2007-03-02 23:07:44 · 4 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
Without proofs, we could assert that anything was true, and mathematics could not distinguish truth from falsity. So, the very notion of proof is essentially the same as the notion of "mathematical truth". A statement is mathematically true if and only if it has a valid proof.
Yes, that's a highly restricted concept of "truth", but then nobody should claim that mathematics captures the essence of truth in all its glory. In fact, Kurt Gödel proved (mathematically) that there will always be things that are true in any given mathematical framework that cannot be proven _within that framework_. You must expand the framework to get at some truths. Then that expanded framework will suffer from the same defect, and so on ad infinitum. Thus, even mathematics cannot really ever fully capture "truth" and "proof".
Confusing, eh? Well, that's why some of the best mathematical and philosophic minds have been trying to sort this out for a long time, and it ain't easy.
2007-03-03 00:09:27 · answer #1 · answered by DavidL 2 · 0⤊ 0⤋
The proof describes each step taken in arriving at the conclusion. This means that, in the case of homework or an exam, you both demonstrate your understanding and also have the opportunity to get marks even if you make a slight mistake somewhere along the line.
Formal proofs (e.g. differentiation from first principles) provide a detailed description of assumptions and steps for arriving at the conclusion. These steps and assumptions can be examined and validated. Thus providing a sound basis for which further proofs can be derived.
2007-03-03 07:20:35 · answer #2 · answered by davidbgreensmith 4 · 0⤊ 0⤋
The proof shows the work that you did to obtain the answer, if the answer was wrong then you or a teacher can show you where your mistake was. To me anybody can do math on a calculator, but do you know how to put it on paper,
My teacher always wanted the proof to thats starting with the formula and working throught the steps to find the correct answer. His idea the long way was the shortest way and the easiest is obtaining the answer. I use the same principal today. It is to prove yourself correct.
2007-03-03 07:16:16 · answer #3 · answered by idak13 4 · 0⤊ 0⤋
To show the teacher you aren't cheating...
2007-03-03 07:09:36 · answer #4 · answered by Anonymous · 2⤊ 0⤋