Aces to God

Aces to God

An investigation of probability, information and knowledge

Buckle up! 

We are about to depart on a whirlwind trip in which an armchair doubt about the way experts figure the probabilities of a card trick forces us to reconsider the 20th century view of probability* (and subsequently information)—leading us to reject the prevailing interpretation among particle physicists of the nature of quarks, question basic assumptions of artificial intelligence researchers, and doubt the existence of God;—oh yes, and completely change our view of what we are doing when we think thoughts and make decisions.

That, in a nutshell, is one ponderous claim in one ponderous sentence, and if you’re not in the mood for ponderousness—or for pondering—now is your best opportunity to crack the gate and hightail-it away.

Since you’ve decided to linger, let me start the introductions. To your right (my left) meet Martin Gardner, our resident expert and highly qualified representative of the modern interpretation of what probability is and how it works. His credentials? Nothing less than 25 years of authoring the famous, very respected Mathematical Games column in Scientific American. Next to Martin, serving as his over-qualified backup, the commonly anointed “smartest person in the world” and Parade Magazine columnist, Marilyn vos Savant. Give them a big hand.

To your left (my right) meet Mr. Jones. No, he does not look familiar. You have not seem him on tv (with the possible exception of all you in Wyoming). Jack Jones is a card shark**, and if we need any cards dealt, he’ll do the dealing. He’s a magician by trade, I am told, so I think we better keep an eye on him. 

The last person in the room—besides you, dear reader—is me, sitting comfortably in my armchair.

The card trick Martin will soon ask Jack to perform can be found in the August 1998 issue of Scientific American (page 69). However, tricks—or “problems” as the experts prefer to call them—of this type have a history. Martin introduced the first version of it (involving prisoners not cards) in his October 1959 column. Marilyn, smiling at his left, introduced another version (involving doors) in her Parade column in 1990. 

However, we have no doors, we have no prisoners. We have Jack. And a deck of cards.

But before we proceed, let me make a confession. Martin and Marilyn are not really here in the room with us. You’re only imagining it—with my help. And later on when they speak and discuss, it is only me dreaming what words might flow from them. They are not actually here to say the sentences I will put in their mouths, and consequently it would be blasphemous—no, wrong word—it would be libelous on your part to attribute to them the particular viewpoints, much less the specific words, I am about to fancifully put in their mouths. 

And it’s not about them anyway, is it? It’s about us, and playing with some ideas, and charging blindly over the hill to see what’s there. 

But enough! Let’s proceed.

Jack and the Ace of Hearts

Martin nods at Jack, and Jack begins. First, we watch him extract three cards from the deck. He shows them to us: the ace, king, and queen of Hearts. He sets the rest of the deck aside, and takes the three cards and begins to bedazzle us with the most rapid, grandiose, and over-blown shuffling of three measly cards as any of us have ever seen. As suddenly as he started, he stops, and places the three cards face down neatly in front of me.

“If you can pick the ace, I’ll pay you $200,” Jack announces, staring intensely at my eyes. “But if you guess wrong, you pay me $100.”

I’m glad when he glances away to Martin and Marilyn. Personally, I find it uncomfortable when someone stares me directly in the eyes like that, unless they’re the opposite sex. For the moment I relax, and begin to consider his offer.

It seems fair—after all, there are three cards, so I have only a 1/3 chance of picking right. And Jack has a 2/3 chance that I’ll pick wrong. Over the long run, he would win twice for my once, so it is only right that I should get double what he gets for winning.

I tap my finger on the middle card. That’s my pick.

Jack carefully peeks at each of the cards I didn’t choose, looks up at me poker-faced, staring directly at my uncomfortable eyes again, and then super casual-like flips over the card on the left. It is the king of Hearts.

“Wait!” Martin jumps up, excited. “What is now the probability that your middle card is the ace?” he asks me.

I glance at the table. Only two cards remain hidden: my card, and the one on the right.

“It’s 50–50,” I say innocently.

“Wrong!” Martin crows triumphantly. “There is still only a 1/3 chance that your card is the ace. So there is now a 2/3 chance that the card on the right is the ace.”

Evidently, I look puzzled.

“Let me explain it,” says Marilyn. “Of the two cards you didn’t pick, at least one will always be a face card, since there is only a single ace. So under every circumstance, Jack will always be able to turn up a face card. Consequently, the original odds have not changed a bit. You still have only a 1/3 chance of having the ace.”

“Why did he turn over the face card?” I ask Marilyn. To me it’s an important question.

“I told him to,” Martin explains. “That’s how the trick is performed. Jack’s instructions are to always reveal a face card.”

“I didn’t know that,” I reply.

Red Herring Blue Herring

Sitting in my armchair, I thought for a moment. If Jack is going to search out a face card to flip up, then the fact that he flipped a face card is a red herring. As Marilyn and Martin said, he’ll always have at least one of his two cards to reveal as not being the ace. But I had a thought.

“Jack looked at both cards before he found a face card to turn up,” I explain. “That means that the odds that the card on the right is the ace is not 2/3 as you say, but 100%. Jack looked at it, but had to move to the other card in order to find a face card to turn over.”

“No, no,” Martin responds. “He is instructed to always look at both cards before turning a face card up.”

“I didn’t know that,” I reply.

“You’re right about one thing,” Martin continued. “If Jack now gives you the opportunity to switch your bet to the other card, you’d be wise to take him up on it. Right now it’s twice as likely to be the ace as the card you picked.”

“Hold on!” I say, after pondering this for a moment. “Jack looked at the cards, so he knows with 100% certainty which card is the ace. If he offers me the option to switch choices, it will be because he realizes that my original pick was correct after all, and he’s about to be out $200 unless he pulls something over on me. I’d be wise not to take him up on any offer like that.”

Martin shakes his head sadly, “Don’t you get it? The basic mathematics hasn’t changed. Your card only has a 1/3 chance, same as ever. So it follows that the odds are now 2/3 that the card on the right is the ace.”

“Not for Jack, it doesn’t,” I rejoin. “His odds differ from mine because he knows more. For him, the odds are 100% that the ace is the card that he knows it is. I don’t know which card that is, but he does. His odds are quite different from my odds. Unless I can figure out his motives, in which case I increase my odds considerably because I increase how much I know.”

I pause for a moment, and when they don’t respond, I sum it up: “Probability depends upon the subjective amount of knowledge of each person thinking in probabilities.”

This strikes a nerve.

“Not at all. Not at all. Probability is not subjective. It’s objective. That’s a very well-established scientific fact.”

“The probabilities for me,” I persist, “are different in this case from the probabilities for Jack. He has looked at the cards.”

“No, no, no.” they respond.

When is an Ace Not an Ace?

Martin and Marilyn are wrong, but they haven’t realized it yet. Let’s show a little politeness and not put them through the agony of being proved wrong in public. So we’ll drop them out of the picture a bit, and continue on in a more natural style. 

But first, let’s talk a bit about the issue here. The experts believe that probabilities tell us something about the world, that is, that probabilities are inherent in the nature of things. Whereas I think that probabilities are about us and our subjective level of knowledge: you might call it a thinking strategy we employ when we have missing information of an appropriate sort. 

And my argument is simple. If probabilities were inherent in things, it shouldn’t matter whether the cards were dealt face up or face down, whether they were marked or not, for they are the same three cards, and the only difference is our level of knowledge, which is much greater when they are marked or face up. Furthermore, if probabilities are inherent in things, they must remain the same regardless of who is in the room or what they know: the odds should not differ from one observer to another. Yet they do. Martin and Marilyn have to try to deny this, or their expert house of cards flails and falls. But it can’t be denied.

If odds are a thinking strategy, as I maintain, then they are individual for each observer, each thinker. If instead odds are a fact about the world, as the experts maintain, then they absolutely must not differ from observer to observer, for probabilities must exist and be true outside the observer who merely observes them and recognizes them. 

Whereas, in my view, the observer doesn’t discover the odds so much as she or he invents them. The observer makes them up. 

(Later, I will take this point of view beyond the field of probabilities and apply it all mathematical and logical thinking—the larger context. We will see that our 20th century ignorance of what the human thinking process is all about conveniently enables a lazy armchair thinker like you or me to show up the experts.)

But back to our cards.

In our card scenario, Jack knows what each of the three cards is—for him they might as well be face up since he as seen the faces of two of them and can infer the third: for him the probabilities are effectively certain. The rest of us don’t know what Jack knows, and therefore we must calculate the odds differently. And indeed, because we have different ideas about Jack’s motive in turning up the card he did, even the rest of us properly and justifiably differ from each other in our calculations. 

Imagine if Jack had dealt out the cards face up instead of face down, and then offered me $200 to pick the ace. Easy money for me. Yet the only difference between this scenario and the actual one is how much I know concerning which card is which. In short, the odds—for me—depend strictly on my level of knowledge. 

Or imagine a scenario in which the deck is marked, but I alone know that, and I alone know how to read the marks. Once again, the $200 is easy money for me, for I can distinguish the cards virtually as easily as if they were face up on the table. But in this case Jack engages me in the bet because he doesn’t know how much I know.

Nothing more clearly shows the individual nature of probabilities.

Bonobo Motives

But add this: determining the odds that the middle card is the ace depends not only subjectively on the level of knowledge I have about the cards on the table, but also on my interpretation of Jack’s motive when he turned up the King of Hearts. 

For example, if I judge that he did so in order to make me think — as Martin and Marilyn believe I ought to—that the card on my right now has a 66% likelihood of being the Ace, then shouldn’t I conclude he is trying deliberately to deceive me, and judge the odds less than 66%. 

Whereas if I judge Jack’s motives as naïve—he is just following the rules as laid down by Martin—then the odds are as Martin and Marilyn state them.

On the other hand, what if his motive was simply to find out if he had won the bet, so that he could gleefully upturn the ace and collect his $100.00 from me. And when neither card was the ace, his strategy shifted to tricking me into changing my choice. A wise strategy considering the insistence from our two experts that changing my choice would be the correct step to make.

Would Jack have looked at both cards had the first one been the ace? Martin says that Jack is instructed to always look at both cards, and then turn up the one that is not the ace. But if so, that is precisely information that I, the chosen dupe, didn’t have—and I might add didn’t suspect.

Yet that is information which is crucial to interpreting the motives behind Jack’s behavior, and thus, my evaluation of the odds. 

For example, imagine how I might have judged probabilities if a bonobo were dealing the cards, and turning them up, instead of Jack. I probably would have figured the chimp turned over the face card “by chance”—unless of course he happens to have a preference for face cards. Or perhaps this is a trained chimp who is as capable as Jack of deceiving me. The point is that once again it comes down to my estimation of the awareness and motives (or lack of motives) of the dealer. 

What this demonstrates, ad nauseam, is that probabilities are not innate in things, or in the world.

 We shall see later that probabilities involve mathematical tautologies, what A. J. Ayer would call analytic rather than synthetic propositions. Rather than constituting descriptions of the natural world out there, their nature is of the same sort as statements such as 2 + 2 = 4.  They have no factual basis, but are strictly analytical. In other words, they concern how we think, or how we go about thinking, rather than factual bits about the subject of our thinking.

They constitute the “rules” you might say, of thinking rather than the subject matter itself which we are thinking about.

Another way to think about it is to think of a map, and the symbols used to indicate various elements on a map. Probability statements, like all math and logic statements, refer to the rules for using the map symbols, rather than to the subject matter of the map itself. 

 In the broader sense, the practice of thinking about a subject in probabilities, rather than elsewise, is a thinking strategy humans employ when we judge our knowledge is uncertain.  In our example, if the cards were dealt face-up, it would be ludicrous to employ probabilities (1/3 chance for each card) since quite simply our knowledge level is already very high: we can see with certainty which card is which.

There’s No Cheating in Cards

“Alright,” said Jack. “Enough games. Do you want to switch your choice?” 

“The fact that you’re offering me the chance to switch when you have $200 of your hard earned money on the line—it tells me I should stay put.”

“No.” interject the M & M’s. Explaining again that the original 1/3 odds on my middle card haven’t changed. “No one has looked at THAT card.”

“Jack knows the other two cards, so he doesn’t have to look.” I rejoined. “And the fact that he offers me the choice means he wants me to switch, otherwise he would risk his money.”

“No, it’s part of the rules. He’s supposed to offer you the chance to switch,” explained Martin.

I shook my head in amazement. “Why am I always the last one to know about the rules?”

“Look,” said Jack. “This is obviously too complicated for you. So if you’d like, we can drop the bet.”

“Because I’ve chosen the ace, and you can’t talk me into changing cards.”

“Keep your card, I don’t care.”

Impulsively I flipped over my center card.

It was the Queen of Hearts.

Disappointed, I turned over the remaining card. It was the Jack of Hearts. “Where—“ I stammered, “where’s the ace?”

Jack laughed. “Right up my sleeve,” he announced. “You were right all along. I am a card shark.”

“You cheated!” said Marilyn.

“No, I deceived. There’s no cheating in cards.”

“Of course there is! You just did it, right in front of us,” interjected Martin.

“I did everything right in front of you, all right, including slipping the ace up my sleeve,” Jack responded. “Is it my fault that your expectations of what I was going to do prevented you from seeing what I actually did?”

Shuffling the Deck

Now, this turn of events raises even more questions of probabilities.

Were the odds 1/3 that my card was the ace, even though eventually it turned out the ace was up Jack’s sleeve? Or were the odds 1/4 because four cards were actually in play rather than the originally supposed three?

Or were the odds 1/52, since in the larger picture we began with a deck of 52?

Or 1/55 because of the two jokers and the manufacturer’s about-this-deck card? Or should we consider other possible decks, which Jack may or may not have hidden in his pockets?

It isn’t just what we know, but what we choose to consider as significant among all the possible information we have, that matters when we calculate probabilities. (We all knew Jack was a card shark, after all—why didn’t we include that knowledge in our estimation of probabilities?)

These considerations demonstrate that the odds are determined by our human choices in deciding what items to include in our probability calculations. Should we choose to consider only the three cards in play before us—reasonable if our assumptions are that the ace is one of those three cards, that there is only one ace, not two, etc.? In short, the “field of play” is decided in our minds based on our individual assumptions about what is going on and what to expect.

For Jack, those assumptions were different than for the rest of us. He knew with a high degree of certainty that he had removed 4 rather than 3 cards from the deck, and was able to confirm that when he slipped one up his sleeve and still had three cards to deal on the table. To the extent he could be certain he had indeed slipped the right card up his sleeve, he could be certain that the odds on the cards face down were: to the extent he couldn’t be certain, the odds would approach 1/4.

Now, imagine that Jack wasn’t so good at sleight of hand. Imagine he withdrew 4 cards, and could be certain one was the ace, but not certain whether ace went up his sleeve or onto the table. The odds would be one-fourth—unless he could be pretty certain the ace was either the last card he dealt, or up his sleeve, in which case the odds would be one-half. 

No matter how you slice it, no matter how you dice it, the odds for Jack depend entirely upon what Jack feels that he knows with certainty, and what he feels he doesn’t know, or can’t be sure of. And whereas the rules of odd-making are direct and mathematical, the rules of determining what you know and what you can depend on, are neither direct nor mathematical.

Clearly, probabilities are subjective even though the rules of deciding probabilities are direct and objective given a specific set of “facts”. But which facts, that is, which “certainties” and which “uncertainties” are the ones which apply? Determining that is wholly subjective. 

This leads us to a very important conclusion: probabilities are not descriptions of fact about the world.  Resorting to probabilities is a thinking strategy usefully employed in circumstances where there is an appropriate combination of known and unknown factors. They are as fallible as our level of knowledge is fallible, as our choices about what knowledge to count as “facts” to be included in the “field of play” is fallible.

Understanding Understanding

Forgive me for being so exhaustive. I want to make clear that we must discard our old notion about the nature of probabilities. I belabor it because the consequences in other areas are so significant. 

1) in particle physics (Schrodinger’s cat, our understanding of quarks and the nature of matter)

2) in philosophy, in understanding the nature of human thought, and human existence

3) in neural research

4) in fields of artificial intelligence

5) in religion, in understanding questions of God’s existence.

If there are any “inherent” probabilities, they are this: that a card is always 100% itself, and located where it is located. Things are what they are. It is only observers who have varying degrees of knowledge (not surprising, since only observers can have knowledge—knowledge is an experience, not an element of things). There is never, speaking inherently out in the world, a 1/3 chance that the middle of three cards is an ace. An ace is always the card it is, and never the card it is not, and even the information which identifies the card as one of three cards, or the information that identifies one card as an ace and other cards as not an ace, exists not in the cards but only in the experiencing mind or minds of observers. If there is any sense in which there are “probabilities” existing rawly out in the world, they are always 100% or zero, never between.

And yet, between three equally unknown objects, the odds are 1/3 that a specifically chosen but hidden one differs in the way that it differs from the other two rather than in the way that the other two differ. (even to try to say this rule “generally” is difficult to do!). Another way to try to say this is to say that probabilities can only be applied to information abstracted about the world, not to the world itself, and if we choose to abstract different information about the world, the “probabilities” will differ correspondingly. Probabilities, in other words, have the same tautological “truth” and “infallibility”—given the exact information applicable—that all analytic propositions (to use A. J. Ayer’s terminology) have—and like analytic propositions, their subject matter is never factual, they are never about describing the world itself.

Here are some points we believe our analysis leads to:

1) probabilities depend of the amount of information available

2) thus information is located in individual human brains—not out in the world. 

3) There is no information which corresponds to “facts” in the world. Information is an internal experience we have, not a state of affairs.

4) Consequently information is not so much “discovered” as “invented.”  It is a creative map-making type enterprise.

5) This doesn’t invalidate information or human thinking. It doesn’t mean that one theory or viewpoint is as good as another. Thoughts are verified not by reference to “facts” or “information” but by their internal tautological validity and their external usefulness for the task or goal at hand when compared with competing thoughts. (Surprisingly, or rather not surprisingly, this approach to verifying thoughts meshes almost perfectly with the practice of scientists, the scientific method. A scientific hypothesis is not proved true, rather it is proved to be a better predictor of future events or experiments than competing hypotheses.)

6) In sum, this means that “informational facts” are not True, but only more or less useful relative to alternative “informational facts” when considered in light of the particular “uses” we have in mind. 

On to Quarks!

In this section we look at Schrodinger’s cat, at the strange behavior of quarks, and the stranger mental behavior of scientists who study and think about quarks. The “expert” conclusion about quark behavior is that it means that the physical world is ultimately probabilistic in nature. Sitting securely in our armchair, we scoff and laugh, of course. We know that if experts see probabilities it’s because they’re not seeing the quarks at all. They’re stuck in the house of mirrors of their own thinking process.

And what is the actual nature of quarks? Is it deterministic or is it probabillistic?

Neither. 

Let’s face it. Quark behavior is popular because it appears to sink determinism. But determinism doesn’t need sinking—any more than probability needs sinking. Both are part of the currency of how we think, they are mental strategies indigenous to the human mind. The world is neither deterministic nor probabilistic because those are traits that can only apply to information. Information is always deterministic, and at the same time always probabilistic in nature. 

But there is a natural, inevitable, and unavoidable mismatch between information and the world which is its “subject”—always has been, always will be—for the simple reasons that the world is not composed of information. Information is not its nature. 

Information is mental currency. Whatever the world is, mental currency it isn’t. And the mistake of confusing the world (the subject of our thinking) with mental currency itself (the content of our thinking), this we call the mental fallacy.

The Disappearance of God

God disappears if the world is not composed of information. Why? Because, for one thing, the word can only create information. But if information relies on the world for its existence, rather than the world relying on information—and this is exactly what we have argued all along—then the world can’t be created ex nihlo. God can’t do it by uttering a word, but instead he would have to do his creating the human way: by having sex. Or else be content with merely using his hands to cut, mix, shape and otherwise modify existing things. 

But God isn’t of the world. God doesn’t have hands—nor head, nor brain matter, nor any other actual component. God belongs, God has always belonged, to the realm of information, of truth, of conceptual essence. This is why God disappears. All along, God was only the human mental realm idealized. From that dust he came, and to that dust God must disappear again.

For it really is quite simple. If God can’t create the world by thinking it—and we’ve shown that notion has got the whole existence of things backwards—then God can’t be God. 

And if that realm of the word, of ideals and of mental (usually called spiritual) experiencing is the realm of God, then we are forced to recognize that it exists—can only exist—in one place: our heads, our human imaginations. 

This is my no means to denigrate the importance or value of our imaginations. But the realm of our imagination is not some realm above, beyond or outside the world. It is the realm of stories, and fiction, poetry, and fanciful language, true enough. Perhaps it pleases us. But to insist that it is a literal truth about the world is a great self-deception, akin to insisting that Santa and his little reindeer do indeed drop gifts at every home in the world on Christmas eve.

It is a quaint story, enjoyable to tell, but if you believe in it literally, you are mad or childish.

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*According to Hans Christian von Baeyer, in his book QBism: The Future of Quantum Physics, the understanding of probability I’m contesting in Aces to God is known as frequentist probability theory. By the end of the 20th century this was being supplanted by bayesian probability theory—first described by Thomas Bayes in the 18th century and reintroduced in the 20th by the mathematician Frank Ramsey (who died young) and by the quantum physicist, E. T. Jaynes. I was right to think that how we understand probabilities matters for quantum mechanics, even if I felt too ignorant to comprehend or press the point. In fact, QBism—Quantum Bayesianism—is a 21st century reinterpretation of quantum mechanics based on embracing bayesian and rejecting frequentist probability theory. It was developed by Christopher Fuchs from 2002 to 2010 in concert with others. (For an early example of bayesian probability theory applied to quantum mechanics, see Saul Youssef, A Reformulation of Quantum Mechanics, 1991.)

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**I meant cardsharp, as specified in Martin Gardner’s original presentation.

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For my take on the similar Monty Hall Doors problem, see https://atheology.com/2010/06/27/lets-make-a-deal/ or https://thenakedatheist.wordpress.com/2010/06/27/lets-make-a-deal/