This volume aims to establish the starting point for the development, evaluation and appraisal of the phenomenology of mathematics.

We think of logic as objective. We also think that we are reliable about logic. These views jointly generate a puzzle: How is it that we are reliable about logic? How is it that our logical beliefs match an objective domain of logical fact? This is an instance of a more general challenge to explain our reliability about a priori domains. In this paper, I argue that the nature of this challenge has not been properly understood. I explicate the challenge (...) both in general and for the particular case of logic. I also argue that two seemingly attractive responses – appealing to a faculty of rational insight or to the nature of concept possession – are incapable of answering the challenge. (shrink)

My aim in this chapter is to extend the Realist account of the foundations of linguistics offered by Postal, Katz and others. I first argue against the idea that naive Platonism can capture the necessary requirements on what I call a ‘mixed realist’ view of linguistics, which takes aspects of Platonism, Nominalism and Mentalism into consideration. I then advocate three desiderata for an appropriate ‘mixed realist’ account of linguistic ontology and foundations, namely (1) linguistic creativity and infinity, (2) linguistics as (...) a theory of types (and not tokens) and (3) independence but structural respect between language and the linguistic competence thereof. My own brand of mixed realism, what I call ante rem realism, is defended along the lines of an ante rem or non-eliminative structuralism, the likes of which has been offered for mathematics by Resnik (1997) and Shapiro (1997). In other words, grammars describe a mind-independent (but not necessarily unconnected) linguistic reality in terms of linguistic patterns or structures also known as natural languages. I further amend this picture to allow for the possibility of a naturalistic account of language acquisition and evolution by arguing against a particular view of the type-token distinction. (shrink)

The standard argument for the existence of distinctively mathematical objects like numbers has two main premises: some mathematical claims are true, and the truth of those claims requires the existence of distinctively mathematical objects. Most nominalists deny. Those who deny typically reject Quine’s criterion of ontological commitment. I target a different assumption in a standard type of semantic argument for. Benacerraf’s semantic argument, for example, relies on the claim that two sentences, one about numbers and the other about cities, have (...) the same grammatical form. He makes this claim on the grounds that the two sentences are superficially similar. I argue that these grounds are not sufficient. Other sentences with the same superficial form appear to have different grammatical forms. I offer two plausible interpretations of Benacerraf’s number sentence that make use of plural quantification. These interpretations appear not to incur ontological commitments to distinctively mathematical objects, even assuming Quine’s criterion. Such interpretations open a new, plural strategy for the mathematical nominalist. (shrink)

Chomsky's criticism of Bloomfieldian structuralism's conception of linguistic reality applies equally to his own conception of linguistic reality. There are too many sentences in a natural language for them to have either concrete acoustic reality or concrete psychological or neural reality. Sentences have to be types, which, by Peirce's generally accepted definition, means that they are abstract objects. Given that sentences are abstract objects, Chomsky's generativism as well as his psychologism have to be given up. Langendoen and Postal's argument in (...) The Vastness of Natural Languages to show that there are more than denumerably many sentences is flawed. But, with the view that sentences are abstract objects, the flaws can be corrected. Once psychologism and generativism are abandoned, the revolution against Bloomfieldian structuralism can be brought to completion and linguistics can be put on a sound philosophical basis. (shrink)

I present and defend a unified, non-reductive analysis of the a priori and a posteriori. It is a mistake to remove all epistemic conditions from the analysis of the a priori (as, for example, Alvin Goldman has recently suggested doing). We can keep epistemic conditions (like unrevisability) in the analysis as long as we insist that a priori and a posteriori justification admit of degrees. I recommend making the degree to which a belief’s justification is a priori or a posteriori (...) solely dependent on the revisability relations that obtain among the faculties that deliver the belief and all other faculties. (shrink)

In his conceivability argument, Chalmers assumes that all properties have their causal powers contingently and causal laws are also contingent. We argue that this claim conflicts with how conceivability itself must work for the conceivability argument to be successful. If conceivability is to be an effective mechanism to determine possibility, it must work as a matter of necessity, since contingent conceivability renders conceivability fallible for an ideal reasoner and the fallible conceivability of zombies would not entail their possibility. But necessary (...) conceivability must either be governed by necessitating causal processes or by a necessitating non-causal mechanism. We argue that the latter option is untenable or mysterious; whereas, if Chalmers chooses the former and applies it only to conceivability, his solution is ad hoc, but if he accepts necessary causal powers or processes generally, the conceivability argument fails. We conclude that, as it stands, the Conceivability Argument does not establish that physicalism is false. (shrink)

The paper focuses on the Epistemic Challenge for mind-independent accounts of modality. The challenge can be formulated as an inconsistency problem among three premises and, therefore, any strategy to meet the challenge will require the negation of (at least) one of its premises. The aim of the paper is not to offer a positive solution to the challenge, but rather to argue for the claim that to follow a hybrid strategy is probably the best way to meet it. With some (...) qualifications, reasons are given as to why empiricism should be the way to meet the challenge as far as de re modality is concerned, whereas rationalism might be the correct way of addressing it for the case of de dicto modality. (shrink)

This paper explores the need to move beyond epistemological and axiological discussions in philosophy of mathematics education by reframing the inquiry to include an ontological perspective. The main goal of this work is to envision a new relationship between philosophical discourse and mathematics education, one that takes into account ontological assumptions in mathematics and relates it to axiological objectives and epistemological claims. I begin with a description of the dominant view of mathematics education as depicted in U.S. policy reform discourses (...) and contrast it with critical views of philosophy of mathematics education. I show that the deficiency of both perspectives is their lack of attention to the ontological assumptions inherent in mathematics education. Lastly, I provide a preliminary sketch of a new type of philosophy of mathematics education, which brings ontological inquiry to the forefront, and I speculate on how this emerging philosophy of mathematics education might impact the field. (shrink)