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Kindly comments on Open Thread, April 15-30, 2013 - Less Wrong
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The rule isn't that you cannot divide by zero. You need a rule to allow you to divide by a number, and the rule happens to only allow you to divide by nonzero numbers.
There are also lots of things logicians can tell you that you're not allowed to do. For example, you might prove that (A or B) is equivalent to (A or C). You cannot proceed to cancel the A's to prove that B and C are equivalent, unless A happens to be false. This is completely analogous to going from AB = AC to B = C, which is only allowed when A is nonzero.
However, {false, true} - {true} has only one member, and so values from it become constant, whereas ℝ - {0} has many members and can therefore remain significant.