Edward Kanterian - Frege: A Guide for the Perplexed

Frege

Bloomsbury Guides for the Perplexed

Bloomsburys Guides for the Perplexed are clear, concise and accessible introductions to thinkers, writers and subjects that students and readers can find especially challenging. Concentrating specifically on what it is that makes the subject difficult to grasp, these books explain and explore key themes and ideas, guiding the reader towards a thorough understanding of demanding material.

Guides for the Perplexed available from Bloomsbury:

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Pragmatism: A Guide for the Perplexed , Robert B. Talisse and
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Utilitarianism: A Guide for the Perplexed , Krister Bykvist

Wittgenstein: A Guide for the Perplexed, Mark Addis

A GUIDE FOR THE PERPLEXED

Frege

EDWARD KANTERIAN

Bloomsbury Academic
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Edward Kanterian, 2012

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ISBN: HB: 978-0-8264-8763-6
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CONTENTS

Dedicated to the memory of Ana Cocerva

If our language were logically perfect, we would perhaps have no more need of logic, or we might read logic off from the language. (1915)

It was from mathematics that I started out. (1919)

Gottlob Frege (18481925) is considered the greatest logician since Aristotle and one of the most important philosophers of the modern period. As a philosopher, his main contributions were in the philosophy of logic, mathematics and language. He also touched, if marginally, upon issues in epistemology and philosophy of mind. His main focus was not philosophy, however. He was a mathematician by training, and his interest lay in setting mathematics, in particular arithmetic, on a firm scientific foundation, a task of great urgency in his time.

A cursory look at his works demonstrates this. Most of them deal with two topics. The first is mathematics. Here, the focus lies on arithmetic, its nature and logical reconstruction, with which his three books, Begriffsschrift ( Conceptual Notation , 1879), Grundlagen der Arithmetik ( Foundations of Arithmetic , 1884) and Grundgesetze der Arithmetik ( Basic Laws of Arithmetic , 2 volumes, 1893 and 1903) are concerned, as are numerous articles and drafts in his Nachlass . He also devoted several discussions to the foundations of geometry, for instance in his first dissertation and later in his criticism of David Hilbert.

The second major topic is logic, logic understood in the narrow sense of a formal language and calculus (conceptual notation, or concept-script , as it will be called here), and in the broader sense of a theory of truth, concepts, thoughts, judgements, inference and reason. This theory underlies the conception of Freges concept-script. The relation between these two topics, logic and mathematics, is initially one of tool versus object of study. The object of study is the essence of number and arithmetical truth, while the calculus is a tool designed specifically to accomplish this study in the most precise way possible. But in fact the relation between object and tool is more intimate.

Arithmetic is a discipline concerned with its peculiar truths, and to establish its scientific status, to know what numbers really are, is to know what precisely these truths are, i.e. what they are about and what is their ultimate justification. But logic is for Frege also a discipline concerned with its own truths. This raises the question concerning the relation between arithmetical truth and logical truth. Do they belong to the same family? Kants answer was yes and no. Yes because arithmetic and logic express a priori truths. No because arithmetic is synthetic, based on what Kant called pure intuition, while logic is analytic, expressing only purely formal relations between concepts, without any intuitive addition. But for Frege, arithmetic and logic do belong to the same family they both express a priori , analytic truths of the greatest generality, and share the same modes of inference (proof). In fact, they dont only belong to the same family, but arithmetic is a branch of logic. This is the doctrine of logicism , Freges most treasured idea, at least until he was confronted with Russells Paradox in 1902. He wrote early on, in the article On Formal Theories of Arithmetic (1885):