Article abstract: Frege is the founder of modern symbolic logic and the creator of the first system of notations and quantifiers of modern logic.
Friedrich Ludwig Gottlob Frege was the son of the principal of a private, girls’ high school. While Frege was in high school in Wismar, his father died. Frege was devoted to his mother, who was a teacher and later principal of the girls’ school. He may have had a brother, Arnold Frege, who was born in Wismar in 1852. Nothing further is known about Frege until he entered the university at the age of twenty-one. From 1869 to 1871, he attended the University of Jena and proceeded to the University of Göttingen, where he took courses in mathematics, physics, chemistry, and philosophy. By 1873, Frege had completed his thesis and had received his doctorate from the university. Frege returned to the University of Jena and applied for an unsalaried position. His mother wrote to the university that she would support him until he acquired regular employment. In 1874, as a result of publication of his dissertation on mathematical functions, he was placed on the staff of the university. He spent the rest of his life at Jena, where he investigated the foundations of mathematics and produced seminal works in logic.
Frege’s early years at Jena were probably the happiest period in his life. He was highly regarded by the faculty and attracted some of the best students in mathematics. During these years, he taught an extra load as he assumed the courses of a professor who had become ill. He also worked on a volume on logic and mathematics. Frege’s lectures were thoughtful and clearly organized, and were greatly appreciated by his students. Much of Frege’s personal life, however, was beset by tragedies. Not only did his father die while he was a young man but also his children died young, as did his wife. He dedicated twenty-five years to developing a formal system, in which all of mathematics could be derived from logic, only to learn that a fatal paradox destroyed the system. During his life, he received little formal recognition of his monumental work and, with his death in 1925, passed virtually unnoticed by the academic world.
Frege’s first major work in logic was published in 1879. Although this was a short book of only eighty-eight pages, it has remained one of the most important single works ever written in the field. Begriffsschrift: Eine der Arithmetischen Nachgebildete Formelsprache des reinen Denkens (conceptual treatise: a formula language, modeled upon that of arithmetic, for pure thought) presented for the first time a formal system of modern logic. He created a system of formal symbols that could be used more regularly than ordinary language for the purposes of deductive logic. Frege was by no means the first person to use symbols as representations of words, since Aristotle had used this device and was followed by others throughout the history of deductive logic.
Earlier logicians, however, had thought that in order to make a judgment on the validity of sentences, a distinction was necessary between subject and predicate. For the purposes of rhetoric, there is a difference between the statements “The North defeated the South in the Civil War” and “The South was defeated by the North in the Civil War.” For Frege, however, the content of both sentences conveyed the same concept and hence must be given the same judgment. In this work, Frege achieved the ideal of nineteenth century mathematics: that if proofs were completely formal and no intuition was required to judge the correctness of the proofs, then there could be complete certainty that these proofs were the result of explicitly stated assumptions. During this period, Frege began to use universal quantifiers in his logic, which cover statements that contain “some” or “every.” Consequently, it was now possible to cover a range of objects rather than a single object in a statement.
In 1884, Frege published Die Grundlagen der Arithmetik (The Foundations of Arithmetic, 1950), which followed his attempt to apply similar principles to arithmetic as his earlier application to logic. In this work, he first reviewed the works of his predecessors and then raised a number of fundamental questions on the nature of numbers and arithmetic truth. This work was more philosophical than mathematical. Throughout the work, Frege enunciated three basic positions concerning the world of philosophical logic: Mental images of a word as perceived by the speaker are irrelevant to the meaning of a word in a sentence in terms of its truth or falsity. The word “grass” in the sentence “the grass is green” does not depend on the mental image of “grass” but on the way in which the word is used in the sentence. Thus the meaning of a word was found in its usage. A second idea was that words only have meaning in the context of a sentence. Rather than depending on the precise definition of a word, the sentence determined the truth-value of the word. If “all grubs are green,” then it is possible to...
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