A Maths Problem You WILL Understand

Think of a whole number. If it’s even, halve it. If it’s odd, multiply it by three and add one. Keep doing the same. You will find you have a series of numbers which go up and down in value, and if you plotted them on a graph they’d often seem to bounce up and down like hailstones in a hailstorm. For this reason, they’re called “hailstone numbers”.

Here’s an example. 5, 16, 8, 4, 2, 1, 4, 2, 1 . . .

Here’s another: 42, 21, 64, 32, 16, 8, 4, 2, 1 . . .

At first glance you might expect the situation to be as follows; the larger the number, the more steps it takes to reach the final cycle of 4, 2, 1 . . . It seems quite simple, but it isn’t. The number 27, for example, takes a hundred and eleven steps to get to the cycle and goes as high as 9 232. There seems to be no way to predict exactly how long the sequence will last before it gets to the cycle, although it is unsurprisingly true that larger numbers, particularly odd ones, tend to take longer. For instance, 27 is an odd number and the number 2596148429267413814265248164610048 also takes a hundred and eleven steps to reach the loop, and is considerably larger than 27. In case you’re interested, 27’s sequence is:

27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182,
91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395,
1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566,
283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858,
2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616,
2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92,
46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, …

All of this is easy to understand. Any numerate person with a late primary school level of maths could make these calculations and test the sequences. We the not particularly mathematically-skilled public have no problem getting this and doing the work on it to a certain extent. It also seems to be pure mathematics so far as I can tell: there seem to be no applications for it at all, except perhaps for that frondy thing at the top of this post which looks nice. Much of my own ability to do maths is blocked by my failure to get to grips with calculus, as I suspect it is for many other people. I don’t consider the obstacle to be entirely insurmountable but I’ve never actually succeeded in vaulting over it. But there’s another aspect to this set of sequences which seems to stymie everyone, no matter how adept at mathematics they might be. This is known as Collatz’s Conjecture.

In maths, a conjecture is a statement which seems to be true on the basis of preliminary evidence but has never been proven or disproven. A famous example is Goldbach’s Conjecture: every even whole number from four upwards is the sum of two prime numbers. This has been demonstrated for every such number up to at least 1000000000000000000, but there’s still no known proof or disproof.

Collatz’s Conjecture is that every whole number treated in this way will eventually collapse into the sequence “4, 2, 1, . . .” rather than either going on forever or being part of a different sequence. It’s clearly true for any power of two, but they get further and further apart the higher you go, and in their case there’s a simple relationship between their size and the number of steps before it happens. You might therefore think that prime numbers which are one less than a power of two, known as Mersenne Primes, would have some kind of relationship but they don’t seem to have. 127, for example, takes forty-six steps to reach the cycle.

If the distributions of the steps for each number are plotted on a graph with a logarithmic scale to the Y-axis, before they collapse into the cycle the movements are pretty close to looking random, although there’s an algorithm to them so they aren’t. It occurs to me that the paths might be similar to the movements of atoms and molecules in an atmosphere thin enough at the surface of a moon or planet to be a collisionless gas, but maybe not. It might also be that it would work as a pseudorandom number generator with that logarithmic step and the omission of the final falls to the cycle. It’s also remarkably easy to write a program to do this. For instance, in BASIC:

10 LET A=<value>

20 IF A DIV 2 * 2=A THEN LET A=A/2: GOTO 20

30 LET A=3*A+1

40 GOTO 20

It can also be implemented easily in assembler using shifting and adding: the above code is actually unnecessarily complex, and this may not be coincidence. John Conway of “Life” fame managed to generalise this sequence to produce a Turing Machine, i.e. a general computer which can, given time, do anything any digital computer can, and this opens it up to comparison with the Halting Problem. The Halting Problem is whether an arbitrary computer program, given an input, will finish running or continue forever. Alan Turing proved that there is no way to show that this will happen. If these sequences are shown to be sufficiently similar to computer programs, the Collatz Conjecture would therefore be shown to be unprovable. Conway came up with an esolang (esoteric programming language) called FRACTRAN which was Turing-complete in 1987, based on this sequence.

The largest number tested for this is 268, which is 295147905179352825856. Riho Terras was able to prove that almost no number reaches a point below its original value, and limits to the values were arrived at in 1979 and 1994 which showed that the function can rise as slowly as possible.

Although every number tested does gravitate to 4,2,1…, that isn’t enough to prove that it always happens. For instance, it could be that high enough numbers wouldn’t do this, and since there are infinity whole numbers, simply testing every step is impossible. There are two not mutually-exclusive possibilities. One is that there is a number out there somewhere which will never start to fall towards the cycle. The other is that there is a set of numbers which is part of a completely different loop. Both of these could be true, or only one. If there is such a cycle, it’s been proven that it must have at least 17 087 915 members, meaning that it can’t be practically proven by one person doing pen-and-paper calculations. It’s also true that if the calculation is changed to 3x-1, two other cycles appear:

5, 14, 7, 20, 10, 5,…

and:

17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61,
192, 91, 272, 136, 68, 34, 17,…

There might also be more of them.

There is something of an argument which suggests the Collatz Conjecture is true. It goes like this: one more than thrice an odd number must be even, so it will divide by two. There is then a 50-50 chance that the result will also be even, and therefore also divisible by two. The longer the sequence is, the more likely this is to happen. This is a statistical argument, but some numbers in the sequence are omitted completely in these calculations and others crop up a lot more than they “should”, so the neat bell-curve that might be expected might not be forthcoming.

The numbers do, however, obey Benford’s Law. Benford’s Law was first noticed when books of logarithms in university libraries started to get dirtier towards the front of the books than the back. This is because numbers which begin with smaller digits are much more common than those which begin with larger ones. This applies, for example, to the lengths of rivers, electricity bills, stock and house prices. More than thirty percent of such numbers begin with a one, slightly more than a sixth begin with a two, an eighth begin with a three and so on, until only 4.6% begin with a nine. Benford’s Law does approximately apply to Collatz sequences, and it gets closer the more numbers are included. It’s also true of numbers in any base other than binary. It works best when considering data which span several orders of magnitude and is used to detect fraud in elections and accounting. This presumably means there are computer programs out there on the Dark Web or something which use Benford’s Law to befuddle forensic accountancy. Collatz sequences might find a use there.

Although the Collatz Conjecture seems useless as far as direct applications are concerned, it does have educational value. It shows, for example, how simple formulæ can lead to highly complicated systems, and that there are attractors as mentioned in Happy Catastrophe.

If the same thing is done with negative integers, there are three loops. These are:

-1, -2, -1. . .

-5, -14, -7, -20, -10, -5 . . .

-17, -50, -25, -74, -37, -110, -55, -164, -82, -41, -122, -61, -182, -91, -272, -136, -68, -34, -17 . . .

The image at the top of the post is produced using Collatz sequences. One way of thinking of them is as a tree. All known positive integers tested eventually settle down to the sequence 4, 2, 1 . . . , so this can be seen as something like a trunk or a final confluence of tributaries to a river. There are also other confluences further up the tree. If the transition to an odd number is drawn on this directed graph at a 20° angle in one direction and that to an even one as 8° in the other, you end up with what’s shown above. These angles can be adjusted and you end up with various shapes which look like living organisms such as corals, bryozoa or shrubs.

The oddity about this problem is that it jumps so rapidly from a simple issue which can be understood by anyone who knows basic arithmetic into a problem which has never been solved by the most skilled and advanced mathematicians who have applied themselves to it. This raises the question of why we are able to understand almost anything. Even though there are a huge number of mathematical problems which can and have been solved, it isn’t clear how we are able to do that. It seems that it could easily have turned out that we wouldn’t be able to do maths at all because it all turned out to be too hard, and where would that leave us? Why are we able to solve any mathematical problems? Also, most mathematicians consider this problem to be significant but also one which is likely to absorb all someone’s time without them coming up with a useful result, so it’s also the ultimate time-waster, or at least seems to be. Hence this entire post may or may not have been a waste of time. What do you think?

Psychic Powers Part II

Photo by cottonbro on Pexels.com
  • Communication with the dead
  • Psychometry
  • Levitation
  • Teleportation

Yesterday I went into the conceptual structure of what might be termed the more prosaic psi abilities, and although I did stray into the area of anecdotal evidence and the assertion that they exist, I was more focussed on what they amount to. I do believe they exist, of course, but what I’m saying here should, I hope, not depend on a commitment to their reality. This is more about the idea of psi than their existence.

Some of the ones I’m left with amount to what are called siddhis in Yoga, that is, supernatural abilities, and are considerably more radical than the ones I’ve mentioned before. They’re more “showy”, sometimes literally in the sense that they might be faked as part of a stage show and would impress an audience. Yogis warn people against using them (which simply suggests they believe they exist) because of their showiness and because they “are powers in the worldly estate”, i.e. the temptation exists to use them for one’s own material gain, which if you believe in reincarnation for example is hazardous. Buddhists believe that if a human is reincarnated as a deity, the chances are they will succumb to the temptation to use their powers for selfish ends and end up in Hell in their next incarnation. A similar attitude exists among Christians, who often see these powers as real, but as dangerous for the person using them as they’re Satanic in origin. Other Christians see them as abilities humans would have had if sin hadn’t entered the world.

Communication with the dead is a controversial topic. For instance, dreaming of people who have passed away could be understood this way, and happen involuntarily. If it is as it appears to be, it seems to rely on consciousness and identity both surviving death, and therefore on the existence of a soul as an entity with a substance which exists in the same sense that a physical body exists but of different essence. However, it’s also conceivable that information is being obtained in another way. Anyone who’s seen ‘Black Mirror’ will be aware of the idea that a dead person can be simulated convincingly from their online behaviour, but that this will run the risk of eventually becoming unbearably creepy, and in our own interactions with each other we do build up sophisticated models of how our loved ones behave and who they are. If we are communicating with mere simulacra, the question arises of one’s own integrity and authenticity in one’s own life, in that if one wishes to live on in such a manner, for it to be an accurate copy of oneself one will need to conduct onself honestly towards others.

I would actually broaden the concept of communing with the dead here to a wider set of phenomena also including apparent past life memories and hauntings, and would further suggest that these are not all the same thing, some being much more amenable to naturalistic explanations than others. In what I’m reluctantly forced to call “ghost hunting” in the absence of a more dignified and Latinate-sounding term, a distinction may be made between genuine hauntings by spirits and something more akin to traces of events being played back by “stone tape”, as it came to be known. I find the latter more convincing than the former, and although a process whereby that could take place is hard to identify, I have discussed this in the post on the Chronovisor. It’s a well-established fact that traces of incidents are left in inanimate objects in various ways, such as exposure to daylight or heating, which can be “played back”, and there are very clear traces of events left in the form of such things as footprints and fossils.

The Tanakh is quite clearly opposed to the idea of communicating with the dead although it also seems quite inconsistent as it has practically nothing to say about what, if anything, follows death. The so-called “Witch Of Endor” incident is the only reference I’m aware of the dead communicating with the living, where it’s specifically stated that Samuel speaks to Saul from beyond the grave. The story is recounted in 1 Samuel 28, and there’s no suggestion that the medium is being deceitful. Elsewhere it says that the dead are conscious of nothing at all, or know nothing, in Ecclesiastes 9:5. Hence this appears to be inconsistent. What I understand to be the standard Christian view, and to some extent probably also the Jewish one, is that when people die, they cease to be conscious until the Day of Judgement, at which point they are brought back to life in a living body. Consequently in a Jewish or Christian setting the idea of communicating with the dead is right out, but in the case of Christianity the emphasis on Satan as a personification of evil means that demons or the Devil are likely to be seen as a likely source of apparent information from the dead, and that it’s a case of deception and impersonation for manipulative purposes.

Outside the Judæo-Christian context, and for once it seems justified to talk of a joint tradition while noting the rather more negative connotations in Christianity, there is ancestor-worship and the elevation of status of elders into that of deities. It makes sense to suppose that the perceived increasing wisdom of the old will continue to increase after death until they have a superhuman status. However, there’s also the attitude that the dead who do communicate with us have unfinished business on this plane of existence which it would be best to resolve. Since I’m not particularly well acquainted with Spiritism or Spiritualism, I feel I’m venturing onto unfamiliar territory here and would actively welcome someone’s input on this. Spiritism differs from Spiritualism in that the former asserts that reincarnation occurs but the latter is agnostic on the issue. They believe that spirits of the dead maintain their identity and continue to influence the physical world, in other words telekinesis. Some scientists took Spiritism seriously and it could be said to have been founded by scientists in the first place, notably Emmanuel Swedenborg. Everyone in the Spiritist Universe is gradually making progress towards moral perfection, and nobody is ever reincarnated into a lower form of life. Spiritism is also theistic or deistic (I’m not sure which), and is associated with the Brazilian/Afrikan religion Umbanda.

Spiritualism, unlike Spiritism, initially had no sacred texts, and had a strong liberal strand, in which many of the people involved in it supported votes for women, the rights of indigenous peoples and the abolition of slavery. It was much-criticised in the late nineteenth century due to a large number of mediums being accused of fraud, but it occurs to me that there is a tide away from belief in spirits towards a more materialistic belief system, perhaps in more ways than one. I honestly don’t know how sincere mediums were at the time, but I don’t really see any reason to suppose that the majority weren’t acting in good faith whether or not they were actually able to do what they claimed. I could compare it to complementary medicine. Whether or not it’s efficaceous, the majority of practitioners either believe that it is or that it’s of benefit to their clients. Why would the same not be true of Spiritualism?

Spirits of the dead are said to be inclined to communicate with those who are still alive, and to be evolving spiritually. The movement was also associated with the Quakers in the nineteenth century, although judging by the Quakers here in England I know today there must surely have been a drastic divergence in beliefs, because I can’t imagine any of them entertaining such a world view. The “Indian spirit guide” can be seen as an abiding awareness of the genocide practiced against the Native Americans and perhaps a recognition of the unearned mercy some of them might show post mortem. Although there is something of a loose system around Spiritualism, people completely outside any such tradition often claim to be in contact with the dead, and in fact that would include me, as I believe I at least meet an accurate representation of my father-in-law and one of my grandfathers in my dreams. However, I’m not convinced that identity survives death. I think perhaps individual experiences move around and enter the minds of others, and because they are always first person experiences they are labelled as happening to the person reporting them by their consciousness. However, I’m not going to say flat-out that it’s impossible to communicate with the spirits of the dead. I’m not sure what I think about EVP either, although I experimented with it as a teenager.

EVP is “Electronic Voice Phenomenon”, which is the perception of voices in static. It was said to have provided the inspiration for the Chronovisor, although in that case the voices were interpreted as coming from the past rather than being spirits. In a way, EVP is rather like divination such as reading tea leaves, where some kind of arbitrary, pseudo-random process is used as the basis for extracting apparent information, which may in fact be pareidolia. Static on TV has been suggested for the same purpose, and it’s even been said that the digitalisation of media is part of a conspiracy to close off a potential channel of communication with the spirit world, although this sounds seriously paranoid to me, but perhaps almost nostalgically so. In 1959, a Swedish film producer made recordings of bird song. When he played them back, he claimed to have heard the voices of his dead parents. I don’t know the details of this incident, but there is sometimes “print-through” on tape recordings, where a previous recording made on the same tape can still be faintly heard. Actually that isn’t print-through apparently, but it does happen (print through is where nearby tape on a reel induces faint audio patterns in the currently played portion of tape). Also, it’s interesting that once again the more spiritualistic interpretation is made of the phenomenon, that it was the current spirits of his parents he heard rather than a relic of the past when they were still alive. I don’t know how to choose between these alternatives. Is it time travel or paranormal? Both are very marginalised views. I don’t remember how I got the idea to do this. It would’ve been in about 1981 and it followed on from listening to things like numbers stations, over the horizon radar, jamming and Morse signals a couple of years previously. It doesn’t seem to have been learned from anyone else’s experiments with it. I found that I got vivid visual images in my imagination and could hear music after a few minutes of listening to white noise. In 1985, the book ‘The Ghost of 29 Megacycles’ was published, claiming that a particular frequency was particularly liable to this.

An interesting experiment conducted in 1972 involved the invention of a fictional ghost and a gradually induced séance atmosphere, and as this was increased, participants began to experience a sense of presence. I’m afraid that’s all I know about that.

Psychometry, a word which I think is correct but which I’m attempting to recall from reading it once about four decades back, seems to refer to the idea that a personal object in someone’s presence becomes charged with their energy and personality, as if it’s been magnetised. I was vividly aware of this idea when I visited a herb garrett in Bermondsey, where a surgeon’s saw for removing legs was on display. It had been used on numerous occasions to remove limbs which would otherwise have guaranteed the patient’s death, without anæsthetic of course. Although my rational mind said one thing, it was almost impossible to believe in the heat of the moment that that saw had not been imprinted with the immense quantity of agony it must’ve caused. However, on making this observation to a friend who was also there, he suggested the opposite. This tool had saved hundreds of people’s lives. This is the kind of thinking involved in the idea of fetishism in the religious sense, or perhaps for some in the sexual sense. More specifically, relics of saints and the cross carry a similar idea. In the realm of mediums and readers, as I might call them, the idea is that you can hold a personal effect and psychically reconstruct a person’s life and identity from the psychological impression you receive from it. Once again, like a chronovisor, it’s based on the idea that there are natural recording properties in objects which have been in close proximity to certain events or perhaps just generally, and to me at least this idea has immense emotional appeal. I know I’m not alone in the idea that I wanted to save every written note my mother left me as a child because destroying it would be like killing her. This has an obsessive-compulsive element but is probably quite common and needn’t be medicalised. In the late nineteenth century some people believed that psychometry would prove to be as important a branch of science as the study of electricity. It’s just hard to believe that the physical world really is as indifferent as it apparently is, and although disbelief in this is fine and probably correct, the emotional element is important, and we are emotional beings living in an emotional world. Few people would consider the possessions of a loved one to be completely insignificant, and if they were to dispose of them all after their death the chances are that they would be motivated by grief and not wishing to be reminded of their loss rather than lack of sentiment. This is also where the urge to hoard originates. As with several other alleged psionic abilities, mediums have been enlisted to use psychometry to investigate crimes and missing persons. The presence of DNA on such items means that today a similar kind of significance can actually be rigorously pursued with a high degree of confidence.

I’ve been into teleportation previously on this blog, so I’ll only cover it briefly here. There are a number of supposèd incidents of teleportation recorded, notably one which is said to have occurred between Manila in the Philippines and Mexico City in 1593. A soldier guarding the governor’s palace in Manila felt dizzy and faint, and leant against a wall, closing his eyes. When he opened them a few seconds later, he found himself in Mexico City. He was aware of the recent assassination of the governor of the Philippines before the news was able to reach the city, was jailed for desertion and then released when it turned out several months later that this assassination had indeed occurred. This is of course said to be a tall tale. Another incident involved a nun referred to as the Lady In Blue, whose real name was María de Jesús de Ágreda, who was said to teleport regularly from her abbey in Spain to the land of the Jumanos in present day Texas and New Mexico, between 1620 and 1623. When visited by missionaries in 1629, the Jumanos were said to have been very eager to be baptised because of her proselytism. There are a number of other examples, and Qephitzat Ha-Derekh is the Hebrew name for the phenomenon. Most teleportation today would be considered to be a science fiction device like the transporters in ‘Star Trek’, and teleportation is a bit different from the other examples of psionic powers because scientists have succeeded in relocating the information of a microscopic object’s quantum state without using any kind of physical signalling mechanism or moving the object itself. However, teleportation of the Qephitzat Ha-Derekh kind is another matter entirely, and a common question asked about teleportation is of whether it amounts to death followed by the creation of a clone in another place with intact memories or is a genuine method of transportation.

Finally, there is levitation, famously promoted by the Natural Law Party in the 1992 General Election here and also elsewhere in the world. I could dilate on the political party, which is associated with George Harrison and Maharishi Mahesh Yogi and has actually won some elections, but will confine myself to making the observation that levitation in the context of the NLP is considered a Siddhi in the Yoga tradition and the party itself was the target of considerable outrage from my ex at the time. Levitation is another example of something which is practically achievable without paranormal involvement in some situations, as with superconducting magnets and high-intensity sound waves. It’s also said to occur by some Hindus and Christians, in the latter case sometimes but not always being seen as demonic. Levitation is practiced as a trick in some situations, where it seems to involve certain kinds of muscle control enabling someone to “plank” from beside a hand-held staff placed on the ground.

To conclude, then, this has been an attempt to survey psionic abilities, some of which haven’t even been mentioned. Although I have my own beliefs, I’m more attempting to describe what they are than advocating for their possibility or impossibility. I was hoping also to investigate philosophically the intelligibility of the claim that there can be supernatural explanations for phenomena, but unfortunately I’ve run out of time so it’ll have to wait.

Ancient Earthians

The Palenque Slab

Far back in the mists of ancient time, yea, e’en before the Great Nova of Gath, I wanted to start a piece of writing with that phrase, which I think is actually a quote from somewhere, so I can’t. But anyway, a long time ago, according to Erich von Däniken, ancient aliens visited this planet, interbred with the hominids here and gave rise to intelligent humans. Their traces are still evident in the markèd differences between us and the other great apes, in ancient myths and legends, and in the art, objects and buildings of ancient civilisations. The word “ancient” has to be used a lot here, with reverence and an air of mystique. None of this happened of course, but the idea has a surprising history.

One of the first people to make this suggestion was none other than Carl Sagan, who later came across very much as a sceptic, although not precisely in the same mould as James Randi or John Sladek because of his claim that the Galaxy is “teeming with life”. In his 1966 book ‘Intelligent Life In The Universe’, cowritten with Iosif Shklovsky, Sagan claimed that scientists should take the possibility that aliens had visited this planet in prehistoric times seriously, although it should be treated as a hypothesis subject to testing as part of the usual scientific process. Their reasoning was that since the Universe was such a big place with so many star systems apparently suitable for life, the chances of intelligent species leaving their worlds and exploring distant stars were very good indeed, particularly over the billions of years this planet has been in existence. He later regretted saying this because he felt responsible for the whole ancient astronaut debacle that ensued, particularly due to Zechariah Sitchin and Erich von Däniken among other less well-known writers, and of course this ultimately led to the interminable History Channel series ‘Ancient Aliens’. Sagan actually used to appear on documentaries in the ’70s trying to undo the damage. He actually wasn’t the first. In 1940, the editor of ‘Astounding’ John W Campbell had Isaac Asimov insert Greek deities as aliens in Asimov’s story ‘Homo sol’, with racist overtones. Asimov, being Jewish, was unsurprisingly not keen on this at all and proceeded to make sure he avoided non-human intelligent life forms in most of his writings after that, thereby inventing the ‘Humans Only Galaxy’ which still shows its influence today, for instance in Rob Grant’s and Doug Naylor’s ‘Red Dwarf’ series, where there are no genuine species not originating ultimately on Earth.

Erich von Däniken is obviously the worst offender here. Some of his work is based on forgeries and outright falsehoods. For instance, he mentions and illustrates stones carved with heart transplant operations which were in fact contemporary engravings which he commissioned personally. I don’t want to go into too much detail about this because even his reputation is old hat nowadays, but just briefly he was found guilty of repeated fraud, embezzlement and forgery in 1968 on unrelated matters connected to loans and went to prison for three years. Luckily for him, ‘Memories Of The Future’, more usually known by its English title ‘Chariots Of The Gods’, had been published by then and he was able to pay off his debts with the royalties. He wrote the second book, ‘Gods From Outer Space’, in prison.

Before I go on, I want to debunk this.

First of all, the really low-hanging fruit is the issue of aliens interbreeding with humans. The definition of a species is a population able to interbreed and produce viable offspring, although this often doesn’t work very well because there are microspecies, asexual organisms and fossils we have no idea about. In one of his attempts to salvage the situation, Carl Sagan once said it would be easier to mate a human with a petunia than with an alien, because petunias are more closely related to us. The only scenario in which this would not be so, given unaltered biology, would be if the aliens were actually completely human. More on that in a bit. However, this is often evoked as an explanation for perceived human exceptionalism, but fails for two reasons even if it worked. Firstly, it attempts to explain something by simply deferring it. Secondly, it makes humans seem more special than we really are. We’re apes, simple as that. Our intelligence differs to some extent from the others but if assessed, inappropriately of course, by human standards all the other species of great ape would have an IQ above thirty, probably higher, and they are also ahead of humans in cognitive development until eighteen months. They also have cognitive abilities we lack, as I mentioned previously.

As far as anyone has been able to discover, all living things on this planet descend from a common ancestor four billion years ago, because our genomes are all fairly similar, and with a few exceptions among the viruses, all use the same DNA or RNA bases. It is possible that there’s a “shadow biosphere”: undetected life forms whose chemistry is somewhat different and therefore undetected. The phenomenon of desert varnish is sometimes explained by this – a coating on rocks in deserts whose mode of formation is unknown. Most scientists involved would say it’s very unlikely that we wouldn’t have detected them by now. Anyway, it would be enormously difficult for visiting aliens not to leave behind a few microörganisms, but there are no traces of those at all, so it’s also unlikely that this planet has ever been visited at all. That said, the Silurian Hypothesis suggests that there may have been a technological civilisation before us on this planet, the best candidate being some time in the Eocene (see the post for details), although that could in theory have been native. Or hypothetically, rather.

The humanoid nature of the aliens might be worth looking into. The general consensus among xenobiologists (the only science without a subject, as it’s been called) is that there won’t be any human-looking aliens. Using a very slapdash approach, if each step in evolution had only two alternatives and every star in the Galaxy had intelligent life on one planet circling it, it would take only thirty-nine steps for every such species to be different. On the other hand, evolution really comprises order imposed on randomness by the constraints of biology and the influence of the environment, and this frequently leads to similar organisms. For instance, a flower-like body plan crops up over and over again in the animal, protist and plant kingdoms, even without similar pressures. As for humans, well, maybe we’re just what intelligent tool-using life forms end up looking like, but I personally doubt it. There are reasons for supposing the opposite though.

The illustration at the start of this post is notoriously interpreted by many as an astronaut at the controls of a spaceship. It’s from Palenque (I’m not sure about the political significance of this spelling incidentally – at school we were taught the spelling “Palenki” and that may be less culturally imperialistic but I’m not sure how Yucatec Maya uses Latin script). This is in fact a bas relief of Pacal (and again I don’t know how to reproduce Mayan glyphs inline), a ruler of the area in question. The Von Dänikenische interpretation of that carving is that the snakes’ heads at the bottom (right) are the bottom of a space capsule, there are flames coming out of them, the thing to Pacal’s left/above Pacal is a control panel and so on. Here’s a link to an illustration of that interpretation in detail.

Glossing over what was happening on the Other Side at the time, which is in fact VASTLY significant to many people, on 26th November 1977, as well as there being a really good episode of ‘Doctor Who’ on BBC1, BBC2 broadcast a Horizon special called ‘The Case Of The Ancient Astronauts’. There must’ve been something in the water on that day, because a nice throwaway tidy-up line at the end of the documentary ended up hugely influencing my thoughts on the matter for years afterwards:

The achievements of the past tell us nothing about spacemen (sic), but a great deal about the intelligence of our ancestors. And if we are ever to find other intelligent life among the stars, it’ll be because we continue to apply that inventiveness and that questioning spirit which the ancient astronaut theory seeks so strongly to deny.

And it’s true of course. The idea of ancient astronauts guiding every great achievement of the human race makes us look as if we’re unable to take the initiative, and also carries with it the suggestion that primitive people were primitive more than they were people. I may be reading too much into this but there seems to be an overtone of colonialism there, in that if a non-Western civilisation comes up with something technologically sophisticated it is automatically explained via alien intervention. The TV series ‘Ancient Aliens’ may be a little better in this regard for all I know, but that’s only because it’s had to keep dredging the bottom of the well for more source material to shoehorn alien intervention into. I don’t know how seriously viewers have taken that programme but if they have, it hasn’t done their critical thinking skills any good. However, I do have a slightly more sympathetic take on this because I think for some people belief in ancient astronaut intervention is a form of religious expression, which is seen elsewhere in for example the Raelians, the Aetherius Society, Scientology and tragically also the Heaven’s Gate cult. Since I am religious, possibly even pathologically so, I can’t help thinking that there’s a widespread instinctive human need for religion and this is an example of a new set of religious beliefs with pseudoscientific trappings. Like some other religious beliefs, it portrays humans as helpless, and this strong statement made at the end of this documentary made quite an impression on me.

Remember first of all that I was only ten at the time. I took on board the idea of human ingenuity and resourcefulness, but didn’t reject the ancient artefacts. As far as I was concerned, there was still a mystery to be resolved vis à vis the likes of the Baghdad Battery, the stainless steel pillar in India, the nuclear war in the ancient Indus Valley and so on. I even attempted to write my own version of ‘Chariots Of The Gods?’ likening snakes to trains and so forth, which was more for fun than anything I took seriously. It was kind of a parody, which ‘The Burkiss Way’ was doing better at the time with their Eric von Kontrick character, who incidentally was a creation of Douglas Adams – his fingerprints are all over the sketch.

These are the odd conclusions I drew at the time, which turned out to be remarkably durable. It’s a good example of my tendency to believe it’s important to be delusional in order to support one’s mental health:

I started with the discovery of fire by Homo erectus. This species was the first whose cranial capacity overlaps that of today’s humans. Two important sets of remains are the Peking and Java Men, the second of which is more like Homo sapiens than the latter. I have to say that although they’re portrayed as another species, since they’re not around today I don’t think there’s any firm evidence that they could not have formed a breeding population with us, the Neanderthals and the Denisovans, and therefore that their physical differences from us may be due to evolution having occurred within our species. I suspect that we are they. Their earlier members had a cranial capacity of 600-800 cm3 but later on it had exceeded a litre, reaching 1250 cm3 . Archaic H. sapiens are actually smaller than that at 1200. I could introduce a lot of stuff that I’ve learnt since then and changes due to scientific findings, but I’m going to stick to this impression as it’s what I based my thoughts on at the time.

This next bit had its first draft in about 1978 when I wrote an essay on it for RE, which I’ve lost.

In order to resolve the problems presented by out of place artefacts without evoking aliens, it’s possible to assume that from the discovery of fire, Homo erectus continued to advance technologically over a period of a few thousand years from about 800 000 BP (=Before Present, present being 1950 here) to develop firstly a global advanced industrial society, then to venture forth into the solar system and then the Galaxy, settling worlds all over the Milky Way and maintaining a Phase Three civilisation for hundreds of thousands of years. At this time, the now lost continents of Atlantis and Mu still existed and were the most heavily developed landmasses on this planet. During the last Ice Age, a catastrophe befell this civilisation and it returned to Stone Age level, before giving rise to civilisation as we know it today.

A little embarrassingly, I held on to this set of beliefs for many years although I can’t place when I stopped believing it was so except that I was definitely adult at the time. I also recognise that it’s somewhat reminiscent of the ‘Battlestar Galactica’ mythos, but I guarantee I wasn’t aware of that at the time because I came up with it in late 1977 and the first episode was broadcast in September 1978. There does, however, seem to be a tendency for ideas to occur to various people at the same time, which I think is due to most of the pieces being in place for them: what Charles Fort calls “steam engine time”.

You might also be wondering how come nobody has found any trace of this civilisation. The answer is that it had been calculated many years before, in the 1920s, that all traces of human civilisation would have disappeared entirely within 50 000 years. However, to my pre-teen mind the presence of fossilised artefacts such as screws, glass jars and hammers would’ve been enough to justify my belief. I think there’s something about that age which makes us very keen to believe in things like the Loch Ness Monster and flying saucers. We want the world to be outlandish, exciting and exotic, and it’s like an outgrowth of our earlier burgeoning curiosity about life with, perhaps, a bit of rebellion mixed in with it.

There’s a danger in using ideas without experience of what other people have done with them. Doris Lessing’s ‘Canopus In Argus’ series is an example of that, as may be Audrey Niffenegger’s ‘The Time Traveler’s Wife’, although I admit that she did something different with it than Kurt Vonnegut did. I’ve seen one Galactica film at the cinema in 1980 and that’s it. I never watched the TV series or the reboot, although I have watched one episode of ‘Caprica’. Therefore I’m quite ignorant of it, which I may in fact already have exposed by writing what I have here, but my impression is that it’s set in prehistoric times and that humans on Earth are in the future relative to the time its set in, although this might have been retconned.

A major flaw in my ideas was that if civilisation had collapsed, it should have left humans all over the Galaxy unaware of their history, which is fine since we don’t know anything about any possible life forms in the rest of the Milky Way except that that then makes the probability of us just happening to be on the planet humans evolved on very low. If it had happened, the chances are we’d be living on a different planet with no fossil record of earlier humans and two distinct sets of life forms, one related to humans (even if only distantly, such as algæ) and another which had completely different genetic code and possibly even biology.

As a world-building idea, it has possibilities. The less plausible one is just to take it as it is: we are living after the end of an interstellar civilisation without knowing it. This is not sustainable in hard science fiction. The more plausible version would involve positing that humans do, in fact, eventually colonise the stars before their civilisation collapses and on various planets recovers and gradually discovers that they are not the first. Andrew Tomas wrote a book with that title in 1972, but I haven’t read it and it seems to confine itself to this planet and the likes of the Ancient Babylonians having electric current or something. I vaguely associate that with a book called ‘On The Shores Of Distant Worlds’, which seems to be imaginary as I can’t find any trace of it.

The remaining question is, what does it say about me that this belief system was so persistent? Does it mean I was immature or psychotic? All I can say is that it’s along the lines of my general belief that it’s important to have some kind of “give” in a rational mind to enable it to persist in being rational under stress. It’s kind of like escapism I suppose. On the other hand, there are people who believe all sorts of strange things throughout their lives, whole communities of them, so maybe I’m not so unusual.