What Is Axiom?

Any formal system combines a set of axioms and rules of derivation. The rules are used to obtain (derive) theorems from the axioms. We may learn the etymology of the word "axiom" from The Words of Mathematics by S. Schwartzman:

 

axiom (noun), axiomatic (adjective): via Latin, from Greek axioma, "that which is thought fitting; decision; self-evident principle." The Indo-European root is ag- "to drive, to lead." A subsidiary Greek meaning, "to weigh," led to axioma, literally "something weighty." In mathematical terms, axioms are concepts felt weighty enough that you can base a logical system on them.

Until about the second half of the 19th century, the only axiomatic system available was that set up in Euclid's Elements (where axioms are often called postulates). Euclid's axioms were perceived as a truthful reflection on the physical structure of the world and, as such, were judged self-evident (with the exception of the Fifth postulate). Even the development of non-Euclidean geometries did not change this perception as the latter were thought rather as pure creations of the human mind. In the 20th century, A. Einstein's General Relativity had a profound impact on the world view of many: Euclidean axioms could no longer be thought as embodiments of universal truth or physical reality.

The present day view on formal systems imposes only one requirement on a set of axioms, that of consistency: the axioms should not contradict each other. In practice, however, the selection of axioms is governed as much by other (interrelated) criteria, e.g., the esthetic sense, frugality, usefulness. In formal terms, derivations from consistent sets of axioms lead to correct, rather than true, statements called theorems. However, colloquially, the latter are often said to be true.

References

  1. E. J. Borowski & J. M. Borwein, The Harper Collins Dictionary of Mathematics, Harper Perennial, 1991
  2. J. Daintith, R. D. Nelson (eds), The Penguin Dictionary of Mathematics, Penguin Books, 1989
  3. S. Schwartzman, The Words of Mathematics, MAA, 1994

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