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Can Mathematics be reduced to logic?

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Introduction

TOK 24/03/11 630 words 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? ...read more.

Middle

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) ...read more.

Conclusion

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. ...read more.

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