Frege, Gottlob

views updated

FREGE, GOTTLOB

mathematical background
systematization of science and sources of knowledge
the history of frege's project
frege's legacy
bibliography

FREGE, GOTTLOB (1848–1925), German mathematician, logician, and philosopher.

Friedrich Ludwig Gottlob Frege devoted most of his career to a single project: the attempt to provide foundations for arithmetic. What is it to provide foundations for arithmetic? To provide foundations for a science, in Frege's sense, is to systematize it: to list its primitive truths and concepts and to show how these primitive truths and concepts figure in the justification of the truths of the science. The resulting systematization of a science is meant to exhibit the source of the knowledge of its truths. Mathematical work provided him with one model of this sort of systematization, the arithmeticization of analysis.

mathematical background

Analysis originated in the seventeenth century as a response to the needs of physics and astronomy. Its proofs originally exploited techniques of geometry. In the mid-nineteenth century, however, it became apparent that many of the geometrical proofs were not as secure as they seemed. Some apparently good proofs were identified as fallacious. The difficulties were attributable, in part, to confusions about some basic notions of analysis, including those of limit and continuity. The response was to try to show that the foundations of analysis lay in arithmetic: to offer a systematization of analysis in which only terms of arithmetic were used. Arithmetic was also systematized. Julius Wilhelm Richard Dedekind (1831–1916) provided an easily surveyable list of axioms from which, it was thought, all truths of arithmetic were derivable. But Frege was not content with Dedekind's systematization. For Dedekind's axiomatization did not, according to Frege, exhibit the source of our knowledge of the truths of arithmetic. The problem was that the terms of arithmetic (for example, "0") appeared in Dedekind's axioms, and that his axioms included familiar truths of arithmetic (for example, that 0 does not equal 1). Such terms and truths are not primitive, in Frege's sense, because it is not clear what the source of knowledge about numbers is. To see why, it is necessary to look at Frege's views about the sources of knowledge.

systematization of science and sources of knowledge

Sense experience is indisputably a source of some knowledge. The empiricist view can be characterized as the view that sense experience is the source of all knowledge. Frege thought that the German philosopher Immanuel Kant (1724–1804) had refuted this view by showing that some knowledge, including knowledge of Euclidean geometry, does not have the senses as its source. The source of this knowledge is, rather, pure intuition—a faculty underlying perception of objects in space. But Kant erred, according to Frege, in failing to recognize that some substantive knowledge requires neither sense experience nor pure intuition for its justification. Knowledge of arithmetic, Frege believed, is more basic than that of either the special sciences or geometry. The source of this knowledge is something that underlies all knowledge: reason alone. The science of reason, the science of the general laws that underlie all correct inference, is logic. Frege was convinced that the truths of arithmetic and, indeed, of all mathematics other than Euclidean geometry were logical truths. But many truths of arithmetic (e.g., that 0 does not equal 1) seem to be truths about particular objects, not general laws of logic. Frege thought he could show that all arithmetical truths are truths of logic by defining the numbers and the concept of number from purely logical concepts and proving the basic truths of arithmetic using only these definitions and logical laws. The result would be logical foundations for arithmetic: foundations showing that reason alone is the source of the knowledge of the truths of arithmetic.

the history of frege's project

Although Frege was convinced that the truths of arithmetic were logical truths, he was aware that they were not derivable using the logical systems generally accepted in his time. But he was also convinced that these logical systems were inadequate. Thus he began by constructing a new logic. In the 1879 monograph, Begriffsschrift, he set out the first logical system adequate for the expression and evaluation of the arguments that are regarded as logically valid. His next major work, Die Grundlagen der Arithmetik (1884; The Foundations of Arithmetic, 1950), was an informal description of his project, its motivations, and Frege's strategy for accomplishing his goals. The project was to have been completed in his Grundgesetze der Arithmetik (Basic Laws of Arithmetic). The first volume of Basic Laws was published in 1893. In 1902, when the second volume was in press, Frege received a now-famous letter from the mathematician and philosopher Bertrand Russell (1872–1970), demonstrating that the logical system was inconsistent. Frege ultimately concluded that the project, as he had envisioned it, was doomed to failure.

frege's legacy

Although Frege failed to accomplish the task he set himself, he made profound contributions to logic and philosophy. One of these is Frege's new logic. The inconsistency in the logic set out in Basic Laws is easily eliminated by omitting the two basic logical laws that were needed for the definitions of the numbers. The resulting logical system is not only more powerful than earlier logical systems, it is also formal in several important respects. In Frege's logic it is a mechanical task to determine, for any string of symbols in the logical language, whether it is a well-formed name or sentence of the language. It is also a mechanical task to determine whether a sentence is a basic law and whether or not a sentence follows immediately by one of the rules of inferences from other sentences. Thus there is a mechanical procedure for evaluating a purported gapless proof of the argument in the formal language. These formal features make it possible to regard the logical system itself as a mathematical entity. The field of mathematical logic thus has its origin in Frege's new logic.

Another important Fregean legacy comes from his approach to his philosophical problem. Frege believed that he could solve a philosophical problem about the nature of the truths of arithmetic by introducing definitions, using purely logical terms, that could replace numerals in all contexts. The justification of these definitions was provided by an analysis of how certain symbols (the numerals) are used and a demonstration that these symbols can be dispensed with by defining them from other terms. The philosophical question that Frege wanted to answer appears to have nothing in particular to do with language or meaning. Yet he answered the question by engaging in a linguistic investigation. The use of this strategy marks Frege as one of the first (perhaps the very first) to take the so-called linguistic turn that is characteristic of analytic philosophy, the dominant school in Anglo-American philosophy since the middle of the twentieth century.

Finally, many of Frege's writings on specific issues concerning language, logic, and mathematics remain immensely influential in the twenty-first century. A great deal of work in linguistics and the philosophy of language has its origin in his discussions of language. Indeed, a substantial number of early-twenty-first-century philosophers regard themselves as neo-Fregeans. Even the logicist project that Frege regarded as having been decisively refuted has been resurrected and forms an important strand of contemporary philosophical thought about arithmetic. Frege's work attracted only a small audience in his lifetime. But in the years since, his influence on contemporary philosophy, especially on thought about language and logic, has become ubiquitous.

See alsoScience and Technology.

bibliography

Primary Sources

Frege, Gottlob. Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number. Translated by J. L. Austin. 2nd edition. Oxford, U.K., 1980. Translation of Die Grundlagen der Arithmetik.

Beaney, Michael, ed. The Frege Reader. Oxford, U.K., 1997. Collection containing Frege's most well known articles as well as excerpts from his most important books.