Foundations of Mathematics
Foundations and Logic
Was mathematics always based on a logical foundation?
No, not all of mathematics was always based on a logical foundation, but some ancient cultures did develop certain aspects of logic in their thought. The Greeks were probably one of the first cultures to understand logic’s role in mathematics and philosophy, and studied the subject extensively. For example, geometry, as presented by Greek mathematician Euclid (c. 325-c. 270 B.C.E.), had some foundations in logic. Greek scientist and philosopher Aristotle’s (384-322 B.C.E.) rules for syllogisms were also based on logic, and he wrote the first systematic treatise on logic. But his logic works were based on ordinary language—making them a matter of interpretation and subject to ambiguities.
It was not until the development of calculus that most of mathematics was put on a logical foundation. By the 17th century, people like German mathematician Gottfried Wilhelm Leibniz (1646–1716) began to demand a more regular and symbolic way to express logic. Logic truly became a part of mathematics around the mid-19th century, especially with the 1847 publication of English mathematician George Boole’s (1815–1864) The Mathematical Analysis of Logic and English mathematician Augustus De Morgan’s (1806–1871) Formal Logic. Thus, mathematics began to encompass symbolic logic, with precise rules to manipulate those symbols (see below for more about symbolic logic).
Of course, nothing is perfect, although mathematicians in the late 19th and early 20th centuries hoped it would be. They believed that all of mathematics could be described using symbolic logic and made purely formal. But in the 1930s, Austrian-American mathematician and logician Kurt Gödel (1906–1978) put a damper on such an idea by showing that not all truths could be derived by a formal logic system.