An axiom is a statement that is accepted as true without proof. Axioms form the basis of mathematical proofs that are written in order to establish theorems. In order to formally prove conjectures, we must start with given information. Axioms often supply us with this assumed information.
There are two major roles that axioms play in mathematics.
1. They describe undefined terms (line, point, etc.).
2. They provide us with a starting point to prove conjectures.
The Greeks (and majorly Euclid) are known for first using mathematical axioms over 2000 years ago.
Their axiomatic approach involved undefined objects such as points and lines. These axioms are simple forms of statements which cannot be further broken down without philosophical considerations.
The axioms that I am most familiar with are those within Euclidean geometry. Euclid’s famous book The Elements contains his geometric axioms. Axioms are sometimes organized into systems, as Euclid's were. An axiomatic system is a collection of axioms such that each axiom is independent from the others – that is, no axiom can be proven from any other axiom in the set.
In Euclidean geometry there is a system of five essential postulates (which are synonymous to axioms) called Euclid's Postulates. These postulates are the building blocks which are used to prove every theorem of Euclidean geometry.
They are:
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one
endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on
one side is less than two right angles, then the two lines inevitably must intersect each other on
that side if extended far enough.
Euclid also wrote a set of "Common Notions", which are axioms that go beyond geometry and refer to general logic. These notions are:
1. Things which are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
Euclid's axiomatic writing was widely accepted and stable for thousands of years. Then, in the 19th century, mathematicians Ivanovitch Lobachevski, Karl Gauss, and János Bolyai extensively studied Euclid's Postulates and attempted to find contradictions within them. Instead, these men ended up creating non-Euclidean, hyperbolic geometry. In the 20th century, Albert Einstein referenced hyberbolic geometry to write The General Theory of Relativity.
It is evident that axioms (especially Euclid's) are the foundation of mathematical theory and have remained meaningful for over 2,000 years of mathematical discovery. Without axioms, math would not have developed into the beautiful science that it is today!
There are two major roles that axioms play in mathematics.
1. They describe undefined terms (line, point, etc.).
2. They provide us with a starting point to prove conjectures.
The Greeks (and majorly Euclid) are known for first using mathematical axioms over 2000 years ago.
Their axiomatic approach involved undefined objects such as points and lines. These axioms are simple forms of statements which cannot be further broken down without philosophical considerations.
The axioms that I am most familiar with are those within Euclidean geometry. Euclid’s famous book The Elements contains his geometric axioms. Axioms are sometimes organized into systems, as Euclid's were. An axiomatic system is a collection of axioms such that each axiom is independent from the others – that is, no axiom can be proven from any other axiom in the set.
In Euclidean geometry there is a system of five essential postulates (which are synonymous to axioms) called Euclid's Postulates. These postulates are the building blocks which are used to prove every theorem of Euclidean geometry.
They are:
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one
endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on
one side is less than two right angles, then the two lines inevitably must intersect each other on
that side if extended far enough.
Euclid also wrote a set of "Common Notions", which are axioms that go beyond geometry and refer to general logic. These notions are:
1. Things which are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
Euclid's axiomatic writing was widely accepted and stable for thousands of years. Then, in the 19th century, mathematicians Ivanovitch Lobachevski, Karl Gauss, and János Bolyai extensively studied Euclid's Postulates and attempted to find contradictions within them. Instead, these men ended up creating non-Euclidean, hyperbolic geometry. In the 20th century, Albert Einstein referenced hyberbolic geometry to write The General Theory of Relativity.
It is evident that axioms (especially Euclid's) are the foundation of mathematical theory and have remained meaningful for over 2,000 years of mathematical discovery. Without axioms, math would not have developed into the beautiful science that it is today!
