Theory of Knowledge (TOK) Essay
How does the mathematician’s knowledge differ from that of the scientist?
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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 endeavours. 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 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 the 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 to make something up and work from that. Mathematical knowledge may stem from nothing.
Mathematics is self-contained. Much of the mathematical knowledge is not good for anything but mathematics itself. This is why it is sometimes called art. It is indeed fascinating, but what is the good of much of it? It is sometimes a mere struggle for enlightenment. One example of this is the great theorem of Fermat, which for hundreds of years was unproven. Last year a man named Andrew Wiley came up with the proof that, yes, there are numbers such that ax + bx = cx for any positive integer value of x. The theorem as such is useless, but the greatness and beauty of the proof are enough to make a mathematician like Andrew Wiley spend half his life in trying to find it. Here it is the knowledge itself, and not what it means that matters most. A scientist would hardly find the knowledge that two hydrogen atoms combine with one oxygen atom to form water more important than the remarkable properties the compound shows to carry. Sciences are generally not self-contained, they have a reason beyond themselves.
Now you may say, “but mathematics is widely used in all the sciences, how come if the two differ so much?” Yes, it is true that mathematics is a very important part of modern science. That is because it is such a great tool to use when solving problems. But mathematics is never applied directly. First, we must transform the knowledge we have within science into mathematical equality. For example, it is not automatically true by nature that Newton’s second law (F = ma) is valid. It only works because we have defined it to be true when we measure mass, acceleration and force using the units we created for this very purpose. We used mathematics to get around a problem that we experimentally came across, and otherwise could not solve.
This shows the generality of mathematics. If a mathematical concept is true once, it will remain true regardless of external factors. A scientific concept is on the contrary highly dependent on external factors to be true. Take Newton’s second law again. It only works here on earth, where all external factors that it requires are satisfied, because it was found out by means of experimenting scientifically. Its mathematical counterpart, Einstein’s theory of relativity, works universally (as far as we know), because it was calculated mathematically.
Mathematics and sciences are two sharply disparate systems of knowledge. Knowledge is acquired differently, used differently and perhaps most importantly, looked upon differently. Sciences tell us something. Mathematics is more about knowledge for its own sake. Perhaps you could say that mathematics is a generic means for treating knowledge. And knowledge must be well treated.
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