Can Mathematics be reduced to logic?

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24/03/11

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Can Mathematics be reduced to logic?

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. 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)

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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 ...

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