The Usefulness and Limitations of Reasoning in Maths.

1. 95% of professional soccer players are paid more than 1 million dollars.
2. Bruce plays professional soccer.
3. Therefore, the probability that Bruce is makes more than 1 million dollars is 95%

This example is based on a premise and then a conclusion is formed. This statement therefore may not be true. Because even though Bruce is a professional soccer player, he is possibly part of the 5% of professional soccer players that get paid less than 1 million dollars.

The other area of reasoning which is deductive reasoning, distinguishes itself from inductive reasoning because it is based on premises that make the conclusion to be true, in all cases. As appose to inductive reasoning. An example of deductive reasoning:

1. All men are mortal
2. Aristotle is a man

      3. Therefore, Aristotle is mortal (example from wikipedea)

Deductive reasoning talks about “all” meaning that all men (in this case) 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%). All premises have to be valid in order to obtain a valid answer; thus the use of inductive reasoning in math is very limited. One area of math where inductive reasoning is used is when you try to find patterns in geometric images. The conclusion reached when using inductive reasoning is then called a “conjecture”. Inductive reasoning in this case is highly useful and since it is based on a generalization it is impossible to use deductive reasoning in order to obtain this conclusion. The limited use of inductive reasoning in math is due to the idea that a mathematic answer is correct in all situations and there is no room for generalization, which would be possible in history for example. Despite this, if one would use inductive reasoning, one would most certainly end up with the correct result either way.

Deductive reasoning is used in most every area of math. Algebra is a good example of an area were deductive reason is very often used. An example of this is if: X-3 = 0 and X+1 =4, one can see that X=3. 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).