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I think, I don't see there the "proving backwards" trick described. The technique goes as follows: a student starts with something (for example, an equation) th

I think, I don't see there the "proving backwards" trick described. The technique goes as follows: a student starts with something (for example, an equation) they need to prove, and transforms it until they get some trivial identity like 1=1. If they succeed in this procedure, they claim that the original equation they started with was correct too. Everyone does it, and it takes ages to make them unlearn it. If somebody is wondering why the technique is incorrect: starting from a false premise, any statement can be proven, including things like 1=1 or 1=0.

arielby 11y ago
That's why you make sure all operations you do are bidirectional.
- a-nikolaev 11y ago
  If you are careful and each step gives an equivalent statement, this is correct. For example, if you read math books/papers, whenever such transformation is made, the author usually states that it is equivalent. However, the problem is that students apply this technique blindly manipulating the formula in any way they can/want. Such carelessness is the real problem.
robryk 11y ago
As long as your steps are implications in the reverse direction (from the later to the earlier statement) this is perfectly correct. In most cases I've seen people do that the transformations are actually equivalences, which muddies the waters somewhat. What is a problem is when people do use reverse-implications that are not equivalences, arrive at something false, and claim that the original equation is false.
- a-nikolaev 11y ago
  Yes, I know ) But then this whole derivation now lives in a "special world", where you cannot do all math, but only certain subset. Your example is cool. I have not seen many exmaples of such logic, but people probably do it too.. )
  - robryk 11y ago
    The most common example of such logic is multiplying by maybe-zero (transforming a=b into ac=bc) and factoring out a possibly-zero factor (transforming ac!=bc into a!=b).

arielby 11y ago

That's why you make sure all operations you do are bidirectional.

a-nikolaev 11y ago
If you are careful and each step gives an equivalent statement, this is correct. For example, if you read math books/papers, whenever such transformation is made, the author usually states that it is equivalent. However, the problem is that students apply this technique blindly manipulating the formula in any way they can/want. Such carelessness is the real problem.

robryk 11y ago

As long as your steps are implications in the reverse direction (from the later to the earlier statement) this is perfectly correct. In most cases I've seen people do that the transformations are actually equivalences, which muddies the waters somewhat. What is a problem is when people do use reverse-implications that are not equivalences, arrive at something false, and claim that the original equation is false.

a-nikolaev 11y ago
Yes, I know ) But then this whole derivation now lives in a "special world", where you cannot do all math, but only certain subset. Your example is cool. I have not seen many exmaples of such logic, but people probably do it too.. )
- robryk 11y ago
  The most common example of such logic is multiplying by maybe-zero (transforming a=b into ac=bc) and factoring out a possibly-zero factor (transforming ac!=bc into a!=b).