A scrutiny on the mathematics system could help us to understand. We derive our mathematical laws by having axioms first. These are certain assumption and definition in which there are no questions about them. Then we use the rule of logics to solve the problem and find the result. This is known as the theorem. Euclid, the most famous Greek mathematician on geometry, based on the axiom of two points can be joined by a straight line to derive his theorem in his geometry.
In the processing stage from axiom and theorem, we use deductive logic. Thus we could get our problem with absolute certainty. However this must not be confused with absolute truth because the whole process is based on the axioms or the assumptions which can be valid or invalid. For example, I assume that all humans are narrow-minded and all old people are humans. Therefore all old people are narrow-minded. Thus my assumption is clearly invalid and so my conclusion does not reflect the truth. However it is certain in term of my assumption. Similarly, in mathematics, we have an absolutely certain theorem but it does not necessarily reflect the reality of the physical world.
Furthermore, this implies that different axioms will lead to different theorem. The answer for the addition of eleven and four is fifteen. On the other hand, five hours after 11:00 am is 4:00 pm. Same addition is used in both cases but the answers are not identical. This is due to different number system being applied. Thus different axioms are used to solve the problem.
There is a treasure of axioms in the mathematics. Although most of them are uninteresting and obvious, they are used and chosen by different mathematicians to solve the problem. For example, in an experiment, a scientist controls the independent variables in order to see the effect of the dependent variable. In another experiment, the scientist controls other independent variables to see the effect of the other dependent variable.
Therefore this sometimes leads to theorems that are bizarre and not likely to present the reality. One of the most noted example is the Banach-Tarski paradox. A sphere in the size of a tennis ball is split into many little pieces. Without changing their size, they could be re-arranged into another sphere in the size of the earth. This conclusion is odd and hard to accept. However this is derived from the obvious axioms in mathematics.
Sometimes obvious axioms could even lead to contradicting theorem. A German mathematicians David Hilbert started to search a theorem that could prove if a theorem is true or false in a axiomatic system. Thus he wants to find a step-by-step recipe which could simplify the whole process of theorem from the beginning of the first axioms. On the other hand, another mathematician Gödel derived two theorems. They stated that there will always be mathematical theorems which are true. However they are impossible to be proven right or wrong from the beginning of the axioms.
However the axioms in which mathematicians chose are supposed to be true and self-evident. Their consequences still lead to a range of different results. If mathematics discovers the reality of the physical world, this world of nature will be in contradictory. Since there are paradoxes in mathematics, this world is in constant conflict between different theorems.
Furthermore, mathematic is also a creative art like other so-called uncertain subject. A mathematician is encouraged to select axioms to solve problem. There are thousands of obvious axioms but one must see the pattern in which others mathematicians have omitted and missed. This insight is similar to other subject. In world history, a historian may observe the trend and pattern of how the rise of dictators is due to poverty of people. In economy, an economist observes the past performance of an economy and sees the effect of increased government spending. In literature, Charles Dickens had a deep insight into the social inequality in the Victorian Age in his novels. Furthermore the deep insight of the mathematicians may not be appreciated by others. A French mathematician Evariste Galois’s group theory were missed and even probably not read by the prominent mathematicians of the time.
We could arrive a deeper understanding by exploring how we get our axioms in the first place. We observe the nature to arrive the basic characteristics which are then arranged into axioms. These observations are massive and trivial. They are meaningless in themselves until we recognise them into our pattern in our mind. Therefore we try to relate one object to the other one. For example, we prick a balloon with a pin. Then an explosive sound is followed. We will then relate pricking a balloon with the sound. Thus pricking balloon causes an explosive sound. Therefore we establish a cause and effect relation between balloon and pricking.
Thus people select their observation in nature to form axioms. Axioms such as the whole is greater than the part (in Euclid‘s “The Elements”) are based on prolonged observation by the human ancestors. We then use the axioms to establish theorem. The golden ratio that is frequently used by the Renaissance artists is the fruit of long observation of human. This ratio is then used by artist to draw more realistic drawing. For example, in history there are millions of facts that include date and name of people but they are used as building block to answer deeper questions, such as the cause and effect of event on a particular person or country. Thus in the very first place, we must create pattern to create axioms. We then use these axioms to derive more complicated theorem to describe the world. In other words, we create mathematic to describe the physical world that allows us to make the world more meaningful.
On the other hand, the patterns we have created from the observation could be after all what we discover in the nature. Thus those patterns are in fact the reality of the physical world. Thus axioms will then be the reality also. Therefore we have discovered the mathematics in the nature. Physic laws, such as Newton’s second law rely heavily on mathematics and they correctly predict the result.
However, the process of finding scientific truth is based on falsification by peer-review. Also numerous and differed experiments are done to confirm the effect and the theories. Wrong theories, such as phrenology, were discarded. Thus the role of mathematic must not be exaggerated in finding truth in science. For example, Newton’s famous three laws were later all experimented by other scientists in later times before they are accepted being truth in describe the motion.
If the mathematical theorem could not be applied to the nature, then it could not be discovered. Since the mathematical laws should be hidden in the nature in which human discover. Thus we are invited to observe the chess player playing the chess but we are not told what the rules are. However the rules are contradictory. Thus by observing the nature, human used axioms to deduce theorem that are possible to describe the nature. Therefore mathematic is invented by people to describe the world.
