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.
A Box Of Chocolates 