**Gottlob Frege** was a German mathematician who was one of the founders of modern symbolic logic putting forward the view that mathematics is reducible to logic.

- Alexander Frege was the head of a girls' high school in Wismar and it was in that town that Gottlob was born.
- It had been administered by the Mecklenburg-Schwerin state since 1803 but at the time when Gottlob was born there, the town was still claimed by Sweden, the country which had controlled it from the Peace of Westphalia in 1648 until 1803.
- Gottlob grew up in Wismar, attending the local Gymnasium where he was taught by Leo Sachse.
- Frege was proud to live in the state of Mecklenburg, he loved the ducal house of Mecklenburg, and certainly believed in this form of government rather than a democratically elected one.
- At Jena Frege was taught by Ernst Abbe and K Fischer.
- After his two years of study at the University of Jena, Frege continued his education in 1871 entering the University of Göttingen where he studied courses in mathematics, physics, chemistry and philosophy.
- However Rudolf Eucken was a colleague of Frege's for more than 40 years in the faculty of philosophy with whom he had close scientific contacts.
- Frege was one of the founders of modern symbolic logic putting forward the view that mathematics is reducible to logic.
- In 1879 Frege published his first major work Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens Ⓣ(Concept notation, an arithmetic formula modeled language of pure thought) .
- In this work Frege presented for the first time what we would recognise today as a logical system with negation, implication, universal quantification, essentially the idea of truth tables etc., but what would not be recognisable today is the notation which Frege used.
- The publication of the Begriffsschrift was followed in the same year by Frege's promotion, again supported by Abbe, to Extraordinary Professor at Jena but on the whole his remarkable work led to surprisingly little recognition for him.
- It is reasonable to ask what prompted Frege to produce the revolutionary Begriffsschrift Ⓣ(Concept notation).
- This aim makes Frege the first to fully develop the main thesis of logicism, that mathematics is reducible to logic.
- Perhaps it will come as a surprise to readers of this article to learn that all attempts to define "number" before Frege contained logical errors.
- The number "two" is, however, the class of all instances of the "plurality two" and so is a "plurality of pluralities" and the logical error which had been made in not recognising this meant that before Frege's Grundlagen nobody had managed to give a logically correct definition of "number".
- Frege then went on to give his own definitions of the basic concepts of arithmetic based purely on logic, and from these he deduced, again using pure logic, the basic laws of arithmetic.
- was a devastatingly hostile one by Georg Cantor, the mathematician whose ideas were the closest to Frege's, who had not bothered to understand Frege's book before subjecting it to totally unmerited scorn.
- The Grundlagen was a non-technical work, written without symbolism and with only sketches of proofs, which Frege saw as a first step towards the realisation of his goal of defining a precise logical framework in which to set up the basic concepts of arithmetic and to deduce the rules of arithmetic.
- In 1893 Die Grundgesetze der Arithmetik, Volume1 Ⓣ(The basic laws of arithmetic, Volume1) appeared in which Frege set up a formal logical system with more rules of inference than that of his earlier work the Begriffsschrift.
- Frege axiomatized arithmetic with an intuitive collection of axioms, and proofs of number theory results which he had only sketched earlier he now gave formally.
- The main thrust of this volume was to develop the rules of number theory and in the later volumes Frege intended to extend the work to the real numbers.
- Frege, who had not allowed the previous lack of reaction to divert him from the tasks that he had set himself, decided to delay publication of the second of his three proposed volumes.
- During this period Frege was appointed ordinary honorary professor at Jena, a post funded by the Carl Zeiss Foundation with which Abbe was closely associated.
- This second volume gives Frege's development of the real numbers which he constructed straight from the integers without taking the route of first defining the rational numbers.
- After the work was written, but before it was published, Frege discovered that this volume, and Volume 1, were based on inconsistent axioms.
- While Volume 2 of The Basic Laws of Arithmetic was at the printers Frege received a letter (on 16 June 1902) from Bertrand Russell.
- Russell pointed out, with great modesty, that the Russell paradox gave a contradiction in Frege's system of axioms.
- After many letters between the two, Frege modified one of his axioms and explains in an appendix to the book that this was done to restore the consistency of the system.
- However with this modified axiom, many of the theorems of Volume 1 do not go through and Frege must have known this.
- He probably never realised that even with the modified axiom the system is inconsistent since this was only shown by Lesniewski after Frege's death.
- One often sees it stated that Frege's work was worthless because of the inconsistency pointed out by Russell.
- This is far from the truth and one must view Frege as the person who made one of the most important contributions to the foundations of mathematics that has ever been made.
- Frege's influence in the short term came through the work of Peano, Wittgenstein, Husserl, Carnap and Russell.
- In the longer term, however, Frege has become a major influence on the development of philosophical logic and the man who seems to have been largely ignored by his contemporaries has been avidly read by many in the second half of the twentieth century, particularly after his works were translated into English.
- Another statement that one often reads is that Frege was so depressed after Russell's letters that he gave up research.
- Frege, as we have mentioned, was a firm believer of the old style monarchy which operated in the German States before the unification.
- Frege disliked the move to democracy, and detested it even more as the socialists gained power.
- For example Thomae, who also taught at Jena, came in for severe personal attacks from Frege.
- Frege retired from his professorship in Jena in 1917.
- Russell had invited him to address a mathematical congress in Cambridge in 1912 but Frege's reply, declining the invitation, shows his depressed state of mind.
- However, Frege began to publish important articles again in 1918 with contributions to the nature of thoughts.
- In 1923 Frege came to the conclusion that the aim he had set himself throughout most of his career, namely to found arithmetic on logic, was wrong.
- We have quoted many tributes to Frege's genius, but let us end with one more.

Born 8 November 1848, Wismar, Mecklenburg-Schwerin (now Germany). Died 26 July 1925, Bad Kleinen, Germany.

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Analysis, Origin Germany, Set Theory

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive