TOK Essay on Mathematician's Knowledge Versus Scientist's Knowledge
Theory of Knowledge (TOK) Essay
How does the mathematician’s knowledge differ from that of the scientist?
This question implies discussing how the knowledge acquired by mathematicians differs from that acquired by scientists.
Defining mathematics is never easy. Some claim it is an art, others that it is a science, yet others that it is a tool. Mathematics is also hard to place on the map of human endeavors. Should it be placed by the natural sciences, or does it belong together with philosophy among the human sciences? These questions arise from other, more foundational questions: What sort of knowledge does mathematics contain? What distinguishes the knowledge concerned with mathematics? And How does the mathematician’s knowledge differ from that of the scientist? I am not intending to answer all these questions, but am leaving the former questions open to concentrate on the latter, on how the mathematical knowledge differs from scientific knowledge.
The most immediate answer that first pops into your mind is the fact that, while all the sciences are desperately trying to describe what already is in the factual world, mathematics is only concerned with itself and the rather small world built up around it. We only stuff into it what makes sense, thwarting all obscure and incorrect knowledge as erroneous, as non-mathematics. The scientists can not do this. However mysterious and sometimes irrational and insane the recent observations within quantum physics may seem, we cannot simply ignore them. Until better explanations are presented we have to stick to the ones we have because it makes most sense.
Mathematicians rarely deal with what makes sense and what does not. A mathematical theory is either correct or is not. Either the theory is proven to work, or it is quickly discarded. Mathematics has no room for doubts. Even if all mathematicians stand up and say "We do not understand this." if it is possible to prove it, it is a fact.
Which leads us to the process of acquiring knowledge. Scientists usually work by means of observation. They observe, and draw conclusions from what occurred. For the knowledge is already there, waiting to be harvested. Mathematicians, on the contrary, obtain knowledge by making it up as they go. Mathematics is never there until we make it up. With sciences, if there is nothing to observe, no new knowledge can be obtained. In mathematics, however, if there is nothing to start with, it is quite easy...