Axiom
In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. To say the least, not all epistemologists agree that any axioms, understood in that sense, exist.Axiomatic reasoning is today most widely accepted in mathematics, where an axiom has come to be an assumption on which proofs are based.
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2 Modern usage in mathematics 3 Mathematical examples 4 Consequences of separating axioms from "reality" 5 See also 6 External links |
Etymology
The word axiom comes from the Greek word
αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident. The word comes from αξιοειν (axioein), meaning to deem worthy, which in turn comes from αξιος (axios), meaning worthy. Among the philosophers of the ancient Greeks an axiom was a claim which could be seen to be true without any need for proof.
Modern usage in mathematics
As the word axiom is understood in modern mathematics, an axiom is not a proposition that is self-evident. Rather, it simply means a starting point in a logical system. For example, in some
rings, the operation of multiplication is commutative, and in some it is not; those rings in which it is are said to satisfy the "axiom of commutativity of multiplication." Another name for an axiom is postulate. An axiom is an elementary basis for a formal logic system that together with the rules of inference define a logic.
Mathematical examples
For instance, (misquoting Peano) simple arithmetic including addition can be defined and many theorems proven by assuming
- a number called 0 exists
- every number X has a successor called inc(X)
- X+0 = X
- inc(X) + Y = X + inc(Y)
- inc(X) = X + 1
- 1 + 2 = 1 + inc(1) Expansion of abbreviation (2 = inc(1))
- 1 + 2 = inc(1) + 1 Axiom 4
- 1 + 2 = 2 + 1 Abbreviation (2 = inc(1))
- 1 + 2 = 2 + inc(0) Expansion of abbreviation (1 = inc(0))
- 1 + 2 = inc(2) + 0 Axiom 4
- 1 + 2 = 3 Axiom 3 and Use of abbreviation (inc(2) = 3)
Probably the most famous very early set of axioms is the 4+1 postulates of Euclid. This turns out to be incomplete, and many more postulates are necessary to completely characterize his geometry (Hilbert used 23).
4+1 since the fifth postulate (through a point outside a line there is exactly one parallel) was suspected to be derivable from the first 4 for nearly two millennia. Ultimately, the fifth postulate was found to be independent of the first four. Indeed, one can assume that no parallels through a point outside a line exist, that exactly one exists, or that infinitely many exist. These choices give us alternative forms of geometry in which the interior angles of a triangle add up to less than, exactly or more than a straight line respectively and are known as elliptic, Euclidean and hyperbolic geometries. The general theory of relativity is essentially a claim that mass gives space hyperbolic geometry.
Consequences of separating axioms from "reality"
Early mathematicians regarded axiomatic geometry as a model of reality, and obviously there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Galois showed just before his untimely death that these efforts were largely wasted, but that the grand parallels between axiomatic systems could be put to good use, as he algebraically solved many classical geometrical problems. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and modern algebra was born.
In the twentieth century, Gödel's incompleteness theorem showed that no explicit (i.e. recursive) set of axioms sufficiently large to define traditional arithmetic could be both (1) complete (i.e. every statement can be either proved or disproved) and (2) consistent (i.e. no statement can be both proved and disproved).