Lemma axiom: No line of rational agents has a zero probability of extinction. - I speak of finite rational agents. - Being finite, they cannot infallibly predict their existence at a given future time. - Being rational, they must assume that always, in the interval, they face a non-zero probability of extinction. - The consequences of this can be modelled mathematically. - Let ⁠t⁠ be the year in the future of a line of (finite) rational agents; ⁠p⁠ the probability per year of the line’s extinction; ⁠q⁠ the probability per year of its survival, such that ⁠q = 1 - p⁠; and ⁠\mathbf{P}⁠ the overall probability that the line ever goes extinct. - Then the latter equals the probability that it goes extinct in the first year (⁠p⁠), plus the probability it survives the first year and goes extinct in the second (⁠qp⁠), plus probability that it survives the first two years and goes extinct in the third (⁠q^2p⁠), and so on. ･\begin{align} \mathbf{P} & = p + pq + pq^2 + \cdots \\[1ex] & = \sum_{t=0}^\infty pq^t \end{align}･ - This is a Bernoulli-process series. : see https://en.wikipedia.org/wiki/Bernoulli_process - Given the axiomatic constraint that ⁠p > 0⁠ (whereby ⁠q < 1⁠), it is also a geometric series and it has a closed-form expression. : re `axiomatic constraint` see `^*axiom` : re `geometric series` see https://en.wikipedia.org/wiki/Geometric_series : re `closed-form expression` see https://en.wikipedia.org/wiki/Geometric_series#Formulation - From the latter, it evaluates to 1. ･\begin{align} \mathbf{P} & = \sum_{t=0}^\infty ar^t &&a = p,\; r = q \\[1ex] & = \frac a {1 - r} &&\lvert r\rvert < 1 \\[1ex] & = \frac p {1 - q} &&q < 1 \\[1ex] & = 1 \end{align}･ - The line will go extinct. - The only way to get a different result would be to relax the constancy constraint on ⁠p⁠, such that ⁠p⁠ can vary. - There are several ways that ⁠p⁠ could vary, but only one that would yield a different result. - It would not help to randomly vary ⁠p⁠ over a probability distribution, for instance. - That would be the same as setting ⁠p⁠ to the mean of the distribution. - Again the result would be 1. - Nor would it help to decrease ⁠p⁠ over a period of time, then leave it constant (or randomly varying). - However low ⁠p⁠ might eventually be driven, and assuming the line were to survive to that point, then effectively it would be back into another Bernoulli process. - Again the result would be 1. - The only way to get a different result is to engage with the infinite length of the series, decreasing ⁠p⁠ over the whole of it, or over some unbounded part. : cf. `^*Conrad.+Lord Jim` @ ~/work/ethic/._/05/sources.brec : ‘The way is to the destructive element submit yourself, and with the exertions of your hands and feet in the water make the deep, deep sea keep you up.’ p. 685. - Let ⁠\delta⁠ be the annual decay rate of ⁠p⁠; and ⁠\epsilon⁠ the complement of ⁠\delta⁠, such that ⁠\epsilon = 1 - \delta⁠. - Then: ･\begin{align} \mathbf{P} & = p + p\epsilon (1 - p) + p\epsilon^2 (1 - p)(1 - p\epsilon) + \cdots \\[1ex] & = \sum_{t=0}^\infty p\epsilon^t \prod_{u=0}^{t-1} (1 - p\epsilon^u) \\[1ex] & = \sum_{t=0}^\infty p\epsilon^t (p;\epsilon)_t \end{align}･ - This is a q-series. : see http://simonrs.com/eulercircle/infiniteseries/steven-qseries.pdf : *Q-series* : see https://faculty.math.illinois.edu/~berndt/articles/q.pdf : What is a Q-series? - I found no closed form for it. : see e.g. https://math.stackexchange.com/q/4692693/1178074 : *Bernoulli-process series with decaying probability ⁠p⁠* - The following results are computed numerically. Table, Survival probability.   % δ     .02 .04 .06 .08 .10 .20 .30 .40 .50 .60   ┏━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┑   ┃ │   .05 ┃ 8.2 28.6 43.4 53.5 60.6 77.9 84.6 88.2 90.5 92.0 │ 4.9   ┃ │   .10 ┃ .6 8.2 18.9 28.6 36.8 60.6 71.6 77.9 81.9 84.6 │ 9.5   % p ┃ │ % p / century   .20 ┃ .7 3.6 8.2 13.5 36.8 51.3 60.6 67.0 71.6 │ 18.1   ┃ │   .40 ┃ .1 .7 1.8 13.5 26.3 36.8 44.9 51.3 │ 33.0   ┃ │   .60 ┃ .1 0.2 5.0 13.5 22.3 30.1 36.7 │ 45.2   ┃ │   .80 ┃ 1.8 6.9 13.5 20.1 26.3 │ 55.2   ┃ │   ┖─────────────────────────────────────────────────────────────┘   2.0 3.9 5.8 7.7 9.5 18.1 26.0 33.0 39.4 45.2     % δ / century Shown is the percent probability of avoiding extinction while subject to an extinction rate (% p) which itself is subject to a rate of decay (% δ). Rates are expressed as percentages both per year (left, top) and per century (right, bottom). The extinction rates shown (left, right) are initial values operative in the first year. Results are computed to the point of convergence using the mathematic model given in the text. Values are rounded to a single decimal place and omitted where they round to zero. : re `mathematic model given in the text` see `^*- Let.+delta.+be the annual decay rate` - For a line of agents to endure, the near-term probability of its extinction (⁠p⁠) must decline and keep declining forever. - Consider what might sustain such a decline. - For a line of rational agents, there are just three possibilities: (a) chance (b) physical necessity (c) rational necessity (a) Could the decline of ⁠p⁠ be sustained by chance? - No; what chance sustained, chance could terminate. - The decline of ⁠p⁠ would itself face extinction, so to speak. - Here the prospects for chance are doomed in a regress on conditions. - With the decline of ⁠p⁠ exposed to its own, near-term probability of extinction (⁠p'⁠), ⁠p'⁠ too would have to decline in order to salvage the decline of ⁠p⁠. - But then the decline of ⁠p'⁠ would be exposed to its own probability of extinction (⁠p''⁠), and so on. !! Chance branching alone might suffice to avoid extinction. : join `^ ${same}$` @ 20_lemma.notes.brec : contra `prospects for chance are doomed in a regress on conditions` - For chance might suffice to branch a line and thereby steadily decrease the near-term probability (⁠p'⁠) of its own extinction, just as deliberate branching would. (b) Could the decline of ⁠p⁠ be sustained by physical necessity? - Then it would be sustained by a law of nature. - Then there might be evidence for it. - Yet there is evidence against it. - The existential risk for humanity has not decreased over the last century, it has increased. - If such law exists, then it must operate at longer time scales. - Consider the physical means by which ⁠p⁠ might decrease indefinitely. - One such means (presently out of reach, but potentially feasible in future) is branching. - A line of agents would split off into existentially-isolated branches, each more-or-less independent in regard to its own existential risk. - Through branch inter-communications, the whole would remain unified as a single line. - The risk of the line’s extinction (⁠p⁠) would then decrease. - All of its branches would have to go extinct in a relatively short period of time; short enough, that is, that branches during the interval failed to split off enough new branches to stem the loss and avoid overall extinction. - It does seem plausible that lines would branch. - Suppose there exists a stochastic law, one that generates a branching tendency sufficient to nudge ⁠p⁠ on a downward trend. - Could that indeed be a law of nature? ⋮ (c) Could the decline of ⁠p⁠ be sustained by rational necessity? ⋮ lemma: No line of rational agents can avoid extinction unless there exists a lineal-autotelic principle of reason. \ Copyright © 2023 Michael Allan.