Unfortunately, my thoughts here are tainted. If I came across the argument as such before another of perhaps two of my only luxury expenditures, I don't remember it. Part I of
Paradoxes in Probability (
SpringerBriefs in Philosophy) by Eckhardt is "Anthropic Fallacies" and the second chapter is "DOOMSDAY!", which has ensured I can never look at the argument without my previous biases resulting from a (highly recommended yet certainly unnecessary) purchase: "The Doomsday argument should be seen as a straightforward application of an uncontroversial theorem (Bayes) to data produced from a highly questionable assumption (HR examined below). For this paradox and the next two, the fallacy is most easily traced in two stages:
randomness in reference class and
retrocausality."
The author wrote a paper published in
Journal of Philosophy I managed to dig up here:
A shooting-room view of doomsday, but I haven't read it (as an excuse I plead the possession of his chapter in his book which post-dates it). The wiki article does a decent job could use some cleaning up, but does a decent job of presenting the argument(s) and counters. A particular area of contentious is liable to be missed, at least in its relative importance compared to any singular an concise formulation: the dependence on a one's birth being a randomly distributed discrete variable (in the probability theory sense). Eckhardt concisely states what I would probably take a post to write so I'll simply quote his statement of one inherent fallacy: "We can validly consider our birth rank as generated by random or equiprobable sampling from the collection of all persons who ever live."
Apart from the entirety of evolutionary theory and in particular that of humans (not to mention choosing the "random" reference class to be "humans" rather than "primates" or "animals" which leads to yet another flaw), we start from a position that most Bayesians (or any who use probability for e.g., machine learning, pattern recognition, classification & clustering, etc.) would idiomatically call a hypothesis. The easiest example I can think of to illustrate is suggestions on Netflix or Amazon. Learning algorithms that are designed to "learn" your preferences for movies/books/CDs/etc. based on previous selections have to begin with the first you make. They take this selection, perform whatever your algorithm dictates, and spit out an estimate that is called a "hypothesis", which is just another way of saying it is the current (hopefully) ideal state given the input to predict future actions AND (initially far more importantly) to UPDATE given more information (more purchases, in this example).
The Doomsday Argument relies on this valid inferential process but rather than an initial hypothesis it spits out a model, fully-formed. It's the difference between Amazon making a guess about your interests based on your first purchase or first few purchases and making a guess about next weeks lottery numbers based upon the numbers you have.
Machine learning and related uses of probability theory and statistical learning/inference generally start where the argument does: assume everything is distributed randomly (in this case, any particular person can be considered as identical in terms of "rank order" of birth among a total population of humans. Here things like the law of large numbers come into play but not well. Select a very large number of total humans like a 100 trillion, assume that births are randomly distributed, and you'll find that given the number of births already extinction is always a lot more probable than one would perhaps suppose. In particular, if we used past estimates of human extinction (at least those based on some sort of model that didn't come down to Jesus' return or a Mayan calendar), and calculated the total number of humans ever born by the time that estimation failed, compared it to another, we'd have evidence against the heart of the Doomsday argument (random distribution). Alternatively, if we took these estimates and then calculated birth rates after them, we'd find that the proportionality between humans born prior to estimated extinction and those when humans were supposed to be extinct, then compared both to the number of humans born after humans were predicted to be extinct, we'd find that assuming random distribution of the total N doesn't just mean a failed prediction but that the proportionality compared to subsequent growth indicates a distribution not only non-random but weighted towards later time periods without any indication of placement along a subsequent predicted total N.
Interestingly, the very arguments
for some doomsday predictions (not the argument), from nuclear holocaust to climate change spelling the end of humanity, are arguments
against the Doomsday Argument. This is because the DA starts from the position of a total N that is either random, based on an estimation of species in general, or based upon current models of human population growth. However, by most current models of human population growth (and its limits), humans would have died over 100,000 years ago. Likewise, if a doomsday event occurred, it is at least somewhat independent of the total N and thus birth rank isn't randomly distributed.
There is also, as mentioned above, the arbitrary selection of our randomness as humans rather than other classes to which we belong. If the logic of the DA is sound, then there is no reason we should not be able to use it to refer to our "random" birth rank amongst all those with a frontal cortex, all those with neurons, all mammals, etc. The selection of the class assumed to be random is misleadingly arbitrary. There is no
a priori reason that it should be true that of the total N of all humans any human X should be distributed randomly (and uniformly), but of the total N of all mammals any mammal X (including a human) should not as well. However, we actually CAN compare population growths and extinctions in references classes to which we belong and we find that they do not hold true of us. Thus either there is something special to humans according to which we must restrict the reference class to humans for the DA to hold true, in which case the DA is false because it assumes no such special considerations exist, or there is no such reason to restrict the reference class to humans, in which case the DA is weakened empirically.
The problems, though, don't end with why this or that N might be appropriate and the distribution of the appropriate reference class given that N. Despite the centrality to Bayesian reasoning to the argument, another line of the same reasoning illustrates yet another problem. I know that there had to have been a certain number of people born in order for me to be. Given any N of total humans that will ever be born, in order to place myself in any birth rank or "location" along the distribution of births from this N, I have to assume something of the distribution of unborn generations. In doing so, I have assumed DOOMSDAY and thus used my conclusion to reach my conclusion. This too relates to doomsday scenarios that counter the argument. Given human extinction from an asteroid collision in a century my birth rank is Xn. If climate change kills all humans in 50 years, though, my birth rank is less than Xn. Whenever and whatever happens in order for humans to become extinct, I am relying on these future events to determine my rank now, and thus I am tacitly or implicitly admitting that future events are causing past events (namely, my birth rank and everybody else who is living or has lived).
