Foundations of Mathematics

Foundations and Logic

What is the historical basis for mathematical logic?

Most mathematicians believe that systematic logic began with Aristotle’s collection of works titled Organon (Tool), in which he introduced his ideas on logic. In particular, Aristotle used general forms to describe logic, such as if all x are y; and all y are z; then all x are z. He presented three laws basic to all valid thought: the law of identity, or A is A (for example, an acorn will always yield a oak tree and nothing else); the law of contradiction, or A cannot be both A and not A (for example, an honest woman cannot be a thief); and the law of the excluded middle, or either or, in which A must either be A or not A (for example, a dog can be brown or not brown). In fact, author Ayn Rand divided her novel Atlas Shrugged into three parts after these three principles, as a tribute to Aristotle.


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