Tuesday, April 11, 2017

P v NP: The Problem - 1

When a math test is given, the immediate attention is towards the answer. However, when an insoluble problem is asked for - it hangs in the air until someone grasps on to the fact that the problem is more important. Euclid, Gauss, Cantor, Gödel, Turing all understood this - but most mathematicians do not. Instead they ask the question which looks like it has an answer - but in reality it is not the question that they should be answering. This is true with this problem as well - before 1 can get to answering the question - one must throw out the outstanding garbage which stands in the way of unimpeded thought.
N = NP problem 1st demands a clear recitation of the problem, and points towards the solution. Because it seems obvious, but in fact is not, hordes of non-thinking get in the way of true wrestling with the problem. Some of this is known to mathematicians - but is obscure to the general public. But some eludes even mathematicians, and causes them to walk by even the hands of an answer. Then there is the problem that one person can prove - but it takes at least one to verify. This is the problem that Cohen had - their were only 3 mathematicians in his field which could verify the answer: one was Turing - who was dead - one was Gödel - who came out of the closet to praise be proof, but did not have enough mathematical intelligence anymore, and was basically a hermit - and Nash - who was at the time paranoid, and saw demons. If he had had a real sounding board, it is quite likely that he would have seen the few defects in his proof. Even though he had only a glimmering of understanding of the significance. So Cohen almost discovered the proof, and I feel confident would have made the full distance.
So what is the actual statement?
1st of all, the statement needs to be recursive - and it is not. 2nd of all, it needs to be reflective - and it is not. 3rd of all, it must have a party which has supernatural strength, so that the proof will be obvious to it, and therefore can be gleaned by those of us who are lesser intellects by intuition - this last part is basically the sum of what I created, the rest of which is already known, but it has not sunk in to the masses or the intelligentsia.
Let me 1st take a problem which bothers the nonmathematical - but does not worry the mathematically inclined. I will cite from Prime Obsession which should be above reproach, but unfortunately is not - even if John Nash praised it. It says that there are 2 divisions of fractions: rational and irrational. And on page 171, he explains the difference. But there are other ways of dividing fractions, and in this case one of the others should be used: computable and incomputable. Of course, Turing was the mathematician who realized, that fractions did not have to worry about whether they were divisible by some are arbitrary integer. This is inconsequential to whether or not they can be used for other purposes. While dividing a number in 2 and even set of units is important for many purposes, Turing showed that an infinite number which could be represented by a series of dots could still be divisible. Even an irrational number can be sometimes.
Why is this important? Because if one is interested in whether or not the fraction is irrational or irrational is hardly the point, the point is can we express the number, even if it requires … sometimes dots look important but are not, one can phrase .5 in several ways which require … but are not necessary. The same is true with pi, and e. in this case we need to use computable rather than rational as our standard. This does not bother the mathematically aware, but does bother people who are interested in numbers, but their main purpose lies elsewhere.
Why this is important be brought up later.

 Next Hard problems which trip up mathematical geniuses, other than those who cannot see the connection between Riemann.
( Now why should you come back when there are 90 odd proves,  which of course prove nothing?  Because there is a proof out there that proves one thing or another -  in this case both.  so it is at least interesting to be the one case which is crazy and different.)