*Macmillan Science Library: Mathematics*. Copyright © 2001-2006 by Macmillan Reference USA, an imprint of the Gale Group. All rights reserved.

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## Axiomatic Systems

To understand Euclid's *Elements,* one must first understand the concept of an **axiomatic system**. Mathematics is often described as being based solely on logic, meaning that statements are accepted as fact only if they can be logically deduced from other statements known to be true.

What does it mean for a statement to be "known to be true?" Such a statement could, of course, be deduced from some other "known" statement. However, there must be some set of statements, called axioms, that are simply

assumed to be true. Without axioms, no chain of deductions could ever begin. Thus even mathematics begins with certain unproved assumptions.

Ideally, in any axiomatic system, the assumptions are of such a basic and intuitive nature that their truth can be accepted without...

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