Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano, Bertrand Russell, and, to some extent, Ludwig Wittgenstein introduced his work to later generations of philosophers. Frege is widely considered to be one of the greatest logicians since Aristotle, and one of the most profound philosophers of mathematics ever.
His contributions include the development of modern logic in the Begriffsschrift and work in the foundations of mathematics. His book the Foundations of Arithmetic is the seminal text of the logicist project, and is cited by Michael Dummett as where to pinpoint the linguistic turn. His philosophical papers "On Sense and Reference" and "The Thought" are also widely cited. The former argues for two different types of meaning and descriptivism. In Foundations and "The Thought", Frege argues for Platonism against psychologism or formalism, concerning numbers and propositions respectively.
Life
Childhood (1848–1869)
Frege was born in 1848 in Wismar, Mecklenburg-Schwerin. His father, Carl Alexander Frege, was the co-founder and headmaster of a girls' high school until his death. After Carl's death, the school was led by Frege's mother Auguste Wilhelmine Sophie Frege ; her mother was Auguste Amalia Maria Ballhorn, a descendant of Philipp Melanchthon and her father was Johann Heinrich Siegfried Bialloblotzky. Frege was a Lutheran.In childhood, Frege encountered philosophies that would guide his future scientific career. For example, his father wrote a textbook on the German language for children aged 9–13, entitled Hülfsbuch zum Unterrichte in der deutschen Sprache für Kinder von 9 bis 13 Jahren , the first section of which dealt with the structure and logic of language.
Frege studied at and graduated in 1869. Teacher of mathematics and natural science Gustav Adolf Leo Sachse, who was also a poet, played an important role in determining Frege's future scientific career, encouraging him to continue his studies at his own alma mater the University of Jena.
Studies at University (1869–1874)
Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the North German Confederation. In the four semesters of his studies, he attended approximately twenty courses of lectures, most of them on mathematics and physics. His most important teacher was Ernst Karl Abbe. Abbe gave lectures on theory of gravity, galvanism and electrodynamics, complex analysis theory of functions of a complex variable, applications of physics, selected divisions of mechanics, and mechanics of solids. Abbe was more than a teacher to Frege: he was a trusted friend, and, as director of the optical manufacturer Carl Zeiss AG, he was in a position to advance Frege's career. After Frege's graduation, they came into closer correspondence.His other notable university teachers were Christian Philipp Karl Snell ; Hermann Karl Julius Traugott Schaeffer ; and the philosopher Kuno Fischer.
Starting in 1871, Frege continued his studies in Göttingen, the leading university in mathematics in German-speaking territories, where he attended the lectures of Alfred Clebsch, Ernst Christian Julius Schering, Wilhelm Eduard Weber, Eduard Riecke, and Hermann Lotze. Many of the philosophical doctrines of the mature Frege have parallels in Lotze; it has been the subject of scholarly debate whether or not there was a direct influence on Frege's views arising from his attending Lotze's lectures.
In 1873, Frege attained his doctorate under Schering.
Frege married Margarete Katharina Sophia Anna Lieseberg on 14 March 1887. The couple had at least two children, who died when young. Years later, they adopted a son, Alfred. Little else is known about Frege's family life, however.
Work as a logician
Though his education and early mathematical work focused primarily on geometry, Frege's work soon turned to logic. His marked a turning point in the history of logic. The Begriffsschrift broke new ground, including a rigorous treatment of the ideas of functions and variables. Frege's goal was to show that mathematics grows out of logic, and in so doing, he devised techniques that separated him from the Aristotelian syllogistic but took him rather close to Stoic propositional logic.In effect, Frege invented axiomatic predicate logic, in large part thanks to his invention of quantified variables, which eventually became ubiquitous in mathematics and logic, and which solved the problem of multiple generality. Previous logic had dealt with the logical constants and, or, if... then..., not, and some and all, but iterations of these operations, especially "some" and "all", were little understood: even the distinction between a sentence like "every boy loves some girl" and "some girl is loved by every boy" could be represented only very artificially, whereas Frege's formalism had no difficulty expressing the different readings of "every boy loves some girl who loves some boy who loves some girl" and similar sentences, in complete parallel with his treatment of, say, "every boy is foolish".
A frequently noted example is that Aristotle's logic is unable to represent mathematical statements like Euclid's theorem, a fundamental statement of number theory that there are an infinite number of prime numbers. Frege's "conceptual notation", however, can represent such inferences. The analysis of logical concepts and the machinery of formalization that is essential to Principia Mathematica, to Russell's theory of descriptions, to Kurt Gödel's incompleteness theorems, and to Alfred Tarski's theory of truth, is ultimately due to Frege.
One of Frege's stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to "intuition". If there was an intuitive element, it was to be isolated and represented separately as an axiom: from there on, the proof was to be purely logical and without gaps. Having exhibited this possibility, Frege's larger purpose was to defend the view that arithmetic is a branch of logic, a view known as logicism: unlike geometry, arithmetic was to be shown to have no basis in "intuition", and no need for non-logical axioms. Already in the 1879 Begriffsschrift important preliminary theorems, for example, a generalized form of law of trichotomy, were derived within what Frege understood to be pure logic.
This idea was formulated in non-symbolic terms in his The Foundations of Arithmetic. Later, in his Basic Laws of Arithmetic, Frege attempted to derive, by use of his symbolism, all of the laws of arithmetic from axioms he asserted as logical. Most of these axioms were carried over from his Begriffsschrift, though not without some significant changes. The one truly new principle was one he called the : the "value-range" of the function f is the same as the "value-range" of the function g if and only if ∀x.
The crucial case of the law may be formulated in modern notation as follows. Let denote the extension of the predicate Fx, that is, the set of all Fs, and similarly for Gx. Then Basic Law V says that the predicates Fx and Gx have the same extension if and only if ∀x. The set of Fs is the same as the set of Gs just in case every F is a G and every G is an F.
In a famous episode, Bertrand Russell wrote to Frege, just as Vol. 2 of the Grundgesetze was about to go to press in 1903, showing that Russell's paradox could be derived from Frege's Basic Law V. It is easy to define the relation of membership of a set or extension in Frege's system; Russell then drew attention to "the set of things x that are such that x is not a member of x". The system of the Grundgesetze entails that the set thus characterised both is and is not a member of itself, and is thus inconsistent. Frege wrote a hasty, last-minute Appendix to Vol. 2, deriving the contradiction and proposing to eliminate it by modifying Basic Law V. Frege opened the Appendix with the exceptionally honest comment: "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion."
Frege's proposed remedy was subsequently shown to imply that there is but one object in the universe of discourse, and hence is worthless, but recent work has shown that much of the program of the Grundgesetze might be salvaged in other ways:
- Basic Law V can be weakened in other ways. The best-known way is due to philosopher and mathematical logician George Boolos, who was an expert on the work of Frege. A "concept" F is "small" if the objects falling under F cannot be put into one-to-one correspondence with the universe of discourse, that is, unless: ∃R. Now weaken V to V*: a "concept" F and a "concept" G have the same "extension" if and only if neither F nor G is small or ∀x. V* is consistent if second-order arithmetic is, and suffices to prove the axioms of second-order arithmetic.
- Basic Law V can simply be replaced with Hume's principle, which says that the number of Fs is the same as the number of Gs if and only if the Fs can be put into a one-to-one correspondence with the Gs. This principle, too, is consistent if second-order arithmetic is, and suffices to prove the axioms of second-order arithmetic. This result is termed Frege's theorem because it was noticed that in developing arithmetic, Frege's use of Basic Law V is restricted to a proof of Hume's principle; it is from this, in turn, that arithmetical principles are derived. On Hume's principle and Frege's theorem, see "Frege's Logic, Theorem, and Foundations for Arithmetic".
- Frege's logic, now known as second-order logic, can be weakened to so-called predicative second-order logic. Predicative second-order logic plus Basic Law V is provably consistent by finitistic or constructive methods, but it can interpret only very weak fragments of arithmetic.