Something About Logic

1. Invalid solutions to quadratic equations: I remember, we had to solve some maths in Chemistry which usually yielded two solutions, one of them would be absurd. I always wondered why one of the mathematically derived solutions should be absurd in reality. One reasoning could be, the mathematical modeling of the reaction scenario is more general than what was needed; so, not all solutions are applicable for the Chemistry scenario. What about those quadratic equations where if you put one of the solutions in, the mathematical equation is not satisfied! It can be understood through logic. Both of the solutions to the quadratic equation are valid only when every next step in the solution-procedure is an 'if and only if' or bi-directional implication. For example, x + 2 = sqrt(4 -x) has an absurd solution where x = -5. Putting x = -5 in the left hand side of the equation gives -3 whereas in the right hand side gives 3. For when we square the equation on both sides, we have a unidirectional implication or the 'if' part only: x+2 = sqrt(4-x) implies (x+2)^2 = 4-x, but not the other way around.
2. Deductive vs. Inductive Reasoning: Deductive Reasoning is based on premises and logically arrived upon conclusions, whereas Inductive Reasoning is based on examples studied so far: 'I have seen this event in 1000 cases and I say it will also happen in the 1001th case.' Mathematicians do not consider Inductive Reasoning as a form of proof in contrast to experimental scientists who are usually reasonably (due to practical limitations) satisfied with Inductive Reasoning backed up by some number of experiments. Yet we should always keep in mind that there is no categorical reason to expect that the 1001th case will be similar.