Elaine Landry

In the Anglophone world, the philosophical treatment of geometry has fallen on hard times. While in Germany as late as the 1920s there were vibrant discussions concerning the nature of geometry— especially in relation to the direction of its development, the role of intuition and the perception of physical space—the rise of logical empiricism largely brought these to a close. Accounts of mathematics in its totality as uniformly reducible to a language such as set theory have led the recipients of logical empiricist doctrines to ignore the thematic contours of modern mathematics. I argue, however, that geometry in all of its many guises flourishes in contemporary mathematics, and that the term ‘geometry’ itself continues to have an important meaning. One way then for philosophy to come to a better understanding of mathematics is to provide an account of modern geometry, including its development of new forms of space, both for mathematical physics and for arithmetic. I argue that we can find a good starting point for this work by returning to the discussions of Weyl and Cassirer on geometry. With the help of modern interpreters, we see that issues salient to these thinkers are very much relevant to us today. I also propose that an effective way to encompass a great part of modern geometry is via (1, 1)-toposes, also known as homotopy toposes, an important development of category theory. Recently an internal language for such categories has been devised, known as ‘homotopy type theory’, a variant of which captures the ‘cohesiveness’ of geometric spaces. With these tools in place, we can now start to see what an adequate philosophical account of current geometry might look like.

Borrowing from the title of Saunders Mac Lane’s seminal work Categories for the Working Mathematician, this book aims to bring the concepts of category theory to philosophers working in areas ranging from mathematics to proof theory to computer science to ontology, from physics to biology to cognition, from mathematical modeling to the structure of scientific theories to the structure of the world. Moreover, it aims to do this in a way that is accessible to a general audience. Each chapter is written by either a category-theorist or a philosopher working in one of the represented areas, and in a way that is accessible and is intended to build on the concepts already familiar to those philosophers working in these areas.

I use my reading of Plato to develop what I call as-ifism, the view that, in mathematics, we treat our hypotheses as if they were first principles and we do this with the purpose of solving mathematical problems. I then extend this view to modern mathematics showing that when we shift our focus from the method of philosophy to the method of mathematics, we see that an as-if methodological interpretation of mathematical structuralism can be used to provide an account of the practice and the applicability of mathematics while avoiding the conflation of metaphysical considerations with mathematical ones.

I argue that if we distinguish between ontological realism and semantic realism, then we no longer have to choose between platonism and formalism. If we take category theory as the language of mathematics, then a linguistic analysis of the content and structure of what we say in and about mathematical theories allows us to justify the inclusion of mathematical concepts and theories as legitimate objects of philosophical study. Insofar as this analysis relies on a distinction between ontological and semantic realism, it relies also on an implicit distinction between mathematics as a descriptive science and mathematics as a descriptive discourse. It is this latter distinction which gives rise to the tension between the mathematician qua philosopher. In conclusion, I argue that the tensions between formalism and platonism, indeed between mathematician and philosopher, arise because of an assumption that there is an analogy between mathematical talk and talk in the physical sciences.

On what basis can we justify the inclusion of mathematical concepts and theories as legitimate objects of philosophical study? One answer to this question is: mathematical concepts and theories ought to be included because they are indispensable for describing the physical world. But what if a particular mathematical concept or theory has no such application? Are its corresponding statements to be counted as meaningless? Are its corresponding objects to be taken as mere linguistic fictions? One way of avoiding these conclusions is provided by the set-theoretic foundationalist: Mathematical concepts, insofar as they reduce to set-theoretic concepts, are meaningful and hence are to be regarded as legitimate objects of philosophical study. The problem with this approach is that it leads to a conflict between truth and meaning: some mathematical statements are true for one interpretation of set theory yet false for another. ;I show that if we shift our philosophical focus from what mathematical theories are about to what mathematical theories say and if we take category theory as the language of mathematics , then we can justify the inclusion of mathematical concepts and theories as legitimate objects of philosophical study. Thus, I argue that the current debate over the status of mathematical objects and statements is resolved by adopting a linguistic approach to the philosophical analysis of mathematical concepts and theories. In particular, I show how category theory can be taken as the language used to specify and organize the common structural features of mathematical discourse. Category theory, then, provides the means for representing and justifying our talk of mathematical existence, meaning and truth from within a given mathematical theory.

This paper explores varieties of scientific structuralism. Central to our investigation is the notion of `shared structure'. We begin with a description of mathematical structuralism and use this to point out analogies and disanalogies with scientific structuralism. Our particular focus is the semantic structuralist's attempt to use the notion of shared structure to account for the theory-world connection, this use being crucially important to both the contemporary structural empiricist and realist. We show why minimal scientific structuralism is, at the very least, a powerful methodological standpoint. Our investigation also makes explicit what more must be added to this minimal structuralist position in order to address the theory-world connection, namely, an account of representation.

Structural realism has rapidly gained in popularity in recent years, but it has splintered into many distinct denominations, often underpinned by diverse motivations. There is, no monolithic position known as ‘structural realism,’ but there is a general convergence on the idea that a central role is to be played by relational aspects over object-based aspects of ontology. What becomes of causality in a world without fundamental objects? In this book, the foremost authorities on structural realism attempt to answer this and related questions: ‘what is structure?’ and ‘what is an object?’ Also featured are the most recent advances in structural realism, including the intersection of mathematical structuralism and structural realism, and the latest treatments of laws and modality in the context of structural realism. The book will be of interest to philosophers of science, philosophers of physics, metaphysicians, and those interested in foundational aspects of science.

In this paper I argue that category theory ought to be seen as providing the language for mathematical discourse. Against foundational approaches, I argue that there is no need to reduce either the content or structure of mathematical concepts and theories to the constituents of either the universe of sets or the category of categories. I assign category theory the role of organizing what we say about the content and structure of both mathematical concepts and theories. Insofar, then, as the structuralist sees mathematics as talking about structures and their morphology, I contend that category theory furnishes a framework for mathematical structuralism

I argue that recollection, in Plato's Meno , should not be taken as a method, and, if it is taken as a myth, it should not be taken as a mere myth. Neither should it be taken as a truth, a priori or metaphorical. In contrast to such views, I argue that recollection ought to be taken as an hypothesis for learning. Thus, the only methods demonstrated in the Meno are the elenchus and the hypothetical, or mathematical, method. What Plato's Meno demonstrates, then, is that we cannot be philosophers if we fail to make use of the mathematician's hypothetical method.

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. The challenge facing scientific structuralists is three-fold: i) to characterize scientific theories in ‘structural’ terms, and to use this characterization ii) to establish a theory-world connection (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.

Recent semantic approaches to scientific structuralism, aiming to make precise the concept of shared structure between models, formally frame a model as a type of set-structure. This framework is then used to provide a semantic account of (a) the structure of a scientific theory, (b) the applicability of a mathematical theory to a physical theory, and (c) the structural realist’s appeal to the structural continuity between successive physical theories. In this paper, I challenge the idea that, to be so used, the concept of a model and so the concept of shared structure between models must be formally framed within a single unified framework, set-theoretic or other. I first investigate the Bourbaki-inspired assumption that structures are types of set-structured systems and next consider the extent to which this problematic assumption underpins both Suppes’ and recent semantic views of the structure of a scientific theory. I then use this investigation to show that, when it comes to using the concept of shared structure, there is no need to agree with French that “without a formal framework for explicating this concept of ‘structure-similarity’ it remains vague, just as Giere’s concept of similarity between models does ...” (French, 2000, Synthese, 125, pp. 103–120, p. 114). Neither concept is vague; either can be made precise by appealing to the concept of a morphism, but it is the context (and not any set-theoretic type) that determines the appropriate kind of morphism. I make use of French’s (1999, From physics to philosophy (pp. 187–207). Cambridge: Cambridge University Press) own example from the development of quantum theory to show that, for both Weyl and Wigner’s programmes, it was the context of considering the ‘relevant symmetries’ that determined that the appropriate kind of morphism was the one that preserved the shared Lie-group structure of both the theoretical and phenomenological models.

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the "algebraic" approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a "foundation", or turning meta-mathematical analyses of logical concepts into "philosophical" ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be algebraic structuralists all the way down.

The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various category-theoretic foundational debates and to point to some common elements found among those who advocate adopting a foundational stance. We then use these elements to argue for the philosophical position that category theory provides a framework for an algebraic in re interpretation of mathematical structuralism. In each context, what we aim to show is that, whatever the significance of category theory, it need not rely upon any set-theoretic underpinning.

Feferman argues that category theory cannot stand on its own as a structuralist foundation for mathematics: he claims that, because the notions of operation and collection are both epistemically and logically prior, we require a background theory of operations and collections. Recently [2011], I have argued that in rationally reconstructing Hilbert’s organizational use of the axiomatic method, we can construct an algebraic version of category-theoretic structuralism. That is, in reply to Shapiro, we can be structuralists all the way down ; we do not have to appeal to some background theory to guarantee the truth of our axioms. In this paper, I again turn to Hilbert; I borrow his distinction between the genetic method and the axiomatic method to argue that even if the genetic method requires the notions of operation and collection, the axiomatic method does not. Even if the genetic method is in some sense epistemically or logically prior, the axiomatic method stands alone. Thus, if the claim that category theory can act as a structuralist foundation for mathematics arises from the organizational use of the axiomatic method, then it does not depend on the prior notions of operation or collection, and so we can be structuralists all the way up.