Difference between revisions of "Gottlob Frege" - New World Encyclopedia

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   school_tradition = [[Analytic philosophy]] |
 
   school_tradition = [[Analytic philosophy]] |
 
   main_interests  = [[Philosophy of mathematics]], [[mathematical logic]], [[Philosophy of language]]|
 
   main_interests  = [[Philosophy of mathematics]], [[mathematical logic]], [[Philosophy of language]]|
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  influences      = [[Edmund Husserl]] |
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   influenced      = [[Giuseppe Peano]], [[Bertrand Russell]], [[Rudolf Carnap]], [[Ludwig Wittgenstein]], [[Michael Dummett]], [[Edmund Husserl]], and most of the [[Analytic philosophy|analytic tradition]] |
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   influenced      = [[Giuseppe Peano]], [[Bertrand Russell]], [[Rudolf Carnap]], [[Ludwig Wittgenstein]], [[Michael Dummett]], [[George Boolos]], [[Crispin Wright]], [[Edward N. Zalta]], [[Edmund Husserl]] |
 
 
   notable_ideas    = [[Predicate calculus]], [[Logicism]], [[Sense and reference]], [[Mediated reference theory]] |
 
   notable_ideas    = [[Predicate calculus]], [[Logicism]], [[Sense and reference]], [[Mediated reference theory]] |
 
}}
 
}}
  
−
'''Friedrich Ludwig Gottlob Frege''' ([[8 November]] [[1848]], [[Wismar]] – [[26 July]] [[1925]], [[Bad Kleinen]]) was a [[Germany|German]] [[mathematics|mathematician]] who became a [[logic]]ian and [[philosophy|philosopher]]. He helped found both modern [[mathematical logic]] and [[analytic philosophy]].
+
'''Friedrich Ludwig Gottlob Frege''' ([[8 November]] [[1848]], [[Wismar]] – [[26 July]] [[1925]], [[Bad Kleinen]], {{IPA2|ˈɡɔtlop ˈfʁeːɡə}}) was a [[Germany|German]] [[mathematics|mathematician]] who became a [[logic]]ian and [[philosophy|philosopher]]. He helped found both modern [[mathematical logic]] and [[analytic philosophy]].
  
 
==Life==
 
==Life==
−
Frege's father was a schoolteacher whose specialty was mathematics. Frege began his studies at the [[University of Jena]] in [[1869]], moving to [[Göttingen]] after two years, where he received his [[Ph.D.]] in mathematics, in [[1873]]. According to Sluga (1980), the nature of Frege's university education in logic and philosophy is still unclear. In 1875, he returned to [[Jena]] as a lecturer. In [[1879]], he was made associate professor, and in [[1896]], professor. Frege had but one student of note, [[Rudolf Carnap]]. His children all having died before reaching maturity, he adopted a son in 1905.
+
Frege's father was a schoolteacher whose specialty was mathematics. Frege began his studies at the [[University of Jena]] in [[1869]], moving to [[Göttingen]] after two years, where he received his [[Ph.D.]] in mathematics, in [[1873]]. According to Sluga (1980), the nature of Frege's university education in [[logic]] and [[philosophy]] is still unclear. In 1875, he returned to [[Jena]] as a lecturer. In [[1879]], he was made associate professor, and in [[1896]], professor. His children all having died before reaching maturity, he adopted a son in 1905. Frege retired in 1918 and wrote until his death.
  
−
Frege's work was not widely appreciated during his life, but the admiration of [[Bertrand Russell]] and [[Ludwig Wittgenstein]], as well as [[Rudolf Carnap|Carnap]], nonetheless guaranteed him significant influence in certain circles. His work became widely known in the English-speaking world only after [[World War II]], in part because of the emigration to the [[United States]] of philosophers and logicians—[[Rudolf Carnap|Carnap]], [[Alfred Tarski]], and [[Kurt Gödel]], for example—who knew and respected Frege's work and the appearance of translations into English of his major writings. Frege's work has since had enormous influence on [[analytic philosophy]].
+
Frege's work was not widely appreciated during his lifetime, although he did debate in print, and correspond with, [[Ernst Schroder]], [[Peano]], [[Husserl]], and [[Cantor]]. [[Bertrand Russell]] discovered Frege's writings around 1900 and became an admirer, as did [[Ludwig Wittgenstein]] somewhat later. These admirers assured Frege's influence and reputation in certain restricted circles. Frege had but one student of note, albeit a consequential one: [[Rudolf Carnap]]. Frege's work became widely known in the English-speaking world only after [[World War II]]; English translations of Frege's writings, which first appeared in 1950, came to have an enormous influence on [[analytic philosophy]]. Frege also became better known thanks to the emigration to the [[United States]] of central European philosophers and logicians who knew and respected Frege's work, including [[Rudolf Carnap|Carnap]], [[Alfred Tarski]], and [[Kurt Gödel]].
  
 
==Logician==
 
==Logician==
 
{{Main|Begriffsschrift}}
 
{{Main|Begriffsschrift}}
−
Frege is widely regarded as a logician on par with [[Aristotle]], [[Kurt Gödel]], and [[Alfred Tarski]]. His revolutionary ''[[Begriffsschrift]]'', or ''Concept Script'' (1879) marked the beginning of a new epoch in the history of logic. The ''Begriffsschrift'' broke much new ground, including a clean treatment of [[function (mathematics)|function]]s and [[variable]]s. Frege attempted to show mathematics as an extension of Aristotelian logic. He invented and [[axiomatization|axiomatized]] [[predicate logic]], thanks to his discovery of [[quantification|quantified variables]], which subsequently became ubiquitous in [[mathematics]] and solved the medieval [[problem of multiple generality]]. Hence the logical machinery essential to [[Bertrand Russell]]'s [[theory of descriptions]] and ''[[Principia Mathematica]]'' (with [[Alfred North Whitehead]]), to [[Kurt Gödel|Gödel's]] famous proof of the [[Gödel's incompleteness theorem|incompleteness theorem]], was ultimately due to Frege.
+
Frege is widely regarded as a logician on a par with [[Aristotle]], [[Kurt Gödel]], and [[Alfred Tarski]]. His 1879 ''[[Begriffsschrift]]'' (''Concept Script'') marked a turning point in the history of logic. The ''Begriffsschrift'' broke much new ground, including a clean treatment of [[function (mathematics)|function]]s and [[variable]]s. Frege wanted to show that mathematics grew out of [[Aristotelian logic]], but in so doing devised techniques that took him far beyond that logic. In effect, he invented [[axiomatization|axiomatic]] [[predicate logic]], in large part thanks to his invention of [[quantification|quantified variables]], which eventually became ubiquitous in [[mathematics]] and logic, and solved the medieval [[problem of multiple generality]]. Hence the logical machinery essential to [[Bertrand Russell]]'s [[theory of descriptions]] and ''[[Principia Mathematica]]'' (with [[Alfred North Whitehead]]), and to [[Kurt Gödel|Gödel's]] [[Gödel's incompleteness theorem|incompleteness theorems]], is ultimately due to Frege.
  
−
Frege was a major advocate of the view that [[arithmetic]] is reducible to logic, a view known as [[logicism]]. In his ''Grundgesetze der Arithmetik'' (1893, 1903), published at its author's expense, he attempted to explicitly derive the laws of arithmetic from what he took to be logical axioms. Most of these were taken over from his ''[[Begriffsschrift]]'', though that had undergone significant changes, as well. The one really new principle was Frege's Basic Law V, which said that the 'value-range' of a function f(x) is the same as the 'value-range' of the function g(x) if, and only if ∀x(fx = gx). As the second volume was about to go to press, Frege learned from [[Bertrand Russell]] that [[Russell's paradox]] could be derived from Basic Law V. Hence, the formal system of ''Grundgesetze'' was inconsistent.  Frege gave a derivation of the contradiction in a last-minute appendix to volume two and attempted to remedy his system by modifying his Basic Law V, which was responsible for the contradiction.  Frege's remedy to Basic Law V has been shown to be inconsistent (or, more precisely, to imply that there is only one object).
+
Frege was a major advocate of the view that [[arithmetic]] is reducible to logic, a view known as [[logicism]]. In his ''Grundgesetze der Arithmetik'' (1893, 1903), published at its author's expense, he attempted to derive 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'')]. In modern notation and terminology, let {''x''|''Fx''} denote the [[extension]] of the [[predicate]] ''Fx'', and similarly for ''Gx''. Then Basic Law V says that the predicates ''Fx'' and ''Gx'' have the same extension [[iff]] ∀x[''Fx'' ↔ ''Gx''].
  
−
Recent work has shown that much of Frege's work can nonetheless be salvaged, in several different ways.  
+
As Vol. 2 of the ''Grundgesetze'' was about to go to press in 1903, [[Bertrand Russell]] wrote to Frege, showing how to derive [[Russell's paradox]] from Basic Law V. (This letter and Frege's reply thereto are translated in [[Jean van Heijenoort]] 1967.) Hence the system of the ''Grundgesetze'' was 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's proposed remedy was subsequently shown to imply that there is but one object in the [[universe of discourse]], and hence is worthless.
−
# Basic Law V can be weakened in various ways that restore the consistency of the system. The best-known of these is due to [[George Boolos]]. Say that a 'concept' F is "small" if the objects falling under F cannot be put in 1-1 correspondence with the universe, that is, if: ¬∃R[R is one-one & ∀x∃y(xRy & Fy)]. Now replaces Law V with the weaker claim, "New V", that a 'concept' F and a 'concept' G have the same 'extension' if, and only if neither F nor G is small or ∀x(Fx ≡ Gx). New V can be shown to be consistent if [[second-order arithmetic]] is and sufficient to allow proofs of the axioms of [[second-order arithmetic]].
 
−
# Replace Basic Law V 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 in one-one correspondence with the Gs. Again, this principle can be shown to be consistent if [[second-order arithmetic]] is and sufficient to allow proofs of the axioms of [[second-order arithmetic]]]. This result is known as [[Frege's Theorem]].
 
−
# The logic Frege uses, [[second-order logic]], can be weakened to so-called [[predicative logic|predicative second-order logic]]. Such theories can be shown to be consistent by [[finitism|finitistic]] or [[Mathematical constructivism|constructive]] reasoning. Only very weak fragments of arithmetic can be interpreted in such systems, however.
 
  
−
Frege's work in logic was little recognized in his own day, in considerable part because his peculiar diagrammatic notation had no antecedents; it has since had no imitators. His ideas spread chiefly through those he influenced, particularly [[Bertrand Russell]] and [[Rudolf Carnap]].
+
Recent work has shown that much of the ''Grundgesetze'' can be salvaged in any of several ways:
 +
* Basic Law V can be weakened in ways that restore the consistency of Frege's system. The best-known way is due to [[George Boolos]]. A "concept" ''F'' is "small" if the objects falling under ''F'' cannot be put in 1-to-1 correspondence with the [[universe of discourse]], that is, if: ¬∃''R''[''R'' is 1-to-1 & ∀''x''∃''y''(''xRy'' & ''Fy'')]. 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''(''Fx'' ↔ ''Gx''). V* is consistent if [[second-order arithmetic]] is, and suffices to prove the axioms of [[second-order arithmetic]].
 +
* Replace Basic Law V with [[Hume's Principle]], which says that the number of ''F''s is the same as the number of ''G''s if and only if the ''F''s can be put into a one-to-one correspondence with the ''G''s. This principle too is consistent if [[second-order arithmetic]], and suffices to prove the axioms of [[second-order arithmetic]]. This result is anachronistically termed [[Frege's Theorem]]. On Hume's Principle and Frege's Theorem, see [http://plato.stanford.edu/entries/frege-logic/].
 +
* Frege's logic, now known as [[second-order logic]], can be weakened to so-called [[predicative]] second-order logic. However, this logic, although provably consistent by [[finitism|finitistic]] or [[Mathematical constructivism|constructive]] methods, can interpret only very weak fragments of arithmetic.
 +
 
 +
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, 1910-13, the dominant approach to [[mathematical logic]] was that of [[George Boole]] and his descendants, especially [[Ernst Schroder]]. His logical ideas nevertheless spread through the writings of his student [[Carnap]] and of other admirers, particularly [[Bertrand Russell]].
  
 
==Philosopher==
 
==Philosopher==
 
{{Main|On Sense and Reference}}
 
{{Main|On Sense and Reference}}
−
Frege is regarded as one of the founding fathers of analytic philosophy, mainly because of his conceptual contributions to the [[philosophy of language]], such as his:
+
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]];
+
*[[Function]]-argument analysis of the [[proposition]];
 
*Distinction between the [[sense and reference]] (''Sinn und Bedeutung'') of a proper name (''Eigenname'');
 
*Distinction between the [[sense and reference]] (''Sinn und Bedeutung'') of a proper name (''Eigenname'');
−
*Advocacy of a [[mediated reference theory]];
+
*[[Mediated reference theory]];
 
*Distinction between [[concept and object]] (''Begriff und Gegenstand'');
 
*Distinction between [[concept and object]] (''Begriff und Gegenstand'');
−
*Advancement of the [[context principle]].  
+
*[[Context principle]];
−
*Formulation of the principle of [[compositionality]].
+
*Principle of [[compositionality]].
 +
 
 +
As a philosopher of mathematics, Frege loathed appeals to [[psychologism|psychologistic]] or "mental" explanations for meanings (such as idea theories of meaning). His original purpose was very far from answering 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 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 and the first collections of his writings appeared only after WWII. Hence despite [[Bertrand Russell]]'s generous praise, 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.
  
−
As a philosopher of mathematics, Frege loathed appeals to [[psychologism|psychologistic]] or "mental" explanations for meanings (such as idea theories of meaning). His original purpose was very far from answering questions about meaning; he wanted to use modern logic to further develop the foundations of arithmetic. He first undertook to answer the question "What is a number?" or "What objects do number-words ("one", "two", etc.) refer to?" But in pursuing these matters, he eventually faced the task of analysing and explaining what meaning is, and came to several major conclusions.
+
==Thought==
 +
A great deal of Frege's writings were not translated into English until the 1950s.  "Thought, is one of those such writings. It was published in 1918 as the first part of a series of three papers entitled 'Logical Investigations'. Along with 'On Sense and Reference', it is one of Frege's most influential and widely discussed papers" (Beaney, 1997).
  
−
Frege, despite Bertrand Russell's generous praise, was little known as a philosopher during his lifetime. Here too, his ideas spread chiefly through those he influenced, including [[Ludwig Wittgenstein]] and [[Rudolf Carnap]]. He also studied, corresponded, and debated [[Edmund Husserl]]'s works in print.  Among some of the debates they had, Frege persuaded Husserl to abandon psychologism [http://www.humboldt.edu/~essays/pikerev.html], while Husserl criticized the way Frege used some of Leibniz's philosophy [http://perso.wanadoo.fr/rancho.pancho/Sub.htm].
+
'Thought' is filled with great quotes and is a treasure chest for anyone interested in Frege. Here is one of those quotes:
 +
 
 +
1) "Just as 'beautiful' points the way for aesthetics and 'good' for ethics, so do words like 'true' for logic."
 +
 
 +
==Sinn and Bedeutung==
 +
The distinction between Sinn and Bedeutung (usually but not always translated [[Sense and Reference]]) was an innovation of the German philosopher and mathematician Gottlob Frege in his 1892 paper Über Sinn und Bedeutung (On Sense and Reference), which is still widely read today. According to Frege, sense and reference are two different aspects of the meaning of at least some kinds of terms (Frege applied "Bedeutung" mainly to proper names and, to a lesser extent, sentences). Roughly, a term's reference is the object it refers to and its sense is the way in which it refers to that object.
  
 
==References==
 
==References==
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===Secondary===
 
===Secondary===
−
* [[George Boolos]], 1998. ''Logic, Logic, and Logic''. MIT Press. Contains several influential papers on Frege's philosophy of arithmetic and logic.
+
* Anderson, D. J., and Edward Zalta, 2004, "Frege, Boolos, and Logical Objects," ''Journal of Philosophical Logic 33'': 1-26.
−
* [[Michael Dummett]], 1973. ''Frege: Philosophy of Language''. Harvard University Press.
+
* Boolos, George, 1998. ''Logic, Logic, and Logic''. MIT Press. Contains several papers on Frege's philosophy of arithmetic and logic.
−
* [[Michael Dummett]], 1991. ''Frege: Philosophy of Mathematics''. Harvard University Press.
+
* Diamond, Cora, 1991. ''The Realistic Spirit''. MIT Press. Ostensibly about Wittgenstein, but contains several valuable articles on Frege.
−
*Demopoulos, William, 1995.  "Frege's Philosophy of Mathematics". Harvard University Press.  A nice collection that explores the significance of Frege's theorem, and his mathematical and intellectural background.
+
* Dummett, Michael, 1973. ''Frege: Philosophy of Language''. Harvard University Press.
−
*Gillies, Douglas A., 1982. ''Frege, Dedekind, and Peano on the foundations of arithmetic''. Assen, Netherlands: Van Gorcum.
+
* Dummett, Michael, 1981. ''The Interpretation of Frege's Philosophy''. Harvard University Press.
−
*[[Ivor Grattan-Guinness]], 2000. ''The Search for Mathematical Roots 1870-1940''. Princeton Uni. Press. Fair to the mathematician, less so to the philosopher.
+
* Dummett, Michael, 1991. ''Frege: Philosophy of Mathematics''. Harvard University Press.
−
* Hatcher, William, 1982. ''The Logical Foundations of Mathematics''. Pergamon. Uses natural deduction to rederive Peano's axioms from the ''Grundgesetze'' system, recast in modern notation.
+
* Demopoulos, William, 1995.  "Frege's Philosophy of Mathematics". Harvard Univ. Press.  Explores the significance of Frege's theorem, and his mathematical and intellectural background.
−
*Hill, C. O., and Rosado Haddock, G. E., 2000. ''Husserl or Frege: Meaning, Objectivity, and Mathematics''. Open Court. The Frege-Husserl-Cantor triangle.
+
* Ferreira, F. and Wehmeier, K., 2002, "On the consistency of the Delta-1-1-CA fragment of Frege's ''Grundgesetze''," ''Journal of Philosophic Logic 31'': 301-11.  
−
*[[Hans Sluga]], 1980. ''Gottlob Frege''. Routledge.
+
* Grattan-Guinness, Ivor, 2000. ''The Search for Mathematical Roots 1870-1940''. Princeton Uni. Press. Fair to the mathematician, less so to the philosopher.
 +
* Gillies, Douglas A., 1982. ''Frege, Dedekind, and Peano on the foundations of arithmetic''. Assen, Netherlands: Van Gorcum.
 +
* Hatcher, William, 1982. ''The Logical Foundations of Mathematics''. Pergamon. Devotes an entire chapter to the system of the ''Grundgesetze'', recast in modern notation, and derives [[Peano axioms]] in this system using [[natural deduction]].
 +
* Hill, C. O., 1991. ''Word and Object in Husserl, Frege and Russell:  The Roots of Twentieth-Century Philosophy''. Athens OH: Ohio University Press.
 +
*------, and Rosado Haddock, G. E., 2000. ''Husserl or Frege: Meaning, Objectivity, and Mathematics''. Open Court. On the Frege-Husserl-Cantor triangle.
 +
* Sluga, Hans, 1980. ''Gottlob Frege''. Routledge.
 +
* Weiner, Joan, 1990. ''Frege in Perspective''. Cornell University Press.
 +
* Wright, Crispin, 1983. ''Frege's Conception of Numbers as Objects''. Aberdeen University Press. Written from the viewpoint of a modern philosopher of language and logic, contains a systematic exposition and limited (i.e. scope-restricted) defense of Frege's ''Grundlagen'' conception of numbers.
  
 
==External links==
 
==External links==
−
*[http://www.ocf.berkeley.edu/~brianwc/frege/ A comprehensive guide to Fregean material available on the web; by Brian Carver.]
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{{wikisource author|Gottlob Frege}}
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* [http://plato.stanford.edu/entries/frege/ Stanford Encyclopedia of Philosophy: Gottlob Frege.]
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*[http://www.ocf.berkeley.edu/~brianwc/frege/ A comprehensive guide to Fregean material available on the web] by Brian Carver.
−
* [http://plato.stanford.edu/entries/frege-logic/ Stanford Encyclopedia of Philosophy: Frege's Logic.]
+
* [[Stanford Encyclopedia of Philosophy]]:
−
* [http://www.utm.edu/research/iep/f/frege.htm Internet Encyclopedia of Philosophy: Gottlob Frege.]
+
**"[http://plato.stanford.edu/entries/frege/ Gottlob Frege]" — by [[Edward Zalta]].
−
* [http://www.utm.edu/research/iep/f/freg-lan.htm Internet Encyclopedia of Philosophy: Frege and Language.]
+
** "[http://plato.stanford.edu/entries/frege-logic/ Frege's Logic, Theorem, and Foundations for Arithmetic]" — by Edward Zalta
 +
* [[Internet Encyclopedia of Philosophy]]:
 +
** [http://www.iep.utm.edu/f/frege.htm Gottlob Frege] — by Kevin C. Klement.
 +
** [http://www.utm.edu/research/iep/f/freg-lan.htm Frege and Language] — by Dorothea Lotter.
 +
*Metaphysics Research Lab: [http://mally.stanford.edu/frege.html Gottlob Frege.]
 
* [http://www.formalontology.it/fregeg.htm Frege on Being, Existence and Truth.]
 
* [http://www.formalontology.it/fregeg.htm Frege on Being, Existence and Truth.]
 
* {{MacTutor Biography|id=Frege}}
 
* {{MacTutor Biography|id=Frege}}
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* [http://ctan.org/tex-archive/macros/latex/contrib/begriff/ Begriff,] a [[LaTeX]] package for typesetting Frege's logic notation.
  
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Revision as of 03:04, 20 September 2006

Western Philosophy
19th-century philosophy,
200px
Name: Friedrich Ludwig Gottlob Frege
Birth: November 8, 1848
Death: 26 July, 1925
School/tradition: Analytic philosophy
Main interests
Philosophy of mathematics, mathematical logic, Philosophy of language
Notable ideas
Predicate calculus, Logicism, Sense and reference, Mediated reference theory

Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen, IPA: [ˈɡɔtlop ˈfʁeːɡə]) was a German mathematician who became a logician and philosopher. He helped found both modern mathematical logic and analytic philosophy.

Life

Frege's father was a schoolteacher whose specialty was mathematics. Frege began his studies at the University of Jena in 1869, moving to Göttingen after two years, where he received his Ph.D. in mathematics, in 1873. According to Sluga (1980), the nature of Frege's university education in logic and philosophy is still unclear. In 1875, he returned to Jena as a lecturer. In 1879, he was made associate professor, and in 1896, professor. His children all having died before reaching maturity, he adopted a son in 1905. Frege retired in 1918 and wrote until his death.

Frege's work was not widely appreciated during his lifetime, although he did debate in print, and correspond with, Ernst Schroder, Peano, Husserl, and Cantor. Bertrand Russell discovered Frege's writings around 1900 and became an admirer, as did Ludwig Wittgenstein somewhat later. These admirers assured Frege's influence and reputation in certain restricted circles. Frege had but one student of note, albeit a consequential one: Rudolf Carnap. Frege's work became widely known in the English-speaking world only after World War II; English translations of Frege's writings, which first appeared in 1950, came to have an enormous influence on analytic philosophy. Frege also became better known thanks to the emigration to the United States of central European philosophers and logicians who knew and respected Frege's work, including Carnap, Alfred Tarski, and Kurt Gödel.

Logician

Frege is widely regarded as a logician on a par with Aristotle, Kurt Gödel, and Alfred Tarski. His 1879 Begriffsschrift (Concept Script) marked a turning point in the history of logic. The Begriffsschrift broke much new ground, including a clean treatment of functions and variables. Frege wanted to show that mathematics grew out of Aristotelian logic, but in so doing devised techniques that took him far beyond that logic. 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 medieval problem of multiple generality. Hence the logical machinery essential to Bertrand Russell's theory of descriptions and Principia Mathematica (with Alfred North Whitehead), and to Gödel's incompleteness theorems, is ultimately due to Frege.

Frege was a major advocate of the view that arithmetic is reducible to logic, a view known as logicism. In his Grundgesetze der Arithmetik (1893, 1903), published at its author's expense, he attempted to derive 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)]. In modern notation and terminology, let {x|Fx} denote the extension of the predicate Fx, and similarly for Gx. Then Basic Law V says that the predicates Fx and Gx have the same extension iff ∀x[Fx ↔ Gx].

As Vol. 2 of the Grundgesetze was about to go to press in 1903, Bertrand Russell wrote to Frege, showing how to derive Russell's paradox from Basic Law V. (This letter and Frege's reply thereto are translated in Jean van Heijenoort 1967.) Hence the system of the Grundgesetze was 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's proposed remedy was subsequently shown to imply that there is but one object in the universe of discourse, and hence is worthless.

Recent work has shown that much of the Grundgesetze can be salvaged in any of several ways:

  • Basic Law V can be weakened in ways that restore the consistency of Frege's system. The best-known way is due to George Boolos. A "concept" F is "small" if the objects falling under F cannot be put in 1-to-1 correspondence with the universe of discourse, that is, if: ¬∃R[R is 1-to-1 & ∀x∃y(xRy & Fy)]. 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(Fx ↔ Gx). V* is consistent if second-order arithmetic is, and suffices to prove the axioms of second-order arithmetic.
  • Replace Basic Law V 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, and suffices to prove the axioms of second-order arithmetic. This result is anachronistically termed Frege's Theorem. On Hume's Principle and Frege's Theorem, see [1].
  • Frege's logic, now known as second-order logic, can be weakened to so-called predicative second-order logic. However, this logic, although provably consistent by finitistic or constructive methods, can interpret only very weak fragments of arithmetic.

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, 1910-13, the dominant approach to mathematical logic was that of George Boole and his descendants, especially Ernst Schroder. His logical ideas nevertheless spread through the writings of his student Carnap and of other admirers, particularly Bertrand Russell.

Philosopher

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 the sense and reference (Sinn und Bedeutung) of a proper name (Eigenname);
  • Mediated reference theory;
  • Distinction between concept and object (Begriff und Gegenstand);
  • Context principle;
  • Principle of compositionality.

As a philosopher of mathematics, Frege loathed appeals to psychologistic or "mental" explanations for meanings (such as idea theories of meaning). His original purpose was very far from answering 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 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 and the first collections of his writings appeared only after WWII. Hence despite Bertrand Russell's generous praise, 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.

Thought

A great deal of Frege's writings were not translated into English until the 1950s. "Thought, is one of those such writings. It was published in 1918 as the first part of a series of three papers entitled 'Logical Investigations'. Along with 'On Sense and Reference', it is one of Frege's most influential and widely discussed papers" (Beaney, 1997).

'Thought' is filled with great quotes and is a treasure chest for anyone interested in Frege. Here is one of those quotes:

1) "Just as 'beautiful' points the way for aesthetics and 'good' for ethics, so do words like 'true' for logic."

Sinn and Bedeutung

The distinction between Sinn and Bedeutung (usually but not always translated Sense and Reference) was an innovation of the German philosopher and mathematician Gottlob Frege in his 1892 paper Über Sinn und Bedeutung (On Sense and Reference), which is still widely read today. According to Frege, sense and reference are two different aspects of the meaning of at least some kinds of terms (Frege applied "Bedeutung" mainly to proper names and, to a lesser extent, sentences). Roughly, a term's reference is the object it refers to and its sense is the way in which it refers to that object.

References
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Primary

  • Online bibliography of Frege's works and their English translations.
  • 1879. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S.: Louis Nebert. Translation: Concept Script, a formal language of pure thought modelled upon that of arithmetic, by S. Bauer-Mengelberg in Jean Van Heijenoort, ed., 1967. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press.
  • 1884. Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner. Translation: J. L. Austin, 1974. The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number, 2nd ed. Blackwell.
  • 1891. "Funktion und Begriff." Translation: "Function and Concept" in Geach and Black (1980).
  • 1892a. "Über Sinn und Bedeutung" in Zeitschrift für Philosophie und philosophische Kritik 100: 25-50. Translation: "On Sense and Reference" in Geach and Black (1980).
  • 1892b. "Über Begriff und Gegenstand" in Vierteljahresschrift für wissenschaftliche Philosophie 16: 192-205. Translation: "Concept and Object" in Geach and Black (1980).
  • 1893. Grundgesetze der Arithmetik, Band I. Jena: Verlag Hermann Pohle. Band II, 1903. Partial translation: Furth, M, 1964. The Basic Laws of Arithmetic. Uni. of California Press.
  • 1904. "Was ist eine Funktion?" in Meyer, S., ed., 1904. Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage, 20. Februar 1904. Leipzig: Barth: 656-666. Translation: "What is a Function?" in Geach and Black (1980).
  • Peter Geach and Max Black, eds., and trans., 1980. Translations from the Philosophical Writings of Gottlob Frege, 3rd ed. Blackwell.

Frege intended that the following three papers be published together in a book titled Logical Investigations. The English translations thereof were so published in 1975.

  • 1918-19. "Der Gedanke: Eine logische Untersuchung (Thought: A Logical Investigation)" in Beiträge zur Philosophie des Deutschen Idealismus I: 58-77.
  • 1918-19. "Die Verneinung" (Negation)" in Beiträge zur Philosophie des deutschen Idealismus I: 143-157.
  • 1923. "Gedankengefüge (Compound Thought)" in Beiträge zur Philosophie des Deutschen Idealismus III: 36-51.

Secondary

  • Anderson, D. J., and Edward Zalta, 2004, "Frege, Boolos, and Logical Objects," Journal of Philosophical Logic 33: 1-26.
  • Boolos, George, 1998. Logic, Logic, and Logic. MIT Press. Contains several papers on Frege's philosophy of arithmetic and logic.
  • Diamond, Cora, 1991. The Realistic Spirit. MIT Press. Ostensibly about Wittgenstein, but contains several valuable articles on Frege.
  • Dummett, Michael, 1973. Frege: Philosophy of Language. Harvard University Press.
  • Dummett, Michael, 1981. The Interpretation of Frege's Philosophy. Harvard University Press.
  • Dummett, Michael, 1991. Frege: Philosophy of Mathematics. Harvard University Press.
  • Demopoulos, William, 1995. "Frege's Philosophy of Mathematics". Harvard Univ. Press. Explores the significance of Frege's theorem, and his mathematical and intellectural background.
  • Ferreira, F. and Wehmeier, K., 2002, "On the consistency of the Delta-1-1-CA fragment of Frege's Grundgesetze," Journal of Philosophic Logic 31: 301-11.
  • Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press. Fair to the mathematician, less so to the philosopher.
  • Gillies, Douglas A., 1982. Frege, Dedekind, and Peano on the foundations of arithmetic. Assen, Netherlands: Van Gorcum.
  • Hatcher, William, 1982. The Logical Foundations of Mathematics. Pergamon. Devotes an entire chapter to the system of the Grundgesetze, recast in modern notation, and derives Peano axioms in this system using natural deduction.
  • Hill, C. O., 1991. Word and Object in Husserl, Frege and Russell: The Roots of Twentieth-Century Philosophy. Athens OH: Ohio University Press.
  • ------, and Rosado Haddock, G. E., 2000. Husserl or Frege: Meaning, Objectivity, and Mathematics. Open Court. On the Frege-Husserl-Cantor triangle.
  • Sluga, Hans, 1980. Gottlob Frege. Routledge.
  • Weiner, Joan, 1990. Frege in Perspective. Cornell University Press.
  • Wright, Crispin, 1983. Frege's Conception of Numbers as Objects. Aberdeen University Press. Written from the viewpoint of a modern philosopher of language and logic, contains a systematic exposition and limited (i.e. scope-restricted) defense of Frege's Grundlagen conception of numbers.

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