Axioms Part 5: Commentary on Mathematical Axioms
Different sets of axioms exist in mathematics and act as the basis for sub-disciplines within that subject such as geometry, number theory and set theory. The different sets of axioms make different mathematical structures logically permissible. The ZFC axiomatisation in set theory, for example, is composed of 9 axioms that recognise particular properties of sets. I note below that even within mathematics axiomatics set can change over time to serve different types of analysis.
In common with logical and descriptive axioms, mathematical axioms need to declare the existence of something. These things are sometimes referred to as ‘primitives’ in mathematics.
The Greek mathematician Euclid of Alexandria wrote his very famous and influential mathematical treatise called the Elements, more than 2,000 years ago. In this book, he created the basis for a geometry by describing a series of theorems and axioms. As these axioms were taught as the basis of geometry in schools for a very long period they are perhaps that most frequently taught in all of human culture outside of religion. Euclidean Geometry describes the existence of lines, angles, areas, 3-dimensional solids and space. This axiomatic set requires the existence primitives that we normally refer to as points, straight lines, curved lines, planes, angles and so on.
a) From an axiomatic perspective, we can see that points are the simplest and most primitive entities in geometry.
b) By definition, a geometric point has a position but no dimensions.
c) We then need to define what is needed to declare the existence of lines (or a way to construct them). A 1-dimensional line has length without breadth.
d) At least 2 points are required to define the existence of lines, and at least 3 points are required to describe the orientation of a flat plane.
e) The meaning of angles (or a way to construct them) needs to be specified using lines.
f) If we wish to state what an angle is, we need the connection or overlap of straight lines.
g) If we wish to define a right angle (i.e. 90 degrees) we need to the two lines to form 2 or 4 equal angles.
This leads us to a definition of what the word ‘parallel’ means.
h) Two lines emerging at right angles from a third line are parallel. If the lines are not parallel and extended indefinitely they will meet.
i) In addition, we need to know what a circle is.
In Hilbert’s axiomatization of Euclidean geometry, there exist additional functional primitives, such as ‘connectedness’ that declare the existence of an abstract entity that can connect points to form a line. We can then use the intersection of lines to create angles.
In order to proceed from Euclidean Geometry to analytic geometry, where the existence of lines and shapes of can be described by algebraic formulae, we need to declare in addition the existence of a coordinate system that subdivides the space into intervals.
We can go beyond these defining ideas and consider Non-Euclidean geometries, such as spherical or hyperbolic geometry, in which different axioms apply. Consider the situation where lines and shapes are drawn on the surface of a sphere or some other curved surface. The properties of these features are different from the situation where they are drawn on a flat surface. The angles of a triangle drawn on a 2-deminsional spherical plane do not add up to 180 degrees as they do on a flat plane. On the earth’s surface the lines of longitude, which depart from the equator at right angles to the equator, meet at the poles. By contrast, if 2 or more lines depart at right angles from a straight line on a flat surface they are parallel and so never meet. Interestingly the sum of angles of a triangle is said to change because the parallel postulate no longer applies. In a hyperbolic space very strange consequences can be deduced such as the sum of angles of a triangle being less than 180 degrees.
If we instead proceed to a geometry within an Affine Space we would also abandon measures of distance and angle and ‘forget’ the concept of an origin although we would maintain the parallel postulate and proportional lengths of such lines.
These considerations demonstrate that context in which axioms are said to apply is completely defined by a set of definitive postulates.
The 19th-century Italian mathematician Giuseppe Peano created a set of axioms (the Peano axioms) that tells us how to create the set of natural numbers (0, 1, 2, 3… and so on). There has to be at least one number to start with so its existence is simply declared. This primitive number is now taken to be zero. There then needs to be some logical operation that leads to the construction of other numbers. This is usually referred to as a successor operation. We can think of this operation as ‘add one’.
If our axioms only allow us to create the set of natural numbers are we forever restricted to those? The answer is obviously not. If we admit (or declare) the existence of the square root of -1, (and give it the identity i), we can then create the series of imaginary numbers, that have proved so useful in physics. We can go further and create the complex plane, as a concept that provides a geometric representation of these numbers.
It is striking that the existence or creation of entities such as natural numbers, which seem so fundamental to us, can be axiomatised. However, we have to ask what is the point of doing so and what does it achieve? The very short answer is that it helps to create an explicitly logical (although incomplete) basis for mathematics.