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Does category theory provide a framework for mathematical structuralism?
 PHILOSOPHIA MATHEMATICA
, 2003
"... Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves ..."
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Cited by 10 (3 self)
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Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell’s “manytopoi” view and modalstructuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out relative to such domains; puzzles about “large categories ” and “proper classes ” are handled in a
The Interactivist model
 Synthese
"... A shift from a metaphysical framework of substance to one of process enables an integrated account of the emergence of normative phenomena. I show how substance assumptions block genuine ontological emergence, especially the emergence of normativity, and how a process framework permits a thermodynam ..."
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Cited by 7 (5 self)
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A shift from a metaphysical framework of substance to one of process enables an integrated account of the emergence of normative phenomena. I show how substance assumptions block genuine ontological emergence, especially the emergence of normativity, and how a process framework permits a thermodynamicbased account of normative emergence. The focus is on two foundational forms of normativity, that of normative function and of representation as emergent in a particular kind of function. This process model of representation, called interactivism, compels changes in many related domains. The discussion ends with brief attention to three domains in which changes are induced by the representational model: perception, learning, and language.
The Language of Mathematics
, 2009
"... The accompanying thesis is part of a longterm project to enable computers to do mathematics in the same way that humans do. I will sketch something of the nature of mathematics and the project, and then turn to role of the thesis. Mathematics Mathematics arises from the interaction of two dissimila ..."
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The accompanying thesis is part of a longterm project to enable computers to do mathematics in the same way that humans do. I will sketch something of the nature of mathematics and the project, and then turn to role of the thesis. Mathematics Mathematics arises from the interaction of two dissimilar modes of reasoning: a ‘soft ’ side, dealing with ideas and analogies, and a ‘hard ’ side, dealing with verification. The ‘hard ’ side is easier to pin down. It consists primarily of formal ‘proofs’, each consisting of a series of assertions. A mathematician can verify that a proof is correct by following it, step by step, checking that each step follows from previous ones via facts already proved to be correct. The ‘soft ’ side is less easily described. It consists of intuitions about the formal objects constructed in mathematical proofs; ideas that one piece of mathematics may analogically correspond to another piece of mathematics; or even analogies between mathematics and objects in the physical world.
Is the Continuum Hypothesis a definite mathematical problem?
"... [t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is gr ..."
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[t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is greater than א0, conjectured that it is א1. An equivalent proposition is this: any infinite subset of the continuum has the power either of the set of integers or of the whole continuum. This is Cantor’s continuum hypothesis. … But, although Cantor’s set theory has now had a development of more than sixty years and the [continuum] problem is evidently of great importance for it, nothing has been proved so far relative to the question of what the power of the continuum is or whether its subsets satisfy the condition just stated, except that … it is true for a certain infinitesimal fraction of these subsets, [namely] the analytic sets. Not even an upper bound, however high, can be assigned for the power of the continuum. It is undecided whether this number is regular or singular, accessible or inaccessible, and (except for König’s negative result) what its character of cofinality is. Gödel 1947, 516517 [in Gödel 1990, 178]
What does it mean to say that logic is formal?
, 2000
"... Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and ..."
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Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topicneutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought. Though these three notions of formality are by no means equivalent, they are frequently run together. The reason, I argue, is that modern talk of the formality of logic has its source in Kant, and these three notions come together in the context of Kant’s transcendental philosophy. Outside of this context (e.g., in Frege), they can come apart. Attending to this
particles and structural realism
"... Even if we are able to decide on a canonical formulation of our theory, there is the further problem of metaphysical underdetermination with respect to, for example, whether the entities postulated by a theory are individuals or not... We need to recognise the failure of our best theories to determi ..."
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Even if we are able to decide on a canonical formulation of our theory, there is the further problem of metaphysical underdetermination with respect to, for example, whether the entities postulated by a theory are individuals or not... We need to recognise the failure of our best theories to determine even the most fundamental ontological characteristic of the purported entities they feature... What is required is a shift to a different ontological basis altogether, one for which questions of individuality simply do not arise. Perhaps we should view the individuals and nonindividuals packages, like particle and field pictures, as different representations of the same structure. There is an analogy here with the debate about substantivalism in general relativity. (Ladyman, 1998) In his paper “What is Structural Realism? ” (1998) James Ladyman drew a distinction between epistemological structural realism (ESR) and metaphysical (or ontic) structural realism (OSR). In recent years this distinction has set much of the agenda for philosophers of science interested in scientific realism. It has also led to the emergence of a related discussion in the philosophy of physics that concerns the alleged difficulties of interpreting general relativity that revolve around the question of the ontological status of spacetime points. Ladyman drew a suggestive analogy between the perennial debate between substantivalist and relationalist interpretations of spacetime on the one hand, and the debate about whether quantum mechanics treats identical particles as individuals or as ‘nonindividuals ’ on the other. In both cases, Ladyman’s suggestion is that a structural realist interpretation of the physics—in particular, an ontic structural realism—might
Towards a Coherent Theory of Physics and Mathematics: The TheoryExperiment Connection
, 2004
"... The problem of how mathematics and physics are related at a foundational level is of much interest. One approach is to work towards a coherent theory of physics and mathematics together. Here steps are taken in this direction by first examining the theory experiment connection. The role of an implie ..."
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The problem of how mathematics and physics are related at a foundational level is of much interest. One approach is to work towards a coherent theory of physics and mathematics together. Here steps are taken in this direction by first examining the theory experiment connection. The role of an implied theory hierarchy and use of computers in comparing theory and experiment is described. The main idea of the paper is to tighten the theory experiment connection by bringing physical theories, as mathematical structures over C, the complex numbers, closer to what is actually done in experimental measurements and computations. The method replaces C by Cn which is the set of pairs, Rn, In, of n figure rational numbers in some basis. The properties of these numbers are based on the type of numbers that represent measurement outcomes for continuous variables. A model of space and time based on Rn is discussed. The model is scale invariant with regions of constant step size interrupted by exponential jumps. A method of taking the limit n → ∞ to obtain locally flat continuum based space and time is outlined. Possibly the most interesting result is that Rn based space is invariant under scale transformations which correspond to expansion and contraction of space relative to a flat background. Also the location of the origin, which is a space and time singularity, does not change under these transformations. Some properties of quantum mechanics, Qmn based on Cn and on Rn space are briefly investigated. 1
Russell’s Absolutism vs.(?)
"... Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And ..."
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Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of Russell’s insights. Problems of absolutism plague some versions, and, interestingly, Russell’s critique of Dedekind can be extended to one of them, ante rem structuralism. This leaves modalstructuralism and a category theoretic approach as remaining nonabsolutist
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