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.
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 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.
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 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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Papers by Elaine Landry