Steps towards a minimalist account of numbers [Mind]

Travaux de Logique, 2007

This paper presents and discuss a logicist reduction of Peano's Arithmetic to S. Lesniewski's extensional calculus of names (the system called 'Ontology' by Lesniewski).

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.

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 .

Dilbilim Araştırmaları, 2022

Numerals participate in the expression of a wide range of operations, including mass, volume, degree, ordering, counting, and arithmetic calculations. This raises the questions of what they denote semantically and how they are derived morpho-syntactically. Although a number of theories have been advanced regarding their semantics, studies on the syntactic side are rather scarce. Further, the syntactic accounts of numerals date back to GB period, calling for a reinterpretation of their conclusions under Minimalist considerations. This study attempts to develop a syntactic account of numerals under Minimalist desiderata. It is proposed that numerals are number-denoting type n objects, derived from two primitives: saturated DIGITs of type n, and unsaturated BASEs of type <n,n>, instrumental in the derivation of simplex and complex numerals, respectively. This view is demonstrated to account for a wide range of distributional and interpretive possibilities of numerals as well as provide principled reasons for why some plausible forms are consistently unattested across languages.

Grazer Philosophische Studien, 2006

Stewart Shapiro and John Myhill tried to reproduce some features of the intuitionistic mathematics within certain formal intensional theories of classical mathematics. Basically they introduced a knowledge operator and restricted the ways of referring to numbers and to finite hereditary sets. The restrictions are very interesting, both because they allow us to keep substitutivity of identicals notwithstanding the presence of an epistemic operator and, especially, because such restrictions allow us to see, by contrast, which ways of reference are not compatible with the simultaneous maintenance of substitutivity of identicals and the classical notions of truth and knowledge. In this paper the difference between the restricted and the unrestricted kind of reference is put in relation with Russell&#39;s ideas on naming and it is argued that the latter as well is compatible with a certain Russellian conception of the understanding of sentences. Then it is discussed whether and how numbe...

Grazer Philosophische Studien, 1998

Axiomathes, 2011

The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in the mind is based on the linear number representation. This classical conception is rejected and a competitive hypothesis is formulated according to which the basic mature representational system of cognitive arithmetic is a structure composed of many numerical axes which possess a common constituent, namely, the numeral zero. Arithmetic of indexed numbers is just a formal tool for modelling the basic mature arithmetic competence. The third task is to develop a standpoint called temporal pluralism, which is motivated by neo-Kantian philosophy of arithmetic.

Trends in Cognitive Sciences, 2008

Number concepts must support arithmetic inference. Using this principle, it can be argued that the integer concept of exactly ONE is a necessary part of the psychological foundations of number, as is the notion of the exact equality -that is, perfect substitutability. The inability to support reasoning involving exact equality is a shortcoming in current theories about the development of numerical reasoning. A simple innate basis for the natural number concepts can be proposed that embodies the arithmetic principle, supports exact equality and also enables computational compatibility with real-or rational-valued mental magnitudes.

Sentence pairs like 'The number of moons of Mars is two' and 'Mars has two moons' give rise to a certain puzzle. On the one hand they seem to be truth-conditionally equivalent. On the other hand, though, it is puzzling how this could be the case. For on Frege's influential analysis of the former as an mathematical identity statement its truth requires the existence of numbers. And linguistic theory tells us that the truth of the latter does not require the existence of numbers, but only that of Mars and its two moons. How, then, can these two sentences be equivalent given that their truth of only one of them requires the existence number? Recently, it has been argued that, pace Frege, sentences like 'The number of moons of Mars is two' are to be analysed as so-called specificational sentences and that, thus analysed, their truth does not require the existence of numbers but, in fact, imposes the same requirements as 'Mars has two moons'. If so, the aforementioned puzzle would be resolved. In this paper, I show that the specificational analysis proposed by Katharina Felka fails to have the desired ontologically deflating consequence. Thus, even if her analysis was linguistically superior to Frege's, the equivalence of 'The number of moons of Mars is two' and 'Mars has two moons' would still be no less puzzling as it is on Frege's own.