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This was 2 decades after the death of Poincaré, so after the quote above, by the logician Kurt Gödel in 1931. Unfortunately for us, It had been proven that “we can’t prove that a system of axioms will not lead to contradictions”. So, in our case, we need to prove that the area of a unit square is 1 by examining all what it leads to and proving that there is no contradiction with all the possible consequences. He says we would prove that an axiom is true only when we prove that it doesn’t lead to any contradiction. I think this is off-topic, but maybe if you read Henri Poincaré’s quote you might notice a problem. So for example instead of taking the area of a unit square as axiom you can take the area of a rectangle as axiom, and compute the area of a unit square from it, this will be a theorem instead of an axiom. The definition will therefore be justified, from a purely logical point of view, only when it has been demonstrated that it does not lead to any contradiction, neither in terms nor with the truths previously admitted. La définition ne sera donc justifiée, au point de vue purement logique, que quand on aura démontré qu’elle n’entraîne pas de contradiction, ni dans les termes, ni avec les vérités antérieurement admises. In his book Sciences Et Méthodes, the french mathematician Henri Poincaré says: In mathematics, we can take as axioms any truths we want provided that they don’t lead to a contradiction. Remember when you got a zero for a question where you forgot to add the unit? The length of any polygon is always understood to be its. So make certain you can identify the two linear measurements of the rectangle. ![]() If you do not know what the length of a rectangle is, though, that formula will not do you much good. When we want to bring the equality to the real world we place the same object in the left and the right hand side. The formula for the area of a rectangle is: area length × width a r e a l e n g t h × w i d t h. 1 + 1 = 2 is true for any kind of object, we don’t need to write it for each object. Early mathematicians invented numbers to abstract the notion of counting for any object. Mathematics is mostly about abstraction and generalization. For rectangles its the ratio area/(length x width) which is constant. For example, a property that characterizes circles is that the value circonference/diameter is a constant, meaning that if you take any 2 circles with different diameters and compute this ratio for both of them, you will get the same value. The most important thing is that the area of a unit square inside this definition is something universal for all rectangles, which means that it is the same if you take any 2 rectangles. There is no absoute unit square, it’s us who choose it, so we choose the one that has area equals to 1. If the area of a unit square was equal to 2 then we can divide this into 2 squares, and take these instead as unit squares. In fact even if this is an axiom, if the area of a unit square was equal to another value, nothing will change! Why ? Because it is up to us to choose the unit square. What if it was equal to 2 or 3 or Pi, … Will the mathematics suddenly crash ? If you start with 0 truths then you can demonstrate nothing.īut how do we know that the area of a unit square is 1 without proving it. A demonstration aims to establish a truth (a theorem) from known truths. Why don’t we demonstrate axioms in math ? Well we need to remember what a demonstration is in the first place. I let you search a proof that the area of a unit square is 1… Probably you won’t find it ! You will rather find it as an axiom, which means a mathematical truth that doesn’t need to be demonstrated. Then the area of the rectangle will be w * l * 1 = w * l and we will be done. If we prove that the area of a unit square is equal to 1. ![]() First, a rectangle of width w and length l, can be divided into w x l unit squares.
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