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Exoplanets, 4 billion-year-old life, Fermi’s paradox and zero-one laws « Math Scholar

Exoplanets, 4 billion-year-old life, Fermi’s paradox and zero-one laws

What do exoplanets, four-billion-year-old life, Fermi’s paradox and zero-one laws of probability theory have to do with each other? Quite a bit, actually. Let us review these developments, one by one:

New exoplanet discoveries

Depiction of the seven exoplanets of the TRAPPIST-1 system. Courtesy NASA.

Depiction of the seven exoplanets of the TRAPPIST-1 system. Courtesy NASA.

On 22 February 2017, a consortium of NASA and European astronomers announced that there are not just one but seven planets that potentially could harbor life, all orbiting a yellow dwarf star named TRAPPIST-1, about 40 light-years (235 trillion miles or 378 trillion km) from earth. This is clearly a remarkable discovery, adding seven to the staggering number (over 3400) of exoplanets currently known, and, more importantly, adding seven to the list of planets (over 30) that are potential habitable homes of life as we know it.

These discoveries were made by carefully analyzing the transit of these planets in front of the host star; conveniently, the planets are in a plane that is in a direct line between the star and earth. What’s more, by careful data analysis the astronomers were even able to deduce the size and mass of these planets, confirming that all seven are roughly the size of earth. While all seven are possible harbors for life, three seem particularly plausible, namely e, f and g in the figure.

Four-billion-year-old life

Iron-rich chert, containing ancient fossils. Courtesy D. Papineau and NY Times.

Iron-rich chert, containing ancient fossils. Courtesy D. Papineau and NY Times.

In another remarkable development, on 1 March 2017 an international (U.K., U.S., Canada, Australia) team of researchers announced discovery of what appear to be primitive fossilized microorganisms that are at least 3.77 billion and possibly up to 4.28 billion years old. These were likely deposited in hydrothermal vents, which are known to be habitats for microbes and may have been the original cradles for the origin of life on earth.

Some researchers dispute these findings. But earlier studies have already confirmed the existence of life 3.5 billion years ago. So either way it is clear that life formed on earth almost “immediately” after its surface solidified, or, in other words, within a few million years (an eye-blink in cosmic time) after the earliest epoch that life possibly could have arisen.

Fermi’s paradox

If life formed so quickly here on earth, surely it has also formed on many of the estimated 100 billion other planets surrounding stars in the Milky Way. And if life has arisen on numerous other planets, surely, after several billion years of evolution, at least some of these planets are now homes to full-fledged technological civilizations, at least as advanced as ours; in fact, almost certainly they are far more advanced, since it is highly unlikely that after billions of years that they are exactly at our level. With their advanced technology, surely they have been able not only to witness the rise of life and civilization on our planet, but also to contact us or otherwise disclose their existence, deliberately or inadvertently. Yet decades of determined high-tech searches by the SETI project and other groups have come up empty-handed. WHERE IS EVERBODY?

Fermi’s paradox, as this conundrum is known, has been analyzed in great detail by many writers since 1950 when Fermi first posed it. Astronomer Frank Drake, for example, proposed his now-famous Drake Equation:

N = R* fp ne fl fi fc L

which estimates the number of technological civilizations in the galaxy. For additional information on the scientific debate, see our previous blogs (Blog A and Blog B), or books by Stephen Webb (2002), John Gribbin (2011) and Paul Davies (2011).

The bottom line is that there are no good answers. Here is a quick summary of some of the proposed explanations and common rejoinders:

  1. They are under strict orders not to disclose their existence. Rejoinder: This explanation (often termed the “zookeeper’s hypothesis”) falls prey to the inescapable fact that it just takes one small group in just one extraterrestrial civilization to dissent and break the pact of silence. It seems utterly impossible that a ban of this sort could be imposed, without a single exception over millions of years, on a vast galactic society of advanced civilizations, each with billions of individuals (if earth society is any guide), dispersed over many star systems and planets.
  2. They exist, but are too far away. Rejoinder: Such arguments ignore the potential of rapidly advancing technology. For example, once a civilization is sufficiently advanced, it could send “von Neumann probes” to distant stars, which could scout out suitable planets, land, and then construct additional copies of themselves, using the latest software beamed from the home planet. In one recent analysis, researchers found that 99% of all star systems in the Milky Way could be explored in only about five million years, which is an eye-blink in the multi-billion-year age of the Milky Way. Already, scientists are planning to send fleets of nanocraft to visit nearby stars such as Alpha Centauri. And astronomers hope to soon be able to detect signatures of life on nearby exoplanets. So given that alien civilizations almost certainly have far more advanced technology than we do, why hasn’t the earth been visited and/or contacted, many times over?
  3. They exist, but have lost interest in interstellar communication and/or exploration. Rejoinder: Given that Darwinian evolution, which is widely believed to be the mechanism guiding the development of biology everywhere in the universe, strongly favors organisms that explore and expand their dominion, it is hardly credible that each and every individual, in each and every distant civilization forever lacks interest in space exploration, or (as in item #1 above) that a galactic society is 100% effective, over many millions of years and over many billions of individuals in numerous civilizations, in enforcing a ban against those who wish to explore the galaxy or communicate with emerging societies such as ours.
  4. They are not interested in making contact with such a primitive species as us. As with #1 and #3, while the majority of individuals and civilizations might not be interested in a species such as us, surely among this vast society at least some individuals and some civilizations are interested. By analogy, while most humans and even most scientists are not at all interested in ants, a few researchers are very interested, and they study ant species all over the world in great detail. Also, we are now learning how to communicate with many species on earth, great and small.
  5. They are calling, but we do not yet recognize the signal. Rejoinder: While most agree that the SETI project still has much searching to do, this explanation doesn’t apply to signals that are sent with the express purpose of communicating to a newly technological society such as us, in a form that we could easily recognize (e.g., by microwave or light signals). And as with item #1, #3 and #4, it is hard to see how multiple galactic societies could forever enforce, without any exceptions, a global ban on such targeted communications. Why are they making it so hard for us to find them?
  6. Civilizations like us invariably self-destruct. From human experience we have survived 200 years of technological adolescence, and have not yet destroyed ourselves in a nuclear or biological apocalypse. In any event, within a decade or two human civilization will spread to the Moon and to Mars, and then its long-term existence will be largely impervious to calamities on earth.
  7. They visited earth and planted DNA. Rejoinder: Although the notion that life began elsewhere (i.e., “directed panspermia”) has been proposed by some scientists, detailed analyses of DNA have found no evidence of anything artificial, and, what’s more, this does not solve the problem of the origin of life — it just pushes it to some other star system.
  8. WE ARE ALONE, at least within the Milky Way galaxy and possibly beyond. Rejoinder: This hypothesis flies in the face of the “principle of mediocrity,” namely the presumption, dominant since the time of Copernicus, that there is nothing special about earth or human society. More importantly, it also flies in the face of virtually all of the recent discoveries in this general arena — extrasolar planets, ancient life, molecular biogenesis and more — which suggest that life is pervasive in the universe.

Zero-one laws of probability

Those who have studied probability theory will recall various “zero-one” laws. Colloquially speaking, if an event has nonzero probability, then eventually, under independent repetitions, it certainly will occur (i.e., it will occur with probability one). Conversely, if an event has zero probability, then no matter how many independent repetitions are performed, it will never occur (i.e., it will occur with probability zero). There is no other logical probability other than zero or one.

Other specific examples of zero-one laws include the Borel-Cantelli lemmas and the Hewitt-Savage law. The first Borel-Cantelli lemma says (under appropriate conditions) that if the sum of the probabilities of a sequence of events is finite, then the probability that infinitely many of them occur is zero. The second Borel-Cantelli lemma says (again, under appropriate conditions) that if the sum of the probabilities diverges and the events are independent, then the probability that infinitely many of them occur is one.

Zero-one laws seem to be analogous to Fermi’s paradox. If, as seems reasonable, there are many alien civilizations (or even just one), each with billions of individuals (as with the human species), having arisen from Darwinian evolution and thus imbued with a drive to expand and explore, then it seems exceedingly improbable that not a single individual or group of individuals from even one of these civilizations has visited the earth, attempted to make contact, or has even merely disclosed their own existence, deliberately or inadvertently. In other words, with billions of possibilities, the probability should be unity that one or more would contact us or at least permit their existence to be disclosed to a civilization such as ours.

The other possibility is, of course, rather disquieting: For reasons we evidently cannot yet fathom, we represent the end of a long string of exceedingly unlikely events (possibilities include the origin of life, the origin of complex multicellular life, the origin of intelligent life, the avoidance of various planetary catastrophes, the maintenance of a hospitable environment over billions of years, etc.), whose probability of simultaneously occurring is virtually zero. For example, there may be some great filter (e.g., a gamma-ray burst or the like) that invariably ends societies such as ours before they advance to the technology level or venture to the stars or colonize the galaxy, and we somehow have miraculously avoided this common fate so far. Either way, we are alone — the first and only technological civilization in the Milky Way if not beyond.

So which is it, probability one or probability zero? Is intelligent life pervasive in the universe, or is humanity a highly improbable freak of nature?


What is the answer to Fermi’s paradox? The present author certainly does not know. But it is clear that the research topics behind Fermi’s paradox (exoplanets, origin of life, SETI, etc.) are among the most significant scientific questions that our species has addressed.

I, for one, hope that I live to see the day that this question is finally resolved.

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