The ongoing name calling and thousands of words resulting from my posting a rather simple mathematical proof which tends to contradict Cant ideas has gone as far as it is going. I will post this and am done with it.
Blackjack has many elements and possibly the least relevant to winning is the math element (now that pure mathematics has shown blackjack is beatable and how to beat it at least risk but such things can be memorized or taken from a program and after being memorized or found and applied have little relation to winning) but that is what happens to most interest me about the game. It was in that context I innocently answered a question on RGE saying the floating advantage was something that happened in blackjack but the effect was mild and often should be ignored but that it would make sense to drop levels of betting one true count when less than a deck remained in six decks because the advantage would be in the range of .5%, the approximate value of one true count. This rather modest post resulted in a spate of virtirputude from Cant, the magnitude of which I have never seen anywhere except from trolls like Cant and Puiu.
I have allowed myself to be baited by both Cant and Puiu and others a number of times and doing such things have been my mistakes. Senseless arguments with trolls do nothing to enlighten and much to confuse and sharing of accurate information, not sharing confusion, is what the Internet and sites like this ought to be about.
It is my concern now my getting into this pissing contest might trivialize the power and IMHO the beauty of formal mathematical proofs. To the extent Cant�s thousands of words might tend to make persons with nonmathematical backgrounds skeptical of the power and legitimacy of a mathematical proof , the board is being ill-served. To the extent the thousands of words might tend make persons with nonmathematical background skeptical of the use of symbols to denote numbers in a mathematical proof, the board is being ill-served.
So this post is meant to accomplish three things: One is to explain more fully and in terms of the notation, definitions, theorems, postulate, and formal mathematical steps in this proof the power in general of mathematical proofs and to explain how the hypothetical in mathematical proofs relate to real things with a hope people with nonmathematical backgrounds will not be misled into thinking formal proofs are irrelevant. Second is to meet Cant�s objections as much as I have understood his fractured prose as objections. Third is to get me out of this pissing contest. After this post I will not address Cant again on this or any other topic.
Proofs, depending on the circumstances, can take various forms. But, whatever the form, they have things in common. A proof such as this one uses the device of the undefined value which is different from the substance of a geometric proof but not different in form because both use some basic propositions to show the truth or falsity of another proposition.
The reason it uses this device is because the device is suited to showing equalities and inequalities. The aim of this proof was to show an inequality between two values, in this instance the value of E(i,n) and the value of E(i,m) which is the definition of the floating advantage if the value of E(i,m) is greater than the value of E(i,n). A way inequalities may be shown is to take an equation which has both of the values contained in the equation and manipulate the equations to simplify them to a point where the relationship between the two values can be shown. So long as the equality is maintained, one is just oing the same thing as solving an algebraic equation. That is the basic this proof and similar proofs encompass. For some reason or another Cant has said the method I used was inconsistent with the method used in the true count theorem but he is wrong. Exactly the same thing was done. Jalib started with the equation for the expected value of an arbitrary future true count at a starting point of an arbitrary specific true count and showed that equation simplified was the same value as the starting arbitrary true count.
And that is where the power of a proof using the notation of undefined variables lies. One can start from any arbitrary point and, starting from that point, one can show a relationship between two different values exist. Jalib showed the value of any arbitrary true count at a starting point was identical to the expected value of future true counts. It matters not where one starts so long as relationships hold and equations stay equal on both sides.
I am now going to attempt to explain how those theoretical terms I used relate to real life and why the Cant criticisms were misplaced again in the hopes I am underlining the power a mathematical proof has to real life to persons not trained in math.
E(i,n) and the others stand for something: E(i,n) is the numerical value of the expectation for basic strategy for an arbitrary stack of cards of consisting of a specific number of cards. The definition of the floating advantage starts with one stack of cards, not many stacks of cards. Many stacks of cards result if you have access to enough cards to replace cards from the original stack maybe billions of time and enough space to lay out all possibilities of smaller stacks of cards in piles. The beauty of mathematical analysis is you can do that without going to the effort and expense of physically doing it.
Without doubt that starting stack has an expectation, a numerical value of how often a player will win and lose if the player plays basic strategy when dealt from that stack without cards being exhausted. The tools of calculation of combinatorial analysis or sims we have now may not be able to calculate precisely to the 100th decimal place but we know there is some accurate value to the largest decimal place we can imagine. Again, the power of the undefined value is demonstrated. It is accurate to the largest decimal place we can imagine for the purposes of a specific mathematical proof.
Now let us buy billions of decks of cards, hire millions of workers, and rent a building about the size of California. On the floor of that building we will lay down some billions of piles of a size smaller than n, let us call the size number m. Each of these piles will consist of every combination of every way the original stack which has an expectation of value E(i.n) might have been arranged because the expectation, E(i,n), is not based on a specific arrangement of cards but all arrangements . Next you can calculate the expectations of all the stacks of m size and average them. Hardly practical in real life but, again, the power of the undefined value raises its face. We can do that complicated laying out by typing a few lines on a word processor and accomplish the same thing for the purposes of a mathematical proof. I did that with Theorem I without all the bother. We know we will have the same value of expectation because the same cards in the same frequencies are involved.
Next we could have gathered up all the stacks, counted them, and sorted them by the true counts they happened to have. We then could have calculated the expectation of each stack of a specific true count and averaged those, getting one specific value for each. But we would notice there were more stacks of true count i than of any other count and there would be less stacks as the true count departed from i. That would happen because the true count theorem is proven. So to get the true expectation of all the piles, we will have to correct for frequency piles of different true counts occur or we will not get the correct value for the overall expectation of the piles, an averaging process. That is Theorem II without as much work or expense.
After going to all that trouble, we have one number, one value which represents the expectation of the original stack of n cards being equal to the exact same value of the expectation of the cards stacked and sorted in the building the size of California of m size. Or, much easier, we express that in terms of the start of the proof.
Cant has said means have been switched in the proof. So far there have been averaging processes used. E(i,m) or the more general E((i +,-,j),m) are numbers derived from averaging the expectations of all of the stacks of specific true counts on the floor of that building the size of California. That they consist of those stacks cannot imply they are not specific numbers which still average out to keep the equation of Theorem II exact. Nothing will have been changed so far as numbers nor exactness are involved. Nothing has been switched.
In the proof reordering occurs. In step two billions of stacks are carried to place i+j and i-j stacks in proximity. Or, in mathematical terms, the cumutative law is applied which is considerably easier. Now we can be done with our millions of stacks. We could calculate the number that is the expectation for each of the i+j and i-j stacks and replace them with a piece of paper with the number written on it and the percentage of times a specific true count occurs. Since we knew the i+j stacks and i-j stacks have the same frequency, we could simplify again by averaging the two numbers representing each of these millions of steps and replace each of the pieces of paper with the average value of the expectation of the i+j and i-j stacks. That is step three.
We can now quit paying rent on the big building. We can rent a right small room and lay the pieces of paper with numbers written on them down on the floor.
The proof has gotten us to this point. Of course, since we started out with an undefined value in the proof, we cannot define the numbers written down either. What we know is they are sheets of papers with numbers on them and every piece of paper is included as one of the terms the proof. The equality remains. Nothing has been changed. Nothing is left out. However, not knowing the exact value of any of the numbers is possibly not required because all we need to find out is whether that first number, E(i,n) is greater than the number E(i,m). Perhaps if we know something about the relationship of E(i,m) and the other numbers in the room we may be able to find this out.
We know of a hypothesis, a conjecture, about the relationship between the number E(i,m) and each of the other numbers lying on the floor and realize if that conjecture is true, we can ascertain whether E(i,m) is greater than E(i,n). If and only if all of the other numbers are smaller than E(i,m) can we know E(i,m) is greater than E(i,n). If that happens to be true, the frequency E(i,m) occurs multiplied by E(i,m) and the frequency of all the others which are written down multiplied by the average expectation of the others averages to the number E(i,n) and for two quantities average to another and one is larger than the other, the larger quantity must be larger than the average. Perhaps the conjecture can tell us that much about the numbers without having to do any more stacking and comparing.
Since the relationship is a conjecture and since we cannot read all the pieces of paper and because a mathematician can never be a know it all and can assume nothing, the fair mathematician doing something like this will start with a conditional, not a definite statement. The mathematician will assert in such a proof �if the conjecture is true, then the conclusion is true� and let others decide if the conjecture is true. The mathematician realizes the conclusion is only valid if the conjecture is valid.
That a bow effect exists in the values of E(i,m) and E((i+,-j),m) was the conjecture in this proof which led to the use of a conditional statement in this proof. The thousands of words Cant has used in response to the proof have all been assertions with no supporting background as to truthfulness and he has made no conditional statements. The attitude that saying something makes it true is not a mathematical concept.
I did assert the truthfulness of the postulate on the bow effect and will not apologize for that. I believe it to be true and gave my reasons for the belief in the body of the proof but still couched the conclusion of the proof in a conditional statement because, up front, I said it was conditional. For the conclusion of the proof to be valid, E(i+1,m)-E(i,m) must be smaller than E(i,m)-E(i-1,m) for all i�s. when the various expectations of an array of subsets of true counts are created from a set of cards.
And if that assertion as to the bow effect is true, each piece of paper has a number written on it smaller than the number E(i,m) and E(i,m) has a value larger than E(i,n).
Mathematical proofs are powerful and show relations not readily seen when the mathematics in the proofs is accurate, the premises are accurate, and any hypotheses are clearly stated as hypotheses. I published one I thought might be of some general interest and might shed some light on a Griffin observation effects of removal do not account for all the differences in expectation among different numbers of decks. I made modest claims, pointing out I believed the bow effect occurs the way I defined it and gave reasons for that but the final statement in the proof was only a conditional one.
And for doing that, posting a tentative, neutral speculation, Cant has called me a liar, a person of no honor, a person totally unversed and unskilled in math, a dishonest lawyer, a syncopate, possibly insane, and other things. And he has not just done it on this board but has called me those things on at least two other boards.
What produces a person of such mean spirit that no one can post anything which might be of some general interest without garbage being thrown all over them? What produces a person who delights in throwing garbage? I have no idea.
But I do know one thing. I do not have to deal with garbage throwers. I do not have to deal with scum. And I will not again. I have learned to ignore Puiu and have now learned it is best to ignore Cant. I recommend everyone ignore him and recommend the garbage he throws be busted on sight.