Mathematics is of key importance to most aspects of modern life.
Due to the great diversity and nature of mathematics it is a subject that is hard to define. Over the years great mathematicians have given there own definitions of mathematics. In general we can define it as ” a group of related sciences, including algebra, geometry, and calculus, concerned with the study of number, quantity, shape and space and there interrelationships using a specialized notation. ” Maths has often been described as the language of science because it is often used by scientists to express new theories. Unlike science though, maths is based on a set of axioms and postulates and not on experimentation or observation. Axioms and postulates are statements that are assumed to be true without being proven.
For example “the whole is greater than the part. ” An axiom is a statement common to all sciences whereas a postulate is a statement peculiar to the particular science being studied. Other statements or theorems must be logically implied by the set of postulates and axioms. The theorem is considered valid if it is consistent with itself and the mathematical system that it is a part and does not create any contradictions within the system. If something is mathimatically true it just means that it is valid.
Mathematics can be divided into two main areas, Pure mathematics and Applied mathematics. Applied mathematicians concern themselves with maths that can be applied to the real world like engineering. To consider a theorem true it must work in the outside world. Pure mathematicians are concerned with abstract ideas and the logical process that is taken to prove these ideas. Absolute certainty of results in pure maths comes from developing theorems from axioms by logical analysis.
There is disagreement between mathematicians over the relationship between maths and reality and whether mathematical objects are real. There are three different groups that have oposing ideas on the subject. One, the Platonist, says that mathematical objects are real and exist independent of our knowledge of them. So mathematicians discover mathematical theories and formulas. Formalists on the other hand argue that there are no mathematical objects and that mathematicians just create them.
Constructivists disagree with both and say that genuine mathematics is only what can be obtained by a finite construction. The set of real numbers or any other infinite set cannot be obtained. According to formalism mathematics consists of axiom postulates and formulas, but they are not about anything. When the formulas or theories are applied to the physical world then they acquire meaning and can either be true or false. But by itself as a purely mathematical formula it has no real meaning or truth value. To a formalist there is no real number system, except as we choose to create it by creating the appropriate axioms to describe it.
The mathematician is free to change it for whatever reason but neither system will correspond better to reality than the other because there is no reality. A good example of this argument is the study of geometry. For years Euclidean geometry was thought to describe the world around us. This was until the 1830s when Bernhard Riemann and Nikolay Ivanovich Lobachevsky with Janos Bolyai developed two new geometric systems. They did this by changing Euclid’s fifth postulate about parallel lines and then making all new deductions based on the new set of axioms. Both geometries were just as valid as Euclid’s and so would any other as long as it was consistent and did not lead to any contradiction within its set of axioms and postulates.
It was now apparent that there were almost an infinite number of geometric systems. It was also unclear which geometry described the outside world. A formalist’s view towards pure maths is that it is just a meaningless game where mathematicians never know what they are talking about or what they are saying is true. In many ways this is true, but pure maths has also been shown to have practical applications. The ancient Greeks for example described the ellipse and the parabola.
Galileo found the parabola to be the path of projectiles and Kepler used the ellipse to describe planetary orbits. Boolean algebra was used in computers and circuitry and in his theory of relativity Einstein used an obscure branch of mathematics called tensor calculus, developed five years earlier by G. Ricci and Tulio Levi-Civita. How is it that theories developed with no consideration of any practical purposes can be found years later to be perfect in describing a new scientific theory or application? To a Platonist the only explanation for this is that all maths is empirical and has and always will exist whether we discover it or not, the mathematician cannot invent anything because it is all there.
From this point of view all branches of maths can be considered applied maths we just haven’t discovered yet how it applies to the real world. Whether mathematics is invented or discovered is an impossible question to answer because it is impossible to prove or disprove and it will probably remain so no matter how far our mathematical knowledge advances in the future. There will always be maths that can be applied to the physical world and maths that seems to be just made up by someone. Though there is evidence to support both the Formalists and the Platonists neither can be absolutely sure the other is wrong. Maybe both are right.
Does it really matter? Whether maths is real or just a product of our imaginations it will continue to be developed and applied to different areas of our lives and maybe one day we will come close to answering this question.