introduction to the position.

Mathematical pluralism notes that there are many different kinds of pure mathematical structures—notably those based on different logics—and that, qua pieces of pure mathematics, they are all equally good. Logical pluralism is the view that there are different logics, which are, in an appropriate sense, equally good. Some, such as Shapiro, have argued that mathematical pluralism entails logical pluralism. In this brief note I argue that this does not follow. There is a crucial (...) distinction to be drawn between the preservation of truth and the preservation of truth-in-a-structure; and once this distinction is drawn, this suffices to block the argument. The paper starts by clarifying the relevant notions of mathematical and logical pluralism. It then explains why the argument from the first to the second does not follow. A final section considers a few objections. (shrink)

Theories about the foundations of mathematics often encounter a problem similar to the traditional demarcation problem in science. In this context, it is pertinent to examine the first candidate for the identifying property of mathematical pluralism: reduction within a structure. As I argue here, this notion is insufficient for a coherent definition of structure within the plurality. In the end, demarcating a plurality of mathematics can be as problematic as demarcating a unitary mathematics. -/- .

IntroductionThis paper focuses on a radical feminist engagement with mathematical pluralism. Radical Feminists are interested in a project of methodological re-tooling. Mathematical pluralism appears to be a possible source of help in this direction.Materials and MethodsWith this aim in view the article examines the contributions of Mihir Chakraborty and Amita Chatterjee. By using the method of philosophical argument their theses have been judged from a feminist perspective.ResultsSome very interesing results have been arrived at in terms of (...) accommodating vagueness and pluralism into feminist methodology by the help of mathematical pluralism.DiscussionBy the way of an ongoing discussion the paper states the contours of a radical feminist road map and specifies the kind help they seek from mathematical pluralism. (shrink)

In The Consistency of Arithmetic and elsewhere, Meyer claims to “repeal” Goedel’s second incompleteness theorem. In this paper, I review his argument, and then consider two ways of understanding it: from the perspective of mathematical pluralism and monism, respectively. Is relevant arithmetic just another legitimate practice among many, or is it a rival of its classical counterpart—a corrective to Goedel, setting us back on the path to the (One) True Arithmetic? To help answer, I sketch a few worked (...) examples from relevant mathematics, to see what a non-classical (re)formulation of mathematics might look like in practice. I conclude that, while it is unlikely that relevant arithmetic describes past and present mathematical practice, and so might be most acceptable as a pluralist enterprise, it may yet prescribe a more monistic future venture. (shrink)

We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context.

In this paper I will review the developments in the foundations of mathematics in the last 150 years in such a way as to show that they have delivered something of a rather different kind: mathematical pluralism.

I claim that a relatively new position in philosophy of mathematics, pluralism, overlaps in striking ways with the much older Jain doctrine of anekantavada and the associated doctrines of nyayavada and syadvada. I first outline the pluralist position, following this with a sketch of the Jain doctrine of anekantavada. I then note the srrong points of overlaps and the morals of this comparison of pluralism and anekantavada.

The pluralist sheds the more traditional ideas of truth and ontology. This is dangerous, because it threatens instability of the theory. To lend stability to his philosophy, the pluralist trades truth and ontology for rigour and other ‘fixtures’. Fixtures are the steady goal posts. They are the parts of a theory that stay fixed across a pair of theories, and allow us to make translations and comparisons. They can ultimately be moved, but we tend to keep them fixed temporarily. Apart (...) from considering rigour of proof as a fixture, I discuss fixed models, invariant notions and fixed information about objects across theories. There are other fixtures, but it is enough to start with these. (shrink)

Mathematical platonism is the view that abstract mathematical objects exist. Ontological pluralism is the view that there are many modes of existence. This paper examines the prospects for...

A belief one has is safe if either (i) it could not easily be false or (ii) in any nearby world in which it is false, it is not formed using the method one uses to form one’s actual belief. It seems our mathematical beliefs are safe if mathematical pluralism is true: if, loosely put, almost any consistent mathematical theory is true. It seems, after all, that in any nearby world where one’s mathematical beliefs differ (...) from one’s actual beliefs, one would believe some other true, consistent theory. Focusing on Justin Clarke-Doane’s recent discussion, I argue the thesis that mathematical beliefs are safe given pluralism faces some obstacles. I argue (i) is true of mathematical belief given pluralism only if we deny plausible claims about the interpretation of non-pluralists who many of us could easily be. Unless strong metasemantic theses are true, it is plausible many of us could easily deny or refuse to believe a consistent and true mathematical theory we actually believe. Since philosophical arguments and controversies permeate the methodology of foundational mathematics, I argue we cannot confidently distinguish the methods we use in mathematics between worlds, thus raising doubts about (ii). (shrink)

Contribution to a review symposium on Marc Lange's Because without cause: Non-causal explanation in science and mathematics. Oxford: Oxford University Press, 2017.

MethodologyA new hypothesis on the basic features characterizing the Foundations of Mathematics is suggested.Application of the methodBy means of it, the several proposals, launched around the year 1900, for discovering the FoM are characterized. It is well known that the historical evolution of these proposals was marked by some notorious failures and conflicts. Particular attention is given to Cantor's programme and its improvements. Its merits and insufficiencies are characterized in the light of the new conception of the FoM. After the (...) failures of Frege's and Cantor's programmes owing to the discoveries of an antinomy and internal contradictions, respectively, the two remaining, more radical programmes, i.e. Hilbert's and Brouwer's, generated a great debate; the explanation given here is their mutual incommensurability, defined by means of the differences in their foundational features.ResultsThe ignorance of this phenomenon explains the inconclusiveness of a century-long debate between the advocates of these two proposals. Which however have been so greatly improved as to closely approach or even recognize some basic features of the FoM.Discussion on the resultsYet, no proposal has recognized the alternative basic feature to Hilbert's main one, the deductive organization of a theory, although already half a century before the births of all the programmes this alternative was substantially instantiated by Lobachevsky's theory on parallel lines. Some conclusive considerations of a historical and philosophical nature are offered. In particular, the conclusive birth of a pluralism in the FoM is stressed. (shrink)

We propose a novel approach to logical pluralism based on algebra-valued models of set theory, systematically demonstrating how both classical and non-classical set theories can be constructed within a unified mathematical framework. Our approach extends Shapiro’s eclectic pluralism to the specific setting of algebra-valued models. This leads us to formulate two notions of logical pluralism: liberal pluralism, which recognizes as legitimate any logic that underpins a set theory capable of capturing a significant portion of (...) class='Hi'>mathematical practice, and strict pluralism, which further requires that the resulting set theory satisfies a convergence constraint, ensuring its mathematical expressiveness is comparable to that of classical set theory. We argue that this latter notion of pluralism represents a departure from previous accounts, as it is based on convergence rather than divergence among classical and non-classical mathematical theories. (shrink)

A plurality of approaches to foundational aspects of mathematics is a fact of life. Two loci of this are discussed here, the classicism/constructivism controversy over standards of proof, and the plurality of universes of discourse for mathematics arising in set theory and in category theory, whose problematic relationship is discussed. The first case illustrates the hypothesis that a sufficiently rich subject matter may require a multiplicity of approaches. The second case, while in some respects special to mathematics, raises issues of (...) ontological multiplicity and relativity encountered in the natural sciences as well. (shrink)

In the nineteenth and twentieth centuries many mathematicians referred to intuition as the indispensable research tool for obtaining new results. In this essay we will analyse a group of mathematicians who interacted with Luitzen Egbertus Jan Brouwer in order to compare their conceptions of intuition. We will see how to the same word “intuition” very different meanings corresponded: they varied from geometrical vision, to a unitary view of a demonstration, to the perception of time, to the faculty of considering concepts (...) that habitually occur in our thinking separately. Furthermore, we will discover that these different meanings had a philosophical, very relevant counterside: they passed from a racial characterization of mathematics to a pluralistic view of it. (shrink)

The paper discusses Husserl's phenomenology of mathematics in his Formal and Transcendental Logic (1929). In it Husserl seeks to provide descriptive foundations for mathematics. As sciences and mathematics are normative activities Husserl's attempt is also to describe the norms at work in these disciplines. The description shows that mathematics can be given in several different ways. The phenomenologist's task is to examine whether a given part of mathematics is genuine according to the norms that pertain to the approach in question. (...) The paper will then examine the intuitionistic, formalistic, and structural features of Husserl's philosophy of mathematics. (shrink)

Lakatos’ (Lakatos, 1976) model of mathematical conceptual change has been criticized for neglecting the diversity of dynamics exhibited by mathematical concepts. In this work, I will propose a pluralist approach to mathematical change that re-conceptualizes Lakatos’ model of proofs and refutations as an ideal dynamic that mathematical concepts can exhibit to different degrees with respect to multiple dimensions. Drawing inspiration from Godfrey-Smith’s (Godfrey-Smith, 2009) population-based Darwinism, my proposal will be structured around the notion of a conceptual (...) population, the opposition between Lakatosian and Euclidean populations, and the spatial tools of the Lakatosian space. I will show how my approach is able to account for the variety of dynamics exhibited by mathematical concepts with the help of three case studies. (shrink)

Truth pluralists say that truth-bearers in different “discourses”, “domains”, “domains of discourse”, or “domains of inquiry” are apt to be true in different ways – for instance, that mathematical discourse or ethical discourse is apt to be true in a different way to ordinary descriptive or scientific discourse. Moreover, the notion of a “domain” is often explicitly employed in formulating pluralist theories of truth. Consequently, the notion of a “domain” is attracting increasing attention, both critical and constructive. I argue (...) that this is a red herring. First, I identify the theoretical role for which pluralists appeal to domains, which is to answer what I call the “Individuation Problem”: saying what determines the way in which a particular truth-bearer is apt to be true. Second, I argue that pluralists need not appeal to domains for this purpose. I thus conclude that, despite the usual way of glossing the view, there is no role for the notion of a “domain” to play in the pluralist’s theory of truth. I argue that this defuses the “Problem of Mixed Atomics” and allows the pluralist to sidestep potentially intractable disputes about the nature of domains. (shrink)

A thriving literature has developed over logical and mathematical pluralism – i.e. the views that several rival logical and mathematical theories can be equally correct. These have unfortunately grown separate; instead, they both could gain a great deal by a closer interaction. Our aim is thus to present some novel forms of abstractionist mathematical pluralism which can be modeled on parallel ways of substantiating logical pluralism (also in connection with logical anti-exceptionalism). To do this, (...) we start by discussing the Good Company Problem for neo-logicists recently raised by Paolo Mancosu (2016), concerning the existence of rival abstractive definitions of cardinal number which are nonetheless equally able to reconstruct Peano Arithmetic. We survey Mancosu’s envisaged possible replies to this predicament, and suggest as a further path the adoption of some form of mathematical pluralism concerning abstraction principles. We then explore three possible ways of substantiating such pluralism—Conceptual Pluralism,Domain Pluralism,Pluralism about Criteria—showing how each of them can be related to analogous proposals in the philosophy of logic. We conclude by considering advantages, concerns, and theoretical ramifications for these varieties of mathematical pluralism. (shrink)

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, (...) I argue that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

Inspired by Bas van Fraassen’s Stance Empiricism, Anjan Chakravartty has developed a pluralistic account of what he calls epistemic stances towards scientific ontology. In this paper, I examine whether Chakravartty’s stance pluralism can exclude epistemic stances that licence pseudo-scientific practices like those found in Scientology. I argue that it cannot. Chakravartty’s stance pluralism is therefore prone to a form of debilitating relativism. I consequently argue that we need (1) some ground or constraint in relation to which epistemic stances (...) can be ranked by degrees, and (2) some way to demarcate science from pseudo-science so that we know what epistemic stances are about. Regarding (1), I argue that empirical detectability can serve as the ground in relation to which epistemic stances are ranked by degrees. Regarding (2), I argue for ranking sciences on a continuum according to established institutional criteria, rather than attempting to draw a strict demarcation. (shrink)

Truth pluralists say that the nature of truth varies between domains of discourse: while ordinary descriptive claims or those of the hard sciences might be true in virtue of corresponding to reality, those concerning ethics, mathematics, institutions might be true in some non-representational or “anti-realist” sense. Despite pluralism attracting increasing amounts of attention, the motivations for the view remain underdeveloped. This paper investigates whether pluralism is well-motivated on ontological grounds: that is, on the basis that different discourses are (...) concerned with different kinds of entities. Arguments that draw on six different ontological contrasts are examined: concrete versus abstract entities; mind-independent versus mind-dependent entities; sparse versus merely abundant properties; objective versus projected entities; natural versus non-natural entities; and ontological pluralism. I argue that the additional premises needed to move from such contrasts to truth pluralism are either implausible or unmotivated, often doing little more than to bifurcate the nature of truth when a more theoretically conservative option is available. If there is a compelling motivation for pluralism, I suggest, it’s likely to lie elsewhere. (shrink)

We are logical pluralists who hold that the right logic is dependent on the domain of investigation; different logics for different mathematical theories. The purpose of this article is to explore the ramifications for our pluralism concerning normativity. Is there any normative role for logic, once we give up its universality? We discuss Florian Steingerger’s “Frege and Carnap on the Normativity of Logic” as a source for possible types of normativity, and then turn to our own proposal, which (...) postulates that various logics are constitutive for thought within particular practices, but none are constitutive for thought as such. (shrink)

It is demonstrated that the reduction of a physical theory S to another one, T, in the sense that S can be derived from T holds in general only for the mathematical framework. The interpretation of S and the associated central terms cannot all be derived from those of T because of the qualitative differences between the cognitive levels of S and T. Their cognitively autonomous status leads to an epistemic as well as an ontological pluralism. This (...) class='Hi'>pluralism is consistent with the unity of nature in the sense of a substantive monism. (shrink)

Women's mating strategies have typically been characterised as restrictive or coy. However, recent research on sociosexual behaviour suggests that the frequency of women's extra-pair copulations is a function of an unrestricted personality. While agreeing with the general thrust of Gangestad & Simpson's strategic pluralism theory we suggest that it is more likely a matter of finely calculated reproductive opportunism.

In this paper I present a pluralist view of truth of a special kind: correspondence-pluralism. Correspondence-pluralism is the view that to fulfill its function in knowledge, truth requires correspondence principles rather than mere coherence, pragmatist, or deflationist principles. But these correspondence principles do not need to be the naive principles of traditional correspondence: copy, mirror image, direct isomorphism. Furthermore, these correspondence principles may vary, in certain disciplined ways, from one field of knowledge to another. This combination of correspondence (...) and pluralism enables us to set high standards of truth for all fields of knowledge while allowing sufficient flexibility to adjust these principles to the special conditions of different fields. In so doing, it provides us with new tools for addressing old as well as new questions about truth: Is there correspondence-truth in mathematics? In ethics? Correspondence with what? What patterns of correspondence? The paper is divided into four parts: (I) Why correspondence? What kind of correspondence? (II) Why pluralism? What kind of pluralism? (III) Applications: mathematics and ethics. (IV) Avoidance of criticisms of other types of truth-pluralism. (shrink)

In this paper, we develop a mathematical model of awareness based on the idea of plurality. Instead of positing a singular principle, telos, or essence as noumenon, we model it as plurality accessible through multiple forms of awareness (“n-awareness”). In contrast to many other approaches, our model is committed to pluralist thinking. The noumenon is plural, and reality is neither reducible nor irreducible. Nothing dies out in meaning making. We begin by mathematizing the concept of awareness by appealing to (...) the mathematical formalism of higher category theory. The beauty of higher category theory lies in its universality. Pluralism is categorical. In particular, we model awareness using the theories of derived categories and (infinity, 1)-topoi which will give rise to our meta-language. We then posit a “grammar” (“n-declension”) which could express n-awareness, accompanied by a new temporal ontology (“n-time”). Our framework allows us to revisit old problems in the philosophy of time: how is change possible and what do we mean by simultaneity and coincidence? Another question which could be re-conceptualized in our model is one of soteriology related to this pluralism: what is a self in this context? A new model of “personal identity over time” is thus introduced. (shrink)

Husserl and Mathematics explains the development of Husserl's phenomenological method in the context of his engagement in modern mathematics and its foundations. Drawing on his correspondence and other written sources, Mirja Hartimo details Husserl's knowledge of a wide range of perspectives on the foundations of mathematics, including those of Hilbert, Brouwer and Weyl, as well as his awareness of the new developments in the subject during the 1930s. Hartimo examines how Husserl's philosophical views responded to these changes, and offers a (...) pluralistic and open-ended picture of Husserl's phenomenology of mathematics. Her study shows Husserl's phenomenology to be a method capable of both shedding light on and internally criticizing scientific practices and concepts. (shrink)

This book discusses the problem of mathematical knowledge, and its broader philosophical ramifications. It argues that the problem of explaining the (defeasible) justification of our mathematical beliefs (‘the justificatory challenge’), arises insofar as disagreement over axioms bottoms out in disagreement over intuitions. And it argues that the problem of explaining their reliability (‘the reliability challenge’), arises to the extent that we could have easily had different beliefs. The book shows that mathematical facts are not, in general, empirically (...) accessible, contra Quine, and that they cannot be dispensed with, contra Field. However, it argues that they might be so plentiful that our knowledge of them is intelligible. The book concludes with a complementary ‘pluralism’ about modality, logic and normative theory, highlighting its revisionary implications. Metaphysically, pluralism engenders a kind of perspectivalism and indeterminacy. Methodologically, it vindicates Carnap’s pragmatism, transposed to the key of realism. (shrink)

Differences among scientific, mathematical, and ethical subject matters motivate a pluralism where distinct domains of subject matter are associated with distinct truth properties and logics. However, it is unclear how such pluralism might accommodate potentially attractive epistemic norms, such as that one ought to believe only what is true, and that one ought to believe what is logically true. In this paper, I show how such pluralism can accommodate such norms by supplementing the account developed in (...) Yu (2017a,b) with epistemic, doxastic, and deontic operators. The upshot is that pluralism is compatible with, even if not committed to, monism about epistemic norms. (shrink)

In this paper, I outline and defend a novel approach to alethic pluralism, the thesis that truth has more than one metaphysical nature: where truth is, in part, explained by reference, it is relational in character and can be regarded as consisting in correspondence; but where instead truth does not depend upon reference it is not relational and involves only coherence. In the process, I articulate a clear sense in which truth may or may not depend upon reference: this (...) involves distinguishing semantic denotation from pragmatic speaker reference and claiming that there may or may not exist a metasemantic connection between these two notions. Finally, I argue that reference is not in general inscrutable—that this metasemantic connection does exist in the case of our ordinary discourse about present macroscopic concrete objects—but that it is in pure mathematics, where reference cannot be secured, and which therefore plays no role in accounting for truth. In this manner, alethic pluralism is upheld. (shrink)

Explaining the behaviour of ecosystems is one of the key challenges for the biological sciences. Since 2000, new-mechanicism has been the main model to account for the nature of scientific explanation in biology. The universality of the new-mechanist view in biology has been however put into question due to the existence of explanations that account for some biological phenomena in terms of their mathematical properties (mathematical explanations). Supporters of mathematical explanation have argued that the explanation of the (...) behaviour of ecosystems is usually provided in terms of their mathematical properties, and not in mechanistic terms. They have intensively studied the explanation of the properties of ecosystems that behave following the rules of a non-random network. However, no attention has been devoted to the study of the nature of the explanation in those that form a random network. In this paper, we cover that gap by analysing the explanation of the stability behaviour of the microbiome recently elaborated by Coyte and colleagues, to determine whether it fits with the model of explanation suggested by the new-mechanist or by the defenders of mathematical explanation. Our analysis of this case study supports three theses: (1) that the explanation is not given solely in terms of mechanisms, as the new-mechanists understand the concept; (2) that the mathematical properties that describe the system play an essential explanatory role, but they do not exhaust the explanation; (3) that a non-previously identified appeal to the type of interactions that the entities in the network can exhibit, as well as their abundance, is also necessary for Coyte and colleagues’ account to be fully explanatory. From the combination of these three theses we argue for the necessity of an integrative pluralist view of the nature of behaviour explanation when this is given by appealing to the existence of a random network. (shrink)

PurposeIn this article, we aim to present and defend a contextual approach to mathematical explanation.MethodTo do this, we introduce an epistemic reading of mathematical explanation.ResultsThe epistemic reading not only clarifies the link between mathematical explanation and mathematical understanding, but also allows us to explicate some contextual factors governing explanation. We then show how several accounts of mathematical explanation can be read in this approach.ConclusionThe contextual approach defended here clears up the notion of explanation and pushes (...) us towards a pluralist vision on mathematical explanation. (shrink)

MATHEMATICAL RESOLUTIONS OF ZENO’s PARADOXES of motion have been offered on a regular basis since the paradoxes were first formulated. In this paper I will argue that such mathematical “solutions” miss, and always will miss, the point of Zeno’s arguments. I do not think that any mathematical solution can provide the much sought after answers to any of the paradoxes of Zeno. In fact all mathematical attempts to resolve these paradoxes share a common feature, a feature (...) that makes them consistently miss the fundamental point which is Zeno’s concern for the one-many relation, or it would be better to say, lack of relation. This takes us back to the ancient dispute between the Eleatic school and the Pluralists. The first, following Parmenide’s teaching, claimed that only the One or identical can be thought and is therefore real, the second held that the Many of becoming is rational and real.1 I will show that these mathematical “solutions” do not actually touch Zeno’s argument and make no metaphysical contribution to the problem of understanding what is motion against immobility, or multiplicity against identity, which was Zeno’s challenge. I would like to point out at this stage that my contention. (shrink)

Abstractabstract:This paper revisits debates on a tension in Cassirer's philosophy of culture. On the one hand, Cassirer describes a plurality of symbolic forms and claims that each needs to be assessed by its own internal standards of validity. On the other hand, he ranks the symbolic forms in terms of a developmental hierarchy and states that one form, mathematical natural science, constitutes the highest achievement of culture. In my paper, I do not seek to resolve this tension. Rather, I (...) aim to arrive at a better understanding of how it arises, and of the different options that it presents for understanding the development of culture. I discuss three recent attempts at resolving the tension, put forward by Sebastian Luft, Samatha Matherne, and Simon Truwant, respectively. Based on a reconstruction of Cassirer's system of symbolic forms that centralizes the concept of function, I show that the most promising of these attempts, formulated by Truwant, is not successful. I then turn to Cassirer's philosophy of the cultural sciences, the implications of which for the present problem have not yet been sufficiently explored. I argue that in this context, Cassirer develops the contours of an alternative to the function-based view of cultural development. I conclude that this alternative does not resolve the tension either, but that it allows for a reconceptualization of the teleology of culture as open. (shrink)

A number of those actively involved in the physical sciences anticipate the creation of a unified approach to all human knowledge based on reductionism in physics and Platonism in mathematics. We argue that it is implausible that this goal will ever be achieved, and argue instead for a pluralistic approach to human understanding, in which mathematically expressed laws of nature are merely one way among several of describing a world that is too complex for our minds to be able to (...) grasp in its entirety. (shrink)