Frege

A Critical Introduction

Harold W. Noonan

• Publisher: Polity
• Year: 2013
• ISBN:
  9780745668017
• Edition: 1
• Topic: Philosophie
• Subtopic: Histoire et théorie de la philosophie

This new book offers a comprehensive and accessible introduction to Frege's remarkable philosophical work, examining the main areas of his writings and demonstrating the connections between them.

Frege's main contribution to philosophy spans philosophical logic, the theory of meaning, mathematical logic and the philosophy of mathematics. The book clearly explains and assesses Frege's work in these areas, systematically examining his major concepts, and revealing the links between them. The emphasis is on Frege's highly influential work in philosophical logic and the theory of meaning, including the features of his logic, his conceptions of object, concept and function, and his seminal distinction between sense and reference.

Frege will be invaluable for students of the philosophy of language, philosophical logic, and analytic philosophy.

1

Introduction:
Frege’s Life and Work

Biography

Friedrich Ludwig Gottlob Frege was the founder of modern mathematical logic, which he created in his first book, Conceptual Notation, a Formula Language of Pure Thought Modelled upon the Formula Language of Arithmetic (Begriffsschrift, eine der arithmetischen nachgebildete Formalsprache des reinen Denkens (1879), translated in Frege 1972). This describes a system of symbolic logic which goes far beyond the two thousand year old Aristotelian logic on which, hitherto, there had been little decisive advance. Frege was also one of the main formative influences, together with Bertrand Russell, Ludwig Wittgenstein and G. E. Moore, on the analytical school of philosophy which now dominates the English-speaking philosophical world. Apart from his definitive contribution to logic, his writings on the philosophy of mathematics, philosophical logic and the theory of meaning are such that no philosopher working in any of these areas today could hope to make a contribution without a thorough familiarity with Frege’s philosophy. Yet in his lifetíme the significance of Frege’s work was little acknowledged. Even his work on logic met with general incomprehension and his work in philosophy was mostly unread and unappreciated. He was, however, studied by Edmund Husserl, Bertrand Russell, Ludwig Wittgenstein and Rudolf Carnap and via these great figures he has eventually achieved general recognition.

Frege’s life was not a personally fulfilled one (for more detailed accounts of the following see Bynum’s introduction to Frege 1972 and Beaney’s introduction to Frege 1997). His wife died twenty years before his own death in 1925 (he was survived by an adopted son, Alfred) and, ironically, his life’s work in the philosophy of mathematics, to which he regarded all the rest of his efforts as subordinate, that is, his attempted demonstration that arithmetic was a branch of logic (the ‘logicist thesis’ as it is now called), was dealt a fatal blow by Bertrand Russell, one of his greatest admirers, who showed that it entailed the inconsistency that now bears his name (‘Russell’s Paradox’). Nevertheless, Frege perhaps gained some comfort from the respect accorded to him by Russell and by Wittgenstein, who met Frege several times and revered him above all other philosophers. In retrospect, indeed, many would perhaps say that in philosophy generally, as distinct from the narrower branches of logic and the philosophy of mathematics, Frege’s greatest contribution was the advance in the philosophy of logic and language which made Wittgenstein’s work possible.

Little is known of Frege’s personality and life outside philosophy. Apparently his politics and social views, as recorded in his diaries, reveal him to have been, in his later years, extremely right-wing, strongly opposed to democracy and to civil rights for Catholics and Jews. Frege’s greatest commentator, Michael Dummett, expresses great shock and disappointment (1973: xii) that someone he had revered as an absolutely rational man could have been imbued with such prejudices. But a more generous view is the one expressed by another great Frege scholar, Peter Geach. Geach writes that while Frege was indeed imbued with typical German conservative prejudices, ‘to borrow an epigram from Quine, it doesn’t matter what you believe so long as you’re not sincere. Nobody can really imagine Frege as an active politico devoted to some course like Hitler’s’ (1976c: 437).

We have, however, a presentation of the more attractive side of Frege in an account Wittgenstein gives of his encounters with him:

I was shown into Frege’s study. Frege was a small, neat man with a pointed beard who bounced around the room as he talked. He absolutely wiped the floor with me, and I felt very depressed; but at the end he said ‘You must come again,’ so I cheered up. I had several discussions with him after that. Frege would never talk about anything but logic and mathematics, if I started on some other subject, he would say something polite and then plunge back into logic and mathematics. He once showed me an obituary on a colleague, who, it was said, never used a word without knowing what it meant; he expressed astonishment that a man should be praised for this! The last time I saw Frege, as we were waiting at the station for my train, I said to him ‘Don’t you ever find any difficulty in your theory that numbers are objects?’ He replied: ‘Sometimes I seem to see a difficulty but then again I don’t see it.’ (Included in Anscombe and Geach 1961)

Rudolf Carnap, who attended Frege’s lectures in 1914, also presents a vivid image:

Frege looked old beyond his years. He was of small stature, rather shy extremely introverted. He seldom looked at his audience. Ordinarily we saw only his back, while he drew the strange diagrams of his symbolism on the blackboard and explained them. Never did a student ask a question or make a remark, whether during the lecture or afterwards. The possibility of a discussion seemed to be out of the question. (Carnap 1963: 5)

Frege was born in 1848 in Wismar on the German Baltic coast. He attended the Gymnasium in Wismar for five years (1864–9), passed his Abitur in the spring of 1869 and then entered Jena University.

There Frege spent two years studying chemistry, mathematics and philosophy. He then transferred to the University of Göttingen (perhaps influenced by one of his mathematics professors, Ernst Abbe), where he studied philosophy, physics and mathematics.

In 1873 Frege presented his doctoral dissertation, On a Geometrical Representation of Imaginary Figures in a Plane (in Frege 1984:1–55), which extended the work of Gauss on complex numbers, and was granted the degree of Doctor of Philosophy in Göttingen in December of that year.

Frege then applied for the position of Privatdozent (unsalaried lecturer), at the University of Jena. Among the documents he supplied in support of his application was his Habilitationsschrift (postdoctoral dissertation required for appointment to a university teaching post), Methods of Calculation Based upon an Amplification of the Concept of Magnitude’ (in Frege 1984: 56–92). In this piece there first emerges Frege’s interest in the concept of a function which, as we shall see, was to play an absolutely central role throughout his philosophy.

Frege’s work was judged acceptable by the Jena mathematics faculty, and in a prescient report Ernst Abbe speculated that it contained the seeds of a viewpoint which would achieve a durable advance in mathematical analysis. Frege was therefore allowed to proceed to an oral examination, which he passed, though he was judged to be neither quick-witted nor fluent. After a public disputation and trial lecture in May 1874 he was appointed Privatdozent at Jena, where he remained for the rest of his career.

Initially Frege had a heavy teaching load and he only published four short articles (see Frege 1984: 93–100), three of them reviews and one an article on geometry, before 1879, when Conceptual Notation was published. Nevertheless, these were probably the happiest years of his life. He was young, ambitious, with a plan of his life’s work (as we see from the Preface to Conceptual Notation) already formed. He was, moreover, well thought of by the faculty and by the best mathematics students at Jena. The description of his ‘student friendly’ lecturing style quoted from Carnap earlier fits with Abbe’s evaluation of Frege for the university officials in 1879. Abbe reported that Frege’s courses were little suited to please the mediocre student ‘for whom a lecture is just an exercise for the ears’. But ‘Dr Frege, by virtue of the great clarity and precision of his expression and by virtue of the thoughtfulness of his lectures is particularly fit to introduce aspiring listeners to the difficult material of mathematical studies – I myself have repeatedly had the opportunity to hear lectures by him which appeared to me to be absolutely perfect on every fundamental point’ (quoted in Frege 1972: 8).

Absolute perfection on every fundamental point was indeed the aim – and the achievement – of Frege’s Conceptual Notation, which he conceived as the necessary starting point of his logicist programme. It appeared in 1879 and partly as a result Frege was promoted to the salaried post of special (ausserordentlicher) Professor. The promotion was granted on the strength of a recommendation by Frege’s mentor Ernst Abbe, who wrote with appreciation of Conceptual Notation. His remarks were again prescient. He thought that mathematics ‘will be affected, perhaps very considerably, but immediately only very little, by the inclination of the author and the content of the book’. He continued by noting that some mathematicians ‘will find little that is appealing in so subtle investigations into the formal interrelationships of knowledge’, and ‘scarcely anyone will be able, offhand, to take a position on the very original cluster of ideas in this book… it will probably be understood and appreciated by only a few’ (quotations from Frege 1972: 16).

Abbe’s pessimism about the immediate reception of Frege’s work was wholly justified. It received at least six reviews, but none showed an appreciation of the book’s significance, even though some of the reviewers were eminent logicians. The reviews by Schröder in Germany and Venn in England must have been particularly bitter disappointments. Frege’s work was judged inferior to the Boolean logic of his leading contemporaries and his ‘conceptual notation’ dismissed as ‘cumbrous and inconvenient’ (by Venn) and ‘a monstrous waste of space’ which ‘indulges in the Japanese custom of writing vertically’ (by Schröder).

It was an unfortunate outcome but neither without precedent, nor, in retrospect, surprising. The extent of Frege’s achievement was something that could not possibly have been expected by a reviewer asked to give an initial assessment of his work. One is reminded of the similar reception of David Hume’s Treatise of Human Nature which, likewise, as Hume famously put it, ‘fell dead-born from the press’. And, as also in the case of Hume, the poor reception of Frege’s work was partly his own fault – arising from the ‘manner rather than the matter’ of presentation, to use Hume’s words – and something that could have been anticipated. Frege did not explain clearly and thoroughly the purpose of Conceptual Notation and did not justify and illustrate the advantages of his bizarre-looking two-dimensional notation and its superiority to those available at the time. One can thus sympathize with the first reviewers. As a recent commentator has put it: The odds that Frege’s work was the production of a genius rather than a crackpot may have seemed long indeed to his colleagues and contemporaries’ (Boolos 1998: 144).

As a result of the poor reception of Conceptual Notation, Frege postponed his plan, announced in its preface, to proceed immediately to the analysis of the concept of number. Instead he attempted to answer his critics. He wrote two papers comparing his logical symbolism with that of Boole. The first, ‘Boole’s Logic Calculus and the Concept-Scripf (now published in Frege 1979: 9–46) was rejected by three journals. The second, a much shorter version of the first, ‘Boole’s Logical Formula Language and my Conceptual Notation’ (now in Frege 1979:47–52) was also rejected. Finally Frege managed to get published a more general justification of his conceptual notation, ‘On the Scientific Justification of a Conceptual Notation’ (now in Frege 1972: 83–9), and was able to deliver a lecture, also subsequently published, at a meeting of the Jenaischüe Gesellschaft fur Medicin und Naturwissenschaft, in which he compared his symbolism with Boole’s (‘On the Aim of the Conceptual Notation’, now in Frege 1972: 90–100).

The disappointing reviews of Conceptual Notation thus sidetracked Frege into a frustrating episode of self-justification. But they also had the effect of making him more aware of how he must present his work if it was to be appreciated. Instead of proceeding straight from Conceptual Notation to a formal demonstration, in his symbolic notation, of the derivability of arithmetic from logic, as anticipated in the Preface to Conceptual Notation, Frege decided to produce an informal sketch of his derivation in ordinary German, set out against the background of a critique of traditional (including Kantian and empiricist) views of number. The result was his masterpiece, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number (Die Grundlagen der Arithmetik: eine logische mathematische Untersuchung über den Begriff der Zahl) published in 1884 (Frege 1968).

Once again, as in the case of Conceptual Notation, Frege viewed this only as a preliminary stage in his logicist project. He thought that he had made the ‘analytic character of arithmetical propositions’ (i.e. their derivability from logical laws by definition) ‘probable’, but to prove his thesis he needed to produce ‘a chain of deductions with no link missing’ using principles of inference all of which could be recognized as purely logical (Frege 1968:102).

Frege could have hoped that after Foundations his achievement of this project would have been eagerly awaited by scholars. For Foundations is indeed, as Frege intended, a brilliantly written exposition of his views, both negative and positive. In fact, it received only three reviews, all of them hostile (one, by Cantor, criticizing Frege on the basis of the misunderstanding that he took numbers to be sets of physical objects), and remained largely unread and unnoted for nearly twenty years. A partial explanation of this situation is perhaps the poor reception of Conceptual Notation, which could not have added to Frege’s reputation or predisposed mathematicians and philosophers to think his subsequent work worthy of the effort needed to understand it. But whatever the case, the result was that Frege had no choice but to persevere with his logicist project unacknowledged and unsupported by any encouragement from his peers.

The next stage in this project appeared as volume 1 of The Basic Laws of Arithmetic (Grundgesetze der Arithmetik) in 1893 (see Frege 1962, translated in part in Frege 1964). However, in the intervening nine years Frege’s views on the underlying philosophy of language and logic of Foundations developed rapidly, necessitating a complete rewriting of a large preliminary manuscript for Basic Laws. It was in this period that he published, in the early 1890s, his three best known papers ‘Function and Concept’ (Funktion und Begriff), ‘On Sense and Reference’ (‘Über Sinn und Bedeutung’) and ‘On Concept and Object’ (‘Über Begriff und Gegenstand’) (all in Frege 1969). All three of these are now regarded as classic works in the philosophy of language, and the second, in particular, must be read by anyone who wishes to understand twentieth-century analytic philosophy at all, but their importance for Frege was that they set out the changes in his views from the time of Foundations and prepared their readers for Basic Laws.

In this period, also, notice began to be taken notice of Frege’s works when the Italian logician Peano cited them in print and Husserl began to correspond with Frege.

With volume 1 of Basic Laws written, Frege should now have been able to look forward to its publication and the recognition his work had for so long gone without. However, so poorly had his previous work been received that no publisher would print the lengthy manuscript as a whole. Frege eventually got an agreement from Hermann Pohle of Jena, who had published ‘Function and Concept’, to publish it in two volumes, with the publication of the second volume being conditional on the success of the first. In this way volume 1 was eventually printed in 1893.

Frege evidently anticipated that his work was likely, once more, to fail to gain the recognition it deserved. He acknowledged that:

An expression cropping up here or there, as one leafs through these pages, may easily appear strange and create prejudice… Even the first impression must frighten people off: unfamiliar signs, pages of nothing but alien looking formulas … I must relinquish as readers all those mathematicians who, if they bump into logical expressions such as ‘concept, ‘relation’, ‘judgement’, think: metaphysica sunt, non leguntur, and likewise those philosophers who at the sight of a formula cry: mathematica sunt, non leguntur; and the number of such persons is surely not small. Perhaps the number of mathematicians who trouble themselves over the foundations of their science is not great, and even those frequently seem to be in a great hurry until they have got the fundamental principles behind them. And I scarcely dare hope that my reasons for painstaking rigour and its inevitable lengthiness will persuade many of them. (Frege 1964: xi–xii)

For this reason Frege made great efforts to make his work more accessible to his readers. He gave hints in the Preface as to how to read the book to achieve a speedy understanding and in the text he prefaced his proofs with rough outlines to bring out their significance. He also attempted to provoke other scholars to respond to his work by attacking rival theories.

It was all to no avail. Volume 1 of Basic Laws received just two reviews, both unfavourable, one of only three sentences, and was otherwise ignored. As a result the publisher refused to publish the remainder of Frege’s work and volume 2 eventually had to be published a decade later by Frege at his own expense.

Nevertheless, publication of volume 1 ...

Table of contents

• Cover Page
• Title Page
• Copyright
• Contents
• Preface
• Acknowledgements
• 1 Introduction: Frege’s Life and Work
• 2 Logic
• 3 Number
• 4 Philosophical Logic
• 5 Theory of Meaning
• Appendix
• Bibliography
• Index
• Back Page