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The Usefulness and Limitations of Reasoning in Maths.

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Introduction

´╗┐Eduardo Salazar TOK Mr. Edwards Jan, 06, 2013 The Usefulness and Limitations of Reasoning in Math Humans use reasoning for a wide range of everyday occurrences; in many cases reasoning is used in the process of predicting the future and in making assumptions. The type of reasoning one chooses to use depends highly on the purpose of what one is investigating. When thinking about the various classes I take, I came to conclude that I use reasoning in mathematics the most. Numerous ideas in math are based upon reasoning and without it math would be seemingly much more difficult. Two key forms of reasoning I will be focusing on are inductive and deductive reasoning. Inductive reasoning evaluates propositions that are derived from specific examples. This can be described as creating generalizations based on individual cases. Inductive reasoning is the type of reasoning that is built on premises that do not necessarily make a correct conclusion, but instead base the conclusion on what is most probable. ...read more.

Middle

are mortal. Because Aristotle is a man, and all men are mortal, it is also true that Aristotle is mortal. This is deductive reasoning and it gives a valid conclusion because it accounts for ?all? cases. As long as the argument follows the structure: 1. P?Q 2. P (Hypothesis stated) 3. Q (Conclusion given) With a valid hypothesis the conclusion is valid. The limitations and usefulness of these concepts of reasoning in math are key. In math the concept of inductive reasoning is not used, since in math, in order to get a correct answer everything has to be valid and one cannot make generalizations. Math is all about specific rules that must be followed step by step in order to uphold the one and only correct answer. However, if generalizations are made in math, the final answer will not always be correct (refer to the example where Bruce could be the 5%). ...read more.

Conclusion

In the previous example the two expressions are two premises. From the two premises one can conclude the answer of the problem (X=3 works for one equation therefore it must work for the other equation). This is only one extremely simple example of many in which one uses deductive reasoning in order to solve a problem or come to a conclusion. The use of reasoning in mathematics is relatively extensive. The use of deductive reasoning is very useful, common, and an important part of the subject. On the other hand, inductive reasoning cannot be used as frequently. Reasoning is a very useful device when studying math, but using the wrong type of reasoning can lead to a student making incorrect assumptions and conclusions. Opting to use deductive reasoning to start with in any problem solving would be the best advice. Inductive reasoning might not be the best tool but could be used under few special circumstances where it can aid in solving a complex problem, by providing a new way of looking at the problem (thinking outside the box). ...read more.

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