Many people fear mathematics and some even loath it, but the fact is that mathematics is such an important human artefact that we can barely survive a day without it, and like economics, it influences our lives from birth to death. Both these subjects are mostly considered to be dull by many. Modern economics, engineering, etc. cannot survive without mathematics, and the same is true of all other sciences. But what is mathematics?
Mathematics is a human, mental artefact and never proofs anything. Mathematics is a mere invention of the human imagination and not a body of universal truths based on common sense. It is therefore only incidental that mathematics sometimes seems to correlate with and describe natural phenomena. Mathematics is a very useful extension of human rational logic, but it is nothing more than that. Because of this fact, mathematics has very severe limitations.
Mathematics is a descriptive language (a case of the difference between a name and the thing named) and abstraction of numbers and is not predictive at all. If mathematics was predictive in any sense then numerology should be considered beyond any doubt as a viable method to make predictions.
When I said that parents should warn their children (tongue in the cheek) not to play with mathematics I was merely underlining the importance of understanding at least basic mathematics. To do this one of the most basic things to understand is then the fact that you can not divide by zero. If you divide by zero you do not get an answer but you simply made a grave error. You can multiply by zero though; the answer will be zero. However, any real number to the power zero will always be one (i.e. a0 = 1). This is not a natural law, but is simply a mathematical convention. If you change this convention you will have to change everything that follows after it.
This has been done in mathematics before. For example, Riemann (1826-1866) and Lobachevski (1792-1856) changed certain basics axioms of Euclidean geometry (the material we study at school) to develop completely different geometries. Non-Euclidian geometry was instrumental to Einstein’s (1879-1955) endeavour to develop his relativity theories.
Euclid (circa 300 BCE) was studying geometry in Alexandria and wrote a thirteen-volume book that compiled all the known and accepted rules of geometry called The Elements, and later referred to as Euclid’s Elements. Because mathematics was a discipline (mathematics is not a science), where every theorem is based on accepted assumptions, Euclid first had to establish some axioms with which to use as the basis of other theorems. He used five axioms as the five assumptions, which he needed to prove all other geometric ideas.
The first four of his axioms (for plane geometry) are fairly straightforward and easy to accept, and no mathematician has ever seriously doubted them. The first four of Euclid’s axioms are:
- One straight line may be drawn from any two points.
- Any terminated straight line may be extended indefinitely.
- A circle may be drawn with any given centre and any given radius.
- All right angles are congruent.
With no concern over the first four axioms, they are regarded as the axioms of all geometries. The fifth and last axiom listed by Euclid stands out a little bit. It is a bit less intuitive and a lot more convoluted. It looks like a condition of the geometry more than something fundamental about it. The fifth axiom is:
- If two straight lines lying in a plane are met by another line, and if the sum of he internal angles on one side is less than two right angles, then the straight lines will meet if the extended on the side on which the sum of the angles is less than two right angles.
The fifth axiom, also known as Euclid’s ‘parallel postulate’ deals with parallel lines. It is equivalent to this slightly clearer statement:
For a given line and point there is only one line parallel to the first line passing through the point.
This statement was first proved equivalent to Euclid’s fifth axiom by John Playfair (1748-1819) in the 18th century. This seems obvious to us because of what we have been taught, but it is far less as intuitive as the first four. Later mathematicians and even Euclid himself were not comfortable with axiom five; it is quite a complicated statement and axioms are meant to be small, simple, and straightforward. Axiom 5 looked more like a theorem than an axiom, and as such, it should have to be proved true and not assumed.
Einstein played with mathematics and the result was some very ‘strange’ theoretical musings on for example gravity. But alas, neither Einstein, or Newton, or anybody else, have explain what exactly gravity is! We have mathematical equations on gravity (e.g. the gravitational constant (G) = 6.693 x 10–11 cubic meters per kilogram second squared), and scientists call gravity a basic force (F = m1.m2/d2 kilogram times metre squared) of the universe, but that is all. What causes gravity – mass? Or is it perhaps the size of the diameter of a body? In fact, we do not really know. The difference is important when you fall into a hole through the centre of the Earth because it will determine where you stop – be it in China or at the centre of the earth! If you want to fall to China this information can proof to be vital!
Willie Maartens
