In recent work on Frege, one of the most salient issues has been whether he was prepared to make serious use of semantical notions such as reference and truth. I argue here Frege did make very serious use of semantical concepts. I argue, first, that Frege had reason to be interested in the question how the axioms and rules of his formal theory might be justified and, second, that he explicitly commits himself to offering a justification that appeals to the (...) notion of reference. I then discuss the justifications Frege offered, focusing on his discussion of inferences involving free variables, in section 17 of Grundgesetze, and his argument, in sections 29-32, that every well-formed expression of his formal language has a unique reference. (shrink)

Against some very common views (e.g. Dummett), this paper argues that Frege did not have a standard interpretation of higher-order logic and for this reason his programme in the foundations of mathematics was a nonstarter.

This paper has three goals: (i) to show that the foundational program begun in the Begriffsschroft, and carried forward in the Grundlagen, represented Frege's attempt to establish the autonomy of arithmetic from geometry and kinematics; the cogency and coherence of 'intuitive' reasoning were not in question. (ii) To place Frege's logicism in the context of the nineteenth century tradition in mathematical analysis, and, in particular, to show how the modern concept of a function made it possible for Frege to pursue (...) the goal of autonomy within the framework of the system of second-order logic of the Begriffsschrift. (iii) To address certain criticisms of Frege by Parsons and Boolos, and thereby to clarify what was and was not achieved by the development, in Part III of the Begriffsschrift, of a fragment of the theory of relations. (shrink)

This influential 1888 publication explained the real numbers, and their construction and properties, from first principles.

Matthias Schirn has argued on a number of occasions against the interpretation of Frege's ``objects of a quite special kind'' (i.e., the objects referred to by names like `the concept F') as extensions of concepts. According to Schirn, not only are these objects not extensions, but also the idea that `the concept F' refers to objects leads to some conclusions that are counter-intuitive and incompatible with Frege's thought. In this paper, I challenge Schirn's conclusion: I want to try and argue (...) that the assumption that `the concept F' refers to the extension of F is entirely consistent with Frege's broader views on logic and language. I shall examine each of Schirn's main arguments and show that they do not support his claim. (shrink)

Frege begins Die Grundlagen der Arithmetik, the work that introduces the project which was to occupy him for most of his professional career, with the question, 'What is the number one?' It is a question to which even mathematicians, he says, have no satisfactory answer. And given this scandalous situation, he adds, there is small hope that we shall be able to say what number is. Frege intends to rectify the situation by providing definitions of the number one and the (...) concept of number. But what, exactly, is required of a definition? Surely it will not do to stipulate that the number one is Julius Caesar - that would change the subject. It seems reasonable to suppose that an acceptable definition must be a true statement containing a description that picks out the object to which the numeral '1' already refers. And, similarly, that an acceptable definition of the concept of number must contain a description that picks out precisely those objects that are numbers - those objects to which our numerals refer. Yet, while Frege writes a great deal about what criteria his definitions must satisfy, the above criteria are not among those he mentions. Nor does he attempt to convince us that his definitions of '1' and the other numerals are correct by arguing that these definitions pick out objects to which these numerals have always referred. Yet, while Frege writes a great deal about what criteria his definitions must satisfy, the above criteria are not among those he mentions. Nor does he attempt to convince us that his definitions of ‘1’ and the other numerals are correct by arguing that these definitions pick out objects to which these numerals have always referred. There is, as we shall see shortly, a great deal of evidence that Frege’s definitions are not intended to pick out objects to which our numerals already refer. But if this is so, how can these definitions teach us anything about our science of arithmetic? And what criteria must these definitions satisfy? To answer these questions, we need to understand what it is that Frege thinks we need to learn about the science of arithmetic. (shrink)

Hintikka and Sandu have recently claimed that Frege's notion of function was substantially narrower than that prevailing in real analysis today. In the present note, their textual evidence for this claim is examined in the light of relevant historical and biographical background and judged insufficient.

In his article The Foundations of Mathematics (1925) Ramsey was concerned with the nature of the statements of 'pure mathematics' and the way these statements differ from those in empirical sciences. He thought that the answer given to these questions by Hilbert and the formalist school according to which mathematical statements are meaningless formulas, is unsatisfactory for several reasons, which will not be discussed here. He also expressed serious doubts about the intuitionist program developed by Brouwer and Weyl. It is (...) the logicist school of Frege, Russell and Whitehead, which attempted to reduce mathematics to few logical concepts, that came closest to Ramsey’s views, although he was not completely satisfied with it, either. (shrink)