Can Mathematics be reduced to logic?



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Math is just a language which humans have created to describe size, quantity, and order. Many people think that math is just based on logic, but I think that this is not true because there are some “theorems” that do not make any sense if it is just based on logic. However, math can be based on logic if it’s simple. For example, we know that 8/2 = 4, therefore 4*2 = 8; but when it starting to be harder, many problems will start to arise. In fact, according to the previous example, if 1/0 = 0, does 0*0 = 1?

Divisions by 0 are always been a problem for most of the students throughout the world, me included. Since the primary school, I always asked my-self and my teacher why we can’t divide by zero, but I never had an answer. As I said before, if 1/0 = 0, 0*0 should be equal to 1 and this is not possible. This means that there is no a real answer for this operation and the answer is undefined. This proves that logic is not enough to understand fully mathematics problems; therefore we have to use also reason.

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Another particular division that involves the use of the zero is: 0/0. The answer could be 1 because any number divided by itself is equal to 1, and it seems to be right because if you do an inverse operation, the answer is still valid (1*0 = 0). However, any number multiplied by 0 equals to 0, which means that 0/0 equals any number, and this is impossible. So mathematicians said that the answer for this specific problem is indeterminate. (www. mathmojo. com)

Another famous equation “proves” that 2 = 1 and it can help us to make us believe that math cannot be reduced to logic. Apparently, it may seem to be right using just logic, but you will realize that the following equation is wrong. Let’s suppose that: a = b Multiply both sides by a a2 = a*b Subtract b2 from both sides a2-b2 = a*b-b2 Apply the distributive law to both sides (a+b)(a-b) = b(a-b) Divide both sides by (a-b) (a+b) = b Substitute all a’s for b’s (remember, if a = b you can do this)

a+a = a Regroup the two a’s in the left side, and rename it 2a 2a = a Divide both sides by a 2 = 1 (www. mathmojo. com) Why is this equation wrong? If only logic was used in this problem, the equation must be completely right because all the steps follow the math laws. This is why we need reason to understand why this equation is wrong. If you pay more attention on the point that is written in italics and underlined, you will realize that the equation is divided both sides by 0.

As I showed to you before, we can’t divide any number by 0. This problem is very similar to inductive reasoning. Inductive reasoning is drawing general conclusion from specific examples. (Van de Lagemaat) In fact, the equation “a = b” is valid if it’s multiplied by “a” on both sides; the equation “a2 = a*b” is valid if b2 is subtracted on both sides; the equation “a2-b2 = a*b-b2” is still valid if you apply the distributive law on both sides. Therefore all the equations are valid if they are modified in the same way on both sides.

We realized that this inductive reasoning is no more valid from the fifth step on because they do not have more the same value. As we can see, inductive reasoning is not very reliable and most of the times it leads us to the wrong conclusion. In conclusion, I think that mathematics cannot be reduced to logic.

Bibliography  Van de Lagemaat R (2005) Theory of knowledge, Cambridge Univ Press, Cambridge, p. 121 http://www. mathmojo. com/interestinglessons/division_by_zero/division_by_zero_1. html http://www. mathmojo. com/interestinglessons/1equals2/1equals2. html.

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