In this paper, I explore Bunge’s fictionism in philosophy of mathematics. After an overview of Bunge’s views, in particular his mathematical structuralism, I argue that the comparison between mathematical objects and fictions ultimately fails. I then sketch a different ontology for mathematics, based on Thomasson’s metaphysical work. I conclude that mathematics deserves its own ontology, and that, in the end, much work remains to be done to clarify the various forms of dependence that are involved in (...) class='Hi'>mathematical knowledge, in particular its dependence on mental/brain states and material objects. (shrink)

This thesis starts with three challenges to the structuralist accounts of applied mathematics. Structuralism views applied mathematics as a matter of building mapping functions between mathematical and target-ended structures. The first challenge concerns how it is possible for a non-mathematical target to be represented mathematically when the mapping functions per se are mathematical objects. The second challenge arises out of inconsistent early calculus, which suggests that mathematical representation does not require rigorous mathematical structures. The (...) third challenge comes from renormalisation group (RG) explanations of universality. It is argued that the structural mapping between the world and a highly abstract minimal model adds little value to our understanding of how RG obtains its explanatory force. I will address the first and second challenges from the similarity perspective. The similarity account captures representations as similarity relations, providing a more flexible and broader conception of representation than structuralism. It is the specification of the respect and degree of similarity that forges mathematics into a context of representation and directs it to represent a specific system in reality. Structuralism is treatable as a tool for explicating similarity rela-tions set-theoretically. The similarity account, combined with other approaches (e.g., Nguyen and Frigg’s extensional abstraction account and van Fraassen’s pragmatic equivalence), can dissolve the first challenge. Additionally, I will make a structuralist response to the second challenge, and suggestions regarding the role of infinitesimals from the similarity perspective. In light of the similarity account, I will propose the “hotchpotch picture” as a method-ological reflection of our study of representation and explanation. Its central insight is to dissect a representation or an explanation into several aspects and use different theories (that are usually thought of competing) to appropriate each of them. Based on the hotchpotch picture, RG explanations can be dissected to the “indexing” and “inferential” conceptions of explanation, which are captured or characterised by structural mappings. Therefore, structuralism accommodates RG explanations, and the third challenge is resolved. (shrink)

In this paper, we look at Bourbaki’s work as a case study for the notion of mathematical style. We argue that indeed Bourbaki exemplifies a mathematical style, namely the structuralist style.

What is the ontological status of mathematical structures? Michael Resnic, Stewart Shapiro and Gianluigi Oliveri, are contemporaries of American philosophers on mathematics, they give Platonic answers on this question.

The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Non-eliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the features to an ontology of structural objects conflicts with claims often made by structuralists to the effect that their structuralist theses are versions of Quine’s ontological relativity or Putnam’s internal realism. I describe and argue for an alternative explanation for these features (...) which instead explains the attributes them to the mathematical practice of representing numbers using more concrete tokens, such as sets, strokes and so on. (shrink)

A prominent version of mathematical structuralism holds that mathematical objects are at bottom nothing but "positions in structures," purely relational entities without any sort of nature independent of the structure to which they belong. Such an ontology is often presented as a response to Benacerraf's "multiple reductions" problem, or motivated on hermeneutic grounds, as a faithful representation of the discourse and practice of mathematics. In this paper I argue that there are serious difficulties with this kind of (...) view: its proponents rely on a distinction between "essential" and "nonessential" features of mathematical objects, and there's no good way to articulate this distinction which is compatible with basic structuralist commitments. But all is not lost. For I further argue that the insights motivating structuralism (or at least those worth preserving) can be preserved without formulating the view in ontologically committal terms. (shrink)

Keränen (2001) raises an argument against realistic (ante rem) structuralism: where a mathematical structure has a non-trivial automorphism, distinct indiscernible positions within the structure cannot be shown to be non-identical using only the properties and relations of that structure. Ladyman (2005) responds by allowing our identity criterion to include 'irreflexive two-place relations'. I note that this does not solve the problem for structures with indistinguishable positions, i.e. positions that have all the same properties as each other and exactly (...) the same relations to all objects (including themselves). I conclude that realistic structuralists must compromise and treat some structures eliminativistically. (shrink)

Since Benacerraf’s ‘What Numbers Could Not Be, ’ there has been a growing interest in mathematical structuralism. An influential form of mathematical structuralism, modal structuralism, uses logical possibility and second order logic to provide paraphrases of mathematical statements which don’t quantify over mathematical objects. These modal structuralist paraphrases are a useful tool for nominalists and realists alike. But their use of second order logic and quantification into the logical possibility operator raises concerns. In (...) this paper, I show that the work of both these elements can be done by a single natural generalization of the logical possibility operator. (shrink)

The approach of structuralism came to philosophy from social science. It was also in social science where, in 1950–1970s, in the form of the French structuralism, the approach gained its widest recognition. Since then, however, the approach fell out of favour in social science. Recently, structuralism is gaining currency in the philosophy of mathematics. After ascertaining that the two structuralisms indeed share a common core, the question stands whether general structuralism could not find its way back (...) into social science. The nature of the major objections raised against French structuralism – concerning its alleged ahistoricism, methodological holism and universalism – are reconsidered. While admittedly grounded as far as French structuralism is concerned, these objections do not affect general structuralism as such. The fate of French structuralism thus does not seem to preclude the return of general structuralism into social science, rather, it provides some hints where the difficulties may lie. (shrink)

I explore the possibility of a structuralist interpretation of homotopy type theory (HoTT) as a foundation for mathematics. There are two main aspects to HoTT's structuralist credentials. First, it builds on categorical set theory (CST), of which the best-known variant is Lawvere's ETCS. I argue that CST has merit as a structuralist foundation, in that it ascribes only structural properties to typical mathematical objects. However, I also argue that this success depends on the adoption of a strict typing system (...) which undermines the metaphysical seriousness of this structuralism. Homotopy type theory adds to CST a distinctive theory of identity between sets, which arguably allows its objects to be seen as ante rem structures. I examine the prospects for such a view, and address many other interpretive problems as they arise. (shrink)

In this paper, we argue that, contrary to the view held by most philosophers of mathematics, Bourbaki’s technical conception of mathematical structuralism is relevant to philosophy of mathematics. In fact, we believe that Bourbaki has captured the core of any mathematical structuralism.

Structuralism has recently moved center stage in philosophy of mathematics. One of the issues discussed is the underlying logic of mathematical structuralism. In this paper, I want to look at the dual question, namely the underlying structures of logic. Indeed, from a mathematical structuralist standpoint, it makes perfect sense to try to identify the abstract structures underlying logic. We claim that one answer to this question is provided by categorical logic. In fact, we claim that the (...) latter can be seen—and probably should be seen—as being a structuralist approach to logic and it is from this angle that categorical logic is best understood. (shrink)

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, (...) class='Hi'>mathematical ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies. (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.”.

Causal structuralism is the view that, for each natural, non-mathematical, non-Cambridge property, there is a causal profile that exhausts its individual essence. On this view, having a property’s causal profile is both necessary and sufficient for being that property. It is generally contrasted with the Humean or quidditistic view of properties, which states that having a property’s causal profile is neither necessary nor sufficient for being that property, and with the double-aspect view, which states that causal profile is (...) necessary but not sufficient. Shoemaker’s (1998) and Hawthorne’s (2001) arguments in favor of causal structuralism primarily focus on problematic consequences of the other two views. I argue, however, that causation does not provide an appropriate framework within which to characterize all physical properties for two main reasons. First, there are physical properties that do not have causal profiles and properties whose causal profiles do not exhaust their essences. Second, there is no unified notion of causation across the sciences. After distinguishing between the causal and the nomological, I suggest that what is needed is a structuralist view of properties that is not merely causal but that incorporates a physical property’s higher-order mathematical and nomological properties into its identity conditions. Such a view retains the naturalistic motivations for causal structuralism while avoiding the problems it faces. (shrink)

Drawing an analogy between modal structuralism about mathematics and theism, I o er a structuralist account that implicitly de nes theism in terms of three basic relations: logical and metaphysical priority, and epis- temic superiority. On this view, statements like `God is omniscient' have a hypothetical and a categorical component. The hypothetical component provides a translation pattern according to which statements in theistic language are converted into statements of second-order modal logic. The categorical component asserts the logical possibility of (...) the theism struc- ture on the basis of uncontroversial facts about the physical world. This structuralist reading of theism preserves objective truth-values for theistic statements while remaining neutral on the question of ontology. Thus, it o ers a way of understanding theism to which a naturalist cannot object, and it accommodates the fact that religious belief, for many theists, is an essentially relational matter. (shrink)

Some scientific explanations appear to turn on pure mathematical claims. The enhanced indispensability argument appeals to these ‘mathematical explanations’ in support of mathematical platonism. I argue that the success of this argument rests on the claim that mathematical explanations locate pure mathematical facts on which their physical explananda depend, and that any account of mathematical explanation that supports this claim fails to provide an adequate understanding of mathematical explanation.

This 4-page review-essay—which is entirely reportorial and philosophically neutral as are my other contributions to MATHEMATICAL REVIEWS—starts with a short introduction to the philosophy known as mathematical structuralism. The history of structuralism traces back to George Boole (1815–1864). By reference to a recent article various feature of structuralism are discussed with special attention to ambiguity and other terminological issues. The review-essay includes a description of the recent article. The article’s 4-sentence summary is quoted in full (...) and then analyzed. The point of the quotation is to make clear how murky, incompetent, and badly written the paper is. There is no way to determine from the article whether the editor or referees suggests improvements. (shrink)

Metaphysical graphical structuralism is the view that at some fundamental level the world is a mathematical graph of nodes and edges. Randall Dipert has advanced a graphical structuralist theory of fundamental particulars and Alexander Bird has advanced a graphical structuralist theory of fundamental properties. David Oderberg has posed a powerful challenge to graphical structuralism: that it entails the absurd inexistence of the world or the absurd cessation of all change. In this paper I defend graphical structuralism. (...) A sharper formulation, some theorems about such structures, and careful attention to the interaction of metaphysical and mathematical features, shows that the absurdities depend on assumptions that are not essential to the view and brings to light a surprising fact about the necessary structure of fundamental properties. (shrink)

In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.

This paper introduces a novel object that has less structure than the natural numbers. As such it is a candidate model for the foundation that lies beneath the natural numbers. The implications for the construction of mathematical objects built upon that foundation are discussed.

This work is a conceptual analysis of certain recent developments in the mathematical foundations of Classical and Quantum Mechanics which have allowed to formulate both theories in a common language. From the algebraic point of view, the set of observables of a physical system, be it classical or quantum, is described by a Jordan-Lie algebra. From the geometric point of view, the space of states of any system is described by a uniform Poisson space with transition probability. Both these (...) structures are here perceived as formal translations of the fundamental twofold role of properties in Mechanics: they are at the same time quantities and transformations. The question becomes then to understand the precise articulation between these two roles. The analysis will show that Quantum Mechanics can be thought as distinguishing itself from Classical Mechanics by a compatibility condition between properties-as-quantities and properties-as-transformations. -/- Moreover, this dissertation shows the existence of a tension between a certain "abstract way" of conceiving mathematical structures, used in the practice of mathematical physics, and the necessary capacity to specify particular states or observables. It then becomes important to understand how, within the formalism, one can construct a labelling scheme. The “Chase for Individuation” is the analysis of different mathematical techniques which attempt to overcome this tension. In particular, we discuss how group theory furnishes a partial solution. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

What can we infer from numerical cognition about mathematical realism? In this paper, I will consider one aspect of numerical cognition that has received little attention in the literature: the remarkable similarities of numerical cognitive capacities across many animal species. This Invariantism in Numerical Cognition (INC) indicates that mathematics and morality are disanalogous in an important respect: proto-moral beliefs differ substantially between animal species, whereas proto-mathematical beliefs (at least in the animals studied) seem to show more similarities. This (...) makes moral beliefs more susceptible to a contingency challenge from evolution compared to mathematical beliefs, and indicates that mathematical beliefs might be less vulnerable to evolutionary debunking arguments. I will then examine to what extent INC can be used to flesh out a positive case for mathematical realism. Finally, I will review two forms of mathematical realism that are promising in the light of the evolutionary evidence about numerical cognition, ante rem structuralism and Millean empiricism. (shrink)

One central aim of science is to provide explanations of natural phenomena. What role(s) does mathematics play in achieving this aim? How does mathematics contribute to the explanatory power of science? Rules to Infinity defends the thesis, common though perhaps inchoate among many members of the Vienna Circle, that mathematics contributes to the explanatory power of science by expressing conceptual rules, rules which allow the transformation of empirical descriptions. Mathematics should not be thought of as describing, in any substantive sense, (...) an abstract realm of eternal mathematical objects, as traditional platonists have thought. In Rules to Infinity, this view, which I call mathematical normativism, is updated with contemporary philosophical tools, and it is argued that normativism is compatible with mainstream semantic theory. This allows the normativist to accept that there are mathematical truths, while resisting the platonistic idea that there exist abstract mathematical objects that explain such truths. Furthermore, Rules to Infinity defends a particular account of the distinction between scientific explanations that are in some sense distinctively mathematical – those that explain natural phenomena in some uniquely mathematical way – and those that are only standardly mathematical, and it lays out desiderata for any account of this distinction. Normativism is compared with other prominent views in the philosophy of mathematics such as neo-Fregeanism, fictionalism, conventionalism, and structuralism. Rules to Infinity serves as an entry point into debates at the forefront of philosophy of science and mathematics, and it defends novel positions in these debates. (shrink)

This book develops a new approach to plural arbitrary reference and examines mereology, including considering four theses on the alleged innocence of mereology. The authors have advanced the notion of plural arbitrary reference in terms of idealized plural acts of choice, performed by a suitable team of agents. In the first part of the book, readers will discover a revision of Boolosʼ interpretation of second order logic in terms of plural quantification and a sketched structuralist reconstruction of second-order arithmetic based (...) on the axiom of infinite, a la Dedekind, as the unique non-logical axiom. The work goes on to analyse the pros and cons of the new interpretation, also with respect to Linneboʼs objections to the thesis that second order logic is genuine logic. A theory of concepts that can be labelled as a theory of logical concepts is expounded. In the second part of the book, the authors consider grounding megethology on plural arbitrary reference and argue that the arguments for the ontological innocence of mereology are not conclusive and that – for a certain use of mereology – a thesis of innocence, similar to that of plural arbitrary reference, is defensible. The work proposes a virtual theory of mereology in which the role of individuals is played by plural choices of atoms. This considered work will appeal to scholars from branches of analytic philosophy, logic and the philosophy of mathematics in particular. (shrink)

Two slogans define structuralism: contemporary mathematics studies structures; mathematical objects are places in those structures. Shapiro’s version of structuralism posits abstract objects of three sorts. A system is “a collection of objects with certain relations” between these objects. “An extended family is a system of people with blood and marital relationships.” A baseball defense, e.g., the Yankee’s defense in the first game of the 1999 World Series, is a also a system, “a collection of people with on-field (...) spatial and ‘defensive-role’ relations”. “A structure is the abstract form of a system” ; it consists of “a collection of places [sometimes called positions or offices] and a finite collection of functions and relations on these places”. (shrink)

Grounding is bicollective if it is possible for some truths δ,δ,... to be grounded in the some truths γ,γ,... without its being the case that each δi is grounded in some subcollection of γ,γ,.... In this paper I show how to do develop a hypergraph-theoretic account of bicollective ground, taking the notion of immediate ground as basic. I also indicate how bicollective ground helps with formulating mathematical structuralism.

Employing the ideas of modern mathematics and biology, seen in the context of Ernst Cassirer's "Symbolic Forms, the author presents an entirely new and novel solution to the classical mind-brain problem. This is a "hard" book, I'm sorry, but it is the problem itself, and not me which has made it so. I say that Dennett, and, indeed, the whole of academia is wrong.

“Structuralism, Fictionalism, and the Applicability of Mathematics in Science”. This article has two objectives. The first one is to review some of the most important questions in the contemporary philosophy of mathematics: What is the nature of mathematical objects? How do we acquire knowledge about these objects? Should mathematical statements be interpreted differently than ordinary ones? And, finally, how can we explain the applicability of mathematics in science? The debate that guides these reflections is the one between (...) mathematical realism and anti-realism. On the other hand, the second objective is to discuss the arguments that use the applicability of mathematics in science to justify mathematical realism, and show that none of them reaches its aim. To this end, we will distinguish three aspects of the problem of the applicability of mathematics: the utility of mathematics in science, the unexpected utility of some mathematical theories, and the apparent indispensability of mathematics in our best scientific theories - in particular, in our best scientific explanations. Finally, I argue that none of these aspects constitutes a reason to adopt mathematical realism. (shrink)

My aim in this chapter is to extend the Realist account of the foundations of linguistics offered by Postal, Katz and others. I first argue against the idea that naive Platonism can capture the necessary requirements on what I call a ‘mixed realist’ view of linguistics, which takes aspects of Platonism, Nominalism and Mentalism into consideration. I then advocate three desiderata for an appropriate ‘mixed realist’ account of linguistic ontology and foundations, namely (1) linguistic creativity and infinity, (2) linguistics as (...) a theory of types (and not tokens) and (3) independence but structural respect between language and the linguistic competence thereof. My own brand of mixed realism, what I call ante rem realism, is defended along the lines of an ante rem or non-eliminative structuralism, the likes of which has been offered for mathematics by Resnik (1997) and Shapiro (1997). In other words, grammars describe a mind-independent (but not necessarily unconnected) linguistic reality in terms of linguistic patterns or structures also known as natural languages. I further amend this picture to allow for the possibility of a naturalistic account of language acquisition and evolution by arguing against a particular view of the type-token distinction. (shrink)

The Newman problem is a fundamental problem that threatens to undermine structural assumptions and structural theories throughout philosophy and science. Here, we consider the problem in the context of consciousness science. We introduce and discuss the problem, and explain why it is detrimental not only to structuralist assumptions, but also to theories of consciousness, if left unconsidered. However, we show that if phenomenal spaces, and mathematical structures of conscious experience more generally, are understood in the right way, the Newman (...) problem does not arise. The upshot of our paper is that consciousness science needs to be careful in which definition of structure to consider, but if it is, the Newman problem disappears. (shrink)

Platonists affirm the existence of abstract mathematical objects, and Nominalists deny the existence of abstract mathematical objects. While there are standard arguments in favor of Nominalism, these arguments fail to account for the necessity of Nominalism. Furthermore, these arguments do nothing to explain why Nominalism is true. They only point to certain theoretical vices that might befall the Platonist. The goal of this paper is to formulate and defend a simple, valid argument for the necessity of Nominalism that (...) seeks to precisify the widespread intuition that mathematical objects are somehow ‘spooky’ or ‘mysterious’. (shrink)

In Mathematics is megethology. Philosophia Mathematica, 1, 3–23) David K. Lewis proposes a structuralist reconstruction of classical set theory based on mereology. In order to formulate suitable hypotheses about the size of the universe of individuals without the help of set-theoretical notions, he uses the device of Boolos’ plural quantification for treating second order logic without commitment to set-theoretical entities. In this paper we show how, assuming the existence of a pairing function on atoms, as the unique assumption non expressed (...) in a mereological language, a mereological foundation of set theory is achievable within first order logic. Furthermore, we show how a mereological codification of ordered pairs is achievable with a very restricted use of the notion of plurality without plural quantification. (shrink)

This paper is concerned with the construction of theories of software systems yielding adequate predictions of their target systems’ computations. It is first argued that mathematical theories of programs are not able to provide predictions that are consistent with observed executions. Empirical theories of software systems are here introduced semantically, in terms of a hierarchy of computational models that are supplied by formal methods and testing techniques in computer science. Both deductive top-down and inductive bottom-up approaches in the discovery (...) of semantic software theories are refused to argue in favour of the abductive process of hypothesising and refining models at each level in the hierarchy, until they become satisfactorily predictive. Empirical theories of computational systems are required to be modular, as modular are most software verification and testing activities. We argue that logic relations must be thereby defined among models representing different modules in a semantic theory of a modular software system. We exclude that scientific structuralism is able to define module relations needed in software modular theories. The algebraic Theory of Institutions is finally introduced to specify the logic structure of modular semantic theories of computational systems. (shrink)

In this paper, I will attempt to develop and defend a common form of intuitive resistance to the companions in guilt argument. I will argue that one can reasonably believe there are promising solutions to the access problem for mathematical realism that don’t translate to moral realism. In particular, I will suggest that the structuralist project of accounting for mathematical knowledge in terms of some form of logical knowledge offers significant hope of success while no analogous approach offers (...) such hope for moral realism. (shrink)

The paper investigates Ernst Cassirer’s structuralist account of geometrical knowledge developed in his Substanzbegriff und Funktionsbegriff. The aim here is twofold. First, to give a closer study of several developments in projective geometry that form the direct background for Cassirer’s philosophical remarks on geometrical concept formation. Specifically, the paper will survey different attempts to justify the principle of duality in projective geometry as well as Felix Klein’s generalization of the use of geometrical transformations in his Erlangen program. The second aim (...) is to analyze the specific character of Cassirer’s geometrical structuralism formulated in 1910 as well as in subsequent writings. As will be argued, his account of modern geometry is best described as a “methodological structuralism”, that is, as a view mainly concerned with the role of structural methods in modern mathematical practice. (shrink)

Functional reductionism characterises inter-theoretic reduction as the recovery of the upper-level behaviour described by the reduced theory in terms of the lower-level reducing theory. For instance, finding a statistical mechanical realiser that plays the functional role of thermodynamic entropy allows for establishing a reductive link between thermodynamics and statistical mechanics. This view constitutes a unique approach to reduction that enjoys a number of positive features, but has received limited attention in the philosophy of science. -/- This paper aims to clarify (...) the meaning of functional reductionism in science, with a focus on physics, to define both its place with respect to other approaches to reduction and its connection to ontology. To do so, we develop and explore two alternative frameworks for functional reduction, called Syntactic Functional Reductionism and Semantic Functional Reductionism. They represent two different possible stances on the view that expand and improve the basic functional reductionist approach along different lines, and make clear how the approach works in practice. The former elaborates on David Lewis’ account, is connected with the syntactic view of theories, is committed to a logical characterisation of functional roles, and is embedded within Nagelian reductionism. The latter adopts a semantic view of theories, spells out functional roles mainly in terms of mathematical roles within the models of theories, and is expressed in terms of the related structuralist approach to reduction. The development of these frameworks has the final goal of advancing functional reductionism, to make it a fully-fledged alternative account for reduction in science. (shrink)

This essay explores the possibility of constructing a structural realist interpretation of spacetime theories that can resolve the ontological debate between substantivalists and relationists. Drawing on various structuralist approaches in the philosophy of mathematics, as well as on the theoretical complexities of general relativity, our investigation will reveal that a structuralist approach can be beneficial to the spacetime theorist as a means of deflating some of the ontological disputes regarding similarly structured spacetimes.

Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first- and second-order) for formulating set theory. By bringing considerations (...) of perturbations in modal space to bear on the debate, we will suggest that a promising option for representing current set-theoretic thought is given by formulating set theory using quasi-weak second-order logic. These observations indicate that the usual division of structures into \particular (e.g. the natural number structure) and general (e.g. the group structure) is perhaps too coarse grained; we should also make a distinction between intentionally and unintentionally general structures. (shrink)

The paper explicates the stages of the author’s philosophical evolution in the light of Kopnin’s ideas and heritage. Starting from Kopnin’s understanding of dialectical materialism, the author has stated that category transformations of physics has opened from conceptualization of immutability to mutability and then to interaction, evolvement and emergence. He has connected the problem of physical cognition universals with an elaboration of the specific system of tools and methods of identifying, individuating and distinguishing objects from a scientific theory domain. The (...) role of vacuum conception and the idea of existence (actual and potential, observable and nonobservable, virtual and hidden) types were analyzed. In collaboration with S.Crymski heuristic and regulative functions of categories of substance, world as a whole as well as postulates of relativity and absoluteness, and anthropic and self-development principles were singled out. Elaborating Kopnin’s view of scientific theories as a practically effective and relatively true mapping of their domains, the author in collaboration with M. Burgin have originated the unified structure-nominative reconstruction (model) of scientific theory as a knowledge system. According to it, every scientific knowledge system includes hierarchically organized and complex subsystems that partially and separately have been studied by standard, structuralist, operationalist, problem-solving, axiological and other directions of the current philosophy of science. 1) The logico-linguistic subsystem represents and normalizes by means of different, including mathematical, languages and normalizes and logical calculi the knowledge available on objects under study. 2) The model-representing subsystem comprises peculiar to the knowledge system ways of their modeling and understanding. 3) The pragmatic-procedural subsystem contains general and unique to the knowledge system operations, methods, procedures, algorithms and programs. 4) From the viewpoint of the problem-heuristic subsystem, the knowledge system is a unique way of setting and resolving questions, problems, puzzles and tasks of cognition of objects into question. It also includes various heuristics and estimations (truth, consistency, beauty, efficacy, adequacy, heuristicity etc) of components and structures of the knowledge system. 5) The subsystem of links fixes interrelations between above-mentioned components, structures and subsystems of the knowledge system. The structure-nominative reconstruction has been used in the philosophical and comparative case-studies of mathematical, physical, economic, legal, political, pedagogical, social, and sociological theories. It has enlarged the collection of knowledge structures, connected, for instance, with a multitude of theoreticity levels and with an application of numerous mathematical languages. It has deepened the comprehension of relations between the main directions of current philosophy of science. They are interpreted as dealing mainly with isolated subsystems of scientific theory. This reconstruction has disclosed a variety of undetected knowledge structures, associated also, for instance, with principles of symmetry and supersymmetry and with laws of various levels and degrees. In cooperation with the physicist Olexander Gabovich the modified structure-nominative reconstruction is in the processes of development and justification. Ideas and concepts were also in the center of Kopnin’s cognitive activity. The author has suggested and elaborated the triplet model of concepts. According to it, any scientific concept is a dependent on cognitive situation, dynamical, multifunctional state of scientist’s thinking, and available knowledge system. A concept is modeled as being consisted from three interrelated structures. 1) The concept base characterizes objects falling under a concept as well as their properties and relations. In terms of volume and content the logical modeling reveals partially only the concept base. 2) The concept representing part includes structures and means (names, statements, abstract properties, quantitative values of object properties and relations, mathematical equations and their systems, theoretical models etc.) of object representation in the appropriate knowledge system. 3) The linkage unites a structures and procedures that connect components from the abovementioned structures. The partial cases of the triplet model are logical, information, two-tired, standard, exemplar, prototype, knowledge-dependent and other concept models. It has introduced the triplet classification that comprises several hundreds of concept types. Different kinds of fuzziness are distinguished. Even the most precise and exact concepts are fuzzy in some triplet aspect. The notions of relations between real scientific concepts are essentially extended. For example, the definition and strict analysis of such relations between concepts as formalization, quantification, mathematization, generalization, fuzzification, and various kinds of identity are proposed. The concepts «PLANET» and «ELEMENTARY PARTICLE» and some of their metamorphoses were analyzed in triplet terms. The Kopnin’s methodology and epistemology of cognition was being used for creating conception of the philosophy of law as elaborating of understanding, justification, estimating and criticizing legal system. The basic information on the major directions in current Western philosophy of law (legal realism, feminism, criticism, postmodernism, economical analysis of law etc.) is firstly introduced to the Ukrainian audience. The classification of more than fifty directions in modern legal philosophy is suggested. Some results of historical, linguistic, scientometric and philosophic-legal studies of the present state of Ukrainian academic science are given. (shrink)

In this article I give an overview of some recent work in philosophy of science dedicated to analysing the scientific process in terms of (conceptual) mathematical models of theories and the various semantic relations between such models, scientific theories, and aspects of reality. In current philosophy of science, the most interesting questions centre around the ways in which writers distinguish between theories and the mathematical structures that interpret them and in which they are true, i.e. between scientific theories (...) as linguistic systems and their non-linguistic models. In philosophy of science literature there are two main approaches to the structure of scientific theories, the statement or syntactic approach -- advocated by Carnap, Hempel and Nagel -- and the non- statement or semantic approach --advocated, among others, by Suppes, the structuralists, Beth, Van Fraassen, Giere, Wojcicki. In conclusion, I briefly review some of the usual realist inspired questions about the possibility and character of relations between scientific theories and reality as implied by the various approaches I discuss in the course of the article. The models of a scientific theory should indeed be adequate to the phenomena, but if the theory is 'adequate' to (true in) its conceptual (mathematical) models as well, we have a model-theoretic realism that addresses the possible meaning and reference of 'theoretical entities' without relapsing into the metaphysics typical of the usual scientific realist approaches. (shrink)

Nineteen fifty-eight was an extraordinary year for cultural innovation, especially in English literature. It was also a year in which several boldly revisionary positions were first articulated in analytic philosophy. And it was a crucial year for the establishment of structural linguistics, of structuralist anthropology, and of cognitive psychology. Taken together these developments had a radical effect on our conceptions of individual creativity and of the inheritance of tradition. The present essay attempts to illuminate the relationships among these developments, and (...) to explain the foundational role played by mathematical, logical and information theory in all of them. (shrink)

Numbers without Science opposes the Quine-Putnam indispensability argument, seeking to undermine the argument and reduce its profound influence. Philosophers rely on indispensability to justify mathematical knowledge using only empiricist epistemology. I argue that we need an independent account of our knowledge of mathematics. The indispensability argument, in broad form, consists of two premises. The major premise alleges that we are committed to mathematical objects if science requires them. The minor premise alleges that science in fact requires mathematical (...) objects. The most common rejection of the argument denies its minor premise by introducing scientific theories which do not refer to mathematical objects. Hartry Field has shown how we can reformulate some physical theories without mathematical commitments. I argue that Field’s preference for intrinsic explanation, which underlies his reformulation, is ill-motivated, and that his resultant fictionalism suffers unacceptable consequences. I attack the major premise instead. I argue that Quine provides a mistaken criterion for ontic commitment. Our uses of mathematics in scientific theory are instrumental and do not commit us to mathematical objects. Furthermore, even if we accept Quine’s criterion for ontic commitment, the indispensability argument justifies only an anemic version of mathematics, and does not yield traditional mathematical objects. The first two chapters of the dissertation develop these results for Quine’s indispensability argument. In the third chapter, I apply my findings to other contemporary indispensabilists, specifically the structuralists Michael Resnik and Stewart Shapiro. In the fourth chapter, I show that indispensability arguments which do not rely on Quine’s holism, like that of Putnam, are even less successful. Also in Chapter 4, I show how Putnam’s work in the philosophy of mathematics is unified around the indispensability argument. In the last chapter of the dissertation, I conclude that we need an account of mathematical knowledge which does not appeal to empirical science and which does not succumb to mysticism and speculation. Briefly, my strategy is to argue that any defensible solution to the demarcation problem of separating good scientific theories from bad ones will find mathematics to be good, if not empirical, science. (shrink)

Recent activities in virtually all fields engaged in consciousness studies indicate early signs of a structural turn, where verbal descriptions or simple formalisations of conscious experiences are replaced by structural tools, most notably mathematical spaces. My goal here is to offer three comments that, in my opinion, are essential to avoid misunderstandings in these developments early on. These comments concern metaphysical premises of structuralist approaches, overlooked assumptions in regard to isomorphisms, and the question of what structure to consider on (...) the side of consciousness in the first place. I will also explain what, in my opinion, are the great promises of structural methodologies and how they might impact consciousness science at large. (shrink)

In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as (...) class='Hi'>mathematical. In this paper, I will argue that Dedekind’s approach can be seen as a precursor to modern structuralism and as such, it enjoys many advantages over Frege’s logicism. I also show that from a modern perspective, Frege’s criticism of abstraction and psychologism is one-sided and fails against the psychological processes that modern research suggests to be at the heart of numerical cognition. The approach here is twofold. First, through historical analysis, I will try to build a clear image of what Frege’s and Dedekind’s views on arithmetic were. Then, I will consider those views from the perspective of modern philosophy of mathematics, and in particular, the empirical study of arithmetical cognition. I aim to show that there is nothing to suggest that the axiomatic Dedekind approach could not provide a perfectly adequate basis for philosophy of arithmetic. (shrink)

The explanation of nature in theoretical terms was first postulated and initiated by Ancient Greek philosophers. With the rise of monotheistic religions, however, curiosity about our transient world was widely regarded as contributing nothing to salvation. There was a decline in natural philosophy, which lasted for several centuries and was then reversed both in Islamic philosophy and in Christian theology in the Middle Ages. At this point, the "Book of Nature" was recognized as a complement to the Book of Revelation. (...) Originally, Aristotelian philosophy played the leading part in this process, but only after Aristotelian physics was substituted by experimental mathematical mechanics did it become possible for modern science to develop. General laws of physics form the basis of the explanation of events in space and time, including life processes. However, modern science has also revealed its own limitations. The scope and limits of scientific knowledge allows for different philosophical, cultural and religious interpretations of human beings and the universe. The history of science demonstrates erratic and persistent developments, but the long-term retrospective view discredits extreme forms of relativism and structuralism. Science appears as a construct of the human mind, and yet it is capable of generating valid explanations of nature. In the book more attention than usual is given to Cusanus (Nikolas de Cusa). -/- . (shrink)

Ladyman and Ross argue that quantum objects are not individuals and use this idea to ground their metaphysical view, ontic structural realism, according to which relational structures are primary to things. LR acknowledge that there is a version of quantum theory, namely the Bohm theory, according to which particles do have denite trajectories at all times. However, LR interpret the research by Brown et al. as implying that "raw stuff" or haecceities are needed for the individuality of particles of BT, (...) and LR dismiss this as idle metaphysics. In this paper we note that Brown et al.'s research does not imply that haecceities are needed. Thus BT remains as a genuine option for those who seek to understand quantum particles as individuals. However, we go on to discuss some problems with BT which led Bohm and Hiley to modify it. This modified version underlines that, due to features such as context-dependence and non-locality, Bohmian particles have a very limited autonomy in situations where quantum effects are non-negligible. So while BT restores the possibility of quantum individuals, it also underlines the primacy of the whole over the autonomy of the parts. The later sections of the paper also examine the Bohm theory in the general mathematical context of symplectic geometry. This provides yet another way of understanding the subtle, holistic and dynamic nature of Bohmian individuals. We finally briefly consider Bohm's other main line of research, the "implicate order", which is in some ways similar to LR's structural realism. (shrink)