In fancy math-talk, we can say apples are a semimodule over the semiring of natural numbers.

• You can add two bunches of apples through the well-known "glomming-together" operation.
• You can multiply a bunch of apples by any natural number.
• Multiplication distributes over both natural-number addition and glomming-together.
• Multiplication-of-apples is associative with multiplication-of-numbers.
• 1 is an identity with regard to multiplication-of-apples.

You could quibble that there is a finite supply of apples out there, so that (3 apples) + (all the apples) is undefined, but this model ought to work well enough for small collections of apples.

True, and I hope no one thinks it is. So we can conclude that doing bad shows at first is not a strong indicator of whether you have a future as a showman.

I guess I see the quote as being directed at people who are so afraid of doing a bad show that they'll never get in enough practice to do a good show. Or they practice by, say, filming themselves telling jokes in their basement and getting critiques from their friends who will not be too mean to them. In either case, they never get the amount of feedback they would need to become good. For such a person to hear "Yes, you will fail" can be oddly liberating, since it turns failure into something accounted for in their longer-term plans.

Well sure, if you parametrize with a time factor the result will be a periodic function. But you can still de-parametrize and simply have a closed loop described relationally. A parametrization of a circle usually consists of periodic functions, but that doesn't mean the circle itself is periodic. It's just there.

Also remember that "exactly the same configuration" means exactly the same configuration, of everything, including for instance your calendar, your watch, and your brain and its stored memories. So pretty much by definition there would be no record of such a thing happening. We wouldn't need another variable to encode it because we wouldn't need to encode it in the first place.

I agree with most of what you say, but I'm not so sure about the last two. As others have pointed out, there are many, many cases where the primary suspect of a crime is never prosecuted. Given a choice, prosecutors will usually choose "easy" cases. So an alternate explanation for America's high prison population and incredibly high black prison population is that * more criminals are prosecuted and convicted in America, and * jurors are biased and black criminals are therefore easier to convict; and/or prosecutors are biased and therefore prosecute more black criminals.

Now, since I don't think it's actually optimal for everyone who ever breaks a law to be punished, I have no problem saying, for example, "More criminals are prosecuted and convicted here, and that's too bad."

This is off the top of my head, so it may be total bullshit. I find the idea of memory in a timeless universe slippery myself, and can only occasionally believe I understand it. But anyway...

If you want to implement a sort of memory in your 2D space with one particle, then for each point (x0,y0) in space you can add a coordinate n(x0,y0), and a differential relation

dn(x0,y0) = δ(x-x0,y-y0) sqrt(dx^2 + dy^2)

where δ is the Dirac delta. Each n(x0,y0) can be thought of as an observer at the point (x0,y0), counting the number of times the particle passes through. There is no reference to a time parameter in this equation, and yet there is a definite direction-of-time, because by moving the particle along a path you can only increase all n(x0,y0) for points (x0,y0) along that path.

A point in this configuration space consists of a "current" point (x,y), along with a local history at each point. If you don't make any other requirements, these local histories won't give you a unique global history, because the points could have been visited in any order. But if you impose smoothness requirements on x and y, and your local histories are consistent with those smoothness requirements, then you will have only one possible global history, or at most a finite number.

Super-late answer!

If you ask about a configuration X, "Where does this configuration come from?" I will point at a configuration W for which the flow from W to X is very high. If you ask, "Well, where does W come from?" I will point to a configuration V for which the flow from V to W is very high. We can play this game for a long time, but at each iteration I will almost certainly be pointing to a lower-entropy configuration than the last. Finally I may point to A, the one-point configuration. If you ask, "Where does A come from?" I have to say, "There is nowhere it comes from with any significant probability." At best I can give you a uniform distribution over all configurations with epsilon entropy. But all this means is that no configuration has A in its likely future.

The thing is, it doesn't make sense to ask what is the probability of a configuration like A, external to the universe itself: you can only ask the probability that a sufficiently long path passing through some specific configuration or set of configurations will have A in * its future, or * its past. The probability of the former is probably 0, so we don't expect a singularity in the future. That of the latter is probably 1, so we do expect a singularity in the past.

It sounded like a bad idea at first, but if the bet is 1 upvote / 1 downvote vs. 89 upvotes/89 downvotes, it could actually be a good use of the karma system. The only way to get a lot of karma would be to consistently win these bets, which is probably as good an indicator for "person worth paying attention to" as making good posts.

Hmm... I'm not sure. I'd take the word of someone with experience on an admissions committee, if you can get it.

If you do it, I think you'd be better off talking just a little about the character and much more about the community you found. Writing to the prompt is not really important for this sort of thing. (Usually one of the prompts is pretty much "Other," confirming that.)