This chapter contains sections titled: Introduction Platonism in Mathematics Nominalism in Mathematics Conclusion References.

For the past hundred years, mathematics, for its own reasons, has been shifting away from the study of “mathematical objects” and towards the study of “structures”. One would have expected philosophers to jump onto the bandwagon, as in many other cases, to proclaim that this shift is no accident, since mathematics is “essentially” about structures, not objects. In fact, structuralism has not been a very popular philosophy of mathematics, probably because of the (...) hostility of Frege and other influential logicists, and quasi-logicists like Quine. Recently, however, structuralism has finally penetrated the philosophical community and a number of “structuralist” volumes in the philosophy of mathematics have seen the light of day. Michael Resnik is one of the most distinguished representatives of this school, along with Stewart Shapiro, who has also recently published a structuralist tract. (shrink)

This paper compares the statement ‘Mathematics is the study of structure’ with the actual practice of mathematics. We present two examples from contemporary mathematical practice where the notion of structure plays different roles. In the first case a structure is defined over a certain set. It is argued firstly that this set may not be regarded as a structure and secondly that what is important to mathematical practice is the relation that exists between the structure (...) and the set. In the second case, from algebraic topology, one point is that an object can be a place in different structures. Which structure one chooses to place the object in depends on what one wishes to do with it. Overall the paper argues that mathematics certainly deals with structures, but that structures may not be all there is to mathematics. (shrink)

In this paper I argue for the view that structuralism offers the best perspective for an acceptable account of the applicability of mathematics in the empirical sciences. Structuralism, as I understand it, is the view that mathematics is not the science of a particular type of objects, but of structural properties of arbitrary domains of entities, regardless of whether they are actually existing, merely presupposed or only intentionally intended.

In Meaning and Structure, Peregrin argues that recent and contemporary (post)analytic philosophy, as developed by Quine, Davidson, Sellars and their followers, is largely structuralistic in the very sense in which structuralism was originally tabled by Ferdinand de Saussure. The author reconstructs de Saussure's view of language, linking it to modern formal logic and mathematics, and reveals close analogies between its constitutive principles and the principles informing the holistic and neopragmatistic view of language put forward by Quine and his followers. (...) Peregrin also indicates how this view of language can be made compatible with what is usually called 'formal semantics'. Drawing on both the Saussurean tradition and recent developments in analytic philosophy of language, this book offers a unique study of the ways in which the concept of meaning can be seen as consisting in the concept of structure. (shrink)

Words form a fundamental basis for our understanding of linguistic practice. However, the precise ontology of words has eluded many philosophers and linguists. A persistent difficulty for most accounts of words is the type-token distinction [Bromberger, S. 1989. “Types and Tokens in Linguistics.” In Reflections on Chomsky, edited by A. George, 58–90. Basil Blackwell; Kaplan, D. 1990. “Words.” Aristotelian Society Supplementary Volume LXIV: 93–119]. In this paper, I present a novel account of words which differs from the atomistic and platonistic (...) conceptions of previous accounts which I argue fall prey to this problem. Specifically, I proffer a structuralist account of linguistic items, along the lines of structuralism in the philosophy of mathematics [Shapiro, S. 1997. Philosophy of Mathematics: Structure and Ontology. Oxford University Press], in which words are defined in part as positions in larger linguistic structures. I then follow Szabò [1999. “Expressions and Their Representations.” The Philosophical Quarterly 49 (195): 145–163] and Parsons [1990. “The Structuralist View of Mathematical Objects.” Synthese 84: 303–346] in further defining words as quasi-concrete objects according to a representation relation. This view aims for general correspondence with contemporary generative linguistic approaches to the study of language. (shrink)

The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, set-theoretic and category-theoretic, arose within mathematics itself. After exposing a number of problems, the book considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as abstract universals, modal, eliminating structures as objects (...) in favor of freely entertained logical possibilities, and finally, modal-set-theoretic, a sort of synthesis of the set-theoretic and modal perspectives. (shrink)

In this paper we put a thesis that it is possible to perceive mathematics as a science of structures, where the difference between structure as the object of study and theory as something which describes this object is blurred. We discusses the view of set-theoretical structuralism with a special emphasis placed on a certain gradual development of set theory as a formal theory. We proposes a certain view concerning the methodology of formal sciences, which is an attempt (...) at describing precisely and capturing in a scheme the ways of development of deductive sciences in general, set theory including. We show that the standpoint of sui generis structuralism has some features characteristic of purely formal theory. (shrink)

In this paper I argue for the view that structuralism offers the best perspective for an acceptable account of the applicability of mathematics in the empirical sciences. Structuralism, as I understand it, is the view that mathematics is not the science of a particular type of objects, but of structural properties of arbitrary domains of entities, regardless of whether they are actually existing, merely presupposed or only intentionally intended.

It is well known that Felix Klein took a decisive step in investigating the invariants of transformation groups. However, less attention has been given to Klein’s considerations on the epistemological implications of his work on geometry. This paper proposes an interpretation of Klein’s view as a form of mathematical structuralism, according to which the study of mathematical structures provides the basis for a better understanding of how mathematical research and practice develop.

Mathematics as a Science of Patterns is the definitive exposition of a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links (...) this to a defence of realism about the metaphysics of mathematics--the view that mathematics is about things that really exist. Resnik's distinctive philosophy of mathematics is here presented in an accessible and systematic form: it will be of value not only to specialists in this area, but to philosophers, mathematicians, and logicians interested in the relationship between these three disciplines, or in truth, realism, and epistemology. (shrink)

This is Part B of an article that defends non-eliminative structuralism about mathematics by means of a concrete case study: a theory of unlabeled graphs. Part A motivated an understanding of unlabeled graphs as structures sui generis and developed a corresponding axiomatic theory of unlabeled graphs. Part B turns to the philosophical interpretation and assessment of the theory: it points out how the theory avoids well-known problems concerning identity, objecthood, and reference that have been attributed to non-eliminative (...) structuralism. The part concludes by explaining how the theory relates to set theory, and what remains to be accomplished for non-eliminative structuralists. (shrink)

This is Part A of an article that defends non-eliminative structuralism about mathematics by means of a concrete case study: a theory of unlabeled graphs. Part A summarizes the general attractions of non-eliminative structuralism. Afterwards, it motivates an understanding of unlabeled graphs as structures sui generis and develops a corresponding axiomatic theory of unlabeled graphs. As the theory demonstrates, graph theory can be developed consistently without eliminating unlabeled graphs in favour of sets; and the usual structuralist (...) criterion of identity can be applied successfully in graph-theoretic proofs. Part B will turn to the philosophical interpretation and assessment of the theory. (shrink)

While Saunders Mac Lane studied for his D.Phil in Göttingen, he heard David Hilbert's weekly lectures on philosophy, talked philosophy with Hermann Weyl, and studied it with Moritz Geiger. Their philosophies and Emmy Noether's algebra all influenced his conception of category theory, which has become the working structure theory of mathematics. His practice has constantly affirmed that a proper large-scale organization for mathematics is the most efficient path to valuable specific results—while he sees that the question of which (...) results are valuable has an ineliminable philosophic aspect. His philosophy relies on the ideas of truth and existence he studied in Göttingen. His career is a case study relating naturalism in philosophy of mathematics to philosophy as it naturally arises in mathematics. Introduction Structures and Morphisms Varieties of Structuralism Göttingen Logic: Mac Lane's Dissertation Emmy Noether Natural Transformations Grothendieck: Toposes and Universes Lawvere and Foundations Truth and Existence Naturalism Austere Forms of Beauty. (shrink)

In recent philosophy of mathematics avariety of writers have presented ``structuralist''views and arguments. There are, however, a number ofsubstantive differences in what their proponents take``structuralism'' to be. In this paper we make explicitthese differences, as well as some underlyingsimilarities and common roots. We thus identifysystematically and in detail, several main variants ofstructuralism, including some not often recognized assuch. As a result the relations between thesevariants, and between the respective problems theyface, become manifest. Throughout our focus is onsemantic and (...) metaphysical issues, including what is orcould be meant by ``structure'' in this connection. (shrink)

One kind of structuralism holds that mathematics is about structures, conceived as a type of abstract entity. Another denies that it is about any distinctively mathematical entities at all—even abstract structures; rather it gives purely general information about what holds of any collection of entities conforming to the axioms of the theory. Of these, pure structuralism is most plausibly taken to enjoy significant advantages over platonism. But in what appears to be its most plausible—modalised—version, even (...) restricted to elementary arithmetic, it requires defence of a very strong possibility claim: that there could be a completed w-sequence of concrete objects. There are very serious epistemological difficulties in the way of providing the requisite defence. (shrink)

Frege's arithmetical-platonism is glossed as the first step in developing the thesis; however, it remains silent on the subject of structures in mathematics: the obvious examples being groups and rings, lattices and topologies. The structuralist objects to this silence, also questioning the sufficiency of Fregean platonism is answering a number of problems: e.g. Benacerraf's Twin Puzzles of Epistemic and Referential Access. The development of structuralism as a philosophical position, based on the slogan 'All mathematics (...) is structural' collapses: there is no single coherent account which remains faithful to the tenets of structuralism and solves the puzzles of platonism. This prompts the adoption of a more modest structuralism, the aim of which is not to solve the problems facing arithmetical-platonism, but merely to give an account of the 'obviously structural areas of mathematics'. Modest strucmralism should complement an account of mathematical systems; here, Frege's platonism fulfils that role, which then constrains and shapes the development of this modest structuralism. Three alternatives are considered; a substitutional account, an account based on a modification of Dummett's theory of thin reference and a modified from of in re structuralism. This split level analysis of mathematics leads to an investigation of the robustness of the truth predicate over the two classes of mathematical statement. Focussing on the framework set out in Wright's Truth and Objectivity, a third type of statement is identified in the literature: Hilbert's formal statements. The following thesis arises: formal statements concern no special subject matter, and are merely minimally truth apt; the statements of structural mathematics form a subdiscourse - identified by the similarity of the logical grammar - displaying cognitive command. Thirdly, the statements of mathematics which concern systems form a subdiscourse which has both cognitive command and width of cosmological role. The extensions of mathematical concepts are such that best practice on the part of mathematicians either tracks or determines that extension - at least in simple cases. Examining the notions of response dependence leads to considerations of indefinite extensibility and intuitionism. The conclusion drawn is that discourse about structures and mathematical systems are response dependent but that this does not give rise to any revisionary arguments contra intuitionism. (shrink)

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)

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are (...) parts of the physical world and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

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)

In this 1992 book, Professor Koslow advances an account of the basic concepts of logic. A central feature of the theory is that it does not require the elements of logic to be based on a formal language. Rather, it uses a general notion of implication as a way of organizing the formal results of various systems of logic in a simple, but insightful way. The study has four parts. In the first two parts the various sources of the (...) general concept of an implication structure and its forms are illustrated and explained. Part 3 defines the various logical operations and systematically explores their properties. A generalized account of extensionality and dual implication is given, and the extensionality of each of the operators, as well as the relation of negation and its dual, are given substantial treatment because of the novel results they yield. Part 4 considers modal operators and studies their interaction with logical operators. By obtaining the usual results without the usual assumptions this new approach allows one to give a very simple account of modal logic minus the excess baggage of possible world semantics. (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.”.

The thesis is a study of the notion of intuition in the foundations of mathematics which focuses on the case of natural numbers and hereditarily finite sets. Phenomenological considerations are brought to bear on some of the main objections that have been raised to this notion. ;Suppose that a person P knows that S only if S is true, P believes that S, and P's belief that S is produced by a process that gives evidence for it. On (...) a phenomenological view the relevant evidence is provided by intuition , and this should be the case in either ordinary perceptual knowledge or in mathematical knowledge. Intuition is to be understood in terms of fulfillments of intentions. Knowledge is a product of intuition and intention. In the case of mathematical knowledge it is said that there is a construction for a mathematical statement S if and only if the intention expressed by S is fulfilled . Constructions are thus viewed as intuition processes that could actually or possibly be carried out. In elementary parts of mathematics they might be characterized in terms of certain classes of recursive functions. This view is discussed in the case where S is taken to be a singular statement about natural numbers or finite sets, and also where S is taken to be a general statement about such objects. The distinction between intuition of and intuition that is also investigated in this context. ;It is pointed out how on a phenomenological view a number of central problems about mathematical intuition can be avoided: problems about the analogousness of perceptual and mathematical intuition, about causal accounts of knowledge in mathematics, and about structuralism in mathematics. The bearing of the account on issues concerning constructivism and platonism is also discussed. (shrink)

The philosophy of linguistics is a rich philosophical domain which encompasses various disciplines. One of the aims of this thesis is to unite theoretical linguistics, the philosophy of language, the philosophy of science and the ontology of language. Each part of the research presented here targets separate but related goals with the unified aim of bringing greater clarity to the foundations of linguistics from a philosophical perspective. Part I is devoted to the methodology of linguistics in terms of scientific modelling. (...) I argue against both the Conceptualist and Platonist interpretations of linguistic theory by means of three grades of mathematical involvement for linguistic grammars. Part II explores the specific models of syntactic and semantics by an analogy with the harder sciences. In Part III, I develop a novel account of linguistic ontology and in the process comment on the type-token distinction, the role and connection with mathematics and the nature of linguistic objects. In this research, I offer a structural realist interpretation of linguistic methodology with a nuanced structuralist picture for its ontology. This proposal is informed by historical and current work in theoretical linguistics as well as philosophical views on ontology, scientific modelling and mathematics. (shrink)

The structuralist Andre-Weil–Claude-Levi-Strauss transformation formula (CF), initially applied to kinship systems, mythology, ritual, artistic design and architecture, was rightfully criticized for its rationalism and tendency to reduce complex transformations to analogical structures. I present a revised non-mathematical revision of the CF, a general transformation formula (rCF) applicable to networks of complementary semantic binaries in conceptual value-fields of culture, including comparative religion and mythology, ritual, art, literature and philosophy. The rCF is a rule-guided formula for combinatorial conceptualizing in (...) non-representational, presentational mythopoetics and other cultural symbolizations. I consider poststructuralist category-theoretic and algebraic mathematical interpretations of the CF as themselves only mathematical analogies, which serve to stimulate further revision of the logic model of the rCF. The rCF can be used in hypothesis-making to advance understanding of the evolution and prehistory of human symbolic behaviour in cultural space, philosophical ontologies and categories, definitions and concepts in art, religion, psychotherapy, and other cultural-value forms. (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.

The realist versions of mathematical structuralism are often characterized by what I call ‘the insubstantiality thesis’, according to which mathematical objects, being positions in structures, have no non-structural properties: they are purely structural objects. The thesis has been criticized for being inconsistent or descriptively inadequate. In this paper, by implementing the resources of a real-definitional account of essence in the context of Fregean abstraction principles, I offer a version of structuralism – essentialist structuralism – which validates a (...) weaker version of the insubstantiality thesis: mathematical objects have no non-structural essential properties. Next, I show how this rendition of structuralism alleviates a Fregean worry against insubstantiality, which is directed at the explanation of the applicability of mathematics from the structuralist perspective. (shrink)

We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the (...) main questions and issues that have arisen. Namely, elements of different structures are different. A structure and its elements ontologically depend on each other. There are no haecceities and each element of a structure must be discernible within the theory. These consequences are not developed piecemeal but rather follow from our definitions of basic structuralist concepts. (shrink)

This compact volume, belonging to the Cambridge Elements series, is a useful introduction to some of the most fundamental questions of philosophy and foundations of mathematics. What really distinguishes realist and platonist views of mathematics from anti-platonist views, including fictionalist and nominalist and modal-structuralist views?1 They seem to confront similar problems of justification, presenting tradeoffs between which it is difficult to adjudicate. For example, how do we gain access to the abstract posits of platonist accounts of arithmetic, (...) analysis, geometry, etc., including numbers, functions, sets, points, lines, spaces, etc., whether it be such objects themselves or the possibilities of such, as postulated by modal and modal structural accounts?2 What real difference does it make, whether it be the existence of such things or the mathematical possibility of such things? Even fictionalist views seem to confront analogous problems. After all, we rely on mathematics for myriad scientific applications; so such mathematics had better at least be coherent, even if not true. But coherence requires at least formal consistency; so we seem implicitly to be committed to things like formal derivations, viz. the absence of any such having contradictory consequences, framed within the appropriate system of axioms and rules. But derivations are themselves akin to mathematical objects, as derivations are strings of strings of symbols. Moreover, derivations provided by axioms and rules form an infinite class, such that, at any future time, infinitely many will not have been written down. Moreover, such strings, as Gödel famously showed, can be coded as natural numbers. Thus, even fictionalist accounts (such as that of Field in his Science without Numbers [2016]) seem to confront problems very much like those plaguing platonist accounts. (shrink)

Ontic structural realism argues that structure is all there is. In (French, 2014) I argued for an ‘eliminativist’ version of this view, according to which the world should be conceived, metaphysically, as structure, and objects, at both the fundamental and ‘everyday’ levels, should be eliminated. This paper is a response to a number of profound concerns that have been raised, such as how we might distinguish between the kind of structure invoked by this view and mathematical structure in general, (...) how we should choose between eliminativist ontic structural realism and alternative metaphysical accounts such as dispositionalism, and how we should capture, in metaphysical terms, the relationship between structures and particles. In developing my response I shall touch on a number of broad issues, including the applicability of mathematics, the nature of representation and the relationship between metaphysics and science in general. (shrink)

The existence of structures with non-trivial authomorphisms (such as the automorphism of the field of complex numbers onto itself that swaps the two roots of – 1) has been held by Burgess and others to pose a serious difficulty for mathematical structuralism. This paper proposes a model-theoretic solution to the problem. It suggests that mathematical structuralists identify the “position” of an n-tuple in a mathematical structure with the type of that n-tuple in the expansion of the (...) structure that has a name for every element in it. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Husserl and Mathematics by Mirja HartimoAndrea StaitiMirja Hartimo. Husserl and Mathematics. Cambridge: Cambridge University Press, 2021. Pp. 214. Hardback, $99.99.Mirja Hartimo has written the first book-length study of Husserl's evolving views on mathematics that takes his intellectual context into full consideration. Most importantly, Hartimo's historically informed approach to the topic benefits from her extensive knowledge of Husserl's library. Throughout the book, she provides references (...) to texts and articles that Husserl read, marked, and annotated, although regrettably, very few of these sources are quoted explicitly in his published works. The overall picture that emerges from Hartimo's book shows that Husserl continued to read and engage with mathematics throughout his career, and, even though extensive discussions of mathematical issues are not frequent in his later works, some of the ideas he develops, for instance, in The Crisis of the European Sciences and Transcendental Phenomenology, can be read against the backdrop of the developments of mathematics in his time.Chapter 1 sets the method for the rest of the book. Hartimo emphasizes the importance of a methodological device that takes center stage in Husserl's work from the 1920s onward: Besinnung. Hartimo claims that Besinnung is a method to reactivate the "goals and purposes" (21) that originally motivated scientists engaging in a particular line of research. According to Husserl, as specialized research progresses and becomes standard practice, such original goals and purposes might be blurred by other concerns. When applied to mathematics, Besinnung thus amounts to a reflection "on what mathematicians should do, on what the genuine goal of their activities should be" (21–22). Hartimo argues that this kind of approach to mathematics comes close to Penelope Maddy's naturalism, which takes its departure from what mathematicians are actually doing and then proceeds to inquire into the ontological status of mathematical objects and theories. Hartimo identifies particularly in David Hilbert, a colleague of Husserl's at Göttingen, a vivid embodiment of what Husserl considered to be exemplary work oriented toward the ideal goal of mathematics.In chapters 2 and 3, Hartimo expands upon Husserl's conception of what the true goal of mathematics should be. She does so, first, by addressing the classical topic of Husserl's psychologism in his first (and only) work entirely devoted to mathematics, The Philosophy of Arithmetic, and Frege's well-known critical review of it. In Hartimo's rendition, Husserl criticizes Frege for his attempt to provide formal definitions of basic notions, such as equality. For Husserl, such basic notions are intuitively available and require no definition—rather, they should be put to work to clarify vague notions. Frege, in return, attacks what he takes to be Husserl's wishy-washy psychological account of numbers, which revolves around the concrete act of collecting items and abstracting from their specific contents. Ultimately, Hartimo argues, Husserl takes Frege's criticism to heart and in his later Prolegomena no longer pursues a psychological foundation of number, but rather emphasizes the distinction between the subjective and the objective sides of mathematics, with philosophers studying the former and mathematicians focusing on the latter. While Hartimo's verdict on the philosophical yield of the debate is in line with the standard interpretation of Frege's review as prompting Husserl's antipsychologistic turn, she makes a remark that might be worthy of further discussion: "While for Husserl the debate was about logicism, Frege shifts the debate to be about psychologism and here commentators have followed Frege's example" (50). It would be interesting to imagine what a Fregean reply to Husserl's view about the uselessness of formal definitions for basic notions like equality might look like. In any case, Hartimo's reconstruction makes it clear that, however motivated, Husserl's rejection of psychologism does not amount to an acceptance of logicism.In chapter 3, Hartimo puts forward two of her central claims: (1) Husserl was a structuralist about mathematics; and (2) for Husserl, the goal of modern mathematics is encapsulated in the concept of definiteness. According to (1), mathematics is about the formal structures of systems, which can be isolated and compared. Definiteness, in turn, is the property of systems... (shrink)

Several case studies and theoretical reports indicate that the structuralist concept of a constraint has a central role in the reconstruction of physical theories. It is surprising that there is, in the literature, only little theoretical discussion on the relevance of constraints for the reconstruction of social scientific theories. Almost all structuralist reconstructions of social theorizing are vacuously constrained. Consequently, constraints are methodologically irrelevant.In this paper I try to show that there really exist constraint-type assumptions in mathematical (...) modelling in the social sciences. Methodologically constraints have exactly the same role in the context of social mathematical modelling as they have in physical theories. In typical cases of mathematical modelling in the social sciences, the related constraints work as empirical hypotheses and should be tested by statistical means. (shrink)

Recent work in the philosophy of mathematics has suggested that mathematical structuralism is not committed to a strong form of the Identity of Indiscernibles (II). José Bermúdez demurs, and argues that a strong form of II can be warranted on structuralist grounds by countenancing identity properties, or haecceities, as legitimately structural. Typically, structuralists dismiss such properties as obviously non-structural. I will argue to the contrary that haecceities can be viewed as structural but that this concession does not (...) warrant Bermúdez’s version of II but, rather, another easily falsified version. I close with some reflections on reference vis-à-vis structurally indiscernible objects. (shrink)

Plato himself would be pleased at the recent emergence of a certain highly Platonic variety of platonism concerning mathematics, viz., the structuralism of Michael Resnik and Stewart Shapiro. In fact, this species of platonism is so Platonic that it is susceptible to an objection closely related to one raised against Plato by Parmenides in the dialogue of that name. This is the Third Man Argument against a view about the relation of Forms to particulars. My objection is (...) not a TMA against structuralism; the position avoids that objection, but is vulnerable to a different one precisely at the point where it avoids the TMA. The way structuralism avoids the TMA has in fact been considered, as one of Plato’s options, by at least one commentator on the Parmenides, Colin Strang, who explicitly rejects it on logical grounds. In the course of the discussion, I shall clarify the reason that I believe led Strang to reject this option, and shall modify his own statement of that reason. (shrink)

This paper pursues two goals. Its first goal is to clear up the “identity problem” faced by the structuralist interpretation of mathematics. Its second goal, through the consideration of examples coming in particular from the theory of permutations, is to examine cases of misunderstandings in mathematics fit to cast some light on mathematical understanding in general. The common thread shared by these two goals is the notion of setting. The study of a mathematical object (...) almost always goes together with the choice of a particular setting, and the understanding of the workings of mathematical settings is an essential component of mathematical knowledge. It is claimed that the recognition of mathematical settings, as features distinct from both mathematical structures and the systems which instantiate those structures, allows one to classify most of understandable misunderstandings in mathematics, and also to solve the identity problem. (shrink)

The paper outlines a novel version of mathematical structuralism related to invariants. The main objective here is twofold: first, to present a formal theory of structures based on the structuralist methodology underlying work with invariants. Second, to show that the resulting framework allows one to model several typical operations in modern mathematical practice: the comparison of invariants in terms of their distinctive power, the bundling of incomparable invariants to increase their collective strength, as well as a (...) heuristic principle related to the search for new invariants. (shrink)

Structuralism is a view about the subject matter of mathematics according to which what matters are structural relationships in abstraction from the intrinsic nature of the related objects. Mathematics is seen as the free exploration of structural possibilities, primarily through creative concept formation, postulation, and deduction. The items making up any particular system exemplifying the structure in question are of no importance; all that matters is that they satisfy certain general conditions—typically spelled out in axioms defining the structure (...) or structures of interest—characteristic of the branch of mathematics in question. Thus, in the basic case of arithmetic, the famous “axioms” of Richard Dedekind (taken over by Giuseppe Peano, as he acknowledged) were conditions in a definition of a “simply infinite system”, with an initial item, each item having a unique next one, no two with the same next one, and all items finitely many steps from the initial one. (The latter condition is guaranteed by the axiom of mathematical induction.) All such systems are structurally identical, and, in a sense to be made more precise, the shared structure is what mathematics investigates. (In other cases, multiple structures are allowed, as in abstract algebra with its many groups, rings, fields, and so forth.) This structuralist view of arithmetic thus contrasts with the absolutist view, associated with Gottlob Frege and Bertrand Russell, that natural numbers must in fact be certain definite objects, namely classes of equinumerous concepts or classes. (shrink)

James Ladyman has recently proposed a view according to which all that exists on the level of microphysics are structures "all the way down". By means of a comparative reading of structuralism in philosophy of mathematics as proposed by Stewart Shapiro, I shall present what I believe structures could not be. I shall argue that, if Ladyman is indeed proposing something as strong as suggested here, then he is committed to solving problems that proponents of structuralism in (...) philosophy of mathematics such as Shapiro are trying to solve. Attempting to do so, however, brings out a tacit tension in Ladyman's position. I shall argue that the upshot of this is that the ontological import that Ladyman attributes to structures is rather epistemological import properly understood. (shrink)

According to the realist rendering of mathematical structuralism, mathematical structures are ontologically prior to individual mathematical objects such as numbers and sets. Mathematical objects are merely positions in structures: their nature entirely consists in having the properties arising from the structure to which they belong. In this paper, I offer a bundle-theoretic account of this structuralist conception of mathematical objects: what we normally describe as an individual mathematical object is the mereological (...) bundle of its structural properties. An emerging picture is a version of mereological essentialism: the structural properties of a mathematical object, as a bundle, are the mereological parts of the bundle, which are possessed by it essentially. (shrink)

Despite some surface similarities, Charles Peirce’s philosophy of mathematics, pragmaticism, is incompatible with both mathematical structuralism and fictionalism. Pragmaticism has to do with experimentation and observation concerning the forms of relations in diagrammatic and iconic representations ofmathematical entities. It does not presuppose mathematical foundations although it has these representations as its objects of study. But these objects do have a reality which structuralism and fictionalism deny.

In modern physics the notion of structure can be treated as an extension of the notion of law of nature. French and Ladyman’s view concerning the ontological priority of structures over objects is confronted with Psillos’ criticism. This kind of view agrees with the paradigmatic case where the structure is an internal symmetry and the instantiations are elementary particles. An ontological model is proposed which demonstrates the relation between structures and their instantiations in this case. This view which (...) may be categorized as “weak ontic structuralism” is compared with Busch’s treatment of ontic structuralism in the philosophy of mathematics. (shrink)

Recent technical developments in the logic of nominalism make it possible to improve and extend significantly the approach to mathematics developed in Mathematics without Numbers. After reviewing the intuitive ideas behind structuralism in general, the modal-structuralist approach as potentially class-free is contrasted broadly with other leading approaches. The machinery of nominalistic ordered pairing (Burgess-Hazen-Lewis) and plural quantification (Boolos) can then be utilized to extend the core systems of modal-structural arithmetic and analysis respectively to full, classical, polyadic third- (...) and fourthorder number theory. The mathenatics of many structures of central importance in functional analysis, measure theory, and topology can be recovered within essentially these frameworks. (shrink)

Philip Kitcher has proposed an account of mathematical truth which he hopes avoids platonistic commitment to abstract mathematical objects. His idea is that the truth-conditions of mathematical statements consist in certain general structural features of physical reality. He codifies these structural features by reference to various operations which are performable on objects: the world is structured in such a way that these operations are possible. Which operations are performable cannot be known a priori; rather, we hypothesize, conjecture, (...) idealize, and eventually wind up with theories which are true of the world, just as we do in the sciences. Kitcher argues that mathematical and physical knowledge are continuous, in that they concern the same subject matter and are subject to the same epistemological and methodological constraints. (shrink)

Wittgenstein’s philosophy of mathematics is often devalued due to its peculiar features, especially its radical departure from any of standard positions in foundations of mathematics, such as logicism, intuitionism, and formalism. We first contrast Wittgenstein’s finitism with Hilbert’s finitism, arguing that Wittgenstein’s is perspicuous or surveyable finitism whereas Hilbert’s is transcendental finitism. We then further elucidate Wittgenstein’s philosophy by explicating his natural history view of logic and mathematics, which is tightly linked with the so-called rule-following problem and (...) Kripkenstein’s paradox, yielding vital implications to the nature of mathematical understanding, and to the nature of the certainty and objectivity of mathematical truth. Since the incompleteness theorems, foundations of mathematics have mostly lost their philosophical driving force, and anti-foundationalism has become prevalent and pervasive. Yet new foundations have nevertheless emerged and come into the scene, namely categorical foundations. We articulate the foundational significance of category theory by explicating three forms of foundations, i.e., global foundations, local foundations, and conceptual foundations. And we explore the possibility of categorical unified science qua pluralistic unified science, arguing for the categorical unity of science on both mathematical and philosophical grounds. We then turn to an issue in conceptual foundations of mathematics, namely the nature of the concept of space. We elucidate Wittgenstein’s intensional conception of space in relation to Brouwer’s theory of space continua and to the modern conception of space as point-free structure in category theory and algebraic geometry. We finally give a bird’s-eye view of mathematical philosophy from a Wittgensteinian perspective, and further sheds new light on Wittgenstein’s constructive structuralism and his view of incompleteness and contradictions in mathematics. (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)

Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also (...) analyzes the concept of intuition and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite. (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)

Systems Biology and the Modern Synthesis are recent versions of two classical biological paradigms that are known as structuralism and functionalism, or internalism and externalism. According to functionalism (or externalism), living matter is a fundamentally passive entity that owes its organization to external forces (functions that shape organs) or to an external organizing agent (natural selection). Structuralism (or internalism), is the view that living matter is an intrinsically active entity that is capable of organizing itself from within, with purely internal (...) processes that are based on mathematical principles and physical laws. At the molecular level, the basic mechanism of the Modern Synthesis is molecular copying, the process that leads in the short run to heredity and in the long run to natural selection. The basic mechanism of Systems Biology, instead, is self-assembly, the process by which many supramolecular structures are formed by the spontaneous aggregation of their components. In addition to molecular copying and self-assembly, however, molecular biology has uncovered also a third great mechanism at the heart of life. The existence of the genetic code and of many other organic codes in Nature tells us that molecular coding is a biological reality and we need therefore a framework that accounts for it. This framework is Code biology, the study of the codes of life, a new field of research that brings to light an entirely new dimension of the living world and gives us a completely new understanding of the origin and the evolution of life. (shrink)

The article considers structuralism as a philosophy of mathematics, as based on the commonly accepted explicit mathematical concept of a structure. Such a structure consists of a set with specified functions and relations satisfying specified axioms, which describe the type of the structure. Examples of such structures such as groups and spaces, are described. The viewpoint is now dominant in organizing much of mathematics, but does not cover all mathematics, in particular most applications. It does (...) not explain why certain structures are dominant, not why the same mathematical structure can have so many different and protean realizations. ‘structure’ is just one part of the full situation, which must somehow connect the ideal structures with their varied examples. (shrink)