In the last two decades, there has been increasing interest in a re-evaluation of Frege’s stance towards consistency- and independence proofs. Papers by several authors deal with Frege’s views on these topics. In this note, I want to discuss one particular problem, which seems to be a main reason for Frege’s reluctant attitude towards his own proposed method of proving the independence of axioms, namely his view that thoughts, that is, intensional entities are the objects of metatheoretical investigations. This stands (...) in contrast to more straightforward interpretations, which claim that Frege’s hesitancy is mainly due to worries concerning the logical constants or what counts as a logical inference. (shrink)

Peano’s axioms for arithmetic, published in 1889, are ubiquitously cited in writings on modern axiomatics, and his Formulario is often quoted as the precursor of Russell’s Principia Mathematica. Yet, a comprehensive historical and philosophical evaluation of the contributions of the Peano School to mathematics, logic, and the foundation of mathematics remains to be made. In line with increased interest in the philosophy of mathematics for the investigation of mathematical practices, this them...

Peano’s axioms for arithmetic, published in 1889, are ubiquitously cited in writings on modern axiomatics, and his Formulario is often quoted as the precursor of Russell’s Principia Mathematica. Yet, a comprehensive historical and philosophical evaluation of the contributions of the Peano School to mathematics, logic, and the foundation of mathematics remains to be made. In line with increased interest in the philosophy of mathematics for the investigation of mathematical practices, this them...

After the publication of Begriffsschrift, a conflict erupted between Frege and Schröder regarding their respective logical systems which emerged around the Leibnizian notions of lingua characterica and calculus ratiocinator. Both of them claimed their own logic to be a better realisation of Leibniz’s ideal language and considered the rival system a mere calculus ratiocinator. Inspired by this polemic, van Heijenoort (1967b) distinguished two conceptions of logic—logic as language and logic as calculus—and presented them as opposing views, but did not explain (...) Frege’s and Schröder’s conceptions of the fulfilment of Leibniz’s scientific ideal. -/- In this paper I explain the reasons for Frege’s and Schröder’s mutual accusations of having created a mere calculus ratiocinator. On the one hand, Schröder’s construction of the algebra of relatives fits with a project for the reduction of any mathematical concept to the notion of relative. From this stance I argue that he deemed the formal system of Begriffsschrift incapable of such a reduction. On the other hand, first I argue that Frege took Boolean logic to be an abstract logical theory inadequate for the rendering of specific content; then I claim that the language of Begriffsschrift did not constitute a complete lingua characterica by itself, more being seen by Frege as a tool that could be applied to scientific disciplines. Accordingly, I argue that Frege’s project of constructing a lingua characterica was not tied to his later logicist programme. (shrink)

In contemporary historical studies, Peano is usually included in the logical tradition pioneered by Frege. In this paper, I shall first demonstrate that Frege and Peano independently developed a similar way of using logic for the rigorous expression and proof of mathematical laws. However, I shall then suggest that Peano also used his mathematical logic in such a way that anticipated a formalisation of mathematical theories which was incompatible with Frege’s conception of logic.

In contemporary historical studies, Peano is usually included in the logical tradition pioneered by Frege. In this paper, I shall first demonstrate that Frege and Peano independently developed a similar way of using logic for the rigorous expression and proof of mathematical laws. However, I shall then suggest that Peano also used his mathematical logic in such a way that anticipated a formalisation of mathematical theories which was incompatible with Frege’s conception of logic.

One dimension of a comprehensive semantic and semiotic theory is its explanation of how a wide-variety of linguistic expressions designate singular objects. The bulk of scholarship on Peirce's theory of proper names has aligned his theory with the so called new theory of reference by drawing connections between proper names qua rhematic indexical legisigns and various aspects of Kripke's theory of names.2 Recent scholarship has navigated away from indexing Kripke-Peirce affinities and has begun the process of articulating a semiotic or (...) pragmatic theory of names that is distinctly Peircean. For example.. (shrink)

The task of this paper is to give a new, catholic interpretation of Leibniz’s concept of characteristica universalis. In § 2 we shall see that in different periods of his development, Leibniz defined this concept differently. He introduced it as “philosophical characteristic” in 1675, elaborated it further as characteristica universalis in 1679, and worked on it at least until 1690. Secondly, we shall see (in § 3) that in the last 130 years or so, different philosophers have advanced projects similar (...) to that of Leibniz, not always referring to him, at that. These very projects, we claim, threw some light on what Leibniz idea of characteristica universalis could be; or, in more positive sense, on how could it be reconstructed in a more workable way. Unfortunately, they failed to answer the question what exactly Leibniz’s philosophy was. Finally, despite the fact that Leibniz’s concept of characteristica universalis impressed generations of philosophers who tried to make sense of it, the result is that in the more than 300 years after it was introduced, it was never used in the scale its author dreamed of. This fact sets out the next objective which we are going to pursue in this paper: we shall try to find out (in §§ 4 and 5) how this concept can be interpreted in more practical form. (shrink)

I argue that we find the articulation of a problem concerning bodily agency in the early works of the Merleau-Ponty which he explicates as analogous to what he explicitly calls the problem of perception. The problem of perception is the problem of seeing how we can have the object given in person through it perspectival appearances. The problem concerning bodily agency is the problem of seeing how our bodily movements can be the direct manifestation of a person’s intentions in the (...) world. In both cases what, according to Merleau-Ponty, obscures a recognition of the phenomenon in question is a conception of our bodily capacities, i.e. our sensibility and our motility, which reduces these to the workings of mechanisms that are blind to meaning. I argue that both the problem concerning perception and the problem concerning bodily agency can be properly called transcendental. The problem of perception is transcendental because it concerns the very intelligibility of appearances and judgements with empirical content. The problem of bodily agency is transcendental in the sense that it concerns the very intelligibility of our bodily capacity to carry out intentions and by implication the intelligibility of our intentions as such. (shrink)

A comprehensive survey of Martin-Löf's constructive type theory, considerable parts of which have only been presented by Martin-Löf in lecture form or as part of conference talks. Sommaruga surveys the prehistory of type theory and its highly complex development through eight different stages from 1970 to 1995. He also provides a systematic presentation of the latest version of the theory, as offered by Martin-Löf at Leiden University in Fall 1993. This presentation gives a fuller and updated account of the system. (...) Earlier, brief presentations took no account of the issues related to the type-theoretical approach to logic and the foundations of mathematics, while here they are accorded an entire part of the book. Readership: Comprehensive accounts of the history and philosophy of constructive type theory and a considerable amount of related material. Readers need a solid background in standard logic and a first, basic acquaintance with type theory. (shrink)

This entry introduces the reader to the main ideas in Frege's philosophy of logic, mathematics, and language.

It is a common wisdom that whereas consequence or entailment is a semantic concept, provability is a syntactic concept. However, what exactly does this mean? What is provability? In the traditional, intuitive sense, to prove something is to demonstrate its truth, and indeed the Latin word for proof is demonstratio. Hence in this sense, we cannot prove something unless it is true. Now in the course of his well known proof of the incompleteness of arithmetic, Gödel showed that provability within (...) the axiomatic system of Peano arithmetic (PA) can be reconstructed as a number-theoretic predicate Pr and he showed that within the system there exists a statement G so that.. (shrink)