This article gives an epistemological analysis of the reflection process by means of which you ca... more This article gives an epistemological analysis of the reflection process by means of which you can come to know the consistency of a mathematical theory that you already accept. It is argued that this process can result in warranted belief in new mathematical principles without justifying them. * Versions of this article were presented in Salzburg, Oxford (University College), and Hong Kong (Lingnan University). I am grateful for the people who attended these presentations for their comments. Thanks in particular to for helpful questions, challenges, and suggestions.
This article gives an epistemological analysis of the reflection process by means of which you ca... more This article gives an epistemological analysis of the reflection process by means of which you can come to know the consistency of a mathematical theory that you already accept. It is argued that this process can result in warranted belief in new mathematical principles without justifying them.
In this article ideas from Kit Fine’s theory of arbitrary objects are applied to questions regard... more In this article ideas from Kit Fine’s theory of arbitrary objects are applied to questions regarding mathematical structuralism. I discuss how sui generis mathematical structures can be viewed as generic systems of mathematical objects, where mathematical objects are conceived of as arbitrary objects in Fine’s sense.
We investigate how to assign probabilities to sentences that contain a type-free truth predicate.... more We investigate how to assign probabilities to sentences that contain a type-free truth predicate. These probability values track how often a sentence is satisfied in transfinite revision sequences, following Gupta and Belnap's revision theory of truth. This answers an open problem by Leitgeb which asks how one might describe transfinite stages of the revision sequence using such probability functions. We offer a general construction, and explore additional constraints that lead to desirable properties of the resulting probability function. One such property is Leitgeb's Probabilistic Convention T, which says that the probability of ϕ equals the probability that ϕ is true.
It is often alleged that Cantor’s views about how the set theoretic universe as a whole should be... more It is often alleged that Cantor’s views about how the set theoretic universe as a whole should be considered are fundamentally unclear. In this article we argue that Cantor’s views on this subject, at least up until around 1896, are relatively clear, coherent, and interesting. We then go on to argue that Cantor’s views about the set theoretic universe as a whole have implications for theology that have hitherto not been sufficiently recognised. However, the theological implications in question, at least as articulated here, would not have satisfied Cantor himself.
... Now Brouwer and Heyting could still appeal to the argument based on unresolved mathematical p... more ... Now Brouwer and Heyting could still appeal to the argument based on unresolved mathematical problems to protest that classical arithmetic cannot be interpreted as being about the natural numbers conceived as a potential infinity. ... Leon Horsten ...
This article gives an epistemological analysis of the reflection process by means of which you ca... more This article gives an epistemological analysis of the reflection process by means of which you can come to know the consistency of a mathematical theory that you already accept. It is argued that this process can result in warranted belief in new mathematical principles without justifying them. * Versions of this article were presented in Salzburg, Oxford (University College), and Hong Kong (Lingnan University). I am grateful for the people who attended these presentations for their comments. Thanks in particular to for helpful questions, challenges, and suggestions.
This article gives an epistemological analysis of the reflection process by means of which you ca... more This article gives an epistemological analysis of the reflection process by means of which you can come to know the consistency of a mathematical theory that you already accept. It is argued that this process can result in warranted belief in new mathematical principles without justifying them.
In this article ideas from Kit Fine’s theory of arbitrary objects are applied to questions regard... more In this article ideas from Kit Fine’s theory of arbitrary objects are applied to questions regarding mathematical structuralism. I discuss how sui generis mathematical structures can be viewed as generic systems of mathematical objects, where mathematical objects are conceived of as arbitrary objects in Fine’s sense.
We investigate how to assign probabilities to sentences that contain a type-free truth predicate.... more We investigate how to assign probabilities to sentences that contain a type-free truth predicate. These probability values track how often a sentence is satisfied in transfinite revision sequences, following Gupta and Belnap's revision theory of truth. This answers an open problem by Leitgeb which asks how one might describe transfinite stages of the revision sequence using such probability functions. We offer a general construction, and explore additional constraints that lead to desirable properties of the resulting probability function. One such property is Leitgeb's Probabilistic Convention T, which says that the probability of ϕ equals the probability that ϕ is true.
It is often alleged that Cantor’s views about how the set theoretic universe as a whole should be... more It is often alleged that Cantor’s views about how the set theoretic universe as a whole should be considered are fundamentally unclear. In this article we argue that Cantor’s views on this subject, at least up until around 1896, are relatively clear, coherent, and interesting. We then go on to argue that Cantor’s views about the set theoretic universe as a whole have implications for theology that have hitherto not been sufficiently recognised. However, the theological implications in question, at least as articulated here, would not have satisfied Cantor himself.
... Now Brouwer and Heyting could still appeal to the argument based on unresolved mathematical p... more ... Now Brouwer and Heyting could still appeal to the argument based on unresolved mathematical problems to protest that classical arithmetic cannot be interpreted as being about the natural numbers conceived as a potential infinity. ... Leon Horsten ...
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