## Introduction

This article is a response to the 2018 paper ‘Dissolving the Fermi Paradox’ by Anders Sandberg, Eric Drexler, and Toby Ord. The Fermi Paradox is the apparent contradiction between the great size and age of the universe, which seem to imply a high probability of extra-terrestrial life, and the fact that no extra-terrestrial life has been definitively observed. In their paper, Sandberg et al. use the Drake equation to estimate the expected number of alien civilizations in the Milky Way galaxy and in the observable universe. What differentiates their approach from previous efforts to answer this question is that they do not simply make a point estimate for the number of alien civilizations. Instead, they estimate the uncertainty for each parameter in the Drake equation, and then take random draws from the distribution of each parameter to estimate the distribution for the end result. Using this technique, they find that, although the mean expected number of alien civilizations may be high, there is a large tail of very low expected values, which means that overall the probability of Earth being the only planet to harbour civilization in the Milky Way, and indeed even in the observable universe, is fairly high. On this basis, the authors argue that since it is quite likely given our existing knowledge that no other intelligent alien civilizations exist, the Fermi Paradox is not paradoxical at all, and is therefore dissolved.

In this essay, I will argue that the analysis of Sandberg et al. is flawed in a number of key respects, and as a result the Fermi Paradox remains an open question. Here I briefly list the key problems with the Sandberg et al. paper, before proceeding to discuss each in more detail.

- The method used of multiplying uncertainties of many small numbers, most of which have an upper bound of one, is biased towards yielding a result of a high probability of Earth being unique, while also leading to various dubious results.
- The key result of the paper is driven largely by uncertainty in the parameter f
_{l}, which is modelled in an unusual way without clear justification. - Adoption of slightly different (and I believe more plausible) modelling choices and parameter values yields totally different results, which do not result in the Fermi paradox being dissolved. I illustrate this by re-estimating the Sandberg et al. models using different parameters and modelling assumptions.

## Multiplying small numbers

The very nature of the Drake equation, in which seven apparently-independent parameters (four of which are fractions) are multiplied together, means that it has the potential to give very low numbers. Indeed, since the rate of star formation is fairly well-established to be around 1-10, the average longevity of detectable civilizations is ultimately bounded by the age of the universe to around 10^{10} years, and the number of Earth-like planets per star is generally set at one, it follows that the largest possible number detectable civilizations in the Milky Way galaxy is 10^{11}, or about the same as the number of stars in the galaxy. By contrast, there is no lower bound to any of these numbers, especially the four fractions, which can potentially be set to arbitrarily low non-zero values. The potential for abusing multiplication of fractions to give extremely low numbers is well known. A particularly egregious example of this can be found in the work of Christian apologist Tim McGrew, who estimates the prior probability of having the evidence we do pertaining to the resurrection of Jesus at less than 10^{-40}, on the basis of multiplying together supposedly independent probabilities of each of Jesus’ disciples separately experiencing a hallucination. The Drake equation does not reach this level of absurdity, but nevertheless I think it is important to bear in mind that the structure of the equation makes it potentially much easier to get very low values than very high values. This is especially relevant because the methodology adopted by Sandberg et al. involves identifying sources of uncertainty (especially for the parameters f_{l} and f_{i}), and then using any such uncertainty to argue for wider parameter intervals. Wider parameter intervals, however, almost invariably result in higher probabilities for Earth being alone in the Milky Way and the universe. Interestingly, in arguing for the rare Earth hypothesis, Ward and Brownlee explicitly acknowledge this, noting that they “have accumulated a laundry list of potentially low-probability events or conditions necessary for animal life”. Contrary to the intention, such statements reduce my confidence in the method being applied, since it is all too easy to come up with more and more factors that can be multiplied to produce smaller and smaller numbers. This approach can also lead to bizarre consequences. For instance, under this method uncertainties which increase the number of possible avenues by which life may emerge (such as alternative encoding methods), increase our uncertainty about the value of the f_{l} parameter. However, since this parameter is capped at 1, while there is no lower limit to the long tail of very low estimates for f_{l}, in practise this primarily has the effect of *reducing* the estimated probability of life emerging spontaneously, even though it represents an additional pathway by which this could occur.

There are some further problems with the way the Drake equation is defined, and also the way it is applied in this study. First, it is not clear why it is assumed that life can only exist on planets. It seems just as probable that life could emerge on moons, either of rocky planets or of gas giants, as in the proposal that life may exist in the oceans of Europa. Given this consideration, is seems reasonable to consider the average number of habitable planets and moons per solar system. This would potentially increase the maximal bounds on parameter by one or two orders of magnitude (see Table 1) based on the number of moons in our own solar system (there are ten moons with atmospheres). Second, defining the parameter f_{l} as the fraction of planets that develop life amounts to an implicit assumption that life can emerge at most once in the history of each planet. It seems to me, however, that such an assumption is unwarranted. Presumably life could evolve multiple times on the same planet, perhaps in succession following an intervening extinction event, or perhaps in different regions of the same planet that are inaccessible to each other (for example a planet with disconnected oceans). A similar possibility should also be considered with respect to the parameter f_{i}, the fraction of planets with life that becomes intelligent, as there is no obvious reason why multiple intelligent species cannot evolve on the same planet. This might take the form of such species coexisting, or could involve one species evolving after the previous species became extinct, or even a second species evolving after the first had left the home planet to colonise other planets. Third, the Drake equation does not seem well-suited to incorporate either the seeding of life from one planet to another (which is called panspermia), or of speciation following stellar settlement, both of which would result in large numbers of contactable civilizations even if the emergence of intelligent life on a given planet is very unlikely. These latter scenarios are speculative to be sure, but ignoring these factors would lead to a downward-bias in the estimates of the Drake equation.

**Table 1. Parameters used in the Sandberg paper compared to my preferred values.**

Symbol | Meaning of parameter | Sandberg et al. | My preferred | Explanation of my preferred |

R^{*} | The rate of star formation per year | 1 to 100 | 1 to 10 | Based on Robitaille and Whitney, 2010 |

f_{p} | The fraction of stars with planets | 0.1 to 1 | 1 | Evidence of abundance of exoplanets, e.g. Cassan et al., 2012 |

n_{e} | The number of Earth-like (or otherwise habitable) planets per system with planets | 0.1 to 1 | 0.1 to 100 | Incorporating habitable moons |

f_{l} | The fraction of habitable planets with life | 1 – exp(-λVt) λVt ~ LogN(1,50) | 0.01 to 1 | See below in text |

f_{i} | The fraction of planets with life that becomes intelligent | 0.001 to 1 | 0.0001 to 1 | Based on Sandberg et al., but slightly less optimistic. |

f_{c} | The fraction of intelligent civilizations that become detectable/contactable | 0.01 to 1 | 0.1 to 1 | Based on Sandberg et al., but slightly more optimistic. |

L | The average longevity of such detectable civilizations in years | 10^{2} to 10^{10} | 10^{3} to 10^{10} | Based on Sandberg et al., but slightly more optimistic. |

## The origin of life

Sandberg et al. model all of the parameters of the Drake equation using log-uniform distributions except for the f_{l} parameter, for which they use a method by which they estimate the rate parameter λ per unit volume of a given planet undergoing an abiogenesis event per unit time. It is not explained why this method was used for this parameter while all the other parameters were based on values from the cited literature. Indeed, it seems that other parameters could potentially be modelled using similar methods. For example, the development of intelligent life could be modelled as having some probability per extant species per unit of surface area of a planet, or the longevity of a civilization could be modelled as a function of its population and resource availability. Furthermore, the justification for using a log-normal distribution with standard deviation of 50 is unclear. In supplement i, following a discussion of protein folding, the authors assert that “taking LU[λ] > 300 (log uncertainty of λ) as a reference value seems conservative. To claim conﬁdence that LU is less than 300 would be bold, even allowing for a combinatorially large set of alternative AGT (abiotic/genetic transition) paths and outcomes”. However it is unclear how this number was arrived at, or on what basis 300 was chosen instead of any other number. In my view, abstract considerations of protein folding rates are not an appropriate way to estimate the probability of abiogenesis, as early forms of life would have likely relied upon currently unknown mechanisms of catalysis to significantly increase reaction rates.

As the authors note, the main result of the paper can be produced by relying entirely on the uncertainty of f_{l}, even while setting all other parameters equal to their most optimistic values (see Table 3). However, the enormous uncertainty in f_{l} is not only sufficient, but also necessary to drive the result. As I show in Table 3, if the fl parameter is modelled in the same way as f_{i} and the other parameters, namely as a log-uniform with an interval determined by the cited literature (which I set to a range of 10^{-7} to 1, derived from the cited literature but ignoring a few outliers as discussed in Table 2), the probability of Earth being alone in the universe falls to almost zero. This result illustrates that the main effect reported by Sandberg et al., namely that the Fermi observation is not antecedently very improbable given our current knowledge of parameters in the Drake equation, is entirely dependent on their largely unexplained approach to describing the uncertainty in f_{l}. If it is treated like the other parameters in their equation, the effect vanishes.

Another problem with the method adopted by Sandberg et al concerns some of the very low parameter estimates for f_{l} and f_{i} cited in supplement iii. As I show in Table 2 below, none of the very low estimates for these parameters are based on any scientific considerations. I did not investigate the higher estimates of these parameters, and so cannot claim that these are based on any better evidence than the very low values, but this lack of substance inclines me to be highly distrustful of estimates of f_{l} and f_{i} based on aggregations of expert opinion, since much opinion seems to be informed by little to no actual evidence or modelling. Although Sandberg et al. comment that “there is also a clear bias present towards optimistic, high values since SETI or ETI skeptics typically do not give estimates”, it is not obvious that this is true given the significant number of very low values reported here. Furthermore, four of the six parameters that I was able to check were not reported correctly, either because the stated number did not appear in the original source, or because the number had been misinterpreted to refer to f_{l} and f_{i} when it fact it was an estimate of something else. This further reduces my confidence in the method of estimating the parameters based on existing literature.

**Table 2. Summary of the lowest reported literature values of fl and fi.**

Year | Source title | Estimate | Explanation | Accurate |

1965 | Dimensions and probability of life | f 10_{l} =^{-18} | This is a combined estimate of the probability of life and intelligent life. It is based on the probability of ‘man existing in its present form’, based on an ad hoc multiplication of the number of mutations needed to produce modern humans. This is not an estimate of the probability of life or intelligence in general. | Yes |

1980 | Extraterrestrial intelligent beings do not exist | f10_{l} = ^{-10} | This estimate is actually for a combination of fl, fi, and fc, and is based entirely on the fact that probes from alien probes have not been observed. Hence this cannot be used to compute an estimate of the number of civilizations prior to the Fermi observation. | No |

2012 | How Likely is Extraterrestrial Life? | f= 10_{l} ^{-17} | I can’t find this number in the cited text. On page 41, the value 10^-17 is given as the probability of a 120 nucleotide long polymer forming randomly in the lifetime of a single planet, but this is not asserted to be an actual estimate of fl. | No |

2015 | On the history and future of cosmic planet formation | f= 10_{i} ^{-30} | This is not an actual estimate, but just an ‘arbitrarily small value’, based on a set of previous estimates, none of which mention actual numbers. | Yes |

1982 | Atmospheric evolution, the Drake equation, and DNA: sparse life in an infinite universe | f= 10_{l} ^{-30} | I have not been able to find this chapter online, and although I ordered the book it has not yet arrived, so I am not able to assess this estimate. | NA |

2009 | The search for extraterrestrials: intercepting alien signals | f= 10_{l} ^{-7} | This figure is not in the source given; the provided table gives a probability of 10%. Furthermore, the table is provided simply as an illustrative example, not an actual estimate. | No |

2015 | On The History and Future of Cosmic Planet Formation | f= 10_{i} ^{-5} | The number 10^-5 is actually the fraction of habitable planets with communicable civilizations. This value comes from Carl Sagan’s 1963 paper, but Sagan is actually referring to the proportion of all stars that have a communicative civilization. Sagan’s actual estimate for f_{i} is 0.1. | No |

## An alternative modelling approach

While Sandberg et al. argue that observer selection effects prevent the use of the short time taken for life to emerge on Earth as evidence in favour of a high value for f_{l} I believe this is mistaken. As Lineweaver and Davis argue, the observer effect that is imposed is that Earth must have existed for long enough to have allowed life to emerge. However, as far as we know it is equally possible that we could have found ourselves on a 9 billion year old Earth instead of a 4.5 billion year old one, had it been the case that life took a lot longer to develop than it actually did. Even with just this modest constraint, Lineweaver and Davis estimate that the 95% confidence interval for the probability of life emerging on planets similar to Earth during the same time period is at least 0.13. On the basis of this argument, as well as other considerations I mentioned previously concerning the possibility of radically different forms of life, I believe it is reasonable to model f_{l} as log-uniform between 0.01 and 1. Incorporating this estimate along with my other preferred parameter values (see Table 2), I estimate at a 6% probability of Earth being alone in the Milky Way, and an effectively 0% chance of Earth being alone in the universe (see Table 3). These estimates were constructed by first attempting to replicate the original results, and then modifying the parameters and modelling assumptions in accordance with the various cases considered in Table 3. While I cannot defend my preferred parameter choices with great confidence, I do believe they are at the very least no less defensible than those used by Sandberg et al. As such my point is that, as has been the case since its invention, the Drake equation tells one more about the person making the estimates than about the probability of alien life. If one believes that alien life is likely to be rare, then low values can be chosen for the parameters, and a low estimate will be produced. Conversely, if one tends to think alien life is likely to be fairly common, large values can be chosen, and therefore a higher estimate produced.

**Table 3. Summary of results for key models, with mean and median number of alien civilisations.**

Model | Mean | Median | P(N<1) in Milky Way | P(N<1) in Universe |

Main Sandberg model | 2.70E+07 | 0.32 | 52% | 38% |

My re-estimate of main Sandberg model | 2.73E+07 | 2.32 | 47% | 26% |

My re-estimate of main Sandberg with fl like fi | 3.15E+06 | 1 | 50% | 0.004% |

Sandberg model with uncertainty in fl only | Unknown | Unknown | 41% | 32% |

My re-estimate of Sandberg model with fl only | 5.12E+11 | 9.17E+11 | 28% | 15% |

Naive model with literature log-uniform values | 5.07E+03 | 0 | 96% | 51% |

My preferred model | 3.18e+08 | 9.95E+03 | 6.16% | 0.000% |

## Conclusion

In this essay I have argued that the analysis presented by Sandberg et al. concerning the dissolution of the Fermi paradox is multiply flawed. First, the approach of multiplying many parameter intervals with an upper bound at one, but no corresponding lower bound, predisposes the resulting distribution of the number of alien civilisations to exhibit a very long negative tail, which drives the reported result. Second, the method used to estimate the distribution of the parameter fl is insufficiently justified, and the main result of the paper disappears if this parameter is modelled in the same way as the other parameters. Furthermore, the very low estimates of fl and fi reported in the literature review as supporting evidence are mostly misstated from their sources, and are based on vague speculation rather than specific scientific evidence. Third, an alternative modelling approach which incorporates the evidence of the early emergence of life on Earth, and also makes slightly different assumptions regarding the other parameters of the Drake equation, finds a much lower probability that Earth is unique in the Milky Way, and a negligible chance of being alone in the universe. Overall, while Sandberg et al. have made a significant contribution to the study of the Fermi Paradox and the Drake equation by illustrating how computational methods can be used to construct distributions for the probability of various numbers of alien civilisations, I do not believe they have succeeded in showing that the probability of Earth being alone is sufficiently high so as to dissolve the Fermi paradox. Indeed, I present two different set of parameters which yield a negligible probability of Earth being alone in the universe, and hence insomuch as the estimated number of alien civilisations depends crucially on the chosen modelling assumptions, the Fermi Paradox remains very much an open question.

Note: readers interested in experimenting with their own choice of parameters or alternative modelling assumptions are invited to download the Jupyter Notebook at this link.

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