(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?...
** Edit **
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"?