Gottlob Frege (8 November 1848–26 July 1925) was a German mathematician, logician, and philosopher whose work helped establish modern logic and analytic philosophy. His investigations into the foundations of mathematics produced a new logical notation, rigorous methods of proof, and influential accounts of meaning. He sought to derive arithmetic from logical principles, although the formal system developed for that purpose proved inconsistent. (mathshistory.st-andrews.ac.uk)
Life and principal works
Frege was born in Wismar, then in Mecklenburg-Schwerin, and began studying at the University of Jena in 1869. He transferred to Göttingen in 1871, receiving a doctorate in 1873 for a dissertation on geometrical representations of imaginary figures. In 1874 he qualified to teach at Jena, where he spent his academic career. His teaching included geometry, calculus, and mechanics, while his principal publications concerned mathematical foundations and philosophy. He died in Bad Kleinen. (mathshistory.st-andrews.ac.uk)
His major books were Begriffsschrift (“Concept-Script,” 1879), The Foundations of Arithmetic (1884), and The Basic Laws of Arithmetic, published in two volumes in 1893 and 1903. Important essays included “Function and Concept” (1891), “On Sense and Reference” (1892), and “On Concept and Object” (1892). These writings developed interconnected accounts of logical structure, number, and linguistic meaning. (iep.utm.edu)
The reconstruction of logic
The Begriffsschrift introduced a formal system capable of representing complex patterns of generality. Unlike traditional syllogistic analysis, it could distinguish statements corresponding to “Everyone loves someone” and “Someone is loved by everyone.” These differ because the order and scope of their quantifiers differ. Frege’s system included resources now associated with first-order logic and second-order logic, although its two-dimensional notation differs from contemporary symbolism. (plato.stanford.edu)
Frege analyzed propositions through function and argument rather than merely subject and predicate. Extending the notion of a function, he treated a concept as something incomplete, requiring an argument; in his mature account, a concept maps an object to a truth-value. He also distinguished expressing a thought from asserting its truth. Explicit axioms and rules of inference made the steps of a formal proof inspectable, reducing reliance on unexpressed assumptions. (plato.stanford.edu)
Arithmetic and logicism
Frege defended logicism about arithmetic: its fundamental truths should follow from logic and appropriate definitions. Against Immanuel Kant, he maintained that arithmetic was analytic rather than dependent on pure intuition. He also rejected accounts grounding arithmetic in experience or individual mental processes. His opposition to psychologism distinguished objective logical justification from the psychological circumstances in which people form beliefs. (iep.utm.edu)
In The Foundations of Arithmetic, Frege argued that numerical statements concern concepts: saying that there are three objects of a kind attributes a number to the relevant concept. He treated numbers as objects rather than mental images or properties of physical collections. His context principle directed inquiry toward the significance of words within complete propositions rather than in isolation. This approach allowed him to investigate number by examining the logical structure of numerical statements. (iep.utm.edu)
Basic Law V and Russell’s paradox
In The Basic Laws of Arithmetic, Frege introduced Basic Law V, identifying the value-ranges of functions that agree on every argument. For concepts, this yields an identity condition for their extensions. Combined with the system’s unrestricted resources for forming concepts, however, the law produces a contradiction. In 1902, Bertrand Russell informed Frege of the difficulty now known as Russell’s paradox. Frege acknowledged it in an appendix to the second volume, but his proposed repair did not establish a consistent foundation. (plato.stanford.edu)
Later work isolated a significant surviving achievement. Hume’s principle states that two concepts have the same number exactly when their instances can be placed in one-to-one correspondence. With suitable definitions, this principle supports a derivation of second-order Peano arithmetic. Called Frege’s theorem, the result preserves much of his arithmetical reasoning without the inconsistent derivation of Hume’s principle from Basic Law V. (plato.stanford.edu)
Sense, reference, and thought
Frege’s contributions to philosophy of language grew from his concern with thought and logical analysis. His distinction between sense and reference separates how an expression presents something from what it designates. Expressions identifying the morning star and the evening star designate the same planet, but present it differently. Their identity can therefore convey information, unlike an identity that simply repeats the same expression. (iep.utm.edu)
For a complete declarative sentence, Frege identified its sense with the thought expressed and its reference, when available, with a truth-value. Senses are not private images: different speakers can grasp the same thought. In reported speech and belief contexts, expressions may refer indirectly to their customary senses. This explains why replacing one designation with another for the same object need not preserve the truth of a belief report. These distinctions became central topics in semantics. (iep.utm.edu)
Reception and influence
Frege’s writings directly influenced Russell, Ludwig Wittgenstein, and Rudolf Carnap. Their initial reception was limited, partly because his notation was difficult and his purposes were not widely understood. English translations and sustained twentieth-century scholarship subsequently broadened his readership. His work became a major reference point for philosophical logic and investigations of mathematical and linguistic meaning, despite the failure of his original foundational system. (mathshistory.st-andrews.ac.uk)