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.

Ethical Intuition And Homophobia

Back in the ’70s, when I was a child, my mother used to read the Bible to me. This was how I discovered that the written Torah appeared to condemn male homosexual acts. There are other takes on this apparently, but they’ve always seemed to be against the grain, perverse interpretations of what was pretty clearly extreme homophobia. At the time though, I didn’t have an issue with it and it seemed entirely logical that if sex was for reproduction, any form of sex which couldn’t lead to pregnancy was morally wrong. This was the simplistic understanding of a nine-year old.

When I was twelve, my English teacher compared homophobia to racism, and asked us, if we were opposed to racism, why would we be homophobic? It was the same kind of issue as far as he was concerned. This seemed an eminently consistent and sensible view to me, partly because at the time I considered racism to be a particularly terrible evil. One influence on my acceptance of this opinion was probably my own queerness, although I had yet to admit that to anyone. Certainly my White friend who was in the same English class as I and was similarly passionately anti-racist persisted in his homophobia for as long as I was aware of his opinions on the matter, which would’ve been another few years. In my case, I remember another pupil calling me “gay” in September 1981 and replying to him that it was terrible that he even considered it an insult. He too was still openly and strongly homophobic four years later. The one person who was aware of my sexuality and identity issues, which I used to call my “Problem”, once said of my opposition to homophobia that homosexuality was “not your Problem,” so clearly both she and I made a connection between the two.

But this post is not just about queerness and homophobia.

A few years later I went to University and became Christian. Before making a commitment, I expressed concern that I would have lots of questions about the issues the Christian faith raised for me, which were multiple. I was assured that this would not be a problem and that they encouraged questions. So, I converted and after a few months began to ask my questions, which were not all about homosexuality, but that was one major concern for me. So I brought it up, and the replies were varied. One was that it might currently be “fashionable” to tolerate homosexual activity but that God’s standards were unchanging and humans were not designed for that purpose. This was from a medical student by the way. Another homophobic Christian said, and this was more sympathetic, that he couldn’t imagine how bad it would be to find out you were gay and felt very sorry for them, but he was nonetheless still homophobic. But to me, this was just not an option, because by that point it seemed intuitively obvious that homosexual activity was not wrong and that homophobia was. As I’ve expressed it more recently, if the Bible told you that 2+2=5, you would either reject that part of the Bible (and possibly the whole thing) or try to work out why it seemed to you that it was saying that because it would clearly be saying something else, and since the Bible at least appears to condemn homosexual acts, that’s equally absurd and one could be expected to feel a similar motivation to resolve the problem.

This equation between the idea that 2+2=5 and the idea that homosexual acts are always sinful, I think, attempts to draw a parallel between the certainties of mathematics and the hope that ethics can be equally certain. There are positions in both ethics and mathematics which are called “intuitionism”. In maths, intuitionism is the position that maths has no external basis and is simply a creation of the mind. This is a more recent usage of the term in the philosophy of mathematics, preceded by Kant’s and his successors’ belief that intuition reveals the principles of maths as true a priori – they arise from logical deduction rather than observation. This seems counterintuitive (ha!) because to us, the Cosmos seems to run on maths and logic, and it’s also problematic for an externalist such as myself because we see concepts and ideas as external to the mind and having their own independent existence. It doesn’t seem to me that intuitionism and externalism could both be true, but since intuitionism can involve denial of the law of excluded middle (either P or not-P), maybe they could be. But at that point logic seems to have become what Arthur Norman Prior once called a “runabout inference ticket” where you can just conclude what you like from any premises. It doesn’t seem to be ultimately useful. This could, however, be psychoanalysed as a need for a feeling of certainty and solidity of foundations. It may not be that it’s mere logic.

Geometry is a notorious example of something which used to seem purely logical and valid without the need for observation to verify it. Euclidean geometry generally needs to be based on axioms which are intuitively true, such as “a straight line segment is the shortest distance between two points” and “a straight line segment can be extended infinitely as a straight line”. However, the fifth postulate is difficult to state simply. It can be stated thus: “If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.” This amounts to the idea that parallel lines never meet, or meet at an infinite distance, and whereas it certainly seems true, the complexity of stating it rigorously makes it suspicious. In fact, it turns out that the Fifth Postulate is the result of observation rather than deduction, and other geometries are possible based on either assuming that parallel lines diverge or that they converge. The former, known as hyperbolic geometry, can be locally true in this Universe and would be most noticeable near the event horizon of a black hole, and the latter, known as Riemann geometry, is actually real geometry as it applies over most of the Universe, particularly on a large scale. Possibly counterintuitive truths which hold in the real world are, for example, that if you imagine the Earth wrapped in bandages, and you kept wrapping it in ever deeper layers, you would eventually find that you were surrounded by bandages and would be inside the ball rather than outside it, and that there is at any one moment a finite maximum distance between two points after which the direction between them reverses. These facts can be known easily to be true on a spherical surface: our antipodes are a maximum distance between two points on the surface of this bandageless orb, after which the directions between them reverses – go far enough east and you find yourself west of your starting point – and a large enough circle on the Earth’s surface will start to shrink if it “grows” any further.

If it’s possible for that postulate to be cast into doubt, and in fact turn out to be false, what else in mathematics could be? One possibility is that logic is also like this. For instance, truth and falsehood could simply be poles in between which other truth values exist or there could be truth values situated beyond truth, i.e. truth from falsehood could simply be the first step towards a “supertruth” infinitely more true than mere truth itself. If there’s that much play in both geometry and logic, perhaps all mathematics is merely an intuitionistic game. Even so, we do tend to operate on the principle that maths is set in stone and reliable most of the time.

Ethical intuitionism is in a sense the opposite kind of view to mathematical intuitionism. Formulated in response to the perceived failure of utilitarianism, it ran into its own problems later which are also thought to have shaped ethical thought later. As I’ve mentioned before, the Utilitarians attempted to prove the utility principle’s desirability by saying that everyone desires to be happy, which is in any case not true, but also suffers from the problem that one needs to attempt to prove that the greatest happiness of the greatest number is worthy of being desired, which is a problem with the English language: we lack the word “desirandous” or any clearer equivalent and are stuck with the “-able” at the end of “desir(e)”. Consequently, in Edwardian times the philosopher G E Moore sought to establish ethics based on the idea that goodness was a simple, non-natural property which could be intuited by people. There is a big problem with this: cultural and interpersonal relativism, which is why he said it was a simple principle: it could not be analysed into a simpler form. The later philosopher Alasdair Macintyre suggested that this step led to later problems in discussing ethical issues which were then picked up on and influenced newer theories.

As the twentieth century wore on, logical positivism and behaviourism became important. Both of these attempted to tidy up pesky things like religious language and psychological states to things which could be observed by the senses. According to Macintyre, because ethics had come to be discussed in terms of what could be intuited and was considered to be essentially impossible to analyse further, conversations about right and wrong in academic circles tended to get reduced to mere emotional expressions. This was known as emotivism, and in fact more or less amounts to ethical scepticism, although there were two versions of it. One actually attempted to reduce expressions of right and wrong to emotional expressions akin to screaming and laughter, just expressed verbally. Another form of emotivism claimed that ethical statements were simply expressions of approval and disapproval implying exortations to another to do the same. Later still, prescriptivism emerged, which was a revival of Kantian ethics which claimed that to say something was good or right meant that it was universalisable (what if everyone did the same?) and entailed an imperative. The problem with this position is well-known. It depends on how it’s described. “Everyone needs to eat” could be given as a reason for poor people to shoplift food, but “everyone needs to make a living” is a reason shoplifting might be wrong. Again, we could be reduced to merely emotional arguments. An oddity about this period of what’s called “non-cognitivist ethics”, i.e. that actual meaning is not relevant to right or wrong, is that it was held by Bertrand Russell, yet he was a strong campaigner on ethical issues such as free love and nuclear disarmament. He himself commented that he couldn’t reconcile the apparent contradiction.

Some theists, not including me incidentally, see ethics as that which is commanded by God – theological voluntarism. There is no right or wrong except what God chooses to tell us to do or forbids us from doing. This crops up, for example, in some Jewish views of kosher diet. Although explanations have been offered, such as the idea that pork is forbidden to avoid parasites or that the forms of certain species, such as cloven hooves, allow special access to the divine, another explanation is that the rules are completely arbitrary and only exist to ensure that God’s people obey without asking why. Belief in theological voluntarism sometimes leads to the peculiar claim that atheists cannot have a moral compass, when it is in fact a pretty weak form of metaethics. It also gives rise to the moral argument for the existence of God, which is that the awareness of morality as a real thing as opposed to mere custom with no real basis means there must be an ultimate moral authority to back it up. I don’t see things this way at all. I see God as merely reporting on what the right thing to do is from a position of infinite wisdom and knowledge. God might sanction something, for instance, due to positive consequences which we can’t perceive ourselves. In the case of kosher food, for example, it might be that there is a very good reason for it but we cannot understand that reason. In fact I would say that veganism is the “new kosher”, and in fact the “new halal”, so in fact I do use my own reasoning to avoid the negative consequences and associations of deliberately eating animal products. Surprisingly, there are atheist theological voluntarists who claim that ethics would make sense if there was a God, but there isn’t, so it doesn’t!

It certainly seems that any God would be bound by logic and mathematics, although this isn’t always held to be the case. By the same token, God to my mind would be aware of right and wrong, and this means that there is a fact of the matter about these things rather than them being non-cognitivist. Alasdair Macintyre sought to replace previous metaethical theories with ideas of vice and virtue, but I would reject that on the grounds that it seems to lead to judging people directly as essentially good or evil, which seems intuitively wrong to me. And there’s that concept again: intuition.

The essential problem with the idea of a moral sense is cultural relativism, and similarly, circumstances altering cases. Take the campaign against sex robots. Those who oppose them argue that it’s wrong to consume bodies as goods and that sex robots and sex workers have the same undesirable status: humans (let’s face it, probably men) would be using sex workers as means rather than ends and therefore also sex robots. Others disagree, claiming that condemning sex robots is transferring concerns about sexual objectification to actual objects. This is an example of how moral intuition could be questioned. The situation could also be tweaked: what’s the morality of allowing paedophiles to have robot children? These two examples also bring up the issue of “the wisdom of disgust”, something which is often evoked to justify homophobia and which might also explain kashrut. Disgust, culturally mediated in this case, is the reason sanitary towels are advertised using blue rather than red liquid. Presumably on another planet the humanoids all have bright blue blood and red liquid is used to advertise them instead. We have instinctive abhorrence of excrement, which protects us from danger. A teleological view would say that God has made us disgusted by excrement in order to keep us healthy, and likewise has made people disgusted by homosexual activity, thereby justifying homophobia. I would say this is an excellent reason for rejecting the idea of the wisdom of disgust. Research has apparently shown that right wing people are more likely to equate disgust and immorality, which means rhetorically it might be more persuasive to appeal to disgust if your interlocutor is right wing. To me, the idea of there being a strong connection between disgust and ethical judgement is never going to gain any ground because I used to have a button phobia, and it’s clearly absurd for a person disgusted by one specific feature of the world to expect it to be banned or controlled in some way simply because of that disgust. That’s clearly not a good guide to morals.

To return to the history of my opposition to homophobia as an intuition, it does seem to be informed by some kind of reasoning. I have a kind of tangential stake in it, some might say a direct one, but it’s also influenced by the fact that disgust as a guide to ethics is manifestly absurd to me due to the button phobia, and also by a kind of inductive inference from racism. But it’s also very deep-seated, to the extent that the very fact that fundamentalist Christians tend to be recalcitrantly homophobic is sufficient reason to reject their world view, and it’s disappointing that they don’t themselves perceive things that way.

I have to say that in spite of difficulties with it, I find intuitionism the most appealing metaethical theory. Although the biggest problem with it is that it seems to make it impossible to resolve disagreements about right and wrong, moral codes tend to agree broadly across cultures, even when their connections must’ve been in the palaeolithic, and to me this suggests that there does seem to be something like a moral sense. This is metaphorical. I don’t imagine there is a sensory organ of some kind in the brain which responds to “conscience radiation” or something. However, I do think we have a moral instinct, and it makes sense to have an innate conscience which enables society to hold together and operate without individuals being taken advantage of too much, although sadly this seems to fail very often. There’s also a problem with the fact that if you actually do try to extract widespread moral principles from the religious and social codes of the world, many of them are homophobic, sexist and so forth. This is why a deeper set of principles must be used. This was the subject of my first degree dissertation, which wasn’t actually very good. I’m not going into it again here.

An ethical sense would seem to be identical with the conscience and distinct from disgust and charm, both of which are often misleading. For an example away from sexual ethics, disgust could prevent one from treating an illness, performing life-saving surgery or working in sanitation, but all of these are very positive things to do ethically. Conscience has been called “the voice of God”. In a situation where a theist has difficulty with conservative religion because of its homophobia or sexism, their conscience cannot allow them to concede or tolerate that prejudice, and if conscience is the word of God, God would themselves be convicting them to rebel or do something else to act against it.

Although the moral argument for the existence of God doesn’t work for a separate divine being “out there” in the Universe or beyond it, there’s another possible take on this based on Ultimate Concern. The philosopher Paul Tillich manages to separate the issue of theism from religion with this concept, which makes the idea of religion less Westernised as it allows for non-theistic religions, which of course do also exist in the West, for example Spiritualism and the Free Zone. Tillich calls faith “the state of being ultimately concerned”. By this he means whatever one holds sacred. This, I think, is a widespread object in most people’s psyche, including non-religious people. It needn’t be God. It could be love, altruism, rationality, compassion, perhaps even one’s own ego for narcissists, but it’s just as real for most non-religious people as it is for religious people and theists. For a Quaker, it might be the spark of divine in us all, and for atheist Quakers there may be no need to alter that. Conscience could be an Ultimate Concern, in other words one’s God, and because this closes off the concept from argument and questions about the existence of an external deity or not, it could be quite a good one. It’s even ineffable in some ways, because of the inscrutability of ethical intuition.

It is of course problematic to have a set of inaccessible moral principles due to the difficulty in being able to see them collectively in the same way. Coming back to sexual orientation though, this is something which can actually be known because it isn’t so much observed as immediately present to the consciousness, when, for example, one might feel attraction to someone of the same gender. One possible response would be to deny it because it clashes with one’s religious values, and clearly this is a fairly common phenomenon given the large number of people involved in reparative “therapy” who are either openly gay already or admit to it, and pastors who have been stridently homophobic and again turn out to be gay, but this shouldn’t be taken as the rule for homophobia among the religious. There really are people who struggle with the homophobia of the Abrahamic religions and only very reluctantly concede to it. On the other hand, I used to know a man who said he wished he could be as disgusted by other kinds of sin as he was by what he saw as the sin of homosexually expressed love. There is an internal process going on here. In one situation, one is divided against oneself because one knows oneself to be queer but struggles against it. In the other, which rather self-righteously I would claim for myself, one’s awareness of one’s queerness and its incontrovertible nature leads one to reject any understanding of religion which is homophobic, and to be honest, if it turns out homophobia is central to any faith, the voice of God, as it were, would surely lead me to reject that faith.