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jueves, febrero 14, 2008

El concepto de los números negativos

DMM explica el teorema de Gödel para torpes. En uno de los pasos, al explicar el concepto de números negativos, aparece el tema de la existencia de objetos matemáticos - en el caso de los enteros negativos, su existencia procede de combinar las operaciones de contar (enumeración) y restar.

(Aprovecho para expresar mi postura sobre ese debate filosófico: en mi opinión, todos los objetos matemáticos tienen ese tipo de existencia - son el resultado de operaciones posibles, no son "objetos auténticos" en el sentido platónico).

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The statement points out a deficiency in the sorts of numbers we are considering. It points us to the existence of more numbers, with different properties to the sort of numbers we were originally talking about. When negative numbers were first conceived of by mathematicians, they were controversial and regarded by many with scepticism. After all, you can see 3 apples. What does -3 apples look like? It makes no sense! The very concept of -3 is absurd!

To people with a simplistic understanding of numbers and what they can be used for, the concept of negative numbers is indeed strange. We all go through this process at school, when we graduate from counting apples to doing more abstract mathematics involving subtractions. We are often introduced to negative numbers via the concept of the number line. This makes it seem a bit more natural that there are "more numbers" over there, on the left side of the normal counting numbers. And we discover that negative numbers are useful for things like accounting, and keeping track of debts, and so on.

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