TOK essay

TOK essay example. My teacher gave me 32/40. I think part of the reason was that EVERYONE chose the same couple of topics. Everyone was afraid of mine because it contained the word "math." Don't be afraid of TOK math, guys, you're talking to a Math frickin' Studies kid:


9. 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 at least one other area of knowledge.

On the surface, mathematics and natural sciences appear to be the two areas of knowledge most likely to yield absolute certainty. People often assume claims backed by scientific knowledge must be correct. When I encouraged a young friend to have an orange as Vitamin C is healthy, she balked. However, when I pointed out that numerous scientific experiments back up my claim and Vitamin C is necessary to the formation of collagen, she appeared swayed by the scientific evidence. Math is connected with reasoning, which seems to suggest that the proofs of math are as certain as if A=B and B=C then A=C. Both natural science and math are backed by numbers, making them seem more precise than ethics. However, as evidenced by the advent of chaos theory, fuzzy logic and Heisenberg’s Uncertainty Principle, the more that is learned about both areas of knowledge, the clearer it becomes that ambiguity and uncertainty are prevalent. For a multitude of reasons relating to perception and precision, it will be difficult for science to achieve complete certainty. In mathematics, this may be possible in elementary arithmetic, but beyond that, mathematics appears similarly uncertain.

Perception is a major problem in science. Scientific experiments are based on empirical evidence, which usually requires perception. A biologist trying to differentiate between a Javan and an Indian Rhinoceros[1] would rely on visual differences such as size and skin texture. Although perception becomes more reliable through experience and training, Javan rhinoceroses are amongst the rarest large mammals in the world[2] and the scientist could mistake the pattern or colour of the skin, confusing one species for another. Perception is necessary even in situations where the scientist bases data on technology such as the Geiger counter rather than relying on direct observation. It is possible to misread the counter.

Each of these slight imprecisions is not significant and should not alter the results by much, especially when an experiment is carried out many times. After all, it is likely that there will be very minor variance between each experiment, considering the factors that are out of the control of the scientist, such as the mood of the subject being experimented on. However, even the smallest indistinctness prevents the scientist from being completely certain. Thus, science can only be highly precise; it cannot be completely certain.

Emotion and ethics influence science and can thus affect theories and viewpoints more, whereas mathematics is almost independent of them. It is possible to misread a number in a moment of joy or anger, but it is much less likely that a mathematician will refuse to do a question because he feels it would be immoral to do so. Conversely, a scientist may shape his hypothesis and his experimentation around what he believes. Catholic scientist William A. Dembski is a proponent of the Intelligent Design theory. Although he has undoubtedly researched both sides of the argument and decided that it is more likely God exists, his faith probably contributed to his viewpoint. His bias would slant how he looks at material, and his interpretations would be tailored to his teleological beliefs. A mathematician may wish to obtain a certain answer, but he or she is more removed ethically from his or her work.

Science uses the experimental method, meaning that during experimentation the scientist attempts to manipulate one variable while trying to keep all others the same. Although the manipulated variable is chosen based on careful thought and research, it could be that the results of an experiment support a hypothesis even though the hypothesis is false. Experiments in the United States have shown that people of colour are more likely to get Alzheimer’s disease, but perhaps it is not because they are genetically predisposed to it, but because they are simply unaware that there is a cure[3]. A similar problem exists in math, the question of whether a pattern exists in nature, or a mathematician wants the pattern to exist so he or she manipulates the numbers until they do.

Certainty is possible in rudimentary arithmetic. Few doubt that 1+1=2. It is true that in logarithmic systems, 1+1=1, but that is because what is meant is not the sum of two objects, but the sum of log1+log1. One can be certain that 1+1=2 because the definition of two is two ones. For this equation to be false would be to refute the meaning of the terms.

This does not carry over into all areas of mathematics. Although mathematics may use syllogisms like logic, the validity of a deduction is based on the logic of the argument and not the truth of its components. This truth is simply assumed. However, for mathematics, this truth is necessary. It may be that a mathematician produces a logical argument, but uses a proof that is not completely certain. Thus, his own proof cannot be certain, even if it is valid. On other occasions, logic cannot be used to derive an answer. Mathematician Gödel realized that there are some true statements that cannot be proved nor disproved in a mathematical structure. He showed that no consistent computable axiomatic system can be complete and thus, the axiom’s consistency cannot be proved.[4] Thus, one cannot find all the truths, and sometimes when one finds a truth, it cannot be properly proved. These statements have no way of being shown to be certain.

One problem of math is that it is a solitary activity, and the process of checking over a proof is extremely time-consuming. Thomas Hales created a proof to Kepler’s sphere packing conjecture, which stated that the most economic way to pack balls is to use face-centred cubic packing[5], requiring twelve mathematicians to verify. Even with a team, it took four years for them to be 99%[6] sure the proof was correct. The Classification Theorem for Finite Simple Groups is estimated to be between 10,000 to 15,000 pages.[7] Some mathematicians are simply unwilling to spend so much time verifying one proof, or acting as a referee when they could be working on their own theorems. In the case of Hales’s proof, it was published, even though the team was not sure it was correct. Modern mathematicians is more accepting of slight fuzziness than past mathematicians. Although once math adhered to the concept of rigorous proof, modern math has changed. Computers are used to calculate such large numbers, only other computers can check the work – humans cannot do it. This creates a self-referential system that can overcome human limitations, but also cannot be certain because of its circular logic. Some mathematicians believe that instead of proofs, some propositions can be compared with computer-run experiments or real-world phenomena, and mathematician and writer Keith Devlin thinks that proof will be less important within 50 years.[8] Certainty may still be possible without rigorous proof, but as of yet it is too early to tell what flaws and uncertainties lie in using a computer.

This ambiguity in math affects certainty in science as well as many other areas of knowledge. Physicians use math which may be accurate but not completely certain, further preventing them from reaching absolute certainty. There are already barriers to certainty in science, as discussed above; added to that is the uncertainty in math.

Connections between areas of knowledge in general make it difficult to state whether certainty is possible in natural sciences. A biology experiment does not solely require knowledge of biology, but sometimes human sciences, as well. Then the degree of certainty in human sciences and how it affects the experiment must be examined. Even then, it is difficult to precisely pin down to what extent the uncertainty in human sciences will affect the uncertainty in the biology experiment.

Pythagoreans thought that there were only rational numbers. Then, they discovered that the square root of two is irrational, breaking one of their fundamental truths. A similar situation happens in science. No matter how certain one is of a concept, there is the possibility that it will be false. As can be seen, both mathematics and science can result in accurate knowledge. However, they are not as certain as it seems at first glance. In fact, both appear to become more and more uncertain. It is unsure whether further developments will continue this trend, or bring mathematics and the natural sciences closer to complete certainty.

Word Count: 1386

[1] Massicot par. 2
[2] Wilson and Burnie 229
[3] Nauert, Ph.D. screen 1
[4] Casti 371
[5] Plus screen 1
[6] Friedl screen 4
[7] Ibid screen 3
[8] Ibid screen 4