Early View of the Special Issue of Theoria on Linnebo’s Thin Objects: https://doi.org/10.1111/theo.12356
Abstract: In Thin Objects: An Abstractionist Account, Linnebo offers what he describes as a “simple and definitive” solution to the bad company problem facing abstractionist accounts of mathematics. “Bad” abstraction principles can be rendered “good” by taking abstraction to have a predicative character. But the resulting predicative axioms are too weak to recover substantial portions of mathematics. Linnebo pursues two quite different strategies to overcome this weakness in the case of set theory and arithmetic. I argue that neither infinitely iterated abstraction nor abstraction on possible specifications fully resolves the bad company problem.
Proceedings of the Aristotelian Society (2017) 117 (1): 81–101 https://doi.org/10.1093/arisoc/aox003
Delivered to the Aristotelian Society, 28th November 2016
Slides [new tab, .pdf]
Abstract: Absolutism is the view that quantifiers like ‘everything’ sometimes range over an absolutely comprehensive domain. The debate between absolutists and the relativists opposing them comes down, in significant part, to a trade off between generality and collectability. But to reach this conclusion we must dispense with some long-standing concerns over the coherence of the argument from the paradoxes in favour of relativism and a heterodox absolutist response that seeks to reconcile the indefinite extensibility of set with the availability of an absolutely comprehensive domain.
Br J Philos Sci (2016) 67 (2): 579-615. doi: 10.1093/bjps/axu035
Neologicists have sought to ground mathematical knowledge in abstraction. One especially obstinate problem for this account is the bad company problem. The leading neologicist strategy for resolving this problem is to attempt to sift the good abstraction principles from the bad. This response faces a dilemma: the system of ‘good’ abstraction principles either falls foul of the Scylla of inconsistency or the Charybdis of being unable to recover a modest portion of Zermelo–Fraenkel set theory with its intended generality. This article argues that the bad company problem is due to the ‘static’ character of abstraction on the neologicist’s account and develops a ‘dynamic’ account of abstraction that avoids both horns of the dilemma.
J Philos Logic (2013) 42: 697–725. doi:10.1007/s10992-012-9245-3
The final publication is available at www.springerlink.com
Abstract: The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously.
Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis, the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary (although not literally). A modal stage theory, 𝖬𝖲𝖳, is developed in a bimodal language, governed by a tenselike logic. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. It is argued that the modal approach is consonant with mathematical practice and a plausible metaphysics of sets and shown that 𝖬𝖲𝖳 interprets a natural extension of Zermelo set theory less the axiom of Infinity and, when extended with a further axiom concerning the extent of the hierarchy, interprets Zermelo–Fraenkel set theory.