Online research in philosophy

Simon Blackburn posed a dilemma for any realist attempt to identify the source of necessity. Either the facts appealed to to ground modal truth are themselves necessary, or they are contingent. If necessary, we begin the process towards regress; but if contingent, we undermine the necessity whose source we wanted to explain. Bob Hale attempts to blunt both horns of this dilemma. In this paper I examine their respective positions and attempt to clear up some confusions on either side. I come to defend Hale’s conclusion that both horns of the dilemma can be resisted. I end by defending my own account of the source of necessity, and showing why it does not fall victim to Blackburn’s problem.

Chapter 10 of Sosa’s important new book provides an exemplary presentation and discussion of a great dilemma for epistemologists— I’ll call it simply the Dilemma. Here is Sosa’s statement of it.

The argument is a dialectical one. It is not a direct proof that the mind is something more than a machine, but a schema of disproof for any particular version of mechanism that may be put forward. If the mechanist maintains any specific thesis, I show that [146] a contradiction ensues. But only if. It depends on the mechanist making the first move and putting forward his claim for inspection. I do not think Benacerraf has quite taken the point. He criticizes me both for "failing to notice" that my ability to show that the Gödel sentence of a formal system is true "depends very much on how he is given.

I analyze here Benacerraf's criticism of Thomson arguments on the impossibility of w-supertasks. Although Benacerraf's criticism is well founded, his analysis of Thomson's lamp is incomplete. In fact, it is possible to consider a new line of argument, which Benacerraf only incidentally considered, based on the functioning laws of the lamp. This argument leads to a contradictory result that compromises the formal consistency of the w-ordering involved in all w-supertasks.

There are two general questions which many views in the philosophy of mathematics can be seen as addressing: what are mathematical objects, and how do we have knowledge of them? Naturally, the answers given to these questions are linked, since whatever account we give of how we have knowledge of mathematical objects surely has to take into account what sorts of things we claim they are; conversely, whatever account we give of the nature of mathematical objects must be accompanied by a corresponding account of how it is that we acquire knowledge of those objects. The connection between these problems results in what is often called “Benacerraf’s Problem”, which is a dilemma that many philosophical views about mathematical objects face. It will be my goal here to present a view, attributed to Richard Dedekind, which approaches the initial questions in a different way than many other philosophical views do, and in doing so, avoids the dilemma given by Benacerraf’s problem.

A complete philosophy of mathematics must address Paul Benacerraf’s dilemma. The requirements of a general semantics for the truth of mathematical theorems that coheres also with the meaning and truth conditions for non-mathematical sentences, according to Benacerraf, should ideally be coupled with an adequate epistemology for the discovery of mathematical knowledge. Standard approaches to the philosophy of mathematics are criticized against their own merits and against the background of Benacerraf’s dilemma, particularly with respect to the problem of understanding the distinction between pure and applied mathematics and the effectiveness of applied mathematics in the natural sciences and engineering. The evaluation of these alternatives provides the basis for articulating a philosophically advantageous Aristotelian inherence concept of mathematical entities. An inherence account solves Benacerraf’s dilemma by interpreting mathematical entities as nominalizations of structural spatiotemporal properties inhering in existent spatiotemporal entities.

Abstract Robert Hanna has recently advanced a theory of non-conceptual content, the central claim of which is that ?it is perfectly possible for there to be directly referential intuitions without concepts?. Hanna bases this claim in Kant?s account of intuition in the Critique of Pure Reason, and so extends his Kantian non-conceptualism beyond the epistemology of empirical knowledge into the realm of mathematics.?Thus, Hanna has proposed a Kantian non-conceptualist solution to a well-known dilemma set out by Paul Benacerraf in his 1973 paper, ?Mathematical Truth?.?I argue that Hanna is right about Kant?s non-conceptualism, but mistaken in its application to Benacerraf?s Dilemma.