Some students may feel that mathematics and Theory of Knowledge do not have much in common. In fact, the opposite is true. The mere fact that mathematicians use their own ‘language of symbols’ raises interesting TOK questions about the role of language within the methodology of an area of knowledge or discipline.
Mathematical ‘truth’ is considered irrefutable to some, but why is this the case? It is quite remarkable how we can seemingly claim something with such a high degree of certainty within mathematics. Mathematics seems to embody principles and assumptions which are universally valid. This is quite unique compared with other areas of knowledge. Perhaps this is due to the fact that mathematics is heavily based on reason. By creating its own language of symbols, mathematics also aims to reduce cultural or contextual influences in the creation of knowledge. In that sense, it may comes as no surprise that mathematicians across the globe readily agree about the validity of things such as geometry. However, to say that mathematics is completely removed from human experience would perhaps be too hasty. In fact, interestingly, mathematics has been used to prove what some people feel intuitively.
Genuine new knowledge in mathematics is often the product of imagination rather than merely following the rules of reason. Things that are very much part of our human experience and intuition, such as concepts like beauty, can sometimes be explained through mathematics. The infamous ‘golden ratio‘ calculation, for example, can be found in nature. This calculation also illustrates how facial symmetry and harmony in things like architecture are linked to the concept of beauty. Links between mathematics and other areas of knowledge such as the arts (where beauty and aesthetics play a role) can lead to interesting knowledge questions. Sometimes, we use mathematics to offer “proof” and produce knowledge in other areas of knowledge. The applications of mathematical knowledge are not confined to its own discipline. In fact, we like to use mathematics to add value to knowledge in other areas of knowledge such as the natural sciences. We also like to use mathematical calculations or mathematical language to explain behaviour in the human sciences. This utility of mathematics seemingly enhances the credibility of the knowledge it produces. However, we could wonder how useful it actually is to explain, let’s say, human behaviour in mathematical terms. Are there circumstances in which applying mathematical knowledge to other areas of knowledge is not useful? The notion of the applicability of mathematics in the world around us leads to one of the most fundamental philosophical questions about the nature of mathematics.