Frege, Peano and the Construction of a Logical Calculus (original) (raw)
Related papers
Frege, Peano and the Interplay between Logic and Mathematics
Philosophia Scientiae, 2021
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.
Debating (Neo)logicism: Frege and the Neo-Fregeans
Between Logic and Reality, 2011
The paper's aim is to determine and discuss in which sense, if any, Frege's and neo-Fregean logicism are responding to the epistemological challenge concerning our arithmetical knowledge. More precisely the paper analyses what the epistemological significance of Frege's logicist programme amounts to, namely, the objective justificatory connections obtaining between arithmetical and logical statements. It then contrasts this result with the self-understanding of the neo-Fregeans who allegedly follow Frege's steps, but in fact take a rather different direction.
Frege's Conception of Logic: From Kant to Grundgesetze
2003
The last few decades have brought impressive new technical insights regarding Frege's logicism and his "reduction of arithmetic to logic." 1 This paper, however, deals with the complementary but far less investigated question how Frege understood the nature of logical truth and of logical knowledge. I shall examine Frege's conception of logic as it developed and matured, beginning with his early Begriffsschrift from 1879 and following it up through to Grundgesetze I from 1893. 2 I shall make two main claims. My first main claim is that Frege started out with a view of logic that is closer to Kant's than is generally recognized, but that he gradually came to reject this Kantian view, or at least totally to transform it. My second main claim concerns Frege's reasons for distancing himself from the Kantian conception of logic. It is natural to speculate that this change in Frege's view of logic may have been spurred by a desire to establish the logicality of the axiom system he needed for his logicist reduction, including the infamous Basic Law V. I admit this may have been one of Frege's motives. But I shall argue that Frege also had a deeper and more interesting reason to reject his early Kantian view of logic, having to do with his increasingly vehement anti-psychologism.
Frege’s relation to Aristotle and the emergence of modern logic
Bloomsbury Academic eBooks, 2023
Gottlob Frege's works are o en taken to mark the beginning of modern logic. More speci cally, the year 1879, when Frege's Begri sschri was published, is seen as witnessing its birth. 'Modern logic' is here contrasted with traditional Aristotelian logic, especially the theory of syllogism that is central for it. In this chapter, I will reconsider the transition from Aristotelian logic to modern logic in Frege's and related writings. I will con rm that Frege's works do, indeed, contain innovations that contributed to a radical transformation of logic. At the same time, I will argue that the story of that transformation is complex, including the fact that Frege's works contain remnants of Aristotelian ideas that make it not fully modern. e chapter will proceed as follows: In Section 1, I will start with several claims that the publication of Begri sschri constituted a 'revolution' in logic, while noting some immediate challenges to that claim as well. In Section 2, I will turn to Frege's central logical innovations and to logicism as the driving force behind them. Next, I will consider Frege's relationship to Aristotle, both in terms of Aristotelian views Frege rejected explicitly, in Section 3, and features that still tie him to Aristotle's model of science implicitly, in Section 4. In Section 5, nally, I will reconsider the ways in which Frege's contributions were crucial for a 'second birth' of logic, including by being articulated further in the works of later logicians. 1 A Fregean revolution in logic? Since the 1950-1960s, the publication of Frege's Begri sschri has o en been seen as the start of modern, post-Aristotelian logic. Crucial roles in assigning a central part to Frege were played by Alonzo Church, W. V. O. Quine and J. v. Heijenoort. As Quine wrote in 1952, '[l]ogic is only now emerging from a renaissance such as was undergone by physics centuries ago. […] e logical renaissance might be identi ed with the publication of Frege's Begri sschri in 1879'. 1 Or more pointedly in Quine's Methods of Logic (1950): 'Logic is an old subject, but since 1879 it has been a great one. ' 2 e same perspective shapes Jean van Heijenoort's in uential collection, From Frege to Gödel: A
- Work of the Subject Matter »History of Logic« -
- Trabalho da Disciplina História da Lógica -, 2024
Is it possible for a rational individual to defend Mill’s position regarding standard arithmetic validly? Can anyone claim that numbers are empirical facts and that mathematical truths can be based on physical induction? Mill believed in the affirmation of these questions and, not only, thought he could demonstrate the standard arithmetic from empirical observations; Frege already believed, in his own Platonism »and this is something his interpreters find questionable, i.e., that he was a Platonic philosopher«, that this was an absurd thesis and arguments. In the following work, we will conclude that it is possible to have a sympathetic, although not entirely welcoming, critique of the doctrine of logical theses in Mill’s System of Logic. We wil give a brief overview of Frege’s Theory of Sense and Reference – Sinn und Bedeutung –, his definition of numbers and his logicist conception of standard arithmetic – already demonstrated as impossible in the 20th century, even though certain individuals try to revive it -; Mill’s main theses about arithmetic, as he conceives it, will be exposed, so that we can see a new approach to structuralist and mereological standard arithmetic based on his doctrine. And we will see if it is possible to pave the way for a logical-empirical theory of the logic of mathematics.
Comments on Patricial Blanchette's Book: Frege's Conception of Logic
Journal for the History of Analytical Philosophy, 2015
All contributions included in the present issue were originally presented at an 'Author Meets Critics' session organised by Richard Zach at the Pacific Meeting of the American Philosophical Association in San Diego in the Spring of 2014.
KWARTALNIK HISTORII NAUKI I TECHNIKI, 2023
The article aims to answer whether Gottlob Frege's letter to Adolph Mayer, dated 8 July 1896, could help German mathematicians get acquainted with Giuseppe Peano's mathematical work, including his mathematical logic. It is the fi rst publication of this letter in English. At the beginning, I present the main characters of this story. Next, I refer to the letters concerning Peano and his mathematical results. Thus, I show the background of Frege's letter to Mayer. In the last part, I collect information about Peano's contacts with German mathematicians-where he was quoted and by whom, who was quoted by Peano, and in which period of his life. I conclude that Peano was known in Germany before Frege wrote to Mayer in 1896. However, the letter could have helped publish fi ve of Peano's articles in Germany, where Peano's mathematical logic was hardly known then. Undoubtedly, the letter promoted that knowledge.
Annals of Pure and Applied Logic, 2004
According to Frege's principle the denotation of a sentence coincides with its truthvalue. The principle is investigated within the context of abstract algebraic logic, and it is shown that taken together with the deduction theorem it characterizes intuitionistic logic in a certain strong sense.