Online research in philosophy

Suggests that the recent emphasis on Benacerraf's access problem locates the peculiarity of mathematical knowledge in the wrong place. Instead we should focus on the sense in which mathematical concepts are or might be "armchair concepts" – concepts about which non-trivial knowledge is obtainable a priori.

In this article, I discuss Hawthorne’s contextualist solution to Benacerraf’s dilemma. He wants to find a satisfactory epistemology to go with realist ontology, namely with causally inaccessible mathematical and modal entities. I claim that he is unsuccessful. The contextualist theories of knowledge attributions were primarily developed as a response to the skeptical argument based on the deductive closure principle. Hawthorne uses the same strategy in his attempt to solve the epistemologist puzzle facing the proponents of mathematical and modal realism, but this problem is of a different nature than the skeptical one. The contextualist theory of knowledge attributions cannot help us with the question about the nature of mathematical and modal reality and how they can be known. I further argue that Hawthorne’s account does not say anything about a priori status of mathematical and modal knowledge. Later, Hawthorne adds to his account an implausible claim that in some contexts a gettierized belief counts as knowledge.

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.

A response is given here to Benacerraf's (1965) non-uniqueness (or multiple-reductions) objection to mathematical platonism. It is argued that non-uniqueness is simply not a problem for platonism; more specifically, it is argued that platonists can simply embrace non-uniqueness—i.e., that one can endorse the thesis that our mathematical theories truly describe collections of abstract mathematical objects while rejecting the thesis that such theories truly describe unique collections of such objects. I also argue that part of the motivation for this stance is that it dovetails with the correct response to Benacerraf's other objection to platonism, i.e., his (1973) epistemological objection.

This is an introductory survey article to the philosophy of mathematics. I provide a detailed account of what Benacerraf’s problem is and then discuss in general terms four different approaches to ….

One textbook may introduce the real numbers in Cantor’s way, and another in Dedekind’s, and the mathematical community as a whole will be completely indifferent to the choice between the two. This sort of phenomenon was famously called to the attention of philosophers by Paul Benacerraf. It will be argued that structuralism in philosophy of mathematics is a mistake, a generalization of Benacerraf’s observation in the wrong direction, resulting from philosophers’ preoccupation with ontology.

According to quasi-empiricism, mathematics is very like a branch of natural science. But if mathematics is like a branch of science, and science studies real objects, then mathematics should study real objects. Thus a quasi-empirical account of mathematics must answer the old epistemological question: How is knowledge of abstract objects possible? This paper attempts to show how it is possible.The second section examines the problem as it was posed by Benacerraf in Mathematical Truth and the next section presents a way of looking at abstract objects that purports to demythologize them. In particular, it shows how we can have empirical knowledge of various abstract objects and even how we might causally interact with them.

A response is given here to Benacerraf's 1973 argument that mathematical platonism is incompatible with a naturalistic epistemology. Unlike almost all previous platonist responses to Benacerraf, the response given here is positive rather than negative; that is, rather than trying to find a problem with Benacerraf's argument, I accept his challenge and meet it head on by constructing an epistemology of abstract (i.e., aspatial and atemporal) mathematical objects. Thus, I show that spatio-temporal creatures like ourselves can attain knowledge about mathematical objects by simply explaininghow they can do this. My argument is based upon the adoption of a particular version of platonism — full-blooded platonism — which asserts that any mathematical object which possiblycould exist actuallydoes exist.

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.

The view that mathematical objects are indefinite in nature is presented and defended, hi the first section, Field's argument for fictionalism, given in response to Benacerraf's problem of identification, is closely examined, and it is contended that platonists can solve the problem equally well if they take the view that mathematical objects are indefinite. In the second section, two general arguments against the intelligibility of objectual indefiniteness are shown erroneous, hi the final section, the view is compared to mathematical structuralism, and it is shown that a version of structuralism should be understood as embracing the same view.