Euclid’s Elements, a volume of geometry and math theorems derived from only five axioms and five postulates, have served as the foundation of Western mathematics. How did Euclid create this foundation, and why are the axioms and postulates so important?
Who was Euclid?
Euclid of Alexandria taught mathematics in that city sometime during the reign of Ptolemy I of Eqypt.
The dates of his birth and death are uncertain, but are likely to be 330 BCE and 260 BCE respectively.
Some early references to “Euclid” might refer to the earlier philosopher Euclid of Megara, so scholars tend to mistrust bold assertions about his life.
It is likely, however, that Euclid of Alexandria studied philosophy and mathematics in Athens, from Plato or Plato’s disciples.
Euclid would also have had access to the work of prior mathematicians such as Thales and Pythagoras. (Thales is best known for his theorem about circled triangles, and everyone knows the Pythagorean Theorem about the squares adjacent to right-angle triangles.)
Euclid is most famous for his 13-volume Elements, perhaps the first mathematical treatise to develop a vast body of math starting from simple definitions, axioms and postulates.
The Elements start with 23 definitions in Book 1, and then list the postulates and axioms. Euclid then generates proof after proof of proposition upon proposition. Most of the later books include further definitions, but no new axioms or postulates.
For mathematicians, axioms are propositions, or statements, which are stated as the basis of deriving further consequences. In common English usage, axioms are self-evident truths that are accepted without requiring a proof. (The sky is blue.)
The definition for postulate as a noun refers back to axiom, since it carries almost the same meaning. As a verb, however, “to postulate” includes the notion of making an assumption for the sake of further discussion. (Let’s say, for the sake of argument, that the sky is green.)
In contrast, a theorem is a proposition deduced from other propositions, whether axioms, postulates or other theorems. (If the sky were green, I might theorize that the grass on the horizon is merely an extension of the sky.)