A possible interpretation of Frege's definition of number

2009

In Frege’s theory of arithmetic where the definite article serves to classify something as an object, the “definite article premise” is one of the main premises along with the “context principle” and definite article premise provided a tool for Frege to define numbers as logical objects. By this tool he could distinguish linguistically concept of cardinal number and a cardinal number as an object so that they could have different terms which denote different things. I will discuss that the “definite article premise” itself is ineffectual. Frege’s logicist system is undermined for one of the premises in his argumentation is problematic. Particularly, the “definite article premise” cannot classify the concept words for numbers as objects words (proper names). Since his logicist project fails to be completed, it is not possible to secure the referent of number terms as logical objects. However the definition he has suggested can still give the Kantian intuition of individual numbers.

Frege: Importance and Legacy

This paper is in honor of my colleague and friend, Leonard Linsky, on the occasion of his retirement. I presented the earliest version in the Spring of 1992 to a reading group, the other members of which were Leonard Linsky, Steve Awodey, Andre Carus and Mike Price. I presented later versions in the autumn of 1992 to the philosophy colloquium at McGill University and in the autumn of 1993 to the philosophy colloquium at Carnegie-Mellon University. The discussions following these presentations were valuable to me and I would especially like to acknowledge Emily Carson (for comments on the earliest draft), Michael Hallett, Kenneth Manders, Stephen Menn, G.E. Reyes, Teddy Seidenfeld, and Wilfrid Sieg and the members of the reading group for helpful comments. But, most of all, I would like to thank Howard Stein and Richard Heck, who read the penultimate draft of the paper and made extensive comments and corrections. Naturally, none of these scholars, except possibly Howard Stein, is responsible for any remaining defects. 1 Frege (1879). Dedekind (1887) similarly analyzed the ancestral F* in the case of a one-to-one function F from a set into a proper subset. In the preface to the first edition, Dedekind stated that, in the years 1872-78, he had written a first draft, containing all the essential ideas of his monograph.

1987

11 est vrai qu'il y a un infini en nombre. Mais nous ne savons ce quil est.

Siegener Beiträge zur Geschichte und Philosophie der Mathematik, 2018

Some have argued that Frege's definitions of numbers are linguistic stipulations, with no content-preserving or ontological point: they don't capture any determinate content of numerals, as these have none, and don't present numbers as pre-existing objects. I show that this view is based on exegetical and philosophical errors.

2005

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.

The purpose of this article is to discuss Wittgenstein 1933-39 self-criticism of one of the most fundamental tractarian theses: There is one and only one complete analysis of the proposition (3.25). Wittgenstein’s arguments against his previous thesis can be found in some passages of the Philosophical Investigation (PI) as well as in the Big Typescript (BT). As we shall argue, Wittgenstein’s self-critique is directly related to two important ideas proposed by Frege in §§18-24 of The Foundations Arithmetic (FA) with respect to his construal of “number attributions”. Our main goal in this paper will therefore be to show how Wittgenstein's self-criticism of his old conception of the tractarian “analysis” would have been influenced by his concerns in the middle period of his work with these two proposals made by Frege in FA regarding the notion of “numerical attribution”. Our first task will then be to connect Wittgenstein’s self-criticisms to his former conception of “analysis” in the Tractatus Logico-Philosophicus (TLP). To accomplish this first goal, we will have to briefly present a version of how such analysis would look like in TLP. Our next task will be to pinpoint some passages of the PI in which Wittgenstein rejects the completeness and uniqueness involved in his previous construal of the analytical process. We will try to circumscribe clearly a crucial notion involved in Wittgenstein’s self-critique: the idea of “seeing as/seeing aspects”. Our claim is that this notion, as well as Wittgenstein’s self-criticism, were both directly influenced by Frege’s construal of the “number attribution” presented in FA. To accomplish this second task, we will have to revisit and discuss some further concepts from an early period of Wittgenstein’s philosophy, like “internal and internal properties” and the “saying/showing distinction” in TLP. On our way towards accomplishing this goal, we will present a curious complaint by Wittgenstein regarding Russell’s, as well as Frege’s conception of generality, or more specifically, on the usage of the existential quantifier suggested by both these authors.

Balkan Journal of Philosophy, 2014

Was Frege firstly a philosopher of language or a philosopher of mathematics? I try to give an answer to this question in this paper. I argue that Frege's definition of natural number is the right way to reach the answer. Frege had simultaneously two theoretical commitments: one regarding the logicist programme in the foundations of mathematics, the other regarding the conception of logic as a language. Therefore, Frege developed a formal language and tried to define arithmetical concepts in pure logical terms. He did this based on semantical suppositions, because he could not do it any other way as long as he regarded logic as a language. I will argue that Frege used semantic tools in order to solve problems related to philosophy of mathematics.

Frege's forty-year scholar work was submitted to searching an answer to the philosophical-mathematical and logical question of what the number is. It is a well-known fact that Frege developed one conception of a number that referred to equinumerosity, where a statement operator is a number has its particular place. In this paper I shall focus on the six important key statements concerning numbers. Frege thought that working on number is a task that should be dealt both by mathematicians and philosophers. Here philosophical aspect seems all-important.

Acta Analytica, 2016

In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as mathematical. In this paper, I will argue that Dedekind's approach can be seen as a precursor to modern structuralism and as such, it enjoys many advantages over Frege's logicism. I also show that from a modern perspective, Frege's criticism of abstraction and psychologism is one-sided and fails against the psychological processes that modern research suggests to be at the heart of numerical cognition. The approach here is twofold. First, through historical analysis, I will try to build a clear image of what Frege's and Dedekind's views on arithmetic were. Then, I will consider those views from the perspective of modern philosophy of mathematics, and in particular, the empirical study of arithmetical cognition. I aim to show that there is nothing to suggest that the axiomatic Dedekind approach could not provide a perfectly adequate basis for philosophy of arithmetic.