A Critical Introduction to Mathematical Structuralism

2005

The focus of this paper is the recent revival of interest in structuralist approaches to science and, in particular, the structural realist position in philosophy of science 1. The challenge facing scientific structuralists is threefold: i) to characterize scientific theories in 'structural' terms, and to use this characterization ii) to establish a theory-world connection 2 (including an explanation of applicability) and iii) to address the relationship of 'structural continuity' between predecessor and successor theories. Our aim is to appeal to the notion of shared structure between models to reconsider all of these challenges, and, in so doing, to classify the varieties of scientific structuralism and to offer a 'minimal' construal that is best viewed from a methodological stance. 1 Structuralism in Mathematics Since much of what is taken as distinctive of scientific structuralism is tied to mathematical structuralism, we begin first with a brief description of what we mean by this. We take mathematical structuralism to be the following philosophical position: the subject matter of mathematics is structured systems 3 and their morphology, so that Version: 28/01/2005 1 brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by PhilSci Archive • mathematical 'objects' are nothing but 'positions in structured systems', and • mathematical theories aim to describe such objects and systems by their shared structure, i.e., by their being instances of the same kind of structure. For example, the theory of natural numbers, as framed by 4 the Peano axioms, describes the various concrete 5 systems that have a Natural-Number structure. These structured systems are, for example, the von Neumann ordinals, the Zermelo numerals, and so forth; they are models (in the Tarskian sense of the term 6) of the Natural-Number structure. The 'objects' that the theory of natural numbers talks about are then the positions in the various models. For example, the von Neumann ordinal '2' is a position in the model 'von Neumann ordinals'; the Zermelo numeral '2' is a position in the model 'Zermelo numerals'; and the theory of natural numbers describes the number '2' in terms of the shared structure of these, and other, models that have the same kind of structure. If all models that exemplify this structure are isomorphic, then the Natural-Number structure and its morphology are said to present its kinds of objects 7 , i.e., are said to determine its 'objects' only 'up to isomorphism'. As explained by Benacerraf [1965], mathematical structuralism implies that there are no natural numbers as particular objects, i.e., as existing things whose 'essence' or 'nature' can be individuated independently of the role they play in a structured system of a given kind. This is because the relevant criterion of individuation, viz., Leibniz's Principle of the Identity of Indiscernibles, does not hold. For example, in one system of the natural numbers the property 2∈4 holds for the natural number '2' while in another it does not.

Analysis, 2007

According to mathematical structuralism, mathematical objects are defined by their positions in mathematical structures. Structures are understood in the standard sense, as domains over which certain privileged relations are explicitly defined and that may contain identified constant elements and functions. In talking about structures we use a language (for the purposes of this article, and for model theory in general, a first-order language) whose signature contains symbols for the relevant relations, constant elements, and functions. The theory of a structure is the set of sentences true in that structure. To take a familiar example, we specify the structure in a way that makes clear its signature and hence the language that we use to talk about it. The theory of this structure, Th () = {ϕ | ϕ is a sentence of and ϕ}, is complete arithmetic. One objection to mathematical structuralism is that there seem to be facts about the domain of a structure that cannot be 'stated' in terms of the relations, elements, and functions available within the structure. That is, there are facts about elements of the domain that do not seem to be reducible to facts about positions in the domain. Thus John Burgess, commenting on the complex in relation to the version of structuralism proposed in Shapiro (1997):

2007

In my view, structuralism as presented by Shapiro (1991, 1997), Resnik (1991), and elsewhere offers the most plausible philosophy of mathematics: Mathematics is about structures, indeed it is the science of pure structures. Structures have no mysterious ontological status, and hence mathematics is not ontologically mysterious, either. Again, it is no mystery how we can acquire knowledge about structures and thus mathematical knowledge. We find structures everywhere. Hence, if mathematics is about structures, we can apply mathematics everywhere. In this way, structuralism promises to offer straightforward answers to the most pressing problems in the philosophy of mathematics. However, there are not only structures, there are also mathematical objects, numbers, pairs, triangles, sets, etc. Concerning their nature, structuralism tends to metaphorics, the most preferred metaphor being that mathematical objects are places in mathematical structures. Maybe it is not really necessary to sa...

Structuralism is the view that mathematics is the science of structure. It has been noted that category theory expresses mathematical objects exactly along their structural properties. This has led to a programme of categorical structuralism, integrating structuralist philosophy with insights from category theory for new views on the foundations of mathematics. In this thesis, we begin by by investigating structuralism to note important properties of mathematical structures. An overview of categorical structuralism is given, as well as the associated views on the foundations of mathematics. We analyse the different purposes of mathematical foundations, separating different kinds of foundations, be they ontological, epistemological, or pragmatic in nature. This allows us to respond to both the categorical structuralists and their critics from a neutral perspective. We find that common criticisms with regards to categorical foundations are based on an unnecessary interpretation of mathematical statements. With all this in hand, we can describe “schematic mathematics”, or mathematics from a structuralist perspective informed by the categorical structuralists, employing only certain kinds of foundations.

2019

Two related slogans for structuralism in the philosophy of mathematics are that "mathematics is the general study of structures" and that, in pursuing such study, we can "abstract away from the nature of objects instantiating those structures". (As such, structuralism stands in contrast with several other general views about mathematics, including: the traditional view that mathematics is the science of number and quantity; the view that it is an empty formalism used primarily for calculation; and the view that it is the study of a basic set-theoretic universe.) As the present survey aims to show, these slogans, while suggestive, are ambiguous and in need of clarification. Indeed, they have been interpreted in various different, even conflicting ways.

Axiomathes, 2013

There has been considerable discussion in the literature of one kind of identity problem that mathematical structuralism faces: the automorphism problem, in which the structure is unable to individuate the mathematical entities in its domain. Shapiro (Philos Math 16(3):285-309, 2008) has partly responded to these concerns. But I argue here that the theory faces an even more serious kind of identity problem, which the theory can't overcome staying within its remit. I give two examples to make the point. Keywords Identity Á Complex numbers Á Mathematical structuralism Á Number systems Mathematical structuralism is the doctrine that the essence of mathematics is structure: rather than being about mathematical entities, traditionally conceived, it is about structural relations, abstractly conceived. This is a view that goes back to Hilbert's 1899 account of Euclidean geometry, with further antecedents in the work of Dedekind, Grassman, Boole and others. On the one hand this attitude gave birth to the tendency in modern mathematics to remake everything within the domain of algebra. But for philosophers there are meant to be more subtle gains: it relegates mathematical entities to a subsidiary role, thereby easing some of the problems that allegedly face realism in mathematics, or accommodating the complaints of nominalists, depending on one's view. And it connects strongly with modern Model Theory, which may be regarded as having begun with Hilbert's work, and thus with logic. The principal structuralists in philosophy today are Stewart Shapiro, Charles

2009

Analysis, 2007

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.

2003

ii This thesis aims to explain the nature and justification of mathematical knowledge using an epistemic version of mathematical structuralism, that is a hybrid of Aristotelian structuralism and Hellman’s modal structuralism. Structuralism, the theory that mathematical entities are recurring structures or patterns, has become an increasingly prominent theory of mathematical ontology in the later decades of the twentieth century. The epistemically driven version of structuralism that is advocated in this thesis takes structures to be primarily physical, rather than Platonically abstract entities. A fundamental benefit of epistemic structuralism is that this account, unlike other accounts, can be integrated into a naturalistic epistemology, as well as being congruent with mathematical practice. In justifying mathematical knowledge, two levels of abstraction are introduced. Abstraction by simplification is how we extract mathematical structures from our experience of the physical world...

Le Centre pour la Communication Scientifique Directe - HAL - memSIC, 2015

We present the structuralist conception of scientific theories as a Deus ex Machina which allows to resolve the entanglements of theories in Mathematics Education. We illustrate with examples how this conception, which forms a solid and solvent body of knowledge in Philosophy of Science, provides us with tools to perform a careful analysis of a theory, both by itself and in connection with other theories.