Part: Historical Development of Logic

The origins of the logic date back to ancient civilizations, such as the Greeks, the Persians, the Chinese, and the Indians. The Greek Aristotle (384-322 BCE) was the first to establish logic as a formal discipline. He studied the properties of so-called premises, i.e., statements, in which the subject of the first statement is the predicate of the second statement. Aristotle observed that the truth of two such sentences guarantees the truth of a third sentence, their conclusion. Aristotle called his observation syllogism.

Aristotle's method of analyzing and performing logic was dominant until the early 19th century until English mathematicians George Bool, John Venn, and Augustus De Morgan started to abstract from words in logic and replacing them by symbols. Also, the German mathematician Gottfried Wilhelm von Leibniz (1646 - 1716) did so earlier in the 17th century but his work remained unpublished for almost 200 years. Leibniz was the inventor of the law of excluded middle and reductio ad absurdum and a mathematical prooving method.

The German mathematicians Ernst Schröder (1841 - 1902) and Gottlob Frege (1848 - 1925) put forward the view of mathematical logic. They lived in an epoch, in which different disciplines of mathematics quickly developed and in which first patterns began to appear among seemingly incoherent areas. Mathematicians started an attempt to base the whole mathematics on a single framework, from which any mathematical theorem could be derived from. Frege believed that his system, known later as first-order predicate logic, was suitable enough to do the trick. But his system lacked techniques to express numbers, and without numbers, it was impossible to describe mathematics or even simple arithmetics. Frege made extensive use of Cantor's set theory. Frege attempted to construct a logical foundation for mathematics, but he finally failed in doing so after he became aware of Russel's paradox, named after concerning sets (see Set theory). Avoiding his own paradox, also Russel tried to construct a logical foundation for mathematics. Together with Alfred Whitehead (1861 - 1947), Russel managed to prove that $1 + 1=2$ after 752 pages (!) in their work Principia Mathematica published 1912. Despite the unhandy techniques being at their disposal at that time, this was the first book to demonstrate the close ties between mathematics and formal logic.

Another attempt to construct a logical foundation for mathematics was that of David Hilbert (1862 - 1943). Hilbert was interested in the features different mathematical disciplines have in common. One of such common features is the use of the axiomatic method, used throughout BookofProofs. Every mathematical discipline begins with a group of statements (called axioms), which are simply assumed to be true and from which all other statements are then derived. Hilbert believed that using the axiomatic method, it was also possible to create a self-contained, foundational theory of mathematics that would prove its own consistency (being free of contradictions). Moreover, such a foundational theory would enable us to prove every statement that is true. This become known as Hilbert's Programme.

The Austrian mathematician and Hilber's student, Kurt Gödel (1906 - 1978) destroyed Hilbert's dream. He was Hilbert's student, and his discoveries made him one of the greatest logicians of the 20th century. He has shown that Hilbert's foundation of arithmetics (and any other theory derived using the axiomatic method) must be incomplete (i.e., it contains true statements, which cannot be proven using the self-contained theory). Moreover, he has shown that we cannot prove the consistency of a mathematical theory using its own axioms. Hilbert's Programme proved to be a dream that will never come true. This was a new insight: Today, we know that any past and any future attempt to create a foundational theory of mathematics will be limited.

Parts: 1

Thank you to the contributors under CC BY-SA 4.0!




  1. Cryan D., Shatil S., Mayblin B.: "Logic. A Graphic Guide", Icon Books Ltd., London, 2001
  2. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
  3. Kohar, Richard: "Basic Discrete Mathematics, Logic, Set Theory & Probability", World Scientific, 2016