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# 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 that 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 that is completely true as there could be another concept that holds more truth to it. In mathematics, it could be possible to obtain 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 that are assumed to be truly called axioms. For example, a mathematician might present a very logical argument that 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 proven as completely certain. Another example is how the Pythagoreans were disproved about their belief that only rational numbers existed. They always believed that only rational numbers existed but then when irrational numbers were discovered, their belief was disproven and eliminated their certainty.

The Pythagoras theorem has a logically structured proof that it works for a triangle where one angle is a right angle and this has been accepted because no contradictions have been found to this law. In addition, perception can also affect the opinion of mathematicians on certain mathematical concepts. For example, shapes of graphs and other geometric concepts such as drawings lines of best fit can be subject to different opinions, making it impossible to make one answer completely certain as some mathematicians might have valid reasons or proofs for their answer, although none of them could be completely certain.

Concepts in the natural sciences contain less certainty than in mathematics. Science theories and results are based on experimentation because it is through this that concepts are developed. For example, a scientist can only determine the number of calories in a peanut if they actually carry out a test under perfect conditions for determining the energy released. This is different from mathematics where the knowledge is largely based on logic and reasoning and no experimentation is required. It is difficult to provide proof for knowledge based on experimentation because even if experiments are carried out under the same conditions, scientists will not obtain exactly the same answer because there are uncertainties involved, largely due to human error. Therefore, it is not possible to obtain absolute certainty in science because there is no logical structure to proofs as there is in mathematics. This is shown through Newton’s Laws of Motion and Einstein’s Relativity.

When the Laws of Motion were tested, they were seen as certain, but it was later shown that they are only an approximation to Special Relativity which has now been superseded by Quantum Mechanics. This shows that concepts in science are never too certain because they often replaced by new theories. Perception contributes to the uncertainty that revolves around science. Scientific experiments involve perception because some results are visual, such as the colour of a reaction between two chemicals, and hence scientists might possess different opinions. For example, a biologist might try to differentiate between an African elephant and an Asian elephant. He or she would rely on characteristics such as size, weight and features such as the ears in order to classify the species.

A scientist could mistake the size of the ears, which are mainly used to classify the elephants and hence the species are confused for each other. Perception is vital in such situations because it is possible to make an error while some scientists might argue over the features of a species or the colour of a reaction because they see it in different ways. Hence, the possibility of absolute certainty is eliminated when looking at visual observations in science as there is no proof of it due to various scientific perceptions. Although these slight differences in perceptions might not make a big difference to the experiment that is carried out, it shows that science cannot be absolutely certain because of the slight variations, but it can be highly precise if the experiment is repeated several times.

Emotion and ethics also play a part in the certainty of knowledge in science. Theories in science might be affected by emotion because certain observations might be ignored or viewed wrongly in a moment of joy or anger. In mathematics, it is more likely that a mathematician will refuse to do a problem if in a similar mood because they might think it is immoral to do so. In addition, if a mathematician does a problem wrongly, he or she is able to refer to a more certain proof that shows a valid way of solving the problem. However, in science, a scientist has almost no method of referring to a more certain observation because everyone’s perception of an experiment might be different. Moreover, science uses experimental methods to make statements while mathematics largely uses logic, reasoning and assumed statements to form proofs. For example, a study in the United States shows that people of colour are more likely to get Alzheimer’s disease. However, this is only simply because they are unaware that there is a cure.

The experiments show that people of colour are more likely to get Alzheimer’s disease but this cannot be certain because they might not have been made aware of the cure. Although one assumes that once a theory or an observation is proved rigorously, it is completely certain, this is not true. This is because proving something does not make it true. It just means that you have convinced people that any evidence you have supports your conclusion. Hence, although certain concepts in mathematics might be proved rigorously, it does not always mean that they are completely certain because there are always some assumptions and an element of perception that is involved that can create some uncertainty.

In conclusion, the ambiguity in many complex mathematical concepts affects the certainty in science as well as other areas of knowledge because some science concepts are based on numbers such as quantum mechanics in physics. Natural sciences are most affected by mathematics in physics because physicians use math which is not completely certain but is very accurate and hence there is always the possibility of discovering a theory that has more truth to it than the previous one. This adds to the barriers of uncertainty that exist in science. This idea leads to the connections that exist between different areas of knowledge. Experiments in biology might require knowledge of subjects from human sciences and it is difficult to state the extent to which the uncertainty in human sciences will affect the uncertainty of the knowledge in biology.

Works Cited

• “Proof and Science.” Carlton Comprehensive Prince Albert, SK. 20 Jan. 2009 <http://www.carlton.srsd119.ca/chemical/Proof/default.htm>.
• “Truth and proof in science.” 20 Jan. 2009 <http://physics.ucr.edu/~wudka/Physics7/Notes_www/node8.html>.
• “Truth and Proof In Science.” Oracle ThinkQuest Library. 20 Jan. 2009 <http://library.thinkquest.org/C0114565/content.php?id=5>.