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(Note: crossposted from https://math.stackexchange.com/questions/2856241/ where some comments suggested maybe the question isn't appropriate on math.se, although I thought the "statistical-inference" Tag there was more precise. Anyway, there are several other "doomsday argument" posts here, e.g., Many-worlds Interpretation defeats the Doomsday argument?)

I just came across the doomsday argument on tv, https://www.closertotruth.com/series/what-the-doomsday-argument, and then read a little more about it at https://en.wikipedia.org/wiki/Doomsday_argument, and also on several math.se threads, e.g., https://math.stackexchange.com/questions/1889091/, and several threads here, e.g., Many-worlds Interpretation defeats the Doomsday argument?

I didn't notice the following objection that occurred to me, but there's so much written about doomsday that I may have easily missed it. Anyway, my microstate vs macrostate objection goes as follows (for introductory info about microstates and macrostates, see, e.g., https://theory.physics.manchester.ac.uk/~judith/stat_therm/node55.html or https://en.wikipedia.org/wiki/Microstate_(statistical_mechanics) or just google "microstate macrostate" for many more hits)...

First, by analogy, consider the collection of molecules comprising the air in the room you're sitting in. Each individual molecule has three position coordinates (x,y,z) and corresponding three momenta, i.e., six degrees of freedom. And your room probably has something like N~10^24 (some multiple of Avogadro's number) molecules. So there are a total of 6N degrees of freedom describing your room air. And at any instant, such a description is a point in this 6N- dimensional space. That's called a microstate. And the probability of any particular microstate in that huge space is infinitesimal. Nevertheless, the air's always in some particular microstate, whereby it's always in a very,very,very unlikely state.

Contrariwise, a macrostate for your room air is (among other thermodynamic variables) something like temperature. And your room's ambient temperature typically remains pretty stable. That is, whatever very unlikely microstate the air's in, it's nevertheless very,very,very likely that the temperature associated with that microstate is quite close to the ambient temperature.

So, my doomsday objection is that our date-of-birth is analogous to a microstate, whereby there's no reason to expect it's close (within a few standard deviations) to some "likely" or "average" value with respect to any other dates (like the date of human extinction). Rather, there's some macrostate, somehow analogous to temeperature, with respect to which close-to-average should be expected. But what, precisely, is that macrostate??? I haven't been able to conjure up any convincing analogous variable. Any suggestions?...

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After considering @ps's answer below, an entirely different approach to refuting the "doomsday argument" occurred to me. It's based on an analogous argument from that television link above. Click it, and then click on the Nick Bostrum interview, and about 50seconds into that, he describes the general form of the argument, as follows...

Suppose you have a jar filled with N balls numbered 1,...,N, but you have no idea how many balls are in the jar.   Could be 10.   Could be 1,000,000.   However, to help you guess how many, you're allowed to pick one ball from the jar. So you pick a ball, and it happens to be number 7. And you therefore infer that N is >>much<< more likely to be 10 than to be 1,000,000.

Let's call this the "jar argument". So how does the "jar argument" relate to my statistical-mechanical analogy/objection of the "doomsday argument"? It doesn't relate at all!!! I have no objection to the "jar argument" statistical inference whatsoever. It's reasonable and correct. But there's a subtle difference between the "jar argument" and the "doomsday argument" that Nick Bostrum fails to distinguish...

In the "jar argument" you're an independent outside observer, randomly choosing one of the balls from inside the jar, and looking at it. In the "doomsday argument" you >>are<< one of the balls inside the jar, and you're just looking at your own number. You have no opportunity to choose, randomly or otherwise, who you happen to be.

To elaborate that a little more clearly, consider the David Ortiz baseball question, "Who's your daddy?" Well, he is whoever he is. And suppose there were seven billion people on Earth when you were born. That means just one chance in 3.5 billion that this particular guy would, in fact, be your daddy. So can we conclude some astronomically miraculous 1-in-3.5-billion coincidence? Of course not! Your daddy had to be somebody, and it just happened to be him (at least that's what your mom tells him:). No choosing and no randomness, and hence no justifiable statistical inference.

Applied to jar-versus-doomsday, if you're just looking at your own number (doomsday argument), and not randomly choosing it (jar argument), then you can't statistically infer anything about the remainder of the population simply from observing your own number.

Sound convincing? I suppose there's already an abstract mathematical discussion about this jar/doomsday distinction. So what's the general name for this "doomsday argument fallacy"?

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Your temperature analogy rests on equilibrium ideas.

For thermodynamic systems there are a few fairly obvious candidates for indicators that something is unusual, meaning not in equilibrium. For example, if the temperature is changing rapidly over time, or over location, then you have good reason to expect it's not equilibrium. A lump of ice, a flame, some change of state such as all the air in the room turning to liquid, and so on. "Round up the usual suspects."

The reason that equilibrium matters is because it makes the calculation a lot easier. In the case of equilibrium with, on average, the same energy in the system, then you can fairly directly define entropy. And you can predict that the system will be at the maximum entropy consistent with the total energy. And that turns out to correspond to the condition in which there are the most ways to arrange the system with the same energy.

So you think of your air molecules. If all but one o fthe molecules had zero energy, then one molecule would have to have all the energy, and could only be going a single speed. If they all have, on average, the same energy, then they can each be going in their own happy direction. Insert a huge amount of math here. (Round about 4 months of 3rd year physics. Sorry, not typing that much.)

https://en.wikipedia.org/wiki/Entropy

And you can then predict that the maximum entropy will be a distribution of energies for the atoms.

Yada yada, what's that got to do with humans and doomsday.

Are there indicators that could be used to indicate humans were not at equilibrium? Because, if we are drastically far from equilibrium then it is unlikely we are in an ordinary time. Remember the fire, or the ice, or the gas changing to liquid.

And yes, there are some indicators that are worth being concerned about. Not all are bad, but not all are perfectly benign.

Population continues to increase. It may be slowing down. But it is still a concerning parameter.

https://en.wikipedia.org/wiki/Population_growthhttps://en.wikipedia.org/wiki/Population_growth

Literacy is growing. https://en.wikipedia.org/wiki/Literacy

It's fairly easy to pick out other trends that are moving far beyond historical limits. Politics, literature, electronics, medicine, understanding of genetics, etc., many these things are moving along at increasing pace. It is by no means trivial to figure out what will happen in the next 10 years. And to try to figure out what will happen over periods such as several centuries, significant compared to the span of the life of species so far, is probably not possible.

So it seems pretty clear we are not at equilibrium. So the microstate-macrostate type of argument is extremely hard to apply.

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    Thanks for the remarks. I hadn't specifically considered equilibrium (versus non-) thermodynamics; just assumed processes in quasistatic equilibrium (as per page 1 of pretty much every textbook). And my overall birth-date~microstate idea is pretty flimsy to begin with, so I'd be happy with any interpretation whereby population comprises an (equilibrium or non-) canonical ensemble, a la statistical mechanics. And I agree with you that in the end this "doomsday idea" is pretty silly. But lots of people apparently take it seriously, so I was just looking for some rigorous argument disproving it – John Forkosh Jul 20 '18 at 6:11

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