# TOK Mathematics and Sciences Essay

Mathematicians have the concept of rigorous proof, which leads to knowing something with complete certainty. Consider the extent to which complete certainty might be achievable in mathematics and the natural sciences.

At first thought, mathematics and the natural sciences appear to be the two areas of knowledge that are most likely to contain absolute certainty. The concept of rigorous proof which is found in mathematics is a process in which an attempt is made to find contradictions in a mathematical proof. If contradictions are not found, it is then concluded that the final statement that was investigated is completely certain. However, mathematical proofs are often based on some statements which are assumed to be true to begin with. In the natural sciences, it is often assumed that statements which are supported by scientific knowledge have to be correct. For example, if you encouraged someone to eat oranges because the Vitamin C in it is healthy, they will only eat it if numerous scientific experiments have been conducted to support your claim. Mathematics is linked to reasoning, suggesting that mathematical proofs provide complete certainty. In addition, the natural sciences and mathematics are both supported by numbers, which makes them more accurate in terms of certainty than other areas of knowledge such as ethics. However, in science, it is difficult to achieve complete certainty because of various reasons, some linking to perception, as it is difficult to identify a concept which is completely true as there could be another concept which holds more truth to it. In mathematics, it could be possible to obtain the absolute certainty with simple arithmetic concepts, but it is difficult to obtain in other mathematical concepts.

Absolute certainty may be achievable in basic mathematical concepts. For example, almost no one will doubt that 1 + 1 = 2, because it is clearly stating that adding two ‘ones’ together will give you ‘two’, and the definition of the number ‘two’ is two ones. However, it can be argued that in logarithmic terms, 1 + 1 ≠ 2 because log 1 + log 1 = 1. But if one tried to use this to prove the other equation false, it would be going against the meaning of the terms used because in the first equation, it simply means to find the sum of two ones. Therefore, since there are no contradictions to the statement presented earlier, it can be considered as certain as it has gone through the process of rigorous proof.

This concept of absolute certainty does not flow into all mathematical concepts which are more complex because most of these concepts are proved using statements which are assumed to be true called axioms. For example, a mathematician might present a very logical argument which appears true, but the proof that he or she has used might not be certain because he or she might have used statements in his or her proof that might have been assumed to be true. Hence, even if this proof is valid, it might not be certain because the axioms used have not been ...