It has been a long time since I last posted but, I have been pretty busy like always.
Einstein stated that the most simple formulas explain the most phenomena in the the universe or something like that.
What can be more simple than an "If P then Q statement."
Say you wanted to prove anything. You can proceed in this way.
Say you wanted to prove that a set A is equal to a set B.
What can you do?
According to tradition we must prove that A is a subset of B and then prove that B was a subset of A.
Let us put this into an "if p than q" statement:
Say if A is a subset of B and B is a subset of A [this is our P statement] then A = B [this is our Q statement.]
Now we have a reason to prove mathematical statements of this sort in this fashion.
Previously my teacher will say if you want to prove that two sets A and B are equal, you must prove that A is a subset of B and that B is a subset of A and you will be done.
But now I say if you want to prove that two sets A and B are equal, you must clearly see the
whole argument first and then prove that they are equal. Start by defining the mathematical statement for the antecedent P and then by defining the mathematical statement for the conclusion p and then you will see the whole mathematical argument more clearly.
Take for example 2x - 5 = 7
then 2x = 12
then x = 6.
say x = 6
then 2x = 12
then 2x -5 = 7
Where is my mathematical argument?
Say if (2x - 5 = 7 then x = 6 and if x = 6 then 2x -5 = 7 ) Then (x = 6 and 6 = x.) and my understanding is wholly, but my teachers never taught me in this way.
Restating:
If A [2x - 5 = 7] is a subset of B [ x = 6] and B [ x = 6 ] is a subset of A [2x - 5 = 7] then A = B.
Clearly in this situation the two sets are equal.
But more is to be said into this, we must say that the proof is by deduction as opposed by induction. Each statement in the argument is deduced from the previous statement. Deduction is from the particular to the general while induction is from the general to the particular.
One is to conclude from this example that the more proofs we construct in this way that we will have no other option than to conclude by induction that knowledge becomes more easily acquired if proceed in this fashion above. Induction in this sense is more like by "Experimentation." That, is the more we experiment the more we understand about our subject matter.
