A. C. Paseau

In my paper , I noted that Fitch's argument, which purports to show that if all truths are knowable then all truths are known, can be blocked by typing knowledge. If there is not one knowledge predicate, ‘ K’, but infinitely many, ‘ K 1’, ‘ K 2’, … , then the type rules prevent application of the predicate ‘ K i’ to sentences containing ‘ K i’ such as ‘ p ∧¬ K i⌜ p⌝’. This provides a motivated response to Fitch's argument so long as knowledge typing is itself motivated. It was the burden of my paper to explore the case that knowledge typing is as motivated as truth typing by drawing on the parallels between epistemic paradoxes generated by sentences of the kind ‘this sentence is unknown’ and semantic paradoxes generated by sentences such as ‘this sentence is untrue’. Given that typing truth is one of the acknowledged options for solving semantic paradoxes, if the parity argument succeeds it follows that epistemic typing is as well-motivated as truth typing and that the typing response to Fitch's argument is correspondingly strong.Halbach presents an apparent problem for this argument. Let ‘ N’ and ‘ P’, respectively, denote the necessity and possibility predicates, ‘ K 1’ the knowledge predicate of first type, and let ‘γ’ be a K-free sentence such that γ ↔¬ P⌜ K 1⌜γ⌝⌝; we know such a ‘γ’ exists by a standard diagonalization argument. If we assume that γ is knowable at the next knowledge type, that is, at type 1, and that the possibility typing does not interfere with the knowledge typing, a contradiction quickly ensues. Formally: γ ↔¬ P⌜ …

The paper aims to develop a resemblance theory of properties that technically improves on past versions. The theory is based on a comparative resemblance predicate. In combination with other resources, it solves the various technical problems besetting resemblance nominalism. The paper’s second main aim is to indicate that previously proposed resemblance theories that solve the technical problems, including the comparative theory, are nominalistically unacceptable and have controversial philosophical commitments

Although the case for the judgment-dependence of many other domains has been pored over, surprisingly little attention has been paid to mathematics and logic. This paper presents two dilemmas for a judgment-dependent account of these areas. First, the extensionality-substantiality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the substantiality condition (roughly: non-vacuous specification). Second, the extensionality-extremality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the extremality condition (roughly: absence of independent explanation). The paper concludes with a moral concerning the judgment-dependence of a posteriori areas of discourse that emerges from consideration of these two a priori cases

Naturalism in the philosophy of mathematics is the view that philosophy cannot legitimately gainsay mathematics. I distinguish between reinterpretation and reconstruction naturalism: the former states that philosophy cannot legitimately sanction a reinterpretation of mathematics (i.e. an interpretation different from the standard one); the latter that philosophy cannot legitimately change standard mathematics (as opposed to its interpretation). I begin by showing that neither form of naturalism is self-refuting. I then focus on reinterpretation naturalism, which comes in two forms, and examine the only available argument for it. I argue that this argument, the so-called Failure Argument, itself fails. My overall conclusion is that although there is no self-refutation argument against reinterpretation naturalism, there are as yet no good reasons to accept it. Naturalism in mathematics The consistency of mathematical naturalism The failure argument Objections to the failure argument Philosophy as the default.

Contemporary philosophy’s three main naturalisms are methodological, ontological and epistemological. Methodological naturalism states that the only authoritative standards are those of science. Ontological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural. In philosophy of mathematics of the past few decades methodological naturalism has received the lion’s share of the attention, so we concentrate on this. Ontological and epistemological naturalism in the philosophy of mathematics are discussed more briefly in section 6

From title to back cover, a polemic runs through David Corfield's "Towards a Philosophy of Real Mathematics". Corfield repeatedly complains that philosophers of mathematics have ignored the interesting and important mathematical developments of the past seventy years, ‘filtering’ the details of mathematical practice out of philosophical discussion. His aim is to remedy the discipline’s long-sightedness and, by precept and example, to redirect philosophical attention towards current developments in mathematics. This review discusses some strands of Corfield’s philosophy of real mathematics and briefly assesses some of his objections to orthodox philosophy of mathematics.