Kit Fine develops a Fregean theory of abstraction, and suggests that it may yield a new philosophical foundation for mathematics, one that can account for both our reference to various mathematical objects and our knowledge of various mathematical truths. The Limits ofion breaks new ground both technically and philosophically.

As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell’s Paradox being derivable in it.This system is, except for minor differ...

George Boolos; IX*—Saving Frege from Contradiction, Proceedings of the Aristotelian Society, Volume 87, Issue 1, 1 June 1987, Pages 137–152, https://doi.org/10.

George Boolos; IX*—Saving Frege from Contradiction, Proceedings of the Aristotelian Society, Volume 87, Issue 1, 1 June 1987, Pages 137–152, https://doi.org/10.

In this paper, I consider two curious subsystems ofFrege's Grundgesetze der Arithmetik: Richard Heck's predicative fragment H, consisting of schema V together with predicative second-order comprehension (in a language containing a syntactical abstraction operator), and a theory T in monadic second-order logic, consisting of axiom V and 1 1-comprehension (in a language containing anabstraction function). I provide a consistency proof for the latter theory, thereby refuting a version of a conjecture by Heck. It is shown that both Heck and T (...) prove the existence of infinitely many non-logical objects (T deriving,moreover, the nonexistence of the value-range concept). Some implications concerning the interpretation of Frege's proof of referentiality and the possibility of classifying any of these subsystems as logicist are discussed. Finally, I explore the relation of T toCantor's theorem which is somewhat surprising. (shrink)

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)

Controversy remains over exactly why Frege aimed to estabish logicism. In this essay, I argue that the most influential interpretations of Frege's motivations fall short because they misunderstand or neglect Frege's claims that axioms must be self-evident. I offer an interpretation of his appeals to self-evidence and attempt to show that they reveal a previously overlooked motivation for establishing logicism, one which has roots in the Euclidean rationalist tradition. More specifically, my view is that Frege had two notions of self-evidence. (...) One notion is that of a truth being foundationally secure, yet not grounded on any other truth. The second notion is that of a truth that requires only clearly grasping its content for rational, a priori justified recognition of its truth. The overarching thesis I develop is that Frege required that axioms be self-evident in both senses, and he relied on judging propositions to be self-evident as part of his fallibilist method for identifying a foundation of arithmetic. Consequently, we must recognize both notions in order to understand how Frege construes ultimate foundational proofs, his methodology for discovering and identifying such proofs, and why he thought the propositions of arithmetic required proof. (shrink)

In this exciting new collection, a distinguished international group of philosophers contribute new essays on central issues in philosophy of language and logic, in honor of Michael Dummett, one of the most influential philosophers of the late twentieth century. The essays are focused on areas particularly associated with Professor Dummett. Five are contributions to the philosophy of language, addressing in particular the nature of truth and meaning and the relation between language and thought. Two contributors discuss time, in particular the (...) reality of the past. The last four essays focus on Frege and the philosophy of mathematics. The volume represents some of the best work in contemporary analytical philosophy. (shrink)

The book is an attempt at explaining to the nation the ideas of Frege's Grundlagen. It is wordy and trite, a paradigm case of a redundant piece of writing. The reader is advised to steer clear of it.

This paper argues that that Caesar problem had a technical aspect, namely, that it threatened to make it impossible to prove, in the way Frege wanted, that there are infinitely many numbers. It then offers a solution to the problem, one that shows Frege did not really need the claim that "numbers are objects", not if that claim is intended in a form that forces the Caesar problem upon us.