The Internal/External Question

Synthese, 2023

Although number sentences are ostensibly simple, familiar, and applicable, the justification for our arithmetical beliefs has been considered mysterious by the philosophical tradition. In this paper, I argue that such a mystery is due to a preconception of two realities, one mathematical and one nonmathematical, which are alien to each other. My proposal shows that the theory of numbers as properties entails a homogeneous domain in which arithmetical and nonmathematical truth occur. As a result, the possibility of arithmetical knowledge is simply a consequence of the possibility of ordinary knowledge 1 .

Manuscrito, 2004

In §1 I discuss Dedekind and Frege on the logical and structural analysis of natural numbers and present my view that the logical analysis of the notion of number involves a combination of their analyses. In §2 I answer some of the specific questions that Abel raises in connection with Chapter 9 of Logical Forms.

Philosophia Mathematica, 2018

In this paper I argue against both neuropsychological and cognitive accounts of our grasp of numbers. I show that despite the points of divergence between these two accounts, they face analogous problems. Both presuppose too much about what they purport to explain to be informative, and also characterize our grasp of numbers in a way that is absurd in the light of what we already know from the point of view of mathematical practice. Then I offer a positive methodological proposal about the role that cognitive science should play in the philosophy of mathematics.

Logos and Episteme, 2020

The present paper deals with the ontological status of numbers and considers Frege´s proposal in Grundlagen upon the background of the Post-Kantian semantic turn in analytical philosophy. Through a more systematic study of his philosophical premises, it comes to unearth a first level paradox that would unset earlier still than it was exposed by Russell. It then studies an alternative path that, departing from Frege's initial premises, drives to a conception of numbers as synthetic a priori in a more Kantian sense. On this basis, it tentatively explores a possible derivation of basic logical rules on their behalf, suggesting a more rudimentary basis to inferential thinking, which supports reconsidering the difference between logical thinking and AI. Finally, it reflects upon the contributions of this approach to the problem of the a priori.

Melisa Vivanco, 2023

Some of the most influential programs in the philosophy of mathematics started from the philosophical study of natural numbers. On the one hand, our arithmetic intuitions appear earlier and more direct than other mathematical (and non-mathematical) intuitions. On the other hand, while arithmetic admits one of the first axiomatizations with wide acceptance within the mathematical practice, the study of natural numbers sets a methodological precedent that will later seek to be replicated in other areas, in particular, in the study of the most complex numerical structures. This course will address the main issues in the philosophical discussion on arithmetic. Among these topics are the ideas of the various classical doctrines on the foundations of arithmetic, from the milestone of Gödel’s incompleteness theorems to recent doctrines on the semantics of numerical expressions and arithmetic sentences. The class will cover debates about metaphysics and the epistemology of numbers and arithmetic truths.

Synthese, 2001

If abstract mathematical entities do not really exist, then what really exist in human mathematical practices? For a naturalist and nominalist, they can only be human brain activities in doing and applying mathematics. This paper belongs to a research project exploring a naturalistic and nominalistic account of human mathematical practices by treating them as cognitive activities of human brains. I will introduce some basic assumptions about human cognitive architecture and then discuss several aspects of human mathematical practices on that basis.

Theoria, 2023

The field of numerical cognition provides a fairly clear picture of the processes through which we learn basic arithmetical facts. This scientific picture, however, is rarely taken as providing a response to a much-debated philosophical question, namely, the question of how we obtain number knowledge, since numbers are usually thought to be abstract entities located outside of space and time. In this paper, I take the scientific evidence on how we learn arithmetic as providing a response to the philosophical question of how we obtain number knowledge. I reject the view that numbers are abstract entities located outside of space and time and, alternatively, derive from the scientific evidence a novel account of the nature of numbers. In this account, numbers are reifications of the counting procedure and arithmetic statements are seen as describing the functioning of counting and calculation techniques.

The Journal of Philosophy, 2001