What If Nobody Could Count?

This is going to come from a weird direction at first, but bear with me.

The Fermi Paradox is well-known nowadays, and amounts to the question: where are all the aliens? The most obvious solution to this is that there just aren’t any, and there are other possibilities such as the idea that it’s just too difficult to get across interstellar space, that there’s no reason good enough to do so and so on. However, there are also “minority” explanations for the Paradox which are less well-known, such as the idea that science reaches the point where testability of hypotheses becomes impractical or impossible or that, far from the scenario of planets being endlessly pelted by asteroids and comets, preventing life from becoming complex, there actually aren’t enough mass extinctions to stimulate evolution to the point where there’s intelligent life. One of these is that intelligence of our kind might be unlikely to evolve, and that we’ve just stumbled across it. It does in fact seem very strange to me that we evolved on the savannah to gather plants and hunt herbivorous mammals and the like and yet somehow this enables us to do things like discover neutrinos and play chess, so I have some sympathy with this. There’s a more specific version of this. What if the reason we never detect or see any aliens is that they can’t do mathematics? If they can’t do maths, they can’t, for example, do rocket science, although presumably they’d get way beyond that in their journey to the stars anyway, but even that basic thing is beyond them. But why might that be? If we couldn’t do maths, would we be able to do other things? How would it have made the world different?

It’s notable that hunter-gatherer societies, which is what we used to be, tend not to care much about counting. They may only be able to count to four or have three numerical concepts, comprising one, two and more than two. This is presumably because it isn’t that important to their survival or even flourishing, but this raises the question of why we have the ability to conceive of infinity, zero, negative numbers, decimal fractions, imaginary numbers and so forth. We have discovered and invented many things since everyone was a hunter-gatherer, so why should maths be any different? After all, other species are often capable of counting, apparently up to about five, and they can usually tell the difference between something being there and it not being there if they can perceive it in the first place, which is the difference between zero and one. However, counting is not the same as another skill, probably found much more widely, known as subitizing.

Subitizing is one of several faculties which I considered capitalising on when I was home edding in the ’90s CE and ‘noughties. It’s the ability to judge at a glance how many objects of a particular kind there are in one’s visual field. It also applies to touch and perhaps other sensory modalities, although some don’t lend themselves to it. It isn’t the same as counting. Subitizing does take longer the more objects there are, for most people. However, for a few the ability to subitise (I really want to spell it with an S!) extends far beyond this:

Subitising is substantially faster than counting. It takes between forty and a hundred milliseconds longer for most people to recognise each additional object compared to the longer period of time it takes to count them. My impression is that the maximum number of subitisable objects for most adults is five. That’s generally my limit but there are a few exceptions with special categories of objects (and I don’t want to talk about this) where my subitisation goes up to around four hundred. But I wouldn’t be able to subitise how many peas there are in a typical serving on a dinner plate and in most respects I am completely normal with regard to the ability.

Subitising is impossible for some people with injured parietal lobes, which are the ones just behind the crown of the head, and they also lack the ability to perceive more than one item at a time. Positron Emission Tomography (PET) scans show that different parts of the brain are used to subitise than to count. The fact that subitising seems to get to five suggests that a quinary counting system would be easier to use than decimal, and perhaps be less disabling for people with dyscalculia.

The existence of this condition suggests that there is some kind of in-built faculty in most people than enables them to do maths fairly well. The rest of us do something mysterious with numbers, in that we learn to use them properly, associate particular notations with them and can develop our ability to do arithmetic to grasp more arcane concepts such as irrational and transcendental numbers, countable and uncountable infinities and hypercomplex numbers, and of course a load of other things I have no idea about because I’m not a mathematician or particularly good at maths. But I am average at maths. I have an O-level in it, for instance. Other species we know of may not be able to do O-level maths, and not just because there aren’t many exam centres for O-levels any more. However, they often do appear to have at least an approximate number system and also to conceive of when there is more or less of something, which serves the same purpose much of the time. The ability to distinguish between numbers in this way is referred to as numerosity rather than numeracy. On the whole, or at least speaking for myself, with the exception of my peculiar subitisation, I would say people seamlessly link nomerosity and numeracy. Very young children seem to have one without the other.

It’s been established that some corvids have a number sense up to five. This was experimentally found in ravens. I say “some corvids” because choughs and jackdaws probably haven’t been investigated, for example. This isn’t surprising because corvids along with parrots have cognition notably similar to that of humans. Other primates have unsurprisingly been found to be able to subitise and their perception, like perception generally, corresponds to a linear relationship with the stimuli at small quantities and a logarithmic one at higher ones, which is challenging to divorce from counting but can be done.

It’s been considered odd that mathematics is in any way useful. Why should the Universe be amenable to being considered in this way? There are cases of people with doctorates in the sciences who can do algebra but not arithmetic, so the inability to perform in one branch of mathematics doesn’t rule them all out. It’s also the case that logarithms, calculus and trigonometry are to some extent built into our abilities, possibly without there even being particular cognitive modules able to perform them. Logarithms turn up in how we believe varying strengths of stimuli. For instance, before the Christian Era, and therefore around two millennia before Napier, the Greeks classified stars into six brightness categories, which to human vision simply looks like a scale of one to six but once formalised turns out to be a logarithmic scale such that a star of first magnitude is a hundred times brighter than one of the sixth. Although this has been made more precise, the actual perception remains. Likewise with sound volume, the decibel scale is logarithmic, with each three decibel increase being roughly equivalent to a doubling in loudness, but this is not just a kludge but connected to how we actually perceive loudness. A third example is with the perception of weight. We will be more aware of the difference in weight of one kilogramme (that’s actually mass of course) if it’s double the previous weight than if it’s only a dozenth of it, and this is to do with loads on muscles and angles of carrying as much as direct perception, suggesting that this logarithmic nature of perception is not to do with what we’ve got built into our brains or sense organs. Regarding calculus, aiming and catching objects, particularly the latter, seems to involve some kind of instinctive or learnt perception of infinitesimals and limits (I don’t know calculus so this is vague), and depth perception, although it also involves other cues such as mist and focus, is a form of trigonometry involving calculating the distance between your eyes and comparing it to the shift in position against a background. These are unconscious, intuitive ways of using various forms of maths, used, for example, by predators chasing prey, but they are apparently impossible to harness for more general purposes. It reminds me of how in the past a graphics card in a computer can do all sorts of fancy calculations which were, however, not available for use directly in something like a spreadsheet, although more recently that has changed somewhat. The same kind of calculations would be involved as with depth perception in some cases.

There are no units, other anatomical features or physiological functions which are dedicated to doing this kind of maths which can be separated from their other functions, and these abilities are trainable but not transferable. Nevertheless they exist. If a particularly vivid or precise form of visual or perhaps other sensory “imaging” process is available, this could be put to such a purpose. For instance, one might imagine standing in front of a series of sheets of glass with numbered grids on them enabling one to judge the angles of ones eyes and the distances involved, which would enable one to come up with a table of trigonometrical functions. I don’t know if anyone has the ability to do this. It does sound very much like it’s latent in the psyche though, particularly in view of the special abilities which some people have acquired after brain injuries. This means it’s very difficult to work out what we are mathematically capable of.

Nevertheless, it’s instructive to imagine a society without maths, and with no history of maths, although also important to specify exactly what that means. It doesn’t exactly seem to imply one where people can’t count, but maybe it does. When we count, we put things in a sequence and it’s possible that this combination of sequencing and increasing quantity would be the bit that was impossible. For instance, we might be able to recognise up to five objects and even have words for those arrangements, but not recognise many significant relations between those concepts. In fact, taking the ‘Rain Man’ example, maybe we could even subitise into the thousands without recognising any connection. It seems far-fetched that this would be so, but maybe there’s something staggeringly obvious and significant about our own lives which we are equally incapable of grasping but which aliens would be able to perceive immediately.

Whereas there are many dyscalculic people in the world, this situation is not similar to that. It isn’t a question of a few people who are unable to use maths effectively, but an entire species which is highly intelligent and yet can’t. I can imagine a situation where crops are sown at a particular time of year, which might be identified by the appearance of particular flowers or animal migration, or perhaps weather or floods if sufficiently reliable, harvested when some other event takes place and then placed in a grain store of a particular size, which if you know is full beyond a certain level would provide for everyone in the village for that winter. Our bodies don’t need to count to lay down fat stores so we can use them up when food is short, so why would a society need to? Nor do the flowers or migrating animals know the date and month when these things happen. We would be thrown back on subitisation and judging quantities non-numerically.

We might or might not have clocks and calendars. We could be aware of sequences, just not numbers in the usual sense. Our current calendar resorts to numbering from September onward, but in Roman times the numbers began with Quintilis and Sextilis, now known as July and August, and the Anglo-Saxon calendar used to call months things like “wulf monaþ” – “wolf month”. Likewise we can think of the day as consisting of morning twilight, sunrise, noon, sunset, evening twilight and night. Not being able to grasp counting is not the same as being unable to have a calendar. However, the years couldn’t have numbers, although they might have cycles like Chinese animal years or some of the cycles used in Mesoamerican calendars. Therefore there could be a calendar and even something like history, but there would be no dates. “Last June” and “next July” are possibilities, and perhaps even “the June before last” and “the August after next”, and perhaps more than that, but historical dates would end up as something like “during Queen Anne’s reign” or “just before the Norman Conquest”. It would be possible to date things according to memorable or significant events or the lives of particular people, especially relatives, but there would be no numbered years. Nor could there be an institution such as a sabbath or a jubilee, or anniversaries or birthdays.

One of the things which makes it hard to imagine such a society is that although we’ve had examples of hunter-gatherer cultures which don’t have much use for numbers, it isn’t clear how impaired a society would be if it wasn’t hunter-gatherer, or what other abilities people might have to compensate. For instance, agriculture seems possible, as does the invention of the wheel and the plough, but not accounting or money. Nothing seems to stand in the way of writing either, even an alphabetic script, although perhaps not alphabetical order. It feels like nothing could be standard though, or standards would be based on comparisons with something obvious and reliable, so for instance a room would have to be higher than the tallest person likely to stand in it and have an appropriately-sized door, but it seems like there could be no concept of, for example, a two-storey or three bedroom dwelling. There’s no problem with travelling on horseback or on a horse-drawn vehicle, but distances would not be easily measured. “Over the horizon” might be one, or “a day’s travel on horseback”, where that article, “a”, is however never associated with the number one. It would be more like “if you set out from here at dawn and walk until sunset you will probably find yourself near place X”. Nonetheless, people could easily become aware that the world was round because of the existence of the horizon. It’s all rather imponderable.

It seems likely that there would be a lot of surplus and over-engineering. Although a grain store might be able to hold an entire winter’s food, there would be no precise way to judge when it would be full. You wouldn’t be able to say that it held a thousand sacks of corn. Not creating a possible surplus could lead to famine, where after the winter was past a parent might be aware that Ruth and Simon had died, but not that two of their children had and that their previous household of six was now down to four. A numerate observer of such a society would probably feel like banging her head against a wall in frustration fairly soon after starting her visit. It doesn’t rule out meticulous planning though. There’s no reason why these people wouldn’t recognise squares and cubes, and therefore lay out a city in the Roman or American way, with grids of streets, but there would be no house numbers and the streets would have to have individual names. It’s also feasible to build straight roads between settlements like the Romans, although surveying would be near-impossible so far as I can tell. In the market, where there is as I said no money, it would be easy to be short-changed in terms of quantity, as there would be no weights, measures or units of capacity.

Could such a society develop beyond a geocentric world view owing to not being able to measure in the same way as we do? There’s no problem with recognising that the world is round, and presumably making the equation with other heavenly bodies visible as discs in the sky that Earth is a sphere among other spheres like them, and retrograde motion might tip thinkers off that we are not stationary with respect to the Sun, so maybe there is a way, but the laws of motion could never be derived from observation, which means no Newtonian physics and, later on, no Einstein. Projectiles hurled from catapults or longbows in battle could have their distances estimated. Maybe balloons are possible too, but motorised vehicles could run out of fuel unexpectedly unless it was possible to inspect the level of petrol, say. There would be no precision engineering.

All this said, there is another rather peculiar possibility. What if they had maths but it was different? What if, although they couldn’t grasp the concept of counting integers or arithmetic operators, they could grasp other branches of maths more easily than we could? Could they perhaps have the likes of group, graph and knot theory, topology and some kind of geometry and develop these early and easily out of some necessity the absence of arithmetic might force upon them, or just anyway due to different kinds of abilities, and ultimately, in some arcane university, someone discovers the concept of adding 2+2, recognises its link to group theory and yet it remains an obscure and ineffable branch of advanced mathematics which no ordinary person wouldn’t be able to understand without years of intense education? Is it possible to be like that?

Now turn this round. These people are never going to be able to achieve space travel, so they’re stuck on their planet. They might be able to fire rockets beyond the stratosphere and take photos with a heavily armoured camera (a lot of them would explode or shoot out sideways) or venture forth tens of kilometres above the surface in order to draw maps of their continents, but there’s no Yuri Gagarin or Neil Armstrong in this world. But what if they hitched a ride over to us in this parallel universe on the same planet with some dimension-hopping squid family? What would they make of us with our ubiquitous numeracy? What would we make of them with that thing that they have which we can’t even imagine, that they can’t believe anyone could manage without? It may not be in the area of mathematics at all. Alternatively, perhaps they would have mathematics, but it would be of a completely different kind. Does that even make sense though?

One interesting feature of the cognition of species which are closely related to ours, such as chimpanzees, is that they sometimes outperform us in some areas. For instance, when chimpanzees who can count using Western Hindu-Arabic numerals are briefly shown digits from 1 to 9 in random positions on a touchscreen, they will remember what order they were in after they disappear. Most members of our species probably wouldn’t be able to do that. The capacity of our short-term memory is usually about six “chunks”, which is surprisingly different from our usual capacity to subitise. Hence it seems that we’ve been on the path of being able to perform arithmetic, if not actually there already, since the mid-Miocene, and this scenario of us not having that capacity would diverge from our time line in such a way that chimpanzees at least would also lack this ability. We seem to have a poorer short term memory, and it’s been suggested that this is because of the development of a capacity for language.

The resemblance of some widespread mammalian skills to calculus, logarithms and trigonometry without the conscious articulation of these abilities until a long way into human history also suggests another way mathematical skills could have evolved. Praying mantises have good depth perception and can therefore be assumed to use something like trig to do what they do. Is there a way to start with these three skills along with subitising and arrive at mathematics without using arithmetic? Maybe we could’ve seen ancient Egyptian papyri dealing with integration and differentiation with no numerical notation. Is that a nonsensical idea? It isn’t clear what the nature of doing what could equally well be done in this other mathematical ways is. We may not be able to generalise from the special case of aiming a projectile or catching a ball to these precisely expressed methods.

Dyscalculia has already been mentioned here. This may accompany dyslexia and exists on a continuum. Although some of it might be misdiagnosed and be due to issues with how maths has been learnt, or rather not learnt, it also exists in its own right as a kind of neurodiversity. ‘Rain Man’ in fact depicts someone who may have dyscalculia as well as numerical savantry:

Incidentally, I’m aware that there may be issues with this film’s depiction of Raymond as in the autistic landscape but these clips do serve as useful illustrations of the relevant features of the human psyche. Dyscalculia may involve difficulty in understanding place value and zero, which could be related to the sequencing issue in dyslexia. However, one can easily have difficulties in sequencing without this having any bearing on one’s mathematical ability, as with dyspraxia. A procedure such as long division can be beyond them, as incidentally it is me although I’m not dyscalculic. However, what I’m describing here is not dyscalculia as that is associated with a deficit in subitising and, like that when it’s isolated, is associated with part of the parietal lobe. I’m trying to envisage a situation where subitising is intact. Hence the following list may not be that useful, but here it is anyway. Dyscalculia can involve not being able to read an analogue clock, not being able to tell the difference between left and right, limited spatial reasoning, the absence of mental images, difficulty in dancing and a poor sense of direction, among various other things. There are two main theories concerning the cause. One is that the approximate number system found in humans and many other species is visualised as a number line, so people without mental images might be expected not to be able to do arithmetic. In that case, maybe there are other species whereof some can subitise better than others. The other theory is that there’s a deficit in being able to associate number with notation or symbols. Although all this is interesting and important to bear in mind, it doesn’t seem to be directly related to the idea of an entire sentient species which has no ability to do arithmetic or mathematics. It would be interesting to investigate the abilities of elephants, parrots and cetaceans to do maths, and it should also be borne in mind that the inability to perform arithmetic is not the same as the ability to reason mathematically, which is at times entirely different.

The fact that the seeds of mathematical reasoning as a separate ability have been present in the brains of our ancestors since the Miocene doesn’t mean it gave a selective advantage at that time, or that if it did, further developments were not as adaptive in a pre-agricultural society. There is some merit in being able to count tribal members or work out what time of year a fruit is likely to be available or at its best even in a Palæolithic society, or to be able to give each person a bag for collecting food or a spear for hunting, because if there are two dozen people in the group, it might be a waste of time and energy to make too many spears or bags. One thing this illustrates, though, is the order in which evolution occurs, which can be quite counter-intuitive. A trait has to appear and be manifested phenotypically before it confers an advantage. The mutations themselves are quite random, and most of the time confer no advantage, but they can sometimes result in one, so the fact that our ancestors developed mathematical abilities doesn’t imply that it has immediate benefits for survival and propagation of that trait. However, when such a trait is in the situation of not conferring an immediate benefit, it can turn out to be energetically expensive for the organism and be selected against. On the other hand, a trait can often only emerge in certain organisms and can confer indirect benefits because it can show how the individual is so “fit” that they can afford to have something like a fancy pair of antlers or beautiful plumage which serves no purpose as such except to advertise that fact. Applying this to prowess in maths conjures up a rather weird scene of ancient hominids being attracted to nerdishness!

There is, however, also group selection. This has been unpopular compared to the Dawkins-style approach that it’s all about the genes surviving and nothing else. Dawkins in his early years always came across to me as Thatcherite, in the sense that there was almost “no such thing as the species” in the same way as Thatcher claimed “there is no such thing as society”. In an even more atomised sociological view, Dawkins believed that even our individual genes were out for themselves. Group selection is the idea that natural selection takes place among groups rather than individuals. An uncontroversial example is found among social insects because they are all siblings or parents, so in their case individual and group selection amount to the same thing, and even Darwin believed in it to some extent. It also changes the nature of ethics because for Dawkins and others of his ilk, altruism is rarely or never anything more than enlightened self-interest. But there is division of labour in today’s society, and it seems to make sense that tribes might need some people who were good at maths. Again, this leads to an incongruous-seeming situation where every hunter-gatherer tribe has an accountant! However, it is credible to me that there could be someone in a tribe keeping track of bartered items, if barter was ever that widespread, which has been questioned. In fact, some of the earliest examples of writing are accountancy-related, so maybe it isn’t that far-fetched although it seems that agriculture and fairly large settlements would lend themselve more to that than possibly nomadic folk. This in turn raises the possibility that writing itself was stimulated by mathematical ability, although this doesn’t seem to be its only origin.

To conclude then, it’s conceivable that the reason we haven’t noticed any aliens is not because they’re absent but because they’re no good at rocket science. Maybe they just can’t do maths. This is not quite the same as not being able to count, or at least tell how many items there are, and in fact subitising could be at what would be savantry levels for us in such a species, but they continue not being able to add up. But also, maybe there are species with different maths, or which find what we find easy difficult and what is hard for us intuitive. There are a few other intriguing possibilities here, such as the idea that science might just “run out” before it provides us with the means necessary to visit other star systems easily, but for now I’m going to stick with this, and also note that in a way, our ability to do any maths at all and its usefulness in the world is in fact really more than a bit weird.

Vulcan And Vulcan

If you say “Vulcan” to most people nowadays in an Outer Space context, the chances are they’ll think of Spock, and that’s an entirely valid thing to do. However, if you were to say it to anyone with much knowledge of astronomy in the nineteenth century, it would’ve called something completely different to mind: a planet which orbits the Sun even more closely than Mercury. I’m going to cover both in this post.

Firstly, the ‘Star Trek’ Vulcan, whose Vulcan name is Ni’Var. This is reputed to orbit the star 40 Eridani A, a member of a trinary star system also known as ο² Eridani (Omicron-2 Eridani – that isn’t an “O”) sixteen and one quarter light years from here, and therefore also quite close to 82 Eridani, which is said to be one of the most suitable nearby stars for life, around which a possibly habitable planet orbits in real life. Of the stars, A is an orange dwarf, B a white dwarf and C a red dwarf which is also a flare star. Because B would previously have been a red giant and exploded, the chances are that any habitable planets orbiting A would have been sterilised by B’s outburst, and since C is a flare star, this is also unsuitable, although there would be nothing to stop an interstellar civilisation settling a planet in A’s habitable zone, which would of course be Vulcan.

As I’ve mentioned, I don’t pay much attention to either ‘Discovery’ or the new ‘Star Trek’ films, but I’m aware that Vulcan has been destroyed in revenge for the destruction of Romulus. I find this a bit annoying and I’m not sure what the point of it was plot-wise, but it doesn’t alter the in-universe fact that Vulcan was the homeworld of the first species to make open contact with humans when Zefram Cochrane first activated the warp drive. I’m also aware that that is inconsistent with the depiction of Cochrane in TOS. It is interesting, though, that any real planet in the habitable zone of 40 Eridani A would have been severely damaged by the 40 Eridani B supernova.

I understand Vulcan to have no moons, higher gravity than Earth and no surface oceans. I’m also aware that Romulans and Vulcans are the same species. It irritates me that they’re humanoid but also interests me that some of their anatomy and physiology is known, such as their copper-based respiratory pigment. Then again, although the in-universe explanation of widespread humanoid aliens is that we are all descended from humanoid ancestors who existed around the time our own Solar System formed, it’s also conceivable that convergent evolution would lead to similar body forms among sentient tool-using species. Back to Vulcan itself though. It has a thinner atmosphere than Earth’s, which I think justifies the copper-based blood pigment, and the sky and much of the surface is red. There are seas, i.e. large landlocked lakes, rather than oceans continuous with each other. Depending on the total surface coverage of bodies of water, I think this would probably make the planet uninhabitable for humans although clearly not for native life. 40 Eridani A is a K-type star, with a longer lifetime than the Sun’s in terms of being able to support a habitable planet, which, if orbiting at the distance necessary to receive the same quantity of light and head from its primary as we do from our Sun as a planet, would have a mean orbital radius of about 0.68 AU, i.e. sixty-eight percent of Earth’s distance from the Sun, and 223-day year. However, Vulcan is supposed to be hotter than Earth and might therefore be closer to its sun or have more greenhouse gases in its atmosphere, or it could just reflect less heat back into space, and in fact it probably would due to less ice on its surface. The difficult thing to account for with Vulcan is the combined higher gravity and thinner atmosphere, but there is another reason than gravity why a body might lose some of the gas surrounding it, which is consistent with what we “know” about Vulcan. Earth’s strong magnetic field is generated by our own large moon, Cynthia, which raises tides in our iron-nickel core and magnetises it like stroking a bar of iron with a magnet does, and that generates our magnetosphere, which traps ionising radiation from the solar wind which might otherwise reach Earth’s surface and strip away our atmosphere. Hence Vulcan, with no pre-existing satellites, would not have this benefit but would on the other hand still be able to hold on to some atmosphere because of its higher gravity, so maybe that is in fact realistic. Venus has no magnetic field but an extremely dense atmosphere, although not one hospitable to life at the solid surface, due to photolysis – the action of light on rocks releasing carbon dioxide gas. However, we’re basically aware that Vulcan’s atmosphere has enough oxygen to support human life without their own oxygen supply, and not enough carbon dioxide to poison us, which is 0.5% at our own atmospheric pressure. 170 millibars partial pressure of oxygen is required for this and CO₂ cannot be making a significant contribution to the pressure, so we can surmise that the rest of Vulcan’s atmosphere substantially consists of other gases. It isn’t pure oxygen. In fact, it’s quite likely to be nitrogen if Vulcan physiology is anything like ours and their bodies consist partly of protein, as the nitrogen has to come from somewhere, so I’m going to say the mean surface air pressure is about 0.25 bars. I’ve plucked this figure out of the air, so to speak. There probably is no such thing as sea level there because of the various lakes with different presumed depths and heights, so this would be defined as some kind of mean distance from the centre of the planet or a level at which gravitational pull is close to a particular standard. The boiling point of water on Vulcan is therefore about 60°C, but we know from McCoy’s mouth that Vulcan is very hot compared to Earth, so this puts an upper limit on its surface temperature unless it’s so hot at the equator that it causes water to evaporate.

40 Eridani A is orange. The sky is likely to be close to a complementary colour, such as teal, given that, but because of the dusty surface it’s entirely feasible that it would in fact be pinkish due to small particles high in the atmosphere. Also, the general ruddiness of the planet as shown on screen gives the impression of heat and dryness, so artistically that does seem to be a good decision. The same features make some people think of Mars as a hot planet when in fact it’s often colder than Antarctica. Regarding sparse water cover, a thin atmosphere might make sense here too, particularly if water is regularly evaporating from the surface at the equator, since some might then be lost into space.

Vulcan would also lack plate tectonics if it’s like this, since that’s fuelled by water. The planet has no continents as such, but it does have active volcanoes and lava fields, which is to some extent to be expected as it corresponds to the “hot spot” situation in the centre of the Pacific plate on Earth, where magma seems to need to vent. Here, this results in Hawaiʻi, but on Vulcan a mountain range could be expected because there are no oceans. There would be nothing like the Pacific Ring Of Fire, and also no fold mountains because those are caused by the collision of continental plates.

Vulcan’s colour is depicted differently in different manifestations of the series. In TOS and Enterprise, it’s red. In TAS it’s yellower, and in TNG brownish. However, on Mars there is variation in colour from space due to a dust storm season, and this can be imagined on Vulcan too. Maybe one way to think of Vulcan is as a larger, hotter version of Mars.

The real 40 Eridani A does have a planet. This is, as usual, called “b”, and orbits much closer to the star than the inner edge of the habitable zone. It has a roughly circular orbit 0.22 AU from the star and a mass estimated at 8.5 times Earth’s (both those figures are rounded off). At Earth’s density, this would give it a diameter of around 25 000 kilometres, which is a type of planet unknown in our own solar system at any distance from us, and it’s classed as a “Super-Earth”, but it has a period of 43 days and would be like Mercury on its surface during the day, if it rotates at all. It’s also the closest known Super-Earth. Its orbit differs considerably from Mercury’s, which will become relevant later in this post, in being much less elliptical, which to me, in my probable naïveté, suggests there are no planets larger than it in at least the inner solar system.

This brings me to the other Vulcan. In the nineteenth Christian century, the French astronomer Urbain Le Verrier came up with a particularly accurate model of planetary motion within the Solar System. It had been noted that the most recently discovered planet, Uranus, tended to drift slightly behind and ahead of its predicted position given its distance from the Sun and shape of its orbit. From this, Le Verrier calculated mathematically that there was likely to be another planet further out pulling at it, and predicted its position, which turned out to be correct. In fact he almost had it named after him, but they eventually decided to call it Neptune. This established his reputation and consequently, when he turned his attention to the orbit of Mercury, people paid attention and took his views seriously.

Mercury’s orbit is quite unusual compared to the other planets, particularly if you ignore the period of time when Pluto was regarded as one. It’s the most eccentric orbit by a long way compared to the others, with a variation in distance from the Sun of around twenty percent. Le Verrier also noted that the movement of the “points” of the orbit precessed around the Sun much faster even when compared to its year of eighty-eight days than those of other planets. Just as he had with Neptune, Le Verrier proposed that there was either an as-yet undiscovered planet even closer to the Sun or a number of smaller bodies like asteroids within the orbit of Mercury, and since it would’ve been so close and so hot, he called it Vulcan after the Roman god of fire, Vulcanus. The planet’s existence could be confirmed in two ways. Either it could be detected in transit, as most planets are detected at the moment, or it could possibly be glimpsed during a total solar eclipse. A number of astronomers then reported that they had indeed seen this planet transiting the Sun. For instance, Edmond Lescarbault, a doctor, described a tiny black spot moving across the Sun faster than a sunspot, moving with the rotation of the Sun, would, and also lacking a sunspot’s penumbra. The observations even seemed to confirm Le Verrier’s prediction of Vulcan’s size and orbit. However, it was difficult to predict when these transits would occur because that depended on the tilt of Vulcan’s orbit compared to ours. Mercury, for example, can only be seen to transit the Sun in May or November because only then is the tilt of both its and our orbits aligned such that it can get between us and the Sun. The observations did seem to occur fairly randomly, but at first glance Mercury’s do as well, if you didn’t know anything about its movements already.

There was a total eclipse of the Sun in 1883, shortly after Le Verrier’s death in 1877, during which Vulcan was not observed. It was still possible that the planet was either behind or transiting the Sun at the time, but six further such observations, the last in 1908, also failed to turn it up, making it increasingly improbable that the planet existed. However since that time astronomers have claimed that close ups of the Sun’s surface do sometimes show small black dots which are not sunspots, although these may be imperfections of photographic plates, and there are asteroids which approach the Sun more closely than Mercury does, such as Icarus. It strikes me that it’s not only possible but probable that there are asteroids which orbit entirely within the orbit of Mercury, although they would have to be very small and would be difficult to observe or confirm. These are known as Vulcanoids, and would have to be between six kilometres and a couple of hundred metres in diameter. Every region of the Solar System which is not severely perturbed by the gravity of known objects has been found to contain objects like asteroids or comets, so if the innermost region of the system doesn’t have any this must be due to a non-gravitational effect. It is in fact possible that the light from the Sun is so strong at that distance that it would push smaller bodies away from it over a long period of time, so this may be the explanation. This might sound far-fetched, but it’s been proposed that this effect could be used to divert asteroids which would otherwise crash into Earth by painting them white in order that the pressure of light from the Sun would change their orbits, and this is also the principle used in a solar sail. The MESSENGER probe took photographs of the region but this was limited because damage from sunlight needed to be avoided. Much closer in than Mercury, asteroids are likely to vaporise of course.

Vulcan was considered to orbit 26 million kilometres from the Sun, giving it a sidereal period (“year”) of twenty-six days. At another point, observations appeared to show it had a year of 38.5 days. I think it was also supposed to be very small but I can’t track this down: possibly about a thirtieth the mass of Mercury, which with the same density would’ve given it a diameter of around 1 600 kilometres, probably meaning that if it had been found to exist it would’ve been demoted from planethood by now in the same way as Pluto was. In fact, if it did exist, it would indeed have perturbed the orbit of Mercury but the other factors which turned out to be the explanation for this phenomenon would still be in play, meaning that there would’ve been an even greater anomaly unless the planet happened to be exactly the right mass and in exactly the right place, and possibly retrograde. Some kind of pointless immense astroengineering project could probably achieve that to some extent, but why? Possibly to prevent us from being aware of relativity?

The fact is that the planets don’t simply orbit the Sun alone without influencing one another and the Sun. This is the famous three-body problem, that it’s impossible to work out in almost all cases how three bodies would orbit each other, and even more so the much larger number of massive bodies in the Solar System. It’s possible to work out how much gravitational influence the planets would have on each other if they were the only two bodies in the Universe though, and if initial conditions are known. For instance, Venus and Earth approach each other to within fifty million kilometres and have roughly the same mass, so left to themselves they would orbit each other at roughly twenty-five million kilometres from their centre of gravity once in forty-five millennia if I’ve calculated that correctly, and at the closest approach, which would be during a transit of Venus, that’s the gravitational pull we’re exerting on each other – about forty-five thousand times less than the Sun’s. Mercury is the least massive planet, being just over half the mass of Mars, the next smallest. Pluto is of course far lower in mass, and if Cynthia is considered a planet in its own right, that would be considerably less massive. Anyway, this means that Mercury is pulled around a lot by the other planets. Venus approaches it to within about 38 million kilometres but without doing the maths it isn’t clear if that’s the biggest gravitational influence because of Jupiter being so much more massive than the other planets, even though it’s far further away. Jupiter is over three hundred times the mass of Earth but would get within 4.8 AU of Mercury, which actually gives it roughly the same influence as Venus. But this is not the only reason Mercury’s orbit precesses as much as it does.

Albert Einstein listed a number of ways to test his theory of general relativity, one of which was the orbit of Mercury. The pull of the other planets is insufficient to explain precession in Newtonian terms. There’s still a bit left over if you try to do this. It’s at least seven percent larger than it “should” be. The explanation for this was instrumental in getting general relativity accepted. Einstein made three suggestions about how general relativity could be corroborated. One was that light would be red shifted if it passed through a gravity well. Remarkably, although it took something like four decades, the observation of 40 Eridani B eventually showed that this was so, I’m guessing because of the other stars in its system. Gravity stretches light because it distorts space. The second proposition was that stars observed near the Sun during a total solar eclipse (Again! They’re useful things) would appear to be in a different position because their light would be bent by the solar gravity, and this was indeed found to be so a few years later. However, the world had to wait for these two findings. The other one was that Mercury’s orbit would precess at the rate it did having taken into account the perturbations of all the other planets, and again this was found to be so, but in this case it was already known that this happened because Le Verrier had observed it in the previous century and the existence of Vulcan had been refuted. The reason this happens, I have to admit, I don’t really understand, but I can provide a kind of visual model of it which could show this.

The Rubber Sheet Theory is a model of space as if it’s two dimensional left to itself with weights representing stars and planets which, if placed on such a sheet would create dents in it. Obviously this is not an adequate explanation as such of general relativity for several reasons, one of which is that it uses gravity to explain gravity – that’s what’s pulling down the weights. It also makes space appear to be a substance, something which physicists had worked heavily against when they disproved the existence of the luminiferous æther, which since it was supposed to be extremely rigid wouldn’t work in this situation anyway. It shouldn’t be mistaken for Einstein’s theory itself, but it is a useful way of looking at it. In any case, if you imagine the kind of dent which shows up in the title sequence of Disney’s ‘The Black Hole’:

. . . which is like one of those charity coin collection things, space around the Sun is distorted to a limited extent like that, and attempting to do a “wall of death”-style orbit around it, which would in any case be elliptical rather than perfectly circular because the Universe is imperfect like that, would lead to your bike describing a series of ellipses which were not perfectly congruent with each other but were more like a spirograph pattern. Having written that paragraph with its references to a number of very ’70s things makes me wonder if it’s going to make any sense to someone born after Generation X.

Now I can see that this does happen, but I am also puzzled by it. Whereas I’m sure that I couldn’t aim a coin at one of those charity collection things in such a way that it would just circle around at the same level until friction interfered, and that at best if I could make it describe an elliptical path for a few revolutions, the bits of the ellipse furthest from and closest to the hole would precess, I would put that down to the fact that I, and anyone else to a lesser extent, can’t aim perfectly rather than simply due to the geometry of the hole. Nevertheless, this appears to be what I’m being asked to believe with this: that it isn’t only one’s inability to aim perfectly, or for that matter the friction the coin (or ball bearing – let’s take the instability of the coin out of the picture), that leads to this precession. But apparently not. Apparently, if you were to have too much time on your hands and designed some kind of precision ball bearing throwing machine for charity coin collectors, and it wouldn’t be popular because they want coins, not ball bearings, it would do the wobble thing even if it stayed circular enough not to fall down the hole immediately, and it would wobble more the closer it was to the whole. So they say, and this is what got general relativity accepted.

There have been other Vulcans. For instance, one of the many hypothetical planets in Western astrology is the intramercurial Vulcan, seen as the soul ruler of Taurus and orbiting once every twenty days. This Vulcan would go retrograde more often than Mercury. It’s fiery and urges the individual to look for non-physical knowledge, which makes sense given its history in astronomy. It was also suggested in a poll as the name of one of the moons of Pluto, and actually won the most votes but that was then named Kerberos after the Hadean dog, which was the runner up. Vulcan actually doesn’t seem like a very good name for a moon of an icy planet way out in the outer reaches of the Solar System, but I don’t know the reasons it wasn’t used. Maybe the IAU just didn’t want to be reminded of what they might regard as an embarrassing phase in the recent history of their science. In the Second Doctor story ‘The Power Of The Daleks’, there’s a planet called Vulcan which is settled by humans and highly volcanic with pools of fuming mercury on its surface. Doesn’t sound very nice at all really. There does not, however, seem to be an asteroid named Vulcan, which is quite surprising.

I’ve sometimes wondered if there’s a story behind the naming of the ‘Star Trek’ Vulcan and if it’s in any way connected to the hypothetical planet, but I don’t know. How about you?