Absence Perception and the Philosophy of Zero

Zero provides a challenge for philosophers of mathematics with realist inclinations. On the one hand it is a bona fide number, yet on the other it is linked to ideas of nothingness and non-being. This paper provides an analysis of the epistemology and metaphysics of zero. We develop several constraints and then argue that a satisfactory account of zero can be obtained by integrating recent work in numerical cognition with a philosophical account of absence perception

Absence Perception and the Philosophy of Zero

Zero provides a challenge for philosophers of mathematics with realist inclinations. On the one hand it is a bona fide number, yet on the other it is linked to ideas of nothingness and non-being. This paper provides an analysis of the epistemology and metaphysics of zero. We develop several constraints and then argue that a satisfactory account of zero can be obtained by integrating recent work in numerical cognition with a philosophical account of absence perception

Large Cardinals and the Iterative Conception of Set

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. One idea sometimes alluded to is that maximality considerations speak in favour of large cardinal axioms consistent with ZFC, since it appears to be `possible' (in some sense) to continue the hierarchy far enough to generate the relevant transfinite number. In this paper, we argue against this idea based on a priority of subset formation under the iterative conception. In particular, we argue that there are several conceptions of maximality that justify the consistency but falsity of large cardinal axioms. We argue that the arguments we provide are illuminating for the debate concerning the justification of new axioms in iteratively-founded set theory

Absence Perception and the Philosophy of Zero

Zero provides a challenge for philosophers of mathematics with realist inclinations. On the one hand it is a bona fide cardinal number, yet on the other it is linked to ideas of nothingness and non-being. This paper provides an analysis of the epistemology and metaphysics of zero. We develop several constraints and then argue that a satisfactory account of zero can be obtained by integrating (i) an account of numbers as properties of collections, (ii) work on the philosophy of absences, and (iii) recent work in numerical cognition and ontogenetic studies

Large Cardinals and the Iterative Conception of Set

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. One idea sometimes alluded to is that maximality considerations speak in favour of large cardinal axioms consistent with ZFC, since it appears to be `possible' (in some sense) to continue the hierarchy far enough to generate the relevant transfinite number. In this paper, we argue against this idea based on a priority of subset formation under the iterative conception. In particular, we argue that there are several conceptions of maximality that justify the consistency but falsity of large cardinal axioms. We argue that the arguments we provide are illuminating for the debate concerning the justification of new axioms in iteratively-founded set theory

On forms of justification in set theory

In the contemporary philosophy of set theory, discussion of new axiomsthat purport to resolve independence necessitates an explanation of howthey come to bejustified. Ordinarily, justification is divided into two broadkinds:intrinsicjustification relates to how ‘intuitively plausible’ an axiomis, whereasextrinsicjustification supports an axiom by identifying certain‘desirable’ consequences. This paper puts pressure on how this distinctionis formulated and construed. In particular, we argue that the distinction asoften presented is neitherwell-demarcatednor sufficientlyprecise. Instead, wesuggest that the process of justification in set theory should not be thoughtof as neatly divisible in this way, but should rather be understood as a con-ceptually indivisible notion linked to the goal ofexplanation

On Forms of Justification in Set Theory

In the contemporary philosophy of set theory, discussion of new axiomsthat purport to resolve independence necessitates an explanation of howthey come to bejustified. Ordinarily, justification is divided into two broadkinds:intrinsicjustification relates to how ‘intuitively plausible’ an axiomis, whereasextrinsicjustification supports an axiom by identifying certain‘desirable’ consequences. This paper puts pressure on how this distinctionis formulated and construed. In particular, we argue that the distinction asoften presented is neitherwell-demarcatednor sufficientlyprecise. Instead, wesuggest that the process of justification in set theory should not be thoughtof as neatly divisible in this way, but should rather be understood as a con-ceptually indivisible notion linked to the goal ofexplanation

Large Cardinals and the Iterative Conception of Set

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles for the iterative conception, and assert that the length of the iterative stages is as long as possible. In this paper, we argue that whether or not large cardinal principles count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular we argue that there is a conception of maximality through absoluteness, that when given certain technical formulations, supports the idea that large cardinals are consistent, but false. On this picture, large cardinals are instead true in inner models and serve to restrict the subsets formed at successor stages