Online research in philosophy

It has been known for more than thirty years that the degree of a non-standard model of true arithmetic is a subuniform upper bound for the arithmetic sets (suub). Here a notion of generic enumeration is presented with the property that the degree of such an enumeration is an suub but not the degree of a non-standard model of true arithmetic. This answers a question posed in the literature.

We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ₂—theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ₁—theory of multiplication and order is decidable in finite models as well as in the standard model. We show also that the exponentiation function is definable in finite models by a formula of arithmetic with multiplication and that one can define in finite models the arithmetic of addition and multiplication with the concatenation operation. We consider also the spectrum problem. We show that the spectrum of arithmetic with multiplication and arithmetic with exponentiation is strictly contained in the spectrum of arithmetic with addition and multiplication.

In this paper we present a negative solution of counting problems for some classes slightly different from bounded arithmetic (▵ 0 sets). To get the results we study properties of chains of finite automata.

This paper explores the development of mathematics on a quantum logical base when mathematical postulates are taken as necessary truths. First it is shown that first-order Peano arithmetic formulated with quantum logic has the same theorems as classical first-order Peano arithmetic. Distribution for first-order arithmetical formulas is a theorem not of quantum logic but rather of arithmetic. Second, it is shown that distribution fails for second-order Peano arithmetic without extensionality. Third, it is shown that distribution holds for second-order Peano arithmetic (second-order quantum logic) with extensionality. Some remarks about extensions to quantum set theory are made.

This is a dialogue in the philosophy of mathematics that focuses on these issues: Are the Peano axioms for arithmetic epistemologically irrelevant? What is the source of our knowledge of these axioms? What is the epistemological relationship between arithmetical laws and the particularities of number?

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.

Standard accounts of the senses attempt to answer the question how and why we count ?ve senses (the counting question); none of the standard accounts is satisfactory. Any adequate account of the senses must explain the signi?cance of the senses, that is, why distinguishing different senses matters. I provide such an explanation, and then use it as the basis for providing an account of the senses and answering the counting question.

Kant’s theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant’s theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant’s theory of arithmetic can be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or “formulas”—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant’s theory of addition as a form of synthetic operation.

The goal of the research programme I describe in this article is a realist epistemology for arithmetic which respects arithmetic's special epistemic status (the status usually described as a prioricity) yet accommodates naturalistic concerns by remaining fundamentally empiricist. I argue that the central claims which would allow us to develop such an epistemology are (i) that arithmetical truths are known through an examination of our arithmetical concepts; (ii) that (at least our basic) arithmetical concepts are accurate mental representations of elements of the arithmetical structure of the independent world; (iii) that (ii) obtains in virtue of the normal functioning of our sensory apparatus. The first of these claims protects arithmetic's special epistemic status relative, for example, to the laws of physics, the second preserves the independence of arithmetical truth, and the third ensures that we remain empiricists. Preliminaries Justifying and grounding concepts Cameras and filters An epistemology for arithmetic Concluding remarks.

Alberto Casullo ("Necessity, Certainty, and the A Priori", Canadian Journal of Philosophy 18, 1988) argues that arithmetical propositions could be disconfirmed by appeal to an invented scenario, wherein our standard counting procedures indicate that 2 + 2 != 4. Our best response to such a scenario would be, Casullo suggests, to accept the results of the counting procedures, and give up standard arithmetic. While Casullo's scenario avoids arguments against previous "disconfirming" scenarios, it founders on the assumption, common to scenario and response, that arithmetic might be independent of standard counting procedures. Here I show, by attention to tallying as the simplest form of counting, that this assumption is incoherent: given standard counting procedures, then (on pain of irrationality) arithmetical theory follows. 1.