Colin J. Rittberg | Vrije Universiteit Brussel (original) (raw)
Papers by Colin J. Rittberg
In this paper I explore how intellectual humility manifests in mathematical practices. To do this... more In this paper I explore how intellectual humility manifests in mathematical practices. To do this I employ accounts of this virtue as developed by virtue epistemologists in three case studies of mathematical activity. As a contribution to a Topical Collection on virtue theory of mathematical practices this paper explores in how far existing virtue-theoretic frameworks can be applied to a philosophical analysis of mathematical practices. I argue that the individual accounts of intellectual humility are successful at tracking some manifestations of this virtue in mathematical practices and fail to track others. There are two upshots to this. First, the accounts of the intellectual virtues provided by virtue epistemologists are insightful for the development of a virtue theory of mathematical practices but require adjustments in some cases. Second, the case studies reveal dimensions of intellectual humility virtue epistemologists have thus far overlooked in their theoretical reflections.
The Happy Enthusiast, 2018
A contribution to the "huldeboek" to Jean Paul Van Bendegem for his retirement.
The Continuum Problem has inspired set theorists and philosophers since the days of Cantorian set... more The Continuum Problem has inspired set theorists and philosophers since the days of Cantorian set theory. In the last 15 years, W. Hugh Woodin, a leading set theorist, has not only taken it upon himself to engage in this question, he has also changed his mind about the answer. This paper illustrates Woodin’s solutions to the problem, starting in Sect. 3 with his 1999–2004 argument that Cantor’s hypothesis about the continuum was incorrect. From 2010 onwards, Woodin presents a very different argument, an argument that Cantor’s hypothesis is in fact true. This argument is still incomplete, but according to Woodin, some of the philosophical issues surrounding the Continuum Problem have been reduced to precise mathematical questions, questions that are, unlike Cantor’s hypothesis, solvable from our current theory of sets.
This paper inquires the ways in which paper folding constitutes a mathematical practice and may p... more This paper inquires the ways in which paper folding constitutes a mathematical practice and may prompt a mathematical culture. To do this, we first present and investigate the common mathematical activities shared by this culture, i.e. we present mathematical paper folding as a material reasoning practice. We show that the patterns of mathematical activity observed in mathematical paper folding are, at least since the end of the 19 th century, sufficiently stable to be considered as a practice. Moreover, we will argue that this practice is material. The permitted inferential actions when reasoning by folding are controlled by the physical realities of paper-like material, whilst claims to generality of some reasoning operations are supported by arguments from other mathematical idioms. The controlling structure provided by this material side of the practice is tight enough to allow for non-textual shared standards of argument and wide enough to provide sufficiently many problems for a practice to form. The upshot is that mathematical paper folding is a non-propositional and non-diagrammatic reasoning practice that adds to our understanding of the multi-faceted nature of the epistemic force of mathematical proof. We then draw on what we have learned from our contemplations about paper folding to highlight some lessons about what a study of mathematical cultures entails.
- To be published in Synthese (2019)
to appear in Synthese We investigate how epistemic injustice can manifest itself in mathematica... more to appear in Synthese
We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively – we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics – and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept to highlight a potential danger of intellectual enculturation.
In this paper I show that mathematicians can successfully engage in metaphysical debates by mathe... more In this paper I show that mathematicians can successfully engage in metaphysical debates by mathematical means. I present the contemporary work of Hugh Woodin and Peter Koellner. Woodin has proposed intrinsically appealing axiom-candidates which could, when added to our current set theoretic axiom system, resolve the issue that some fundamental questions of set theory are formally unsolvable. The proposed method to choose between these axioms is to rely on future results in formal set theory. Koellner connects this to a contemporary metaphysical debate on the ontological nature of sets. I argue, mathematics is connected to the philosophical debate in such a way that by doing more mathematics an argument in the philosophical debate can be obtained. This story reveals an active connectedness between mathematics and philosophy.
In this short paper I argue that our best modal theories do not imply the impossibility of the ro... more In this short paper I argue that our best modal theories do not imply the impossibility of the round square.
Thesis Chapters by Colin J. Rittberg
In this paper I explore how intellectual humility manifests in mathematical practices. To do this... more In this paper I explore how intellectual humility manifests in mathematical practices. To do this I employ accounts of this virtue as developed by virtue epistemologists in three case studies of mathematical activity. As a contribution to a Topical Collection on virtue theory of mathematical practices this paper explores in how far existing virtue-theoretic frameworks can be applied to a philosophical analysis of mathematical practices. I argue that the individual accounts of intellectual humility are successful at tracking some manifestations of this virtue in mathematical practices and fail to track others. There are two upshots to this. First, the accounts of the intellectual virtues provided by virtue epistemologists are insightful for the development of a virtue theory of mathematical practices but require adjustments in some cases. Second, the case studies reveal dimensions of intellectual humility virtue epistemologists have thus far overlooked in their theoretical reflections.
The Happy Enthusiast, 2018
A contribution to the "huldeboek" to Jean Paul Van Bendegem for his retirement.
The Continuum Problem has inspired set theorists and philosophers since the days of Cantorian set... more The Continuum Problem has inspired set theorists and philosophers since the days of Cantorian set theory. In the last 15 years, W. Hugh Woodin, a leading set theorist, has not only taken it upon himself to engage in this question, he has also changed his mind about the answer. This paper illustrates Woodin’s solutions to the problem, starting in Sect. 3 with his 1999–2004 argument that Cantor’s hypothesis about the continuum was incorrect. From 2010 onwards, Woodin presents a very different argument, an argument that Cantor’s hypothesis is in fact true. This argument is still incomplete, but according to Woodin, some of the philosophical issues surrounding the Continuum Problem have been reduced to precise mathematical questions, questions that are, unlike Cantor’s hypothesis, solvable from our current theory of sets.
This paper inquires the ways in which paper folding constitutes a mathematical practice and may p... more This paper inquires the ways in which paper folding constitutes a mathematical practice and may prompt a mathematical culture. To do this, we first present and investigate the common mathematical activities shared by this culture, i.e. we present mathematical paper folding as a material reasoning practice. We show that the patterns of mathematical activity observed in mathematical paper folding are, at least since the end of the 19 th century, sufficiently stable to be considered as a practice. Moreover, we will argue that this practice is material. The permitted inferential actions when reasoning by folding are controlled by the physical realities of paper-like material, whilst claims to generality of some reasoning operations are supported by arguments from other mathematical idioms. The controlling structure provided by this material side of the practice is tight enough to allow for non-textual shared standards of argument and wide enough to provide sufficiently many problems for a practice to form. The upshot is that mathematical paper folding is a non-propositional and non-diagrammatic reasoning practice that adds to our understanding of the multi-faceted nature of the epistemic force of mathematical proof. We then draw on what we have learned from our contemplations about paper folding to highlight some lessons about what a study of mathematical cultures entails.
- To be published in Synthese (2019)
to appear in Synthese We investigate how epistemic injustice can manifest itself in mathematica... more to appear in Synthese
We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively – we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics – and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept to highlight a potential danger of intellectual enculturation.
In this paper I show that mathematicians can successfully engage in metaphysical debates by mathe... more In this paper I show that mathematicians can successfully engage in metaphysical debates by mathematical means. I present the contemporary work of Hugh Woodin and Peter Koellner. Woodin has proposed intrinsically appealing axiom-candidates which could, when added to our current set theoretic axiom system, resolve the issue that some fundamental questions of set theory are formally unsolvable. The proposed method to choose between these axioms is to rely on future results in formal set theory. Koellner connects this to a contemporary metaphysical debate on the ontological nature of sets. I argue, mathematics is connected to the philosophical debate in such a way that by doing more mathematics an argument in the philosophical debate can be obtained. This story reveals an active connectedness between mathematics and philosophy.
In this short paper I argue that our best modal theories do not imply the impossibility of the ro... more In this short paper I argue that our best modal theories do not imply the impossibility of the round square.