Spatial Reasoning

[diagram of a round peg and a square hole]

Here’s a peg and a hole. Can you fit them together?

The answer should be obvious. And yet, you didn’t need to fit them together to figure it out. You simply created an imaginary peg and hole of the same dimensions in your mind, and tried the experiment there. It’s a remarkable talent, perhaps even… miraculous?

We put our spatial reasoning to best use when creating tools. In order to solve a problem in the physical world, we must be able to mentally break it down into sub-problems, picture an object or technique that solves each of them in turn, and assemble a tool or tools in the real world that behave like our imagined ones.

So if I find another species that uses tools, I’ve also found one capable of spatial reasoning.

Tool Use

That should be tricky. Humans are practically defined by tools. Even the lowliest hunt/er-gatherer never leaves home without a spear or axe. We’ve built entire civilizations around them, from the daggers and shields of the Romans, to the automobiles of the USA. No other animal is as skilled a toolmaker.

Other species still create and use tools, though. I’ve already talked about Betty’s amazing abilities with tools, so I should elaborate on their use in wild crows. Dr Lucas Bluff and others from the Department of Zoology at Oxford University have collected nearly 2,000 hours of video footage in a detailed study of those birds. They noted that wild crows have stopped using their beaks to dig out insect larvae, and instead gather most of their food using tools. The birds are smart enough to match their tools to their job, using long twigs in deep holes and short twigs in shallow holes. They’re also picky in what they’ll use as a tool, presumably because they’ve learned some materials and sizes work better than others.

And as mentioned before, adult crows are much better at using tools than juveniles, a sign that tool use is a learned skill, not an evolved instinct.

Tool use is not limited to land animals, however, or even organisms with a backbone. The Veined Octopus has been filmed gathering coconut shells from the sea floor. After digging out two of them, they’ll combine them into an invincible spherical shelter, perfect for sleeping in or lounging around. Once finished, the satisfied octopus will discard the halves. By itself, that would be notable; the only other animal that has been observed fashioning their own temporary shelter is a human.[53]

But then the Veined Octopus did something that had the researchers doubled over in fits of laughter. As Julian Finn, of Museum Victoria, and colleagues looked on, one octopus positioned itself over top a coconut half, gathered it up in its tentacles, and walked across the ocean floor! Sort of, anyway; the coconuts are roughly twice as big as the poor cephalopod, so its “walk” is more of a drunken stumble. Like any good comedic video it’s been posted to the internet, so you too can enjoy science history by rolling on the floor, laughing.

I can’t leave this topic without mentioning our closest cousin, the chimp. After all, they were the first animal noted to use tools in the wild, by Jane Goodall in 1960, and have remained the best-documented.

Chimpanzees don’t have a single tool, so much as an entire tool kit. Over fifty years of close study, researchers have discovered that chimps have nine different ways of putting tools to use. They’ll use branches to check out-of-reach places, clean themselves with wet bundles of leaves and moss, and even sharpen sticks into spears for hunting.

Play

[All children] shall have full opportunity for play and recreation, which should be directed to the same purposes as education; society and the public authorities shall endeavour to promote the enjoyment of this right.

( Principle 7, “Declaration of the Rights of the Child.” UN General Assembly Resolution 1386, adopted December 10th, 1959 )

Granting a Universal Human Right to the protection of consciousness, free expression of thought, and even medical assistance makes quite a lot of sense. But why should the ability to play get such fundamental protection?

Play allows children to use their creativity while developing their imagination, dexterity, and physical, cognitive, and emotional strength. Play is important to healthy brain development. It is through play that children at a very early age engage and interact in the world around them. Play allows children to create and explore a world they can master, conquering their fears while practicing adult roles, sometimes in conjunction with other children or adult caregivers. As they master their world, play helps children develop new competencies that lead to enhanced confidence and the resiliency they will need to face future challenges. Undirected play allows children to learn how to work in groups, to share, to negotiate, to resolve conflicts, and to learn self-advocacy skills. When play is allowed to be child driven, children practice decision-making skills, move at their own pace, discover their own areas of interest, and ultimately engage fully in the passions they wish to pursue.

(“The Importance of Play in Promoting Healthy Child Development and Maintaining Strong Parent-Child Bonds,” by Kenneth R. Ginsburg. Pediatrics, Vol. 119 No. 1 January 1, 2007. )

Come to think of it, human beings also have a ridiculous amount of free time compared to other species. They blindly focus on sex and defence, while we sharpen our social and mental skills by engaging in play. I’m not sure what sort of skills we’re enhancing when slowly float down a river with a cooler of beer in the boat, but maybe that’s the point; we’re also capable of having mindless fun, in addition to the more beneficial types of play. Is this a divine gift, perhaps?

At the Konrad Lorenz Research Centre, researchers have spotted some bizarre behaviour in wild crows. One of them would fly to the top of a steep, snowy hill, tuck in its wings, and flop over. After it slid down the slope on its back, it would shake itself off then fly back to the top for another go. Another time, they spotted birds grabbing the tail of a wild boar, then letting the larger creature drag them through the snow upside-down.

Crows do not have to know how to slide in order to survive. On the contrary, these behaviours are dangerous to a delicate winged creature, and yet the birds didn’t look distressed or hassled into doing them. After considering all the alternatives, the researchers conceded the best explanation for their behaviour was play. The crows were just stunting for a good time.

Dolphins get their kicks by swimming in a tight circle, then blowing bubbles into the column of rotating water. The resulting bubble ring is usually admired as it floats to the surface, though sometimes the dolphin will give it a bite and delight in the scattered bubbles that rise more quickly. For bigger thrills, they’ll body-surf along waves that crash into shore, or in the big waves of a giant ship.

Lauren Highfill and Stan Kuczaj have been studying porpoise play. In five years, they spotted 317 different games. One young calf had an elaborate game that involved blowing bubbles while upside-down, then chasing them to the surface:

She then began to release bubbles while swimming closer and closer to the surface, eventually being so close that she could not catch a single bubble. During all of this, the number of bubbles released was varied, the end result being that the dolphin learned to produce different numbers of bubbles from different depths, the apparent goal being to catch the last bubble right before it reached the surface of the water.

She also modified her swimming style while releasing bubbles, one variation involving a fast spin-swim. This made it more difficult for her to catch all of the bubbles she released, but she persisted in this behavior until she was able to almost all of the bubbles she released. Curiously, the dolphin never released three or fewer bubbles, a number which she was able to catch and bite following the spin-swim release.

(Behavioral and Brain Sciences, October 2005)

Interestingly, most of these games were developed by young dolphins, something Highfill and Kuczaj suspect may be their contributions to a dolphin “culture.”

Culture

Ah, yes, culture. The collection of little things that we share from one person to another that have little or nothing to do with reproduction. Our great works of art fall into this category, as does our choice of music. We dress like other people, and try to inspire others to take on our fashions; we paint ourselves up, to look good for potential mates but also to fit into a social niche. We add little flourishes to our buildings that follow someone else’s tradition, and we even adjust our slang to fit in. We have so much intelligence that we can afford to waste it on these trivial touches.

Other species lead much duller lives, preferring to bask in the sun instead of accessorize their coats. What could be a better indicator of intelligence?

As it turns out, nearly everything. The banded mongoose has never been called a brainy animal, and yet Corsin Müller has shown they have culture. He presented wild populations with a plastic egg containing fish and rice for half a month, and noted how they treated the egg; did they hold it in their mouth with their paws and crack it open with their teeth, did they smash it against a tree or rock to open it, or did they pass it by? Mongooses are very dogmatic, so once their initial decision was made they stuck with it. They also have a unique way of raising their pups; the young ones don’t spend much time with their parents, but instead grow up with another adult and learn by watching them go about their business. Müller ensured there were some apprentices nearby when he dropped the plastic treats in front of these tutors, but also made sure that only the adults got to handle the egg.

Müller then safely stashed the eggs for two to ten months. After that, he dropped a few by the former apprentices, now mature adults in their own right. Would they follow in the paw prints of their mentor, and adopt the behaviour they had been shown, or invent their own method? As you’ve guessed, they nearly always did what their tutor originally did.

What makes this study stand out is that it was an experiment on a wild population. Most studies of primates, and almost all the ones done on whales, are done to a captive audience. There’s a chance that those animals have picked up on human behaviours and adopted them as their own, but would never consider fluff like culture in the wild.

So it’s amusing that the humble mongoose is our best example of culture in a wild animal.

Whales may not be far behind. We know that wild killer whales can be divided into three populations: one group hangs out in one place and eats nothing but fish, another that wanders the coastline looking for plump seals and otters, and a third that dives deep and does something-or-other.[54] Genetic analysis shows that all three could interbreed, yet they don’t. It strongly suggests a deep cultural divide between wild populations, but no-one has proven it.

We’ve taken captive chimps and taught two different ways to retrieve food from a puzzle to two of them from two different groups. When returned to the cage they immediately enlightened other members of their group how to solve the puzzle. Two months on, each group was using the method they had been taught, even though both shared the same cage and could watch the other group use the other method to do the same task. It experimentally proved that chimps could have culture, but said nothing about their wild behaviour.

[53] Hermit Crabs don’t count. While they’ll freely use shells and even bottles to create a shelter, those are permanent and stay firmly attached to the crab until they are either outgrown or their squatter dies.

[54] This third group was only found recently, so little is known about them. It’s notoriously difficult to study deep-diving creatures; no human has seen a deep-sea squid, and no scientist who’s gone looking has found one, yet their dead bodies will occasionally wash up on a beach. Presumably, some of them prefer a burial at land…

Visual Processing

The same reasoning applies to visual processing.

Object recognition has always been one of our strongest points. Hand us a photo of an object, say a bicycle or a fire hydrant, then ask us to find it in another photo and we’ll have no problems. This simple task is actually incredibly difficult for a computer to pull off, since the target object may be rotated differently, obscured behind another one, differently coloured, or even have a texture overlaid on it. Tomaso Poggio of MIT thinks it’s as difficult a feat as simulating intelligence in general.

He should know: he’s partially cracked that problem.

In 2007, he released two papers that detailed a new method of object recognition. For one of those papers, he did exactly the task I outlined above.[45] The new algorithm was able to track down an object about 97% of the time, though it could range from 93% to 99.8% accuracy depending on the exact task. Unlike most other algorithms, which specialize in finding only one type of object, his method works equally well on a wide variety of objects.

There’s good reason for that: it’s modelled on the human visual system. His second paper demonstrates just how closely that model follows reality, by pitting man against machine.[46] Poggio sat human beings down in front of a computer, flashed them a single image for 1/50th of a second, then asked the humans if there was an animal in the scene or not. We’re literally born for this task; how fast you can tell if that’s a wild animal or a branch dropping towards you determines how likely you are to survive and pop out offspring. Unsurprisingly, human beings do pretty well at this, getting it right around 80% of the time.

Poggio’s algorithm was then used to classify each image. The results? It guessed correctly  80% of the time. Eerily, when the researchers analysed the images the algorithm did poorly on, they discovered that the humans had struggled on those images as well.

This algorithm can still be improved. Our brains use feedback from other parts of the brain to improve our guesses further, something this method doesn’t handle. It also took much longer for this algorithm to come to a conclusion than our brains did, but that’s only a temporary problem. Computers improve in speed much faster than human brains do, and more efficient programming should reduce the number of necessary calculations.

Still, Poggio’s work hints very strongly that there’s no magic to our visual system.

Music

Sounds may be another matter, though. Not just any sound, though, but the semi-repetitive collections of sound we call music. Humans have spent centuries, probably even millennia, creating harmonies and melodies for no other reason than pleasure. No other species can dare make that claim.

Well, except birds. And maybe whales. Oh, and gibbons.

But before we get into the details, we first have to settle what music is. The suboscine branch of the bird family can have elaborate calls, but those are hard-wired into their genes. You can separate them from their parents, play the songs of other birds as often as you want, and they’ll still chirp out their innate tune. Most people would not consider this music; there must be an element of creativity involved, and while genes can produce variation through mutations, that happens on too long a time scale to qualify.

Songbirds are a different feather. Play them a tune at the right age, and they’ll pick it up and use it as their own. Deprive them of music, and they’ll sing poorly or not at all. While better, this still doesn’t quite qualify as a creative act since they’re just copying the songs they heard. Changes will happen over time due to accident, faster than they would through genes, but still not fast enough.

Not all songbirds are born alike, though. The Indigo Bunting will pluck a song out of thin air, with no resemblance to anything it’s heard before, then slowly mix in fragments from nearby competitors until it becomes a variation on a theme. Mockingbirds got their name from a remarkable ability to imitate sounds in their environment, everything from the calls of insects to the ring of a cell phone, which are then incorporated into their songs. Both species can be considered creative.

Both could also be dismissed as too greedy. Birdsong is primarily used to attract mates, warn about predators, and establish territory. It also makes a handy show of fitness; sick birds have difficulty carrying a tune, and it puts them at risk of an attack by predator. The music that human beings make has much purer motives, and is rarely used to show off.

Sorry, but I couldn’t keep a straight face while writing that last line. One of the leading theories of why we make music is that it’s a show a reproductive fitness. Humans can’t sing or play an instrument very well if they’re sick, either, and we frequently use music to set a romantic mood. As Geoffrey Miller of the University of New Mexico has pointed out, musical output peaks and declines with sexual ability, and a whopping 40% of all lyrics relate to sex or romance. Musicians are usually considered sexually desirable.

Consider Jimi Hendrix, for example.  This rock guitarist extraordinaire died at the age of 27 in 1970, overdosing on the drugs he used to fire his musical imagination.  His music output, three studio albums and hundreds of live concerts,  did him no survival favours.  But he did have sexual liaisons with hundreds of groupies, maintained parallel long-term relationships with at least two women, and fathered at least three children in the U.S., Germany, and Sweden.  Under ancestral conditions before birth control, he would have fathered many more.  […] As Darwin realized, music’s aesthetic and emotional power, far from indicating a transcendental origin, point to a sexual-selection origin, where too much is never enough.  Our ancestral hominid-Hendrixes could never say, “OK, our music’s good enough, we can stop now”, because they were competing with all the hominid-Eric-Claptons, hominid-Jerry-Garcias, and hominid-John-Lennons.  The aesthetic and emotional power of music is exactly what we would expect from sexual selection’s arms race to impress minds like ours.

(“Evolution of human music through sexual selection,” by Geoffery Miller. Published in “The origins of music,” edited by N. L. Wallin et al, 2000. )

Most damning of all is the genetic evidence. If music was related to reproduction, we should expect to find genes that control it. Not only do those genes exist, but they’re almost identical to the genes that give birds the ability to create songs.

The identification of FOXP2 as the monogenetic locus of a human speech disorder exhibited by members of the family referred to as KE enables the first examination of whether molecular mechanisms for vocal learning are shared between humans and songbirds. […] In support of this idea, we find that FOXP1 and FOXP2 expression patterns in human fetal brain are strikingly similar to those in the songbird, including localization to subcortical structures that function in sensorimotor integration and the control of skilled, coordinated movement. The specific colocalization of FoxP1 and FoxP2 found in several structures in the bird and human brain predicts that mutations in FOXP1 could also be related to speech disorders.

(“Parallel FoxP1 and FoxP2 Expression in Songbird and Human Brain Predicts Functional Interaction,” by Ikuko Teramitsu et al. The Journal of Neuroscience, March 31, 2004, 24(13):3152-3163)

There’s still an objection to be made, even if we agree with Miller and others. An evolved trait may be used differently at different times. Music in human beings may have started as a show of fitness, but it need not stay that way. After all, very few people actively pursue a career in music; the majority instead write songs privately, for their own enjoyment. Human beings may have once sung for sex, but nowadays we’re more likely to sing for ourselves.

Against this stands the Brown Thrasher. Birdsong comes in roughly five flavours:[47] mating song (“I’m here, and I’m sexy!”), companion calling (“I’m here, where are you my mate/friend?”), begging by young birds (“GIMMMIE FOOOOD NOOOOOOOW!!”), trespass threats (“Get out of my area, you upstart, or else!”), and predator alerts (“I see something dangerous!”). Sometimes the lines can blur a bit (“Everybody, come help me harass this predator!”), and some species have multiple calls within each flavour (“Head’s up, it’s a predator from the sky!”), but a grand total of a dozen or two should be more than enough for most birds. And it is, generally.

So why does the Brown Thrasher have a library of 2,000 calls? To give that a baseline, the average vocabulary of a human being consists of 10,000 words.

 It’s tempting to dismiss all that variation as invention. The Thrasher may be taking a “base” song and improvising new versions of it. If this is the case, we’d expect very few songs to be repeated; instead, Thrashers can recall a song they tweeted nineteen days earlier.

Alternatively, that vast song repertoire may be a way to show off to the opposite sex. There’s a problem, however; most female songbirds are not attracted to males with a giant songbook.[48] One study showed that Brown Thrashers were actually more interested in a limited sample of Thrasher calls than the full collection, provided they displayed more versatility in singing ability.[49]

It’s tough to draw a definitive conclusion from a single study, so I won’t. What I will say is that it’s plausible the Brown Thrasher’s vast library is more for personal kicks than practical use.

Intrapersonal and Self-Awareness

Humans are quite good at understanding the personal. We’ve got an entire branch of science devoted to it, named psychology. Philosophers from Plato onward have valued looking inward, to discover what we really are like. Surely no other species can come close to us here.

So far as we know, none has. It’s hardly their fault, though; we have only two ways to learn about the inner lives of others, by direct communication and indirect brain scanning. With no way to ask other animals how they feel, and quite different brain structures between us, plumbing the depths of other species’ cognition ranges from difficult to impossible.

It doesn’t help that we tend to project ourselves onto other creatures. Alexandra Horowitz conducted a study that asked dog owners to forbid their pets to eat a treat. When the humans left the room, Horowitz randomly fed some of the dogs that forbidden fruit; when they came back, she randomly told some of the owners that the dog had eaten the treat. The humans that were told their pet had broken their order thought their dogs looked guilty, even if they never ate the treat. When punished, the pets that looked most guilty were actually the ones which never got a lick at the prize.[50] Any interpretation of animal behaviour has to be done very, very careful to filter out our inner biases.

We do, however, have a proxy for inner knowledge: knowledge of the self. If you have no concept of “you,” there’s no self to learn about. And once you realize there’s a “you” there, curiosity will drive you to give it a quick once-over, at minimum.

The standard test for self-awareness is pretty simple: place an animal in front of a mirror, and let them get used to it. Then put them to sleep, paint a dot on an area of their body that they can’t normally see, then wake them up and place them by the mirror. If they try to rub off or touch the dot, they must know the animal in the mirror is actually themselves, and must be capable of mapping between the image and themselves. Human children pass this test easily, as do all primates, elephants, dolphins, and European magpies.[51] Other species, such as pigs and pigeons, fail this test but can demonstrate that they know the image in the mirror reflects reality. In the case of pigeons, this can even be used to “train” them for the test, resulting in a pass.[52]

Those last results have been used to criticise the test; perhaps a species just doesn’t care about cleaning off the dot, leading researchers to falsely conclude it isn’t self-aware. Another problem is that self-recognition may not be tied to self-awareness; humans with prosopagnosia cannot recognize themselves, yet clearly are self-aware. Note however that both arguments lead us to conclude there are more self-aware species than we realize, not less.

[45] “Robust Object Recognition with Cortex-Like Mechanisms ,” Thomas Serre et al. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol 29 No 3, March 2007 .

[46]”A feedforward architecture accounts for rapid categorization,” Thomas Serre et al. Proceedings of the National Academy of Sciences, vol. 104 no. 15 6424-6429, April 10, 2007.

[47] http://www.natureskills.com/birds/bird-language/

[48] http://www.sciencedirect.com/science/article/pii/S000334720800496X

[49]Boughey, M. J. and Thompson, N. S. (1981), “Song Variety in the Brown Thrasher (Toxostoma rufum). Zeitschrift für Tierpsychologie, 56: 47–58. doi: 10.1111/j.1439-0310.1981.tb01283.x

[50] http://www.elsevier.com/wps/find/authored_newsitem.cws_home/companynews05_01246

[51] http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.0060202

[52] http://www.sciencemag.org/content/212/4495/695

Problem Solving

In the meantime, let’s start with a big one: problem solving. We pride ourselves on being able to fix situations that wouldn’t occur naturally. The lives of the Apollo 13 astronauts depended on fitting a square carbon dioxide filter into a smaller round hole, otherwise they would suffocate on their own breath. Nothing in that scenario is natural.

Wire isn’t natural, either, yet a crow surprised us by bending it into a hook. New Caledonian crows have been making hooks for some time, actually, but in the wild they use twigs instead. They learn this trick by watching other crows do it, too, and not by figuring out for themselves. For one of them to bend a material they’d never seen before, in a way they’ve never witnessed, is a significantly harder problem.

Betty pulled it off on her first attempt.

Jackie Chappell and company didn’t mean to test that. Betty and Alex, a male crow, were presented with a collection of bent and straight wire, then tested to see if they could use them to grab a tasty treat that was otherwise out of reach. Wire was used because both crows had rarely seen it in their lives. When Alex nabbed the only bent wire and flew off, Betty grabbed a straight piece and bent it into a hook with her beak and feet.

After the researchers picked themselves up off the floor, they devised a new test. Separately, each crow was given a straight wire and a treat that was otherwise out of reach. Out of ten trials, Alex only succeeded once, and even then he cheated. Nine times out of ten, Betty tried to pick up the treat with the straight wire, failed, then bent the wire into a hook using her beak, feet, or the tube containing the food, and succeeded in getting the treat.

Betty had be raised in captivity, so she couldn’t have learned this trick from her wild peers. She had to analyse this new situation, find a solution using what she knew about the materials and herself, then put it in motion. That’s novel problem solving, done in a species with a much smaller brain than ours.

There’s also the case of a feisty octopus at the Sea Star Aquarium in Coburg, Germany. It did not like captivity one bit, and found creative ways to protest. Otto would juggle its non-mobile tank-mates, sometimes hiding them under grate covers, and several times shorted out a bright light by squirting it.

Think about that last one. Octopuses live entirely underwater, where they swim by sucking in and squirting out water. Light bends when it moves from air to water or vice versa, and even we humans take a fair bit of training to compensate for that. So in order to hit that light, Otto had to take an organ it uses for a single purpose and put it to a different use in a habitat it never visits with physics quite different from its home turf, and hit a small target that isn’t where it appears to be.

Nothing about that is natural, either.

Mathematics and Logical Thinking

Neither is calculus, for that matter. And yet human beings have no problems doing complicated arithmetic in their heads, or pondering long chains of subtle logic. Our fellow species can barely count, in comparison.

There is one crucial difference, however: Homo Sapiens Sapiens goes to school. We’re not born math wizards, we have to be taught via long, intensive training sessions. Remove those, and our huge advantage goes with it. Good proof of this comes from the languages of hunter-gatherers. They spent most of their time sleeping, doing chores, gathering food, or fighting. There was no time or need to invent mathematics, so whatever number systems they came up with reflect our uneducated understanding of number.

Their achievements are depressing. Many hunter-gatherers could barely count, usually reaching no higher than one or two before invoking words that mean “few” or “many.” Some didn’t even have the concept of “one”:

In Pirahã, there are two words which prototypically mean ’one’ and ’a couple’ respectively, but it has been checked fairly extensively that their meanings are fuzzy ’one’ and ’two’ rather than discrete quantities (Everett 2005, 2004, Frank et al. 2008). It is not possible to combine or repeat them to denote higher (inexact?) quantities either (Gordon 2004). The Pirahã have the same cognitive capabilities as other humans and they are able to perform tasks which require discerning exact numeration up to the subitizing limit, i.e. about 3 (Gordon 2004). They just do not have normed expressions even for low quantities, and live their life happily without paying much attention to exact numbers.

(Unsupervised Learning of Morphology and the Languages of the World,” chapter Nine. Harald Hammarström , 2009)

The last two sentences of that quote bring up more evidence; our subitizing limit, better known as our working memory capacity, is only three or four items.[44] If you’ve had no training on how to count or do math, that’s the only storage space you have for numbers, and thus it limits how high you can count.

Interestingly, our species’ subitizing limit is on par with other species.

In a study published last summer in the Proceedings of the Royal Society B, Kevin C. Burns of Victoria University of Wellington in New Zealand and his colleagues burrowed holes in fallen logs and stored varying numbers of mealworms (beetle larvae) in these holes in full view of wild New Zealand robins at the Karori Wildlife Sanctuary. Not only did the robins flock first to the holes with the most mealworms, but if Burns tricked them, removing some of the insects when they weren’t looking, the robins spent twice as long scouring the hole for the missing mealworms. “They probably have some innate ability to discern between small numbers” as three and four, Burns thinks, but they also “use their number sense on a daily basis, and so through trial and error, they can train themselves to identify numbers up to 12.”

More recently, in the April issue of the same Royal Society journal, Rosa Rugani of the University of Trento in Italy and her team demonstrated arithmetic in newly hatched chickens. The scientists reared the chicks with five identical objects, and the newborns imprinted on these objects, considering them their parents. But when the scientists subtracted two or three of the original objects and left the remainders behind screens, the chicks went looking for the larger number of objects, sensing that Mom was more like a three and not a two. Rugani also varied the size of the objects to rule out the possibility the chicks were identifying groups based simply on the fact that larger numbers of items take up more space than smaller numbers.

(“More Animals Seem to Have Some Ability to Count,” by Michael Tennesen. Scientific American, September 2009.)

We’ve managed to out-reason other species because we found a very efficient way to gather food, which freed up enough spare time to come up with wonderful systems of math, and because our longer lifespans increased the odds of us stumbling on a technique, or gave us more time to learn it from someone else. No other species has pulled off both feats; elephants and whales rarely use tools to gather food, and wild crows only live eight years.

When you provide both time and training, other species can break past the subitizing limit too.

[Pepperburg] discovered that Alex could accurately add two sets of objects, such as crackers or jelly beans, so long as the total was six or fewer. In related work, Alex learned to order the Arabic numerals 1 through 8 (in the form of multi-coloured refrigerator magnets) in the correct order. She says he then spontaneously learned to equate these symbols with the appropriate number of objects.

In the newly published work, Pepperberg tested whether Alex could correctly add the Arabic numerals and also whether he could sum three sets of objects totalling 6 or less. Both experiments were cut short when Alex died, but Pepperberg says that the parrot did better than chance in both experiments.

In 12 trials of the Arabic numeral addition task, when asked “How many total?” he indicated the correct sum 9 times, demonstrating that 3 + 4 is 7, 4 + 2 is 6, 4 + 4 is 8 and so on. When presented sequentially with three sets of objects hidden under three cups, and asked how many, Alex offered the correct answer eight out of 10 times. He determined, for instance, that one, two and one jelly beans adds up to four.

(“Alex the parrot’s last experiment shows his mathematical genius,” Ewen Callaway. Nature News Blog. )

Even if you don’t agree with the above argument, there’s still the mechanistic one. As I write this, the fastest computer in the world can perform about 8,162,000,000,000,000 math operations per second, to sixteen digits of precision. The computer I’m typing this document on can manage roughly 1,570,000,000, and even my phone does 6,900,000. In comparison, try working out this slightly easier calculation entirely in your head:

 Currently, Marc Jornet Sanz is the fastest multiplier on this planet. He can do the math above in about thirty seconds, without any mechanical aids, which translates to roughly 0.04 calculations per second.

Computers can do more than mundane arithmetic, too. Mathematicians have begun to rely on them for proving theorems. They are commonly used to verify proofs, a tedious and error-prone task, but computers are increasingly generating their own proofs. To name one example, the Robbins conjecture was proven by EQP, a computer program developed at Argonne National Laboratory in the United States.

If mathematics and logic can be done as well, or even better, by a machine, we have no reason to think of them as gifts from a god.

[44] Thanks to a misunderstanding, most people think this number is actually seven. See “Seven plus or minus two,” by Jeanne Farrington. Performance Improvement Quarterly, 23: 113–116.