I play a lot of solitaire on my iPod Touch. It's seriously problematic at times; just ask my fiancee. Matter of fact, the gadget seems to exist mostly for three purposes: listening to podcasts, surfing the net, and playing solitaire.
Over many hours I've become somewhat adept at knowing which cards are still buried toward the end of a game and, knowing that, whether or not I have any chance to win based on what they're buried under. This led me to realize that I was basically claiming knowledge without direct empirical evidence. I can deduce what cards are where without actually looking at them. More accurately, I can deduce probabilities without actually looking at the cards, but when I have only one buried card, I can know exactly what it is without ever turning it over.
This idea crept around my head for a while: as a dyed-in-the-wool skeptic and empiricist, as someone who always demands evidence before I apportion belief, I was claiming certain knowledge of the contents of a hidden object through indirect deduction without ever actually looking at it. I began considering the question "Can we ever reliably know something specific about the world without direct empirical evidence?"
The answer, I think, is "yes." The answer is also "no."
Let's take this one at a time.
The answer is "yes" in certain limited contexts like, for example, a deck of playing cards. The contents of a deck of playing cards are predetermined, and regular, and organized into sets; every deck has the same 52 cards that can be broken down by color, by suit, or by number (defined loosely). If I turn over 51 cards and the only one not showing is the king of hearts, I know without looking that the face-down card is the king of hearts. I don't have to look to confirm. I will always be right. If I pull out all aces and turn over three to find a heart, a spade, and a diamond, I know that the one I can't see is a club.
Of course, when you have more than once face-down card, this isn't quite so accurate. Until you only have one unknown, you can only judge probability. Turning half of the deck face-up allows you to determine with certainty which cards are face-down without looking at them, but your certainty ends there. You only have a 1/26 chance of getting any specific face-down card correct. Let's say you start turning cards face-up at this point, one by one. As the number of face-down cards lowers, your level of certainty (read: "chance of guessing a specific card correctly") increases: 1/25, 1/24, 1/23, and so on until only one face-down card is left, at which point the probability has collapsed to a 1/1.
As far as I can tell, this is how really good blackjack card counters work; they keep track of what has already been played and figure probabilities to determine what move to make. Of course, I'm no blackjack player, so I might be fictionalizing this based on Rainman. Please correct me if I'm wrong.
Incidentally, this kind of collapse of probability as choices are minimized is what is behind the counterintuitive solution to the Monty Hall problem. It's not easy to grasp why you should switch doors when there are only three options, but if we model the problem with a deck of cards, it becomes much more obvious. Monty tells you that if you pick the ace of spades, you win the new car. You choose a card from the deck, giving yourself a 1/52 chance to win and leaving a 51/52 chance that the winner is among the other cards. Monty then reveals the faces of fifty other cards, all of them losers and asks if you want to switch to the other remaining card. At this point, because you chose when all cards were face-down, your pick still only have a 1/52 chance of being the winner. Because Monty always leaves the winner, however, the other face-down card stands in for the rest of the deck and has a 51/52 chance of being the winner. You can judge this probability and make the switch.
Anyway, if Monty asks you to turn over your card (whether you switched or not) and it is not the ace of spades, the one remaining card must be the ace of spades. You can know this with 100% certainty.
Right?
Well, no.
The answer is only "yes" for systems necessarily governed by hard and fast absolute rules. A deck of cards is a system like that, as is a cup-and-ball game (i.e. if two cups are empty, the ball must be under the third) or a multiple-choice test question (i.e. the process of elimination: if you know A, B, and C are wrong, even if you don't know D is right, you know D is right). As long as the standard rules (the axioms) hold in systems like these, you can, in fact, know something about the world with 100% certainty without directly observing it.
The only reason this works, though, is that (to get back to our original example) a deck of cards functions essentially as an axiomatic system: a set of absolute rules governs the contents of a deck of cards, but we created those rules and accept them a prioi as given for any regulation deck. In a conceptual axiomatic system, like mathematics or formal logic, every statement proven true in the confines of the system is true only by virtue of and in relation to the rules of that system. And the rules, to quote MarkCC of Good Math, Bad Math, "aren't rules about the universe - they're self-contained rules about concepts that they describe."
A deck of cards then, can perhaps be called a material axiomatic system. The rules governing the contents of a deck of cards are not facts about the universe; they only apply to decks of cards. More to the point, those rules only apply to decks of cards that follow those rules. Tautology? Yes. But true nevertheless, and very important in a moment.
When you deduce the contents of the final face-down card, what you're doing is figuring out a specific consequence of a set of constructed axioms; you're making a statement about the deck of cards based on the formal rules governing a deck of cards. Of course, you are, in fact, arriving at 100% empirical knowledge without direct observation, but that's simply a consequence of the fact that a deck of cards, unlike numbers or rules of logic, exist in the real world. They form a material axiomatic system.
This, then, is why the answer is "no." You can only know something with 100% certainty if you're making a statement about a system based on the rules governing that system. If the system happens to be represented by material objects, then it appears that your deduction allows you to discern empirical facts without direct observation when you're really only engaging the system's rules. The conclusions only hold so long as the rules hold.
And rules, as they say, are meant to be broken.
Let's say I lay out a deck of cards, 51 of them face-up, a single lonely card face-down. I instruct you to tell me what the face-down card is. Given the rules of a deck of cards, you can tell me exactly what it is without looking. Unless, of course, I'm not using a regulation deck. If you're basing your deduction on the rules of a deck of cards and my deck of cards doesn't follow those rules, then you can't know anything about it without direct observation. Your rules, remember, only apply to decks that follow them (and, more to the point, about deductions about decks that follow them), and my deck doesn't. My deck may be entirely random, which case nothing you do will avail you. Or it might even follow its own set of rules, in which case the rules of a standard deck won't get you anywhere.
Basically, any deduction relies on the rules of a deck of cards and is only useful to describe a deck of cards, but those rules are not immutable. They're constructed. And they can be changed. When they get changed, what would, with a regulation deck, be a watertight deduction becomes, instead, a stab in the dark; you can only draw your 100% empirical conclusion in an extremely limited context. And because you can't know with 100% certainty whether or not you're actually working within that context, you can never be 100% sure about your conclusion.
This is, in a sense, how many card tricks work. The magician is utilizing a different set of rules than the audience. The audience is working within the system of a regulation deck of playing cards with a few tacked-on assumptions, e.g. the deck has been randomized, the magician doesn't know anything about the locations of cards in the deck, the magician is telling the truth (what a silly assumption). The audience's adherence to rules like these is what allows the magician to manipulate them, and its why they're amazed when he does. They think the deck (and the magician) is functioning and must function in one way, but the magician knows it doesn't have to and does something outside their system and thus outside their expectations.
I own a deck of Zener cards. Each is marked subtly no two back corners to indicate what is on the face. If I were to use them for a psychic trick, the audience would be working with the following assumed axioms: the cards are uniform, they are random, and I have no way of knowing what is on the other side. I might even lie to reinforce one or more of these assumptions. I would then work with my own set of axioms: mark A means "star," mark B means "square," and so on. The audience, because they're working under one set of rules, thinks I'm just guessing, but I am in fact making 100% accurate knowledge claims based on the rules of my own system.
Of course, afterward they may begin to question their set of rules. Or they may think I'm psychic.
On a more sinister note, this kind of rule discontinuity, so to speak, is how lots of con games function. Three-card Monte and the shell game are prime examples : they function by causing the mark to operate under a different set of rules than the con artists, rules that lead him to erroneous conclusions because they don't necessarily apply to all situations; he thinks he can draw solid conclusions about some small facet of the universe (which card is the queen, which shell is the ball under) without direct observation, but he doesn't realize that such a conclusion is only really making a statement about the system under whose rules he is operating. And if the dealer is operating under a different system, he'll just take all your money.
All of this gives us some insight into yet another reason why science is so important: the universe, far as we know, is not an axiomatic system, so we can't draw any conclusions about it with any level of confidence unless we actually go look at it. Directly. As often and as rigorously as possible. Sure, the universe has laws, but it doesn't follow them, exactly; that phrasing is just an unfortunate quirk of the English language. Our physical laws are attempts (usually pretty accurate ones) to describe the way the universe behaves. By contrast, the rules of a formal system are prescriptive: they define and govern what must happen in the system.
What we already know (or think we know) about the universe allows us to deduce new possibilities. We don't just accept them, though. We call them "hypotheses" and go out and check with the universe to see whether or not we were right.
Armchair-types (like theologians and many philosophers) like to think we can make accurate statements about the nature of reality from the comfort of our living rooms with only the power of our minds (using, strangely enough, formal logic, which is itself based on a number of axioms without which it doesn't function). The universe, however, is not very amenable to that. It is not a constructed formal system and it doesn't have to follow our rules. The universe goes its own way.
The most we can do is to use our science, meager as it may be when compared to the cosmos, and try our best to keep up.
The action continues below the fold...
Shrink it back down...
This is Stan Lippman. He is a dangerous ideological fanatic. If you see this man, tell him that he is a dangerous ideologue and that he could not possibly be more wrong.
