Suppose we have a long street with building slots officially numbered 0-999, but with the numbers not posted. At numbers 990–994 and 996–999, we have barn facades with no barn behind them. At all the other numbers, we have normal barns. You know all these facts.
I will assume that the barns are sufficiently widely spaced that you can’t tell by looking around where you are on the street.
Suppose if you find yourself at #5 and judge you are in front of a barn. Intuitively, you know you are in front of a barn. But if you find yourself at #995 and judge you are in front of a barn, you are right, but you don’t know it, as you are surrounded by mere barn facades nearby.
At least that’s the initial intuition (it’s a “safety” intuition in epistemology parlance). But note first that this intuition is based on an unstated assumption, that the buildings are numbered in order. Suppose, instead, that the building numbers were allocated by someone suffering from a numeral reversal disorder, so that, from east to west, the slots are:
- 000, 100, 200, …, 900, 010, 110, 210, …, 999.
Then when you are at #995, your immediate neighborhood looks like:
- 595, 695, 795, 895, 995, 006, 106, 206, 306.
And all these are perfectly normal barns. So it seems you know.
But why should knowledge depend on geometry? Why should it matter whether the numerals are apportioned east to west in standard order, or in the order going with the least-significant-digit-first reinterpretation?
Perhaps the intuition here is that when you are at a given number, you could “easily have been” a few buildings to the east or to the west, while it would have been “harder” for you to have been at one of the further away numbers. Thus, it matters whether you are geometrically surrounded by mere barn facades or not.
Let’s assume from now on that the buildings are arranged east to west in standard order: 000, 001, 002, …, 999, and you are at #995.
But how did you get there? Here is one possibility. A random number was uniformly chosen between 0 and 999, hidden from you, and you were randomly teleported to that number. In this case, is there a sense in which it was “easy” for you to have been assigned a neighboring number (say, #994)? That depends on details of the random selection. Here are four cases:
A spinner with a thousand slots was spun.
A ten-sided die (sides numbered 0-9) was rolled thrice, generating digits the digits in order from left to right.
The same as the previous, except the digits were generated in order from right to left.
A computer picked the random number by first accessing a source of randomness, such as the time, to the millisecond, at which the program was started (or timings of keystrokes or fine details of mouse movements). Then a mathematical transformation was applied to the initial random number, to generate a sequence of cryptographically secure pseudorandom numbers whose relationship to the initial source of randomness is quite complex, eventually yielding the selected number. The mathematical transformations are so designed that one cannot assume that when the inputs are close to each other, the outputs are as well.
In case 1, it is intuitively true that if you landed at #995, you could “easily have been” at 994 or 996, since a small perturbation in the input conditions (starting position of spinner and force applied) would have resulted in a small change in the output.
In case 2, you could “easily have been” at 990-994 or 996-999 instead of 995, since all of these would have simply required the last die roll to have been different. In case 3, it is tempting to say that you could easily have been at these neighboring numbers since that would have simply required the first die roll to have been different. But actually I think cases 2 and 3 are further apart than they initially seem. If the first die roll came out differently, likely rolls two and three would have been different as well. Why? Well, die rolls are sensitive to initial conditions (the height from which the die is dropped, the force with which it is thrown, the spin imparted, the initial position, etc.) If the initial conditions for the first roll were different for some reason, it is very likely that this would have disturbed the initial conditions for the second roll. And getting a different result for the first roll would have affected the roller’s psychological state, and that psychological state feeds in a complex way into the way they will do the second and third rolls. So in case 3, I don’t think we can say that you could “easily” have ended up at a neighboring number. That would have required the first die roll to be different, and then, likely, you would have ended up quite far off.
Finally, in case 4, a good pseudorandom number generator is so designed that the relationship between the initial source of randomness and the outputs is sufficiently messy that a slight change in the inputs is apt to lead to a large change in the outputs, so it is false that you could easily have ended up at a neighboring number—intuitively, had things been different, you wouldn’t have been any more likely to end up at 994 or 996 than at 123 or 378.
I think at this point we can’t hold on to the initial intuition that at #995 you don’t know you’re at a barn but at #5 you would have known without further qualifications about how you ended up where you are. Maybe if you ended up at #995 via the spinner and the left-to-right die rolls, you don’t know, but if you ended up there via the right-to-left die rolls or the cryptographically secure pseudorandom number generator, then there is no relevant difference between #995 and #5.
At this point, I think, the initial intuition should start getting destabilized. There is something rather counterintuitive about the idea that the details of the random number generation matter. Does it really matter for knowledge whether the buildin number you were transported to was generated right-to-left or left-to-right by die rolls?
Why not just say that you know in all the cases? In all the cases, you engage in simple statistical reasoning: of the 1000 barn facades, 999 of them are fronts of a real barns, one is a mere facade, and it’s random which one is in front of you, so it is reasonable to think that you are in front of a real barn. Why should the neighboring buildings matter at all?
Perhaps it is this. In your reasoning, you are assuming you’re not in the 990-999 neighborhood. For if you realized you were in that neighborhood, you wouldn’t conclude you’re in front of a barn. But this response seems off-base for two reasons. First, by the same token you could say that when you are at #5, you are assuming you’re not in front of any of the buildings from the following set: {990, 991, 992, 993, 994, 5, 996, 997, 998, 999}. For if you realized you were in front of a building from that set, you wouldn’t have thought you are in front of a barn. But that’s silly. Second, you aren’t assuming that you’re not in the 990-999 neighborhood. For if you were assuming that, then your confidence that you’re in front of a real barn would have been the same as your confidence that you’re not in the 990-999 neighborhood, namely 0.990. But in fact, your confidence that you’re in front of a real barn is slightly higher than that, it is 0.991. For your confidence that you’re in front of a real barn takes into account the possibility that you are at #995, and hence that you are in the 990-999 neighborhood.
