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Friedrich Ludwig Gottlob Frege (8 November 1848 – 26 July 1925) was a German mathematician who became a logician and philosopher. He was one of the founders of modern logic, and made major contributions to the foundations of mathematics. As a philosopher, he is generally considered to be the father of analytic philosophy, for his writings on the philosophy of language and mathematics. Although he was mainly ignored by the intellectual world when he published his writings, it was Giuseppe Peano, and later Bertrand Russell who helped introduce his work to the later generations of logicians and philosophers. Frege was born in 1848 in Wismar, in the state of Mecklenburg-Schwerin (the modern German federal state Mecklenburg-Vorpommern). His father, Karl Alexander Frege, was the founder and headmaster of a girls' high school until his death in 1866. Afterwards, the school was led by Frege's mother, Auguste Wilhelmine Sophie Frege (née Bialloblotzky, apparently of Polish extraction).
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, the first section of which dealt with the structure and logic of language. Frege studied at a gymnasium in Wismar, and graduated in 1869. His teacher Leo Sachse (also a poet) played the most important role in determining Frege’s future scientific career, encouraging him to continue his studies at the University of Jena. Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the North German Federation. 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 Abbe (physicist, mathematician, and inventor). Abbe gave lectures on the theory of gravity, galvanism and electrodynamics, the 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 Zeiss,
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 Karl Snell (subjects: use of infinitesimal analysis in geometry, analytical geometry of planes, analytical mechanics, optics, physical foundations of mechanics); Hermann Schäffer (analytical geometry, applied physics, algebraic analysis, on the telegraph and other electronic machines); and the famous philosopher, Kuno Fischer (history of Kantian and critical philosophy). 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 (analytical geometry), Ernst Christian Julius Schering (function theory), Wilhelm Weber (physical studies, applied physics), Eduard Riecke (theory of electricity), and Rudolf Hermann Lotze (philosophy of religion).
(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 Ernst Schering, with a dissertation under
the title of "Über eine geometrische Darstellung der
imaginären Gebilde in der Ebene" ("On a Geometrical Representation
of Imaginary Forms in a Plane"), in which he aimed to solve such
fundamental problems in geometry as the mathematical interpretation of projective geometry's infinitely distant (imaginary) points. Though his education and early work were mathematical, especially geometrical, Frege's thought soon turned to logic. His 1879 Begriffsschrift (Concept Script) 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 wanted to show that mathematics grew out of logic, but in so doing devised techniques that took him far beyond the
Aristotelian syllogistic and Stoic propositional logic that had come
down to him in the logical tradition. In effect, he invented axiomatic predicate logic, in large part thanks to his invention of quantified variables, which eventually became ubiquitous in mathematics and logic, and 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 pair of sentences
like "every boy loves some girl" and "some girl is loved by every boy"
was able to 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". It
is frequently noted that Aristotle's logic is unable to represent even
the most elementary inferences in Euclid's geometry, but Frege's
"conceptual notation" can represent inferences involving indefinitely
complex mathematical statements. The analysis of logical concepts and
the machinery of formalization that is essential to Bertrand Russell's theory of descriptions and Principia Mathematica (with Alfred North Whitehead), and to 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 mathematical induction, were derived within what Frege understood to be pure logic. This idea was formulated in non-symbolic terms in his Foundations of Arithmetic of 1884. Later, in the Basic Laws of Arithmetic (Grundgesetze der Arithmetik (1893,
1903)), published at its author's expense, 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 Basic Law V: the "value-range" of the
function f(x) is the same as the "value-range" of the function g(x) if and only if ∀x[f(x) = g(x)]. The crucial case of the law may be formulated in modern notation as follows. Let {x|Fx} denote the extension of the predicate Fx, i.e., the set of all Fs, and similarly for Gx. Then Basic Law V says that the predicates Fx and Gx have the same extension iff ∀x[Fx ↔ Gx].
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. (The case is special because what is here being
called the extension of a predicate, or a set, is only one type of
"value-range" of a function.) 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 (indeed, this would make for a contradiction in
Frege's system if he had axiomatized the idea, fundamental to his
discussion, that the True and the False are distinct objects), but recent work has shown that much of the program of the Grundgesetze might be salvaged in other ways: Frege's
work in logic was little recognized in his day, in considerable part
because his peculiar diagrammatic notation had no antecedents. It has
since had no imitators. Moreover, until Principia Mathematica appeared in 1910–13, the dominant approach to mathematical logic was still that of George Boole and his descendants, especially Ernst Schroeder. Frege's logical ideas nevertheless spread through the writings of his student Rudolf Carnap and other admirers, particularly Bertrand Russell and Ludwig Wittgenstein. Frege is one of the founders of analytic philosophy, mainly because of his contributions to the philosophy of language, including the Function-argument analysis of the proposition; Distinction between concept and object (Begriff und Gegenstand); Principle of compositionality; Context principle; Distinction between the sense and reference (Sinn und Bedeutung) of names and other expressions, sometimes said to involve a mediated reference theory. As a philosopher of mathematics, Frege attacked the psychologistic appeal
to mental explanations of the content of judgment of the meaning of
sentences. His original purpose was very far from answering general
questions about meaning; instead, he devised his logic to explore the
foundations of arithmetic, undertaking to answer questions such as
"What is a number?" or "What objects do number-words ("one", "two",
etc.) refer to?" But in pursuing these matters, he eventually found
himself analysing and explaining what meaning is, and thus came to
several conclusions that proved highly consequential for the subsequent
course of analytic philosophy and the philosophy of language. It
should be kept in mind that Frege was employed as a mathematician, not
a philosopher, and he published his philosophical papers in scholarly
journals that often were hard to access outside of the German-speaking
world. He never published a philosophical monograph other than The Foundations of Arithmetic,
much of which was mathematical in content, and the first collections of
his writings appeared only after World War II. A volume of English
translations of Frege's philosophical essays first appeared in 1952,
edited by students of Wittgenstein, Peter Geach and Max Black,
with the bibliographic assistance of Wittgenstein. Hence, despite the generous praise of Russell and
Wittgenstein, Frege was little known as a philosopher during his
lifetime. His ideas spread chiefly through those he influenced, such as
Russell, Wittgenstein, and Carnap, and through Polish work on logic and
semantics. The distinction between Sinn ("sense") and Bedeutung (usually
translated "reference", but also as "meaning" or "denotation") was an
innovation of Frege in his 1892 paper "Über Sinn und Bedeutung"
("On Sense and Reference"). According to Frege, sense and reference are
two different aspects of the significance of an expression. Frege
applied Bedeutung in
the first instance to proper names, where it means the bearer of the
name, the object in question, but then also to other expressions,
including complete sentences, which bedeuten the two "truth values", the true and the false; by contrast, the sense or Sinn associated with a complete sentence is the thought it expresses. The sense of an
expression is said to be the "mode of presentation" of the item
referred to. The
distinction can be illustrated thus: In their ordinary uses, the name
"Charles Philip Arthur George Mountbatten-Windsor", which for logical
purposes is an unanalyzable whole, and the functional expression "the
Prince of Wales", which contains the significant parts "the prince of
ξ" and "Wales", have the same reference, namely, the person best known as Prince Charles. But the sense of
the word "Wales" is a part of the sense of the latter expression, but
no part of the sense of the "full name" of Prince Charles. These distinctions were disputed by Bertrand Russell, especially in his paper "On Denoting"; the controversy has continued into the present, fueled especially by the famous lectures on "Naming and Necessity" of Saul Kripke. Imagine the road signs outside a city. They all point to (bedeuten) the same object (the city), although the "mode of presentation" or sense (Sinn)
of each sign (its direction or distance) is different. Similarly "the
Prince of Wales" and "Charles Philip Arthur George Mountbatten-Windsor" both denote (bedeuten) the same object, though each uses a different "mode of presentation" (sense or Sinn). Like many German intellectuals, Frege was a sympathizer of the National Socialist party in its early stages, a known anti-Semite, and later in life named Adolf Hitler as one of his heroes.
For those who admire Frege's contributions to logic and philosophy,
this may be a shock to learn, as this is often overlooked. As Reuben
Hersh states in What Is Mathematics, Really?, p. 241: "Frege actually died a Nazi. Sluga reports: 'Frege confided in his diary in 1924 that he had once thought of himself as a liberal and was an admirer of Bismarck, but his heroes now were General Ludendorff and
Adolf Hitler. This was after the two had tried to topple the elected
democratic government in a coup in November 1923. In his diary Frege
also used all his analytic skills to devise plans for expelling the
Jews from Germany and for suppressing the Social Democrats.' Michael Dummett tells of his shock to discover, while reading Frege's diary, that his hero was an outspoken anti-Semite (1973)." Prior
to his sympathies with Hitler and the Nazis, Frege held very
conservative political views. He disliked the small steps towards democracy made in the German Empire (created
1871), not the least because it increased the power of the Socialists.
His anti-Semitism fueled his desire to see all Jews expelled from
Germany, or at least deprived certain political rights (notwithstanding the fact that among his students was Gershom Scholem,
who much valued his teacher). In addition to his anti-Semitic
sentiments, his diaries also show a deep hatred of Catholics and of the
French. Frege
was described by his students as a highly introverted person, seldom
entering the dialogue, mostly facing the blackboard while lecturing
though being witty and sometimes bitterly sarcastic. |