For example, the ass stares at the bales for 15 seconds, then it moves towards whichever one it estimates is larger (ignoring variance in estimates). If it turns out that they are exactly equal, it instead picks one at random.
Your problem is that you're using an algorithm that can only be approximated on an analog computer. You can't do flow control like that. If you want it to do A if it has 0 as an input and B if it has 1 as an input, you can make it do A+(B-A)x where x is the input, but you can't just make it do A under one condition and B under another. If continuity is your only problem, you can make it do A+(B-A)f(x), where f(x)=0 for 0<=x<=0.49 and f(x)=1 for 0.51<=x<=1, but f(x) still has to come out to 1/2 when x is somewhere between 0.49<x<0.51.
If you tried to do your algorithm, after 15 seconds, there'd have to be some certainty level where the Ass will end up doing some combination of going left and choosing at random, which will keep it in the same spot if "random" was right. If "random" is instead left, then it stops if it's half way between that and right.
The second objection is that our world is, in fact, not continuous (with the Planck length and whatnot).
I'm not really sure where that idea came from. Quantum physics is continuous. In fact, derivatives are vital to it, and you need continuity to have them. The position of an object is spread out over a waveform instead of being at a specific spot like a billiard ball, but the waveform is a continuous function of position. The waveform has a center of mass that can be specified however much you want. Also, the Planck length seems kind of arbitrary. It means something if you have an object with size one Planck mass (about the size of a small flea), but a smaller object would have a more spread out waveform, and a larger object would have a tighter one.
get it to make true random choices.
That would make it so you can't purposely fool the Ass, but it won't keep that from happening on accident. For example, if you try to balance a needle on the tip outside when there's a little wind, you're (probably) not going to be able to do it by making it stand up perfectly straight. It's going to have to tilt a little so it leans into every gust of wind. But there's still some way to get it to balance indefinitely.
...The second objection is that our world is, in fact, not continuous (with the Planck length and whatnot).
I'm not really sure where that idea came from. Quantum physics is continuous. In fact, derivatives are vital to it, and you need continuity to have them. The position of an object is spread out over a waveform instead of being at a specific spot like a billiard ball, but the waveform is a continuous function of position. The waveform has a center of mass that can be specified however much you want. Also, the Planck length seems kind of arbitrary. It
[Crossposted]
The medieval philosopher Buridan reportedly constructed a thought experiment to support his view that human behavior was determined rather than “free”—hence rational agents couldn’t choose between two equally good alternatives. In the Buridan’s Ass Paradox, an ass finds itself between two equal equidistant bales of hay, noticed simultaneously; the bales’ distance and size are the only variables influencing the ass’s behavior. Under these idealized conditions, the ass must starve, its predicament indistinguishable from a physical object suspended between opposite forces, such as a planet that neither falls into the sun nor escapes into outer space. (Since the ass served Buridan as metaphor for the human agent, in what follows, I speak of “ass” and “agent” interchangeably.)
Computer scientist Leslie Lamport formalized the paradox as “Buridan’s Principle,” which states that the ass will starve if it is situated in a range of possibilities that include midpoints where two opposing forces are equal and it must choose in a sufficiently short time span. We assume, based on a principle of physical continuity, that the larger the bale of hay compared to the other, the faster will the ass be able to decide. Since this is true on the left and on the right, at the midpoint, where the bales are equal, symmetry requires an infinite decision time Conclusion: within some range of bale comparisons, the ass will require decision time greater than a given bounded time interval. (For rigorous treatment, see Buridan’s Principle (1984).)
Buridan’s Principle is counterintuitive, as Lamport discovered when he first tried to publish. Among the objections to Buridan’s Principle summarized by Lamport, the main objection provides an insight about the source of the mind-projection fallacy, which treats probability as a feature of the world. The most common objection is that when the agent can’t decide it may use a default metarule. Lamport points out this substitutes another decision subject to the same limits: the agent must decide that it can’t decide. My point differs from that of Lamport, who proves that binary decisions in the face of continuous inputs are unavoidable and that with minimal assumptions they preclude deciding in bounded time; whereas I draw a stronger conclusion: no decision is substitutable when you adhere strictly to the problem’s conditions specifying that the agent be equally balanced between the options. Any inclination to substitute a different decision is a bias toward making the decision that the substitute decision entails. In the simplest variant, the ass may use the rule: turn left when you can’t decide, potentially entrapping it in the limbo between deciding whether it can’t decide. If the ass has a metarule resolving conflicting to favor the left, it has an extraneous bias.
Lamport’s analysis discerns a kind of physical law; mine elucidates the origins of the mind-projection fallacy. What’s psychologically telling is that the most common metarule is to decide at random. But if by random we mean only apparently random, the strategy still doesn’t free the ass from its straightjacket. If it flips a coin, an agent is, in fact, biased toward whatever the coin will dictate, bias, here, means an inclination to use means causally connected with a certain outcome, but the coin flip’s apparent randomness is due to our ignorance of microconditions; truly random responding would allow the agent to circumvent the paradox’s conditions. The theory that the agent might use a random strategy expresses the intuition that the agent could turn either way. It seems a route to where the opposites of functioning according to physical law and acting “freely” in perceived self-interest are reconciled.
This false reconciliation comes through confusing two kinds of symmetry: the epistemic symmetry of “chance” events and the dynamic symmetry in the Buridan’s ass paradox. If you flip a coin, the symmetry of the coin (along with your lack of control over the flip) is what makes your reasons for preferring heads and tails equivalent, justifying assigning each the same probability. We encounter another symmetry with Buridan’s ass, where we also have the same reason to think the ass will turn in either direction. Since the intuition of “free will” precludes impossible decisions, we construe our epistemic uncertainty as describing a decision that’s possible but inherently uncertain.
When we conceive of the ass as a purely physical process subject to two opposite forces (which, of course, it is), and then it’s obvious that the ass can be “stuck.” What miscues intuition is that the ass need not be confined to one decision rule. But if by hypothesis it is confined to one rule, the rule may preclude decision. This hypothetical is made relevant by the necessity of there being some ultimate decision rule.
The intuitive physics of an agent that can’t get stuck entails: a) two equal forces act on an object producing an equilibrium; b) without breaking the equilibrium, an additional natural law is added specifying that the ass will turn. Rather than conclude this is impossible, intuition “resolves” the contradiction through conceiving that the ass will go in each direction half the time: the probability of either course is deemed .5. Confusion of kinds of symmetry, fueled by the intuition of free will, makes Buridan’s Principle counter-intuitive and objective probabilities intuitive.
How do we know that reality can’t be like this intuitive physics? We know because realizing a and b would mean that the physical forces involved don’t vary continuously. It would make an exception, a kind of singularity, of the midpoint.