I argue that if we distinguish between ontological realism and semantic realism, then we no longe... more 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.
I use my reading of Plato to develop what I call as-ifism, the view that, in mathematics, we trea... more 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.
Christopher Pincock's Mathematics and Scientific Representation is an interesting and important n... more Christopher Pincock's Mathematics and Scientific Representation is an interesting and important new book on the applications of mathematics to science. The book is loaded with fascinating case studies that bring out the varied and interesting ways that mathematics is used in science. Pincock provides a much more nuanced picture of the roles that mathematics plays in empirical science than we usually get from philosophers of mathematics writing on the topic of applications. He describes various dimensions along which different uses of mathematics can differ; for instance, to pick out just one of the dimensions he discusses, we sometimes use mathematics to help us represent the causal structure of a series of events, whereas in other cases, we use it to provide an acausal representation of a system in a static, fixed state. The overall picture we get from Pincock is a sort of taxonomy of different kinds of uses of mathematics. He also draws interesting connections
This Element argues that Plato was not a mathematical Platonist. It shows that Plato keeps a clea... more This Element argues that Plato was not a mathematical Platonist. It shows that Plato keeps a clear distinction between mathematical and metaphysical realism, and the knife he uses to slice the difference is method. The philosopher's dialectical method requires that we tether the truth of hypotheses to existing metaphysical objects. The mathematician's hypothetical method, by contrast, takes hypotheses as if they were first principles, so no metaphysical account of their truth is needed. Thus, we come to Plato's methodological as-if realism: In mathematics, we treat our hypotheses as if they were first principles, and, consequently, our objects as if they existed, and we do this for the purpose of solving problems. Taking the road suggested by Plato's Republic, the author shows that some methodological commitments to mathematical objects are made in light of mathematical practice; some are made in light of foundational considerations; and some are made in light of mathematical applicability.
The aim of this paper is to put into context the historical, foundational and philosophical signi... more 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.
This paper explores varieties of scientific structuralism. Central to our investigation is the no... more 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 realists have made use of category theory in three ways. The first is as a meta-level ... more Structural realists have made use of category theory in three ways. The first is as a meta-level formal framework for a structural realist account of the structure of scientific theories, either syntactic or semantic. The second is an appeal to the category-theoretic structure of some successful, successive or fundamental, physical theory to argue that this is the structure we should be physically committed to, either epistemically or ontically. The third is to use category theory as a conceptual tool to argue that it makes conceptual sense to talk of relations without relata and structures without objects. After a brief overview of structural realism, I consider how each appeal to the use of category theory stands up against the aims of the structural realist.
The Western Ontario series in philosophy of science, 2011
ABSTRACT In this paper I offer an answer to a question raised in (Psillos, 2006): How can one spe... more ABSTRACT In this paper I offer an answer to a question raised in (Psillos, 2006): How can one speak of structures without objects? Specifically, I use category theory to show that, mathematically speaking, structures do not need objects. Next, I argue that, scientifically speaking, this category-theoretic answer is silly because it does not speak to the scientific structuralist’s appeal to the appropriate kind of morphism to make precise the concept of shared structure. Against French et al.’s approach,1 I note that to account for the scientific structuralist’s uses of shared structure we do not need to formally frame either the structure of a scientific theory or the concept of shared structure. Here I restate my (Landry, 2007) claim that the concept of shared structure can be made precise by appealing to a kind of morphism, but, in science, it is methodological contexts (and not any category or set-theoretic framework) that determine the appropriate kind. Returning to my aim, I reconsider French’s example of the role of group theory in quantum mechanics to show that French already has an answer to Psillos’ question but this answer is not found in either his set-theoretic formal framework or his ontic structural realism. The answer to Psillos is found both by recognizing that it is the context that determines what the appropriate kind of morphism is and, as Psillos himself suggests,2 by adopting a methodological approach to scientific structuralism.
ABSTRACT This is a short version of the author’s paper “Logicism, structuralism and objectivity” ... more ABSTRACT This is a short version of the author’s paper “Logicism, structuralism and objectivity” [Topoi 20, No. 1, 79–95 (2001; Zbl 1142.00004)].
ABSTRACT The aim of this paper is to present category theory as a framework for an in re interpre... more ABSTRACT The aim of this paper is to present category theory as a framework for an in re interpretation of mathematical structuralism. The use of the term ‘framework’ is significant. On the one hand, it is used in distinction from the term ‘foundation’. As such, what I propose is that we consider category theory as a philosophical tool that allows us to organize what we say about the shared structure of abstract kinds of mathematical systems. On the other hand, the term ‘framework’ is used in the sense of R. Carnap [in: Philosophy of mathematics. Selected readings. 2nd ed., Cambridge (1983; Zbl 0548.03002)]. That is, category theory is taken as a language used to frame what we say about the shared structure of abstract kinds of mathematical systems, as opposed to being a “background theory” which constitutes what a structure is.
The focus of this paper is the recent revival of interest in structuralist approaches to science ... more 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.
Feferman (1977) argues that category theory cannot stand on its own as a structuralist foundation... more Feferman (1977) 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 (2005), 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 (Hilbert, 1900a) 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. §1. Introduction. Feferman's 1977 paper, "Categorical Foundations and Foundations of Category Theory," has been appealed to often to argue that category theory cannot stand on its own as a structuralist foundation for mathematics (Bell, 1981; Hellman, 2003). Others have argued that a category-theoretic structuralist foundation is still possible by claiming that Feferman misses his mark (
This paper considers the nature and role of axioms from the point of view of the current debates ... more 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, its relation to the "algebraic" 1 approach to mathematical structuralism. I first consider the Frege-Hilbert debate with the aim of distinguishing between axioms as assertions, i.e., as statements that are used to express or assert truths about a unique subject matter, and an axiom system as a schema that is used to provide "a system of conditions for what might be called a relational structure" (Bernays [1967], p. 497) so that axioms, as implicit definitions, are about whatever satisfies the conditions set forth. I then use this inquiry to reevaluate arguments against using category theory to frame an algebraic structuralist philosophy of mathematics. Hellman has argued that category theory cannot stand on its own as a "foundation" for a structuralist interpretation of mathematics because "the problem of the home address remains" (Hellman [2003], pgs. 8 & 15). That is, since the axioms for a category "merely tell us what it is to be a structure of a certain kind" and because "its axioms are not assertory" (Ibid. 7), we need a background mathematical theory whose axioms are assertory, i.e., a theory that assert truths about (possibly or actually) existing systems so structured. 2 With aims similar to mine but with a decidedly different conclusion, Shapiro [2005] has claimed that the Frege-Hilbert debate can be used to show that the current algebraic structuralist debates ought to be concerned with questions that consider the status of meta-mathematical axioms (as opposed to Hellman who considers the status of mathematical axioms). That is, Shapiro argues, even if we agree with the Hilbert-inspired algebraic structuralist that, at the mathematical level, "any given branch is 'about' any system that satisfies its axioms" (Shapiro [2005], p. 74), to give "criteria of acceptability" (of coherence, of consistency, of satisfiability) for such axioms or axiom systems themselves, we need a "foundation", as a background meta-mathematical theory, which is assertory, and so we cannot be algebraic structuralists all the way down. According to Shapiro, our only other option, as proposed by Awodey [2004] is to "kick away the foundational ladder altogether, and take the meta-mathematical set-theory, structure theory, or whatever, itself to be an algebraic theory" (Shapiro [2005], p. 74). This option, however, is presented as a way not to be looked into because it supposedly has the unwanted consequence that mathematical logic is similarly liberated from theories… our theorist can hold… that satisfiability, consistency, or coherence implies existence, but she cannot maintain that any of these notions are mathematical matters (Ibid. 75). The alleged result being that meta-mathematical analyses of these logical concepts are turned into non-mathematical, or, even worse, "philosophical", ones (see Shapiro [2005], pp. 74-75). 3 Against the claims of both Hellman and Shapiro, 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 structure, without requiring either an assertory mathematical or meta-mathematical background theory, and too without 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 Frege-Hilbert Debate As is well known, Frege and Hilbert debated the nature of geometric axioms. Frege held that geometric axioms are assertions; that they are statements used to express or assert truths. On the other hand, Hilbert, like our modern-day algebraic structuralist, maintained that axioms are implicit definitions. Related to these differing views of axioms, Frege and Hilbert further disagreed on at least three points. First, a theory, for Hilbert, is not a set of truths about a "fixed" subject matter. As he explains, … every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and the basic elements can be thought of in any way one likes. If in speaking of points, I think of some systems of things, e.g., the system love, law, chimney-sweep… and then assume all my axioms as relations between these things, then my propositions, e.g., Pythagoras' theorem, are also valid for these things…. [A]ny theory can always be applied to infinitely many systems of basic elements. (Hilbert [1899], pp. 40-41
This paper considers the nature and role of axioms from the point of view of the current debates ... more 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.
The Western Ontario Series in Philosophy of Science, 2011
ABSTRACT In this paper I offer an answer to a question raised in (Psillos, 2006): How can one spe... more ABSTRACT In this paper I offer an answer to a question raised in (Psillos, 2006): How can one speak of structures without objects? Specifically, I use category theory to show that, mathematically speaking, structures do not need objects. Next, I argue that, scientifically speaking, this category-theoretic answer is silly because it does not speak to the scientific structuralist’s appeal to the appropriate kind of morphism to make precise the concept of shared structure. Against French et al.’s approach,1 I note that to account for the scientific structuralist’s uses of shared structure we do not need to formally frame either the structure of a scientific theory or the concept of shared structure. Here I restate my (Landry, 2007) claim that the concept of shared structure can be made precise by appealing to a kind of morphism, but, in science, it is methodological contexts (and not any category or set-theoretic framework) that determine the appropriate kind. Returning to my aim, I reconsider French’s example of the role of group theory in quantum mechanics to show that French already has an answer to Psillos’ question but this answer is not found in either his set-theoretic formal framework or his ontic structural realism. The answer to Psillos is found both by recognizing that it is the context that determines what the appropriate kind of morphism is and, as Psillos himself suggests,2 by adopting a methodological approach to scientific structuralism.
Feferman (1977) argues that category theory cannot stand on its own as a structuralist foundation... more Feferman (1977) 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 (2005), 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 (Hilbert, 1900a) 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 m...
I argue that recollection, in Plato’s Meno, should not be taken as a method, and, if it is taken ... more 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, ap riorior 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. I wouldn’t support every aspect of the argument [for recollection] with particular vigour, but there’s one proposition that I’d defend to the death, if I could, by argument and by action: that as long as we think we should search for what we don’t know we’ll be better people — less faint-hearted and less lazy — than if we were to think that we had no chance of discovering what we don’t know and that there’s no point in even searching for it. [Meno, 86b–c] 1
I argue that if we distinguish between ontological realism and semantic realism, then we no longe... more 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.
I use my reading of Plato to develop what I call as-ifism, the view that, in mathematics, we trea... more 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.
Christopher Pincock's Mathematics and Scientific Representation is an interesting and important n... more Christopher Pincock's Mathematics and Scientific Representation is an interesting and important new book on the applications of mathematics to science. The book is loaded with fascinating case studies that bring out the varied and interesting ways that mathematics is used in science. Pincock provides a much more nuanced picture of the roles that mathematics plays in empirical science than we usually get from philosophers of mathematics writing on the topic of applications. He describes various dimensions along which different uses of mathematics can differ; for instance, to pick out just one of the dimensions he discusses, we sometimes use mathematics to help us represent the causal structure of a series of events, whereas in other cases, we use it to provide an acausal representation of a system in a static, fixed state. The overall picture we get from Pincock is a sort of taxonomy of different kinds of uses of mathematics. He also draws interesting connections
This Element argues that Plato was not a mathematical Platonist. It shows that Plato keeps a clea... more This Element argues that Plato was not a mathematical Platonist. It shows that Plato keeps a clear distinction between mathematical and metaphysical realism, and the knife he uses to slice the difference is method. The philosopher's dialectical method requires that we tether the truth of hypotheses to existing metaphysical objects. The mathematician's hypothetical method, by contrast, takes hypotheses as if they were first principles, so no metaphysical account of their truth is needed. Thus, we come to Plato's methodological as-if realism: In mathematics, we treat our hypotheses as if they were first principles, and, consequently, our objects as if they existed, and we do this for the purpose of solving problems. Taking the road suggested by Plato's Republic, the author shows that some methodological commitments to mathematical objects are made in light of mathematical practice; some are made in light of foundational considerations; and some are made in light of mathematical applicability.
The aim of this paper is to put into context the historical, foundational and philosophical signi... more 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.
This paper explores varieties of scientific structuralism. Central to our investigation is the no... more 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 realists have made use of category theory in three ways. The first is as a meta-level ... more Structural realists have made use of category theory in three ways. The first is as a meta-level formal framework for a structural realist account of the structure of scientific theories, either syntactic or semantic. The second is an appeal to the category-theoretic structure of some successful, successive or fundamental, physical theory to argue that this is the structure we should be physically committed to, either epistemically or ontically. The third is to use category theory as a conceptual tool to argue that it makes conceptual sense to talk of relations without relata and structures without objects. After a brief overview of structural realism, I consider how each appeal to the use of category theory stands up against the aims of the structural realist.
The Western Ontario series in philosophy of science, 2011
ABSTRACT In this paper I offer an answer to a question raised in (Psillos, 2006): How can one spe... more ABSTRACT In this paper I offer an answer to a question raised in (Psillos, 2006): How can one speak of structures without objects? Specifically, I use category theory to show that, mathematically speaking, structures do not need objects. Next, I argue that, scientifically speaking, this category-theoretic answer is silly because it does not speak to the scientific structuralist’s appeal to the appropriate kind of morphism to make precise the concept of shared structure. Against French et al.’s approach,1 I note that to account for the scientific structuralist’s uses of shared structure we do not need to formally frame either the structure of a scientific theory or the concept of shared structure. Here I restate my (Landry, 2007) claim that the concept of shared structure can be made precise by appealing to a kind of morphism, but, in science, it is methodological contexts (and not any category or set-theoretic framework) that determine the appropriate kind. Returning to my aim, I reconsider French’s example of the role of group theory in quantum mechanics to show that French already has an answer to Psillos’ question but this answer is not found in either his set-theoretic formal framework or his ontic structural realism. The answer to Psillos is found both by recognizing that it is the context that determines what the appropriate kind of morphism is and, as Psillos himself suggests,2 by adopting a methodological approach to scientific structuralism.
ABSTRACT This is a short version of the author’s paper “Logicism, structuralism and objectivity” ... more ABSTRACT This is a short version of the author’s paper “Logicism, structuralism and objectivity” [Topoi 20, No. 1, 79–95 (2001; Zbl 1142.00004)].
ABSTRACT The aim of this paper is to present category theory as a framework for an in re interpre... more ABSTRACT The aim of this paper is to present category theory as a framework for an in re interpretation of mathematical structuralism. The use of the term ‘framework’ is significant. On the one hand, it is used in distinction from the term ‘foundation’. As such, what I propose is that we consider category theory as a philosophical tool that allows us to organize what we say about the shared structure of abstract kinds of mathematical systems. On the other hand, the term ‘framework’ is used in the sense of R. Carnap [in: Philosophy of mathematics. Selected readings. 2nd ed., Cambridge (1983; Zbl 0548.03002)]. That is, category theory is taken as a language used to frame what we say about the shared structure of abstract kinds of mathematical systems, as opposed to being a “background theory” which constitutes what a structure is.
The focus of this paper is the recent revival of interest in structuralist approaches to science ... more 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.
Feferman (1977) argues that category theory cannot stand on its own as a structuralist foundation... more Feferman (1977) 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 (2005), 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 (Hilbert, 1900a) 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. §1. Introduction. Feferman's 1977 paper, "Categorical Foundations and Foundations of Category Theory," has been appealed to often to argue that category theory cannot stand on its own as a structuralist foundation for mathematics (Bell, 1981; Hellman, 2003). Others have argued that a category-theoretic structuralist foundation is still possible by claiming that Feferman misses his mark (
This paper considers the nature and role of axioms from the point of view of the current debates ... more 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, its relation to the "algebraic" 1 approach to mathematical structuralism. I first consider the Frege-Hilbert debate with the aim of distinguishing between axioms as assertions, i.e., as statements that are used to express or assert truths about a unique subject matter, and an axiom system as a schema that is used to provide "a system of conditions for what might be called a relational structure" (Bernays [1967], p. 497) so that axioms, as implicit definitions, are about whatever satisfies the conditions set forth. I then use this inquiry to reevaluate arguments against using category theory to frame an algebraic structuralist philosophy of mathematics. Hellman has argued that category theory cannot stand on its own as a "foundation" for a structuralist interpretation of mathematics because "the problem of the home address remains" (Hellman [2003], pgs. 8 & 15). That is, since the axioms for a category "merely tell us what it is to be a structure of a certain kind" and because "its axioms are not assertory" (Ibid. 7), we need a background mathematical theory whose axioms are assertory, i.e., a theory that assert truths about (possibly or actually) existing systems so structured. 2 With aims similar to mine but with a decidedly different conclusion, Shapiro [2005] has claimed that the Frege-Hilbert debate can be used to show that the current algebraic structuralist debates ought to be concerned with questions that consider the status of meta-mathematical axioms (as opposed to Hellman who considers the status of mathematical axioms). That is, Shapiro argues, even if we agree with the Hilbert-inspired algebraic structuralist that, at the mathematical level, "any given branch is 'about' any system that satisfies its axioms" (Shapiro [2005], p. 74), to give "criteria of acceptability" (of coherence, of consistency, of satisfiability) for such axioms or axiom systems themselves, we need a "foundation", as a background meta-mathematical theory, which is assertory, and so we cannot be algebraic structuralists all the way down. According to Shapiro, our only other option, as proposed by Awodey [2004] is to "kick away the foundational ladder altogether, and take the meta-mathematical set-theory, structure theory, or whatever, itself to be an algebraic theory" (Shapiro [2005], p. 74). This option, however, is presented as a way not to be looked into because it supposedly has the unwanted consequence that mathematical logic is similarly liberated from theories… our theorist can hold… that satisfiability, consistency, or coherence implies existence, but she cannot maintain that any of these notions are mathematical matters (Ibid. 75). The alleged result being that meta-mathematical analyses of these logical concepts are turned into non-mathematical, or, even worse, "philosophical", ones (see Shapiro [2005], pp. 74-75). 3 Against the claims of both Hellman and Shapiro, 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 structure, without requiring either an assertory mathematical or meta-mathematical background theory, and too without 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 Frege-Hilbert Debate As is well known, Frege and Hilbert debated the nature of geometric axioms. Frege held that geometric axioms are assertions; that they are statements used to express or assert truths. On the other hand, Hilbert, like our modern-day algebraic structuralist, maintained that axioms are implicit definitions. Related to these differing views of axioms, Frege and Hilbert further disagreed on at least three points. First, a theory, for Hilbert, is not a set of truths about a "fixed" subject matter. As he explains, … every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and the basic elements can be thought of in any way one likes. If in speaking of points, I think of some systems of things, e.g., the system love, law, chimney-sweep… and then assume all my axioms as relations between these things, then my propositions, e.g., Pythagoras' theorem, are also valid for these things…. [A]ny theory can always be applied to infinitely many systems of basic elements. (Hilbert [1899], pp. 40-41
This paper considers the nature and role of axioms from the point of view of the current debates ... more 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.
The Western Ontario Series in Philosophy of Science, 2011
ABSTRACT In this paper I offer an answer to a question raised in (Psillos, 2006): How can one spe... more ABSTRACT In this paper I offer an answer to a question raised in (Psillos, 2006): How can one speak of structures without objects? Specifically, I use category theory to show that, mathematically speaking, structures do not need objects. Next, I argue that, scientifically speaking, this category-theoretic answer is silly because it does not speak to the scientific structuralist’s appeal to the appropriate kind of morphism to make precise the concept of shared structure. Against French et al.’s approach,1 I note that to account for the scientific structuralist’s uses of shared structure we do not need to formally frame either the structure of a scientific theory or the concept of shared structure. Here I restate my (Landry, 2007) claim that the concept of shared structure can be made precise by appealing to a kind of morphism, but, in science, it is methodological contexts (and not any category or set-theoretic framework) that determine the appropriate kind. Returning to my aim, I reconsider French’s example of the role of group theory in quantum mechanics to show that French already has an answer to Psillos’ question but this answer is not found in either his set-theoretic formal framework or his ontic structural realism. The answer to Psillos is found both by recognizing that it is the context that determines what the appropriate kind of morphism is and, as Psillos himself suggests,2 by adopting a methodological approach to scientific structuralism.
Feferman (1977) argues that category theory cannot stand on its own as a structuralist foundation... more Feferman (1977) 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 (2005), 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 (Hilbert, 1900a) 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 m...
I argue that recollection, in Plato’s Meno, should not be taken as a method, and, if it is taken ... more 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, ap riorior 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. I wouldn’t support every aspect of the argument [for recollection] with particular vigour, but there’s one proposition that I’d defend to the death, if I could, by argument and by action: that as long as we think we should search for what we don’t know we’ll be better people — less faint-hearted and less lazy — than if we were to think that we had no chance of discovering what we don’t know and that there’s no point in even searching for it. [Meno, 86b–c] 1
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