Proof from Intelligence (4)

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

Zvan on the Gendered Pay Gap

I have a really nice document about the gendered pay gap buried on a hard drive. To write it, I spent a good few months reading policy documents and research study after research study after research study after research stu– well, you get the point. My favorite of the bunch is this one. The gender breakdown of an industry tends to vary with time, so Emily Murphy and Daniel Oesch looked into whether or not that effected pay.

Both baseline models suggest that moving from a male to a female occupation – or staying within an occupation that feminizes – entails a sizeable wage loss. Adding controls for the workplace (M1) and general human capital (M2) makes no difference: the wage penalty associated with FEM amounts to about 15 per cent for British women, British men and Swiss women, 15 and to about 5 per cent for German women, German men and Swiss men.
If women rush to your occupation, your wages drop… even if you’re a man or a childless woman. This is tough to explain as anything but discrimination.
While I’ve been mulling over how and when to release my document, Stephanie Zvan independently came up with her own version.
Let’s start by noting that at least one person who studies the factors that account for pay gaps says that choice of careers, while a factor in unequal pay, is not the silver-bullet solution that pay–gap critics suggest. It isn’t even the biggest factor driving the difference between men’s and women’s wages. […]
… even though women work fewer paid hours than men, they work the same number of hours overall. The reason women more frequently require constrained work weeks and more flexibility in their schedules is that they do the bulk of the unpaid work that makes our society run, particularly caregiving, both for children and for other adults.
It may not have an excessive number of footnotes, but her version states much the same thing as mine in fewer words and clearer language. Give it a read, in honour of International Why-Isn’t-There-An-International-Men’s-Day Day.

Proof from Intelligence (3)

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:

 

29669907

x

42669080

 

 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.

BBC’s “Transgender Kids, Who Knows Best?” p4: Dirty Sexy Brains

This series on BBC’s “Transgender Kids: Who Knows Best?” is co-authored by HJ Hornbeck and Siobhan O’Leary. It attempts to fact-check and explore the many claims of the documentary concerning gender variant youth. You can follow the rest of the series here:

  1. Part One: You got Autism in my Gender Dysphoria!
  2. Part Two: Say it with me now…
  3. Part Three: My old friend, eighty percent
  4. Part Four: Dirty Sexy Brains

In North America, one of our pet obsessions is dividing everything up according to sex. Gendered toys, gendered clothes, gendered bathrooms, even gendered jobs. And yet if you follow those links, you’ll find these divisions were always in flux: gender-neutral toys used to be common yet are increasingly rare; dresses were gender-neutral, and colours weren’t gendered until roughly World War I; there were no public women’s washrooms in the US until the 1880’s, because women weren’t allowed in public; and computer science flipped from being women’s work to men’s work in the span of a few decades, leading to increased salaries and prestige.

This extends all the way down to our organs.

[Read more…]

A Wee Peek

Ugh, I wish I had more time to sink into this blog. What’s worst, most of what I’m focused on can’t or is too boring to share over here.

Most. There are always exceptions, of course.

[Read more…]

Proof from Intelligence (2)

Divine Gift or Solvable Mystery?

Suppose, for argument’s sake, we claimed long-term memory as definitive proof of the intellectual superiority of human beings over all its peers. Suppose that a few decades from now, scientists crack their current roadblocks and come up with a complicated but complete understanding of the brain processes responsible. Obviously, we’d look like fools for casting our chips on a losing bet, and begin looking for some other mystery of intelligence to claim as a divine gift.

But how can we tell the difference between a divine gift and a solvable mystery?

A divine gift could never be understood by scientific means. A solvable mystery will be, if you don’t mind waiting a while. In the here-and-now, though, both are equally baffling. You can’t tell how long you’d need to wait, because then that mystery wouldn’t be much of a mystery. Turning to holy texts isn’t much help, as I’ll outline in another chapter.

We’re stuck in something like the Prisoner’s Dilemma (see the Morality proof), with four possibilities:

It’s a Divine Gift

It’s a Solvable Mystery

Treat it as Divine

No problem!

You look like an idiot when it gets solved.

Treat it as Solvable

Either you’ll run into proof it’s divine, or it’ll perpetually be studied with no direct answer

No problem, plus you know more about the world!

If we treat this portion of intelligence as divine, we reach a dead end where we no longer study it in detail. You could argue this saves energy, since if we treat it as solvable when it really is divine we’d just grind our gears forever. That won’t happen; the process of evolution ensures all species have a certain level of curiosity, since a little exploration might lead to fertile new areas with no competition. Human beings will always search for answers, via science or some other means, no matter what the question actually is. Instead of saving energy, treating something as divine will simply shift our curiosity elsewhere, and have no net savings.

That assumes everyone treats that problem as divine, of course. If different religions have different ideas of divine, the off-limit topics will get studied anyway. In the case of intelligence and the brain, the Gelung Tibetan Buddhists are willing to give science a try:

In a final decision, the Society [for Neuroscience] will move forward with the Dalai Lama’s lecture at Neuroscience 2005 in Washington, DC, as planned. At its July meeting, the SfN Council expressed overwhelming support for proceeding with the Dalai Lama’s talk on “The Neuroscience of Meditation.”

(Fall 2005 Neuroscience Quarterly, official publication of the SfN)

My confidence in venturing into science lies in my basic belief that as in science, so in Buddhism, understanding the nature of reality is pursued by means of critical investigation. […] If scientific analysis were conclusively to demonstrate certain claims in Buddhism to be false, then we must accept the findings of science and abandon those claims.


(Tenzin Gyatso, the 14th Dalai Lama, leader of the Gelung Tibetan Buddhists)

Treating the components of intellect as solvable mysteries makes more sense, in every case.  Importantly, this reasoning can be used against any claim of a divine gift, not just intelligence. Examples of this include a designed universe (Fine-Tuning) or designed body (Teleological), moral guidance (Morality), or any Miracle.

Cogs in the Machine

But before I get further side-tracked and forget, we should return to memory.

I hope you noticed that some parts of intelligence seem to have little to do with intelligence. A good short-term memory is certainly helpful when solving problems, but only as a helper to some other form of processing. There’s otherwise little special about short-term memory, and it seems widespread across all life. Indeed, an experiment done by Keir G. Pearson[37] suggests that cats can remember a barrier they’ve seen for a few seconds after it goes out of view. Interestingly, if they also step over the barrier with their front legs, they will remember to step over with the back ones even after a ten-minute long distraction. Even goldfish, which urban legends claim have a memory lasting only a few seconds, can actually remember some things for a span of three months.[38]

Long-term memory has been studied to a ridiculous level. We know new permanent memories are formed by creating proteins which decrease the resistance to transmit signals between neurons. Memories are not stored in any single place, but seem to be tied to global patterns of brain activity. There’s an alphabet soup of receptors involved: NMDA, AMPA, CaMKII, PKC, and many more. The interactions between all of them are complicated, and this has kept scientists from understanding the full mechanics of it.

Still, it just doesn’t have the type of specialness that we’d attribute to the actions of a god. Why is that?

Part of the reason may be that it seems easy to expand. If we know, say, elephants can keep the current location of seventeen to thirty family members in their head,[39] we can easily picture another creature that can manage twice as many. That skill is a mere numbers game, and bumping up its capacity is probably as easy as enlarging some part of the brain.

However, I suspect the main reason is that it doesn’t seem mysterious. Decades of research have taken their toll, and even our limited knowledge of memory suggests there’s a good mechanistic explanation of the entire process out there.

Both reasons are variations on the same theme: if a machine can do it, it can’t be special. For instance, Gordon Bell estimates that he could archive all the books, photos, mail, and movies that a typical person encounters in a lifetime in about one terabyte of computer storage.[40] In comparison, I own a computer that can store four lifetimes, and I could add one more in exchange for a day’s wages. This computer can also do math much, much faster than I ever could, has a reaction time that makes mine look positively glacial, and can crunch through more data than my poor brain could ever hope to, all while making fewer mistakes and never getting tired.

This also applies to biology, as well. We understand how human arms and legs work in excellent detail, from the force absorbed by the skeletal structure to the conversion of ATP[41] into mechanical energy. While our artificial versions are not nearly as efficient or flexible, we have no reason to suspect that’ll be permanently true.

Thanks to this, we can cut out a few categories of intelligence. Memory gets chucked completely, as does processing speed, reaction time, and kinesthetic ability. I’ll also invoke the mechanical argument for visual and auditory processing, logic and mathematics, and spatial intelligence, though I want to go into more detail than can comfortably fit in this introduction.

Language

Language needs no introduction. You already know the power of language, because you’re decoding it right now. Without language, we would have no way to indicate we’re planning a big hunt tomorrow, describe the motions of the planets, or enjoy pictures of cats with captions added. Surely no other animal can claim to be nearly as advanced.

Unfortunately, we can’t be sure. We have not decoded the language used by elephants or whales, for instance, so entirely possible that they’re top of the heap. Whales in particular have access to one trick that we’ve only duplicated in the last few decades. They communicate with a series of clicks and yelps that can be heard from thousands of kilometres away. If you’re a fan of submarines, you might know about the “SOund Fixing And Ranging” channel. It’s a layer of water that acts much like an optical fibre; sound waves that enter it never leave, they just bounce around within the layer until they fade out, which can be the length of an entire ocean. We’ve spotted Humpback Whales taking advantage of this, so we know at least one species uses SOFAR on occasion.

It’s a staggering thought. For millions of years before us, whales had access to their own World Wide Web.

Elephant calls can carry pretty far themselves, thanks to their low frequency, but alas there is no SOFAR near the ground.[42] Like many other animals, though, elephants do have specific calls for specific instances. When elephants are menaced by bees, they’ll make a distinctive call that causes other elephants to take defensive measures. Lucy King, and others at Oxford, have tested this call by playing back several different versions of it to wild elephants. Altered calls didn’t result in head-shaking, used to keep bees away from the pachyderm’s face, or the tossing of trunk-fulls of dirt to keep bees from everything else.

Prairie dogs have an elaborate set of calls that warn not only what type of predator is approaching, but what size, shape, colour, and speed it has. All that is strung together in a basic grammar. Based on that information, prairie dogs can actually recognize the predator as an individual, and adapt their responses to it. Different colonies even have different accents, suggesting this chatter is a learned behaviour instead of hard-wired genetics.

C.N. Slohodchikoff wanted to confirm that, so he set up a simple experiment where plywood cutouts of a coyote, skunk, and an oval were randomly brought towards a dog colony. The warning call for the coyote was close to, but not quite the same, as the call for a normal coyote. All three received very different calls, even though two of them weren’t predators. That fact reinforces the theory that this “language” is learned; both the oval and skunk are novel situations, since skunks are nocturnal, so if alarm calls were hard-wired in via evolution you’d expect both calls to be similar.

Admittedly, this proto-language is nowhere near as complicated as ours. That doesn’t prove prairie dogs cannot develop a true language, only that they haven’t had the need to. The verbal skills of non-humans may be dormant, sleeping contentedly until some twist of fate forces them to develop.

Parrots, while ranked as one of the smartest birds to grace the skies, were considered too stupid for complex language. Their brains lack a folded structure called the cerebral cortex, which helps pack a ridiculous amount of neurons into a tiny area and was thought to be necessary for high-level intelligence. Humans, dolphins, and chimps all have this structure.

Irene Pepperberg disagreed, and decided to prove her point by picking up a random parrot from an ordinary pet shop, and trying to teach it our language.

Alex exceeded all expectations. He could recognize 150 words, including five shapes and seven colours, and could string them together in a sensible manner. One memorable day, he was presented with an apple for the first time and asked what it was. His response: “ban-erry.” While he didn’t know what an apple was, he knew about bananas and cherries. This strange fruit seemed to be a cross between the two of them, so he created an appropriate word on the fly.

Another time he was presented with a tray full of coloured blocks. When asked “What colour two?,” for instance, he would examine the tray, find that there were two red blocks, and answer “red.” After Irene had asked him about the “three” blocks several times, and Alex correctly responded with “blue” each time, he started replying “five” instead. Irene tried to coax the correct answer out of him, but eventually gave up in frustration and said “fine, what colour five?” Alex replied “none,” as there were no set of five blocks with the same colour.

He wasn’t confused, merely bored with the exercise, and was acting up like a human child would do in the same situation.

Alex would apologize if he ticked off one of the researchers, though he was never taught this. When he got bored of testing, he would ask to be put back into his cage; again, he was never taught that. He was taught the difference between “I” and “You,” to retrieve any number of any type of object from a tray, and as shown above understood the concept of “none.” He understood relations between objects, like “different” or “smaller.” He sometimes practised his lessons on his own, yet never saw any-one or -thing doing the same, and would help the researchers train other parrots, even though he was never taught to.

Pepperberg estimates he was as intelligent as a five-year old human, or the smartest dolphins and gorillas.

And speaking of primates: recent research by Catherine Hobaiter and others at the University of St. Andrews have shown that wild chimps can communicate with at least sixty-six different gestures.[43] It’s a good reminder that there are more ways to speak than through sound.

[38] http://www.plymouth.ac.uk/pages/view.asp?page=7705

[39] According to a study by Richard Byrne published in Biology Letters, DOI:10.1098/sbl.2007.0529

[40] http://totalrecallbook.com/about-the-authors/

[41] Adenosine-5′-triphosphate, the molecule all Earth life runs on. A human being consumes its body weight of the stuff in a single day!

[42]  A similar layer of air might exist at higher altitudes. A top-secret research project that would have confirmed this was cancelled, unfortunately, but not before one of their test balloons made a big splash near Roswell, New Mexico.

[43]  http://news.bbc.co.uk/earth/hi/earth_news/newsid_9475000/9475408.stm

Proof From Intelligence (1)

Proof from Intelligence

There.

What you just did is quite unique in the history of life. The act of reading, so far as we can tell, was first done within our species. This is rather astonishing, since life has been loafing around for four billion years, and complex land-based multicellular life popped up sometime around four hundred million years ago. The odds of some other species coming up with our neat trick should be pretty high, and yet none did.

We don’t just read, though. We collect reading like crows or octopuses[34] collect shiny toys, enshrining them in large buildings called “libraries.” We then leverage them to do all sorts of stupendous things, like build telescopes or launch hunks of metal into outer space.

This fits into a greater pattern: those books, telescopes, and metal bits all exist to gather knowledge, which we then sift through in search of patterns and more knowledge. This perpetual cycle of gathering and interpreting is a sign of intelligence, something that just seems to be lacking in other animals. It’s richly rewarded us, by doubling our life span, freeing our spare time for more knowledge gathering, and building us some very cool toys.

If it’s been such a help to us, though, why haven’t other animals jumped on the intelligence bandwagon? Perhaps this is something exclusive to our species, something that took a divine touch to bring about.

Definitions

There are two ideas hidden behind this proof. Homo Sapiens Sapiens[35] has intelligence, while other species don’t, so we must be special. And since intelligence couldn’t possibly have evolved via small steps, it can only have come from god. To rebut these, I have to show that other species have some intelligence, and that there’s nothing we have that they don’t in smaller doses.

Note that I don’t have to show that other animals are smarter. The Fangtooth fish may have the longest teeth in proportion to its body size,[36] but no one would argue this proves it was blessed by a deity. In fact, we’d prefer to find a wide variety of intelligence in the animal kingdom; just like comparing other species’ eyes to infer the eye’s evolution, we can trace the development of intelligence by snooping on other intelligent beings.

But before I can show intelligence in other species, we need to get one thing straight: What is intelligence?

This may seem like a simple problem, but it has haunted philosophers and scientists for centuries. Everyone “knows” it when they see it, yet they struggle to describe it:

Individuals differ from one another in their ability to understand complex ideas, to adapt effectively to the environment, to learn from experience, to engage in various forms of reasoning, to overcome obstacles by taking thought. Although these individual differences can be substantial, they are never entirely consistent: a given person’s intellectual performance will vary on different occasions, in different domains, as judged by different criteria. Concepts of “intelligence” are attempts to clarify and organize this complex set of phenomena. Although considerable clarity has been achieved in some areas, no such conceptualization has yet answered all the important questions, and none commands universal assent. Indeed, when two dozen prominent theorists were recently asked to define intelligence, they gave two dozen, somewhat different, definitions.

(Ulric Neisser et al, “Intelligence: Knowns and Unknowns,” American Psychologist , February 1996)

 And without a clear definition, there is ample wiggle room to define “intelligence” in a way that’s convenient for you. An example: I can state as a fact that I believe a god exists. This might seem impossible for an atheist:

  Atheist \A"the*ist\, n. [Gr. ? without god; 'a priv. + ? god: cf. F. ath['e]iste.]
     1. One who disbelieves or denies the existence of a God, or supreme intelligent Being.
     2. A godless person. [Obs.]
     Syn: Infidel; unbeliever.

(Webster’s Dictionary, 1913 edition)

And yet there’s no conflict here. Look up “believe” in a thesaurus, and you’ll find “assume” right next to it. While both words refer to a fact that is true despite a lack of evidence, a “belief” is implied to be absolutely true, while an “assumption” is only relatively true. Assumptions can be easily discarded if proven false, and you’re free to pretend they were false if you’d like. Beliefs should remain true no matter what facts come to light, and thus should never change. Nonetheless, both meanings are close enough to be confused and interchanged, which is exactly what I did.

“God” has traditionally meant a powerful conscious being, but that definition has been expanded over the years. Pantheists reject that meaning, for instance, as well as the concept of souls. Instead, they define “god” as the entirety of the universe. Since the universe is just a definition, as per my remarks in Cosmological, that’s entirely consistent with my world-view.

So while I said

I believe that a god exists,

my true meaning was closer to

I assume that a universe exists,

which is quite compatible with atheism. The lesson is obvious: without a consistent, clear definition for “intelligence,” it’s easy to shift the meaning of the word around to suit your whim and dodge whatever argument you’re facing.

That doesn’t make it impossible to counter-argue, though. Look over the various definitions of intelligence, and you’ll note they’re usually a mix of mental attributes, such as logical thinking and tool use. By examining each potential component separately, I can check most definitions of intelligence without defining the term.

The most popular test of intelligence is the “psychometric approach,” for the simple reason that it can be easily tested. The victim is given a set of problems to work on, and asked to solve as many as possible until the clock runs out. Anything that can be fit on a sheet of paper has been on an intelligence test at some point, ranging from word vocabulary to picture patterns. Today these tests focus on abstract logic and reasoning, mathematics, and solving novel problems.

As implied, they usually split apart intelligence into several sub-components. For instance, the popular Cattell-Horn-Carroll theory uses ten categories:

  • Quantitative Knowledge: In a word, mathematics.
  • Short-Term Memory: The ability to remember things for a few seconds.
  • Long-Term Storage and Retrieval.
  • Reading/Writing: The ability to read and write, and all skills directly related to that.
  • Visual Processing: Dealing with visual patterns, by analysing, remembering or creating them.
  • Auditory Processing: Much like the above, though this also includes speech.
  • Processing Speed: How well someone can repeat a mental task.
  • Reaction Time: How quickly a person can respond to some input.
  • Fluid Intelligence: Reasoning with new problems or information.
  • Crystallized Intelligence: Reasoning with old problems or information, as well as the amount of information already known and the ability to share that with others.

(Most of us would group those last two categories as “problem solving,” and in the interest of keeping my workload down I’ll take full advantage.)

Other researchers have rejected easy tests of intelligence, and broadened the definition still further. Howard Gardner’s theory of multiple intelligences includes:

  • Logic and Mathematics.
  • Linguistic: Reading and writing, plus the ability to use speech.
  • Spatial: The ability to picture and analyse a scene, be it real or a product of the mind.
  • Kinesthetic: How well people can control their own bodies, including how quickly they react and memorize sequences of movement.
  • Musical: Recognizing or producing a pitch or beat, and the ability to play or write music.
  • Intrapersonal: How well someone knows their emotions, desires, and abilities.
  • Interpersonal: The ability to interact with society, including recognizing another person’s intrapersonal content and sharing information with others.
  • Naturalistic: Interacting with other species, and the ability to nurture.

I myself will add on another few categories, to catch a few other talents that have been flagged as unique to our species:

  • Tool Use
  • Play
  • Culture
  • Altruism
  • Lying
  • Long-term Planning
  • Creativity

[34] Yes, that’s really the plural form of “octopus.” While “octopodes” is the correct Greek plural, it’s so rarely used that dictonaries either bury it as the last candidate or don’t mention it at all. “Octopi” is completely wrong.

[35] Homo Sapiens means “Wise Man;” anthropologists have begun tacking on an extra “Wise” to make room for a potential sub- or co-branch of our species, and given our messy origins I think this is a Wise idea.

[36]  They’re so long that if it closed its mouth like other creatures do, its front teeth would stab through the brain and create an impressive set of horns poking out of its forehead.

Proof from Logical Necessity, or the Ontological Proof (3)

Kant-er Arguments

Obviously, I’m not the only one with objections. Immanuel Kant, most notably, spent eleven pages in “Critique of Pure Reason” poking holes in the Ontological proof. He uses much more robust reasoning than I do, so if nothing I’ve said so far has convinced you, I’d recommend you give his arguments a go.[28] I’ll attempt to summarize the entire thing here.

Kant’s critique comes in four separate parts. First off, he points out that “God is something greater than we can think of” has a hidden assumption: god exists. If god did not exist, then we can say anything about it without contradicting ourselves. “Unicorns are made of gold” is just as truthful as “Unicorns are not made of gold,” but only “Coelacanths[29] are not made of Gold” is true. Since we can say anything we want about non-existent beings, we can prove anything we want about them too.

Second, the ontological proof is supposed to be a proof of god’s existence, yet as noted above Anselm had to assume god existed to write his proof. This is allowed in proofs, if you use a  technique known as “reductio ad absurdum;” you assume something, derive a contradiction from that assumption, and are forced to conclude that assumption is false. Of course, any proof that applies “proof from contradiction” to the assumption “god exists” would have to conclude god does not exist, which seems counter-productive in this case.

Third, we don’t toss around “being” and “exist” lightly. We know that coelacanths exist because we’ve found fossils of them, photographed a few of them swimming about, held them in our hands, and even tasted them. [30] We don’t say they exist because they are “beings,” or have a property called “existence.” Anselm calls god a “being” and says he “exists,” but offers no evidence beyond his proof to back that up. God hasn’t earned either of those labels, yet most Ontological proofs assume he has.

Fourth, we can describe what a unicorn is in physical terms, and set up various tests and experiments to try and catch one. Since we could pin them down as “beings” in the same way as we’ve done to the coelacanth, we can debate their existence in a meaningful way even if no-one’s actually seen a unicorn in the wild. The rational god of Avicenna will never jump into a fishing net or be lured out by hay. Not only do we lack any tangible proof of its existence, we could never find any. God will never be a “being,” no matter how badly an Ontological proof wants him to be.

All of the “distilled” proofs provided by the Encyclopaedia of Philosophy trip up one at least one of Kant’s counter-arguments, and all of them trip up on the last two.

Those last two, in fact, apply to all variations of the Ontological proof. You cannot show something exists in the real world without referring to the real world in some way. Remember my kitten example from the introduction? Until I began defining the physical characteristics of a kitten, you had no way to prove its existence and no reason to take the idea seriously. Conceptual ideas can only be defined within a logical system, and only when that system relies on assumptions that are a close match to the laws of reality do those ideas happen to coincide with reality. For instance, zero-order logic is not permitted to use the concept of sets, and those seem to be an essential abstraction for understanding the real world.

As a direct example, take the third “distilled”[31] proof from the Encyclopaedia of Philosophy, which claims any god is not contingent. While this dodges most of the objections outlined above, it treats existence as if it was an arbitrary label and not something justified via tangible evidence. Since existence is contingent, and this proof says a god has the property of existence, god must be contingent after all.

The Ontological proof tries to use concepts and logic alone to prove the existence of something physical and tangible. It’s impossible, plain and simple.

Gödel’s Proof, and the Problem of Infinity

Simplicity is rare in Ontological proofs, though.

As I mentioned in the introduction, long and complicated chains of reasoning seem more impressive than short, simple ones. In reality, long proofs are more likely to suffer from small errors in logic, and less open to cleaning them out.

A perfect example of this is Gödel’s Ontological proof. To start at the beginning, his insistence on positive properties is suspicious. We tend to make negative properties the verbal negation of positive ones because we prefer to think about the positive, not because one method is inherently better; compare “non-corrupt” and “corrupt“ to “just” and “non-just.” If we apply Gödel’s argument to “negative, morally aesthetic properties” instead, we can prove a god’s existence via the same line of reasoning, since if none of the positive properties conflict then neither can the negative ones, but we’re forced to conclude this god is “perfectly corrupt,” “all-weak,” “merciless,” and an “absolutely amoral” deity.[32] The restriction on “positive” properties is in place to ensure Gödel proves the existence of a god he wants to exist, not because it’s necessary for the proof.

Speaking of which, why does Gödel go to great lengths to use the pure, rational logic to formulate his proof, yet use such a loose definition of “morally aesthetic?” That’s like trying to build a rock-solid building on a swamp. A logical proof is no stronger than its weakest part, and the definitions form the bedrock of the entire argument.

Note as well a subtle problem with Definition 1 and Assumption 3:

Definition 1: An object has the “God-like” property if, and only if, that object has every property in P.

Assumption 3: The “God-like” property is in P.

If you’ve read my take on the Cosmological proof, this should twig an alarm bell. I demonstrated that a container of things is not automatically a thing itself. If the “God-like” property is a property, then it was already in P and thus assigning an object the “God-like” property means that it must already have the “God-like” property to begin with! We could also define a “God-God” property, which requires every property in P including “God-like,” a “God-God-God” property via similar means, and so on.

Even if you object to the above lines of reasoning, Gödel’s proof has a gaping hole. The Epicurean Paradox[33] is the same size as that hole:

If God is willing to prevent evil, but is not able to, then He is not omnipotent.
If He is able, but not willing, then He is malevolent.
If He is both able and willing, then whence cometh evil?
If He is neither able nor willing, then why call Him God?

(“Dialogues Concerning Natural Religion,” by David Hume)

So is this paradox:

If God is perfectly just, no-one is punished less than they deserve.
If God is merciful, someone must be punished less than they deserve.
Therefore, God cannot be perfectly just and merciful.

Same here:

If God is omnipotent, can he perform an action that he cannot perform?

Gödel is careful to prevent simple contradictions from derailing his proof, but does nothing to keep out more complicated ones. These conflicts lead to a definition of a god that contradicts itself, rendering it nearly useless.

I say “nearly” because there is one way out; instead of combining multiple attributes into a single god, you could place a strict limit of one property per god. Most believers will reject that outright, at the time of this writing, since most believers are monotheistic. It also denies any composite property such as “good” from being a god, since that would include the contradictory properties “merciful” and “just,” among others. It also suggests that any property we could come up with has a god associated with it, including “fortitude,” “ambidexterity,” “radical-ness,” and “ability to explain mathematics without sounding condescending.” Even polytheists have their limits, and the vast majority would reject thousands of gods, let alone a potentially infinite number.

Ignoring that escape route, believers dismiss the contractions as not applying to a god because they are beyond rational thought, or as proving that the person asking the question doesn’t understand the type of infinity that the gods posses. The first reply also dismisses the Ontological proof, since it relies on the target god being rational. The second instead proves that the believer doesn’t understand what they’re asking for. Here, let me remind you of a few definitions:

Omnipotent \Om*nip"o*tent\, a. [F., fr.L. omnipotens, -entis;
     omnis all + potens powerful, potent. See {Potent}.]
     1. Able in every respect and for every work; unlimited in
        ability; all-powerful; almighty; as, the Being that can
        create worlds must be omnipotent.
     2. Having unlimited power of a particular kind; as,
        omnipotent love. --Shak.
Omniscient \Om*nis"cient\, a. [Omni- + L. sciens, -entis, p. pr.
     of scire to know: cf. F. omniscient. See {Science}.]
     Having universal knowledge; knowing all things; infinitely
     knowing or wise; as, the omniscient God. --
     {Om*nis"cient*ly}, adv.

(Webster’s Revised Unabridged Dictionary, 1913 edition)

Note that no restrictions are placed on the chosen god, in either case. If a god can do any action, then even the contradictory ones must be doable. If the god can know everything, it must know about things you’d rather keep private. If god is infinitely just, then she must punish fairly in every case, no matter how much mercy you’d like him to grant.

There’s only one escape from this quagmire: redefine the words to place a limit on your god. This is quite dishonest, because people expect the words to mean roughly what a dictionary says they do and thus will misunderstand you. It’s far better to use a different phrase to avoid confusion, though I’ll admit “effectively omnipotent” or “really, really, really super powerful” don’t have the same ring.

A limited god is still a god, mind you. The two definitions I outlined in the introduction do not make reference to infinite power, as you’ll recall. We can still bless a god with enough power to create the universe, or do any number of incredible feats. But note that all versions of Ontological make reference to infinity, either by directly describing an unlimited being, or indirectly implying an infinite number of traits that are infinitely more perfect than any other being can claim. You can’t have your infinity and eat it too.

That doesn’t stop Ontological proofs from trying. All the ones I’ve seen merely introduce more errors, above and beyond the pair related to existence.

The Proof that God Does Not Exist

I can’t leave this proof without sharing my favourite variation. Instead of spending most of a chapter developing objections, Douglas Gasking just cuts to the point by using the same reasoning to prove god doesn’t exist:

  1. The creation of everything is the most marvellous achievement imaginable.
  2. The merit of an achievement is the product of (a) its intrinsic quality, and (b) the ability of its creator.
  3. The greater the disability (or handicap) of the creator, the more impressive the achievement.
  4. The most formidable handicap for a creator would be non-existence.
  5. Therefore if we suppose that the universe is the product of an existent creator we can conceive a greater being — namely, one who created everything while not existing.
  6. Therefore, God does not exist.

The implications are pretty clear. If you can prove and disprove something using the same line of thought, there’s something wrong with your line of thought.

[28] “Critique of Pure Reason” has long since dropped out of copyright, so there are a number of translations available online. The same is true of most of the documents I’ve mentioned, so feel free to analyse them for yourself.

[29] A fish thought to have gone extinct with the dinosaurs, until one jumped into an African fishing net in 1938.

[30] From what I’ve read, they’re very fishy.

[31] As quoted from the Encyclopaedia: “These are mostly toy examples. But they serve to highlight the deficiencies which more complex examples also share.”

[32] Not even Satanists would worship this god. They value personal responsibility, knowledge, justice, and individuality.

[33] Ironically, Epicurus never came up with his paradox. A critic of his, Lactantius, incorrectly attributed it to him four centuries later. Epicurus is sometimes labelled an atheist, again thanks to Lactantius, but was more deist.

Proof from Logical Necessity, or the Ontological Proof (2)

Existence is not Great

A core assertion of Ontological is that it’s better to physically exist than to be a concept. I’m not convinced, and to help illustrate the point I propose a simple thought experiment.

  1. Picture a vertical line, which we’ll declare to be one unit high.
  2. Place a circle around it with a diameter of exactly that line; if you’ve pictured this correctly, you’ll have a circle split in two.
  3. Now, mentally undo the part of the circle that touches the bottom of the line, as well as the top part of the line on the left side. Unwrap it counter-clockwise from the top, keeping the left-most portion anchored to the top of the vertical line, until the line is perfectly horizontal and at a right angle to the original line.
  4. You now have two sides of a rectangle. Complete it by extending a horizontal line from the bottom of the vertical line with the same length as the upper horizontal line, and a vertical line from the end of the upper horizontal line with the length of one unit.

The area for a rectangle is the width times the height. We know the circumference, or length around a circle is π times the diameter of that circle, which in this case is π units. So with a width of π units and a height of our circle’s diameter, or one unit, that rectangle has an area of exactly π square units.

If you had difficulty, the following diagram should help you. Conveniently, this diagram will also prove my point:

Creating a rectangle pi units long by unwrapping a circle.

The final rectangle in your mind is exactly π square units big. The rectangle in this diagram is not. In fact, no real-world diagram will ever be exactly π square units.

For argument’s sake, we’ll say the rectangle is exactly 10cm wide, and the diagram is printed at 472dots per centimetre.[26] The width of this rectangle is thus 4,720 dots, and the height is 1502 dots, for an area of 7,089,440 dots. To convert that to “units,” we need to divide by the size of one square unit in dots, which is 1,502×1,502 or 2,256,004 dots. According to my math, that rectangle is about 3.142477 square units in size. This differs from π after the third decimal place.

The reason is pretty clear. π is an irrational number with an infinite number of decimal places; by definition, it can never be represented by one integer divided by another, or by counting a finite number of elements. And yet if we construct this diagram in the real world, both the width and the height must be finite numbers. Even if the width of the rectangle was the size of the visible universe, it would still be a finite number of atoms in area, and our answer would be wrong somewhere around the 36th decimal place.

There’s no escape from this problem, either. No matter how you try to represent π in the real world, you’ll be forced to use a finite number of decimal places to represent a number with an infinite number of them. And yet, what’s impossible in the real world is easy in your head. You don’t need to convert π to a decimal number, you can treat it as a concept and dodge around the problem. As a consequence, the rectangle in your head is exactly π square units in area.

This confirms something I’ve observed as an artist. Like Anselm’s painter, I too have had a vision of a finished drawing or photo in my head; unlike him, the finished product is rarely more than a shadow of what’s in my head. Something always goes wrong; a line falls out of place, a photo has a branch that’s poorly placed, and so on. The exceptions are usually because I accidentally found an improvement while creating the work, and not through perfect execution.

The Triumph of Irrationality

I’ve also got a beef with another assertion of Ontological, that it’s impossible to think of any being greater than god.

If you’ve heard of Buddhism, you’ve likely heard of its most famous variation, Zen. Part of that sect’s seductive quality comes from an emphasis on “kōans,” or questions that have no rational answer but instead are intended to provoke an enlightened train of thought. Here’s the most famous of them:

“Two hands clap and there is a sound. What is the sound of one hand?”

(Hakuin Ekaku)

That question cannot be logically answered,[27] yet Hakuin had no problems asking it and we had no problems contemplating it. Given a little thought, you should be able to come up with your own kōans: What is the colour of an object in perfect darkness? What is the sound of empty space?

Kōans show that we have no problem thinking irrational thoughts. The Ontological proof uses reason to prove god, however. In order to do this, it assumes that god must be rational. If he is not, he could not be described by reason, and if he could not be described by reason, he could not be proven by it either.

But if a god must be rational, and our thoughts don’t need to be, we can contemplate things greater than the gods. Let’s try it: There is something greater than a god. The consequences of that sentence are not logically valid, yet I had no problems writing it and you had no problems understanding it. If we shift our assumptions, we can make it valid; I’ll use this in the Morality proof later on, for instance.

If we’re not bound by rationality, while god is, then we can easily think of something greater than god, and another assertion is shot down.

Anselm realized this loophole, and tried to close it in Chapter 4 of Proslogion. He separates thought into two categories:

For a thing is thought in one way when the words signifying it are thought, and it is thought in quite another way when the thing signified is understood. God can be thought not to exist in the first way but not in the second. For no one who understands what God is can think that he does not exist.

(Chapter 4: “How the Fool Managed to Say in His Heart That Which Cannot be Thought”, as translated by David Burr)

The problem with Anselm’s counter-counter is that it doesn’t address the counter-argument at all. He clearly wants to put “a being greater than god” in the “signified but not understood” category, where he can safely ignore it, but he doesn’t say why it belongs there, let alone why his categories exist in the first place.

In order to decide which category that sentence goes into, you have to understand the sentence first. For instance, which of the two gets “Yd.p. lprxaxnf co br Ire?” That might be a meaningless statement, making it impossible to understand, or it might have meaning in a language or code you’re not familiar with. In contrast, “There is something greater than god” can be easily understood and thus placed in a category. But because you understood it, there’s only one possible placement: things that are “signified” and “understood.”

You may not be aware of all the rational implications of that statement, or you might know them better then I do, but the underlying concepts must be clear to you before you can make a rational decision. Irrational decisions are another beast, but they only prove my point: the irrational trumps the rational.

Anselm’s defence doesn’t work, and my counter-example still stands.

This cuts the other way, as well. Merely being able to state something does not make it rational or logically justified. The fourth “distilled” proof falls into this trap. “The existent perfect being is existent” is true, in the same way that “the existent @#^*$ is existent” is:

  • “The existent ___ is existent” depends on the assumption “a ___ exists.”
  • If that assumption is true, then there’s no need for the proof!
  • If that assumption is false, then the proof is contradictory and we can conclude anything we wish from it, including the existence of whatever we want.

The fifth earns a medal from me for squashing three separate proofs into one. There’s proof from Witness (“if one person is convinced a god exists, god exists”), proof from Popularity (“multiple believers can’t be wrong”), and just a sprig of Ontological (“the word ‘God’ only means something if a god exists”). Unfortunately, it falls flat on that last assertion, as the trio of Faust, Bilbo Baggins and the Jabberwocky will swear to. We’re surrounded by fictional, non-existent things that provide us with meaning, by setting an example or just giving us a good time. This extends to science as well; Niels Bohr expanded on Ernest Rutherford’s model of the atom to create the boringly-named Rutherford-Bohr model, which has very little in common with the real layout of an atom but is easy to teach. And so it is taught.

[26] For readers who think in imperial units, that’s 3 15/16ths inches and 1200dpi respectively.

[27] It’s a common misconception that kōans have no answer. To the contrary, every kōan has an answer, though that answer varies from Buddhist Master to Buddist Master. For instance, Zhàozhōu’s answer to “Does a dog have Buddha-nature or not?” to one of his students was “Wú,” Japanese for “no.” To another, he replied “yes.”

Proof from Logical Necessity, or the Ontological Proof (1)

Proof from Logical Necessity, or the Ontological Proof

I’m going to need your help with this one. Relax and get comfortable.

Now, I’d like you to imagine the most perfect being. One that embodies all the qualities you’d like to be in a conscious entity. A perfectly-balanced sense of judgement, say, or a deep pool of empathy and caring, or a graceful ability to forgive and move on, or a wisdom well beyond its years. Picture an entity that manages all that, better than anything that has or will exist.

It’s a lovely thought, isn’t it? Don’t you wish such an entity existed?

I bet you’d agree that it’s better to exist than not exist, right? A being that has all those traits would be more perfect if it existed than a similar being that did not. And yet, you had no problems picturing that perfect entity, didn’t you? How could you have pictured perfection without including every portion of that perfection, including the ability to exist?

The ease of picturing perfection, therefore, must mean that perfection exists. And since a god matches that perfection to a tee, that must mean the gods exist.

No Really, It’s Quite Popular With Some People

Most of you are scratching your heads right now. “Really? That’s a proof?”

It is, and it’s been a favourite of philosophers for some time. Despite the “-logical” suffix, most scholars do not think this proof was dreamed up by a Greek. I’m not so sure; while it’s true that I can’t find a single example within their works, you’ll note that it bears a strong resemblance to the Cosmological proof, which was quite popular with them. Indeed, my description above is almost identical to variation of Cosmological. Instead of stepping up one degree of perfection, however, I stepped sideways and invoke the property of “existence.” Thus Ontological proofs do not “step outside” the universe, and are immune to that particular counter-argument.

There’s a better argument for this proof’s origins, that a fan of Greek learning came up with this twist. Avicenna[22] was a Persian philosopher who lived roughly 1.5 millennia after the Greeks, but was one of the few people in the world to have access to their written texts. He claims to have memorized the entire Qur’an by age ten and outsmarted his teachers by age fourteen, which sounds unlikely until you start cataloguing his accomplishments. Avicenna pioneered the use of clinical trials, knew the heart worked as a valve, discovered Newton’s first law of motion, reasoned that light had a finite speed, came pretty close to inventing Germ Theory, was the first psychologist, published a book of medicine that was used for 600 years in Europe, and during a moment of boredom invented the refrigerated coil and scented oils. As if to further rub it in, half of his surviving work is
written in verse.

A lover of Aristotle, Avicenna extended the old Greek’s philosophy and applied it to Islam. I can track down several scholars, such as M. E. Marmura and Parviz Morewedge, who make it quite clear that Avicenna’s proof differed from his classical hero by focusing more on logic and existence, independent of physical reality:

It is not in any sense a proof that infers God’s existence from the observation of His handiwork. On this Avicenna is explicit. After giving one version of his proof from contingency, for example, he writes: “Reflect on how our proof for the existence and oneness of the First and His being free from attributes did not require reflection on anything except his existence itself and how it did not require any consideration of His creation and acting, even though the latter [provide] evidential proof (dalīl) for Him. This mode, however, is more reliable and noble, that is, where when we consider the state of existence, we find that existence inasmuch as it is existence bears witness to Him, while He thereafter bears witness to all that comes after Him in existence” (Ešārāt, p. 482).

(M. E. Marmura, Encyclopædia Iranica, http://www.iranica.com/articles/avicenna-iv, retrieved April 30th 2011)

Other scholars dispute Avicenna’s departure, and unfortunately I can’t track down a translated version of his Šefāʾ or Ešārāt to decide for myself.

I can track down the original texts of Anselm of Canterbury, however. His book Proslogion is universally accepted as a true proof from Logical Necessity. Anselm had a rather interesting life, most notably feuding with two kings of England and being forced into exile twice. It’s a shame the topic of this book doesn’t permit me to go into more details, because those decades of political intrigue would be easier to sort out than Anselm’s writing:

In fact, it [God] so undoubtedly exists that it cannot be thought of as not existing. For one can think there exists something that cannot be thought of as not existing, and that would be greater than something which can be thought of as not existing. For if that greater than which cannot be thought can be thought of as not existing, then that greater than which cannot be thought is not that greater than which cannot be thought, which does not make sense. Thus that than which nothing can be thought so undoubtedly exists that it cannot even be thought of as not existing.

(Proslogion, Chapter 3: “That God Cannot be Thought Not to Exist”, as translated by David Burr)

Since I’m not a cruel person, I’ll spare you from having to parse the original and write up my own translation:

Chapter 2: That God Really Exists

1. No being is greater than God. Even a fool who would deny God recognizes this.

2. That fool has only a partial understanding, however, since he does not understand such a being to exist.

3. Consider a painter’s thoughts of his or her next masterpiece, to the finished masterpiece itself. They have a rough sketch of the painting within their minds, but since it does not physically exist they cannot fully understand that painting.

4. It follows from 3. that it is greater for any object to exist that to not exist.

5. But if point 4. is true, then it would be impossible to think of the greatest possible being unless that being existed. Otherwise, there could exist a greater being, having all the attributes you gave to your greatest being plus the attribute of existence.

Chapter 3: That God Cannot be Thought Not to Exist

6. We can think of a being that must exist, and this being must be greater than one that we cannot think of as existing.

7. By that reasoning, if that being from 4. couldn’t exist, it would be lesser than another thing in our thoughts that could exist in reality.

8. Thus, we’ve reinforced point 5: that being must exist in reality.

9. By extension of points 6 through 8, we cannot even think of this being as not existing.[23]

10. We can’t think of anything greater than God.

11. This “being” we’ve been referring to must be God, otherwise we’d contradict point 10.

12. Praise God, isn’t He wonderful, etc. etc. etc.

Philosophers fell in love with this style of proof. The possibility of finding a proof for god using nothing but logic is like finding a long-forgotten civilization without leaving the house. A number of big-names have come up with their own versions, such as Gottfried Leibniz, Kurt Gödel, and René Descartes. The Stanford Encyclopaedia of Philosophy lists five more “distilled” variations on it:

1. God is a being which has every perfection. (This is true as a matter of definition.) Existence is a perfection. Hence God exists.

2. [24]

3. It is possible that God exists. God is not a contingent being, i.e., either it is not possible that God exists, or it is necessary that God exists. Hence, it is necessary that God exists. Hence, God exists.

4. [It is analytic, necessary and a priori that][25] Each instance of the schema “The F G is F” expresses a truth. Hence the sentence “The existent perfect being is existent” expresses a truth. Hence, the existent perfect being is existent. Hence, God is existent, i.e. God exists. (The last step is justified by the observation that, as a matter of definition, if there is exactly one existent perfect being, then that being is God.)

5. The word ‘God’ has a meaning that is revealed in religious experience. The word ‘God’ has a meaning only if God exists. Hence, God exists.

6. I exist. Therefore something exists. Whenever a bunch of things exist, their mereological sum also exists. Therefore the sum of all things exists. Therefore God—the sum of all things—exists.

(http://plato.stanford.edu/entries/ontological-arguments/ , retrieved April 30th 2011)

Missing from that list is Leibniz’s Monad theory, which borrows an ancient Greek idea. A “monad” is a super-atom of sorts; they make up every substance and contain consciousness, yet cannot be changed or destroyed, only divided into infinitely small pieces. No two are alike. They’re classified by the abstract properties they’ve been granted; “Entelechies” do little more than move, “Souls” have the added ability to remember things, and “Spirits” are permitted to reason. While all modads are tied to physical bodies, Leibniz claims there’s a special monad out there that is not attached to a body, and does nothing but continually create new monads. This god monad is stopped from just cranking out “Spirits” by the laws of nature. This system cannot be improved on, according to Leibniz.

None of Leibniz’s contemporaries agreed on that point.

The sixth on that list is a close match for Descartes’ “proof” for God. The scare-quotes are because Descartes would probably object to that argument being called a proof, let alone one of his own. In general, he thought God’s existence was obvious via intuition, instead of rational arguments. As a result he never bothered to write up a formal rationale, but instead randomly scattered bits of logical argument and informal reasons throughout his work. He took full advantage of this sloppiness, by throwing out many variations and forcing his critics to comb through his entire work to clean up his “proofs.” Fortunately for me, his arguments boil down to a mix of proofs that I have or will cover in this book, so I don’t need to elaborate further.

Gödel’s proof, by comparison, is both much better and much worse. Instead of dealing with “perfections” or simplistic “attributes,” his proof works on “positive, morally aesthetic properties;” examples of these include “perfectly just,” “all-powerful,” “merciful,” and “absolutely moral.”

Out of all the Ontologicals I’ve presented, Gödel’s is easily my favourite. It’s much better than Descartes’ “proof” because it is a well-structured logical argument, presented in a formal logical system with well-defined rules. It’s also much worse, for the same reason:

Godel's Ontological Proof

That chart-junk is called “modal logic,” and is a little tough to understand without study. Would another translation help?

Assumption 1: No property in collection P will be the inverse of another property also in P.
Assumption 2: If a property is in P, and if for all objects with that property that implies the existence of some other property in every possible situation, then the collection P also contains the other property.
Theorem 1: If a property is in P, an object might exist with that property.
Definition 1: An object has the “God-like” property if, and only if, that object has every property in P.
Assumption 3: The “God-like” property is in P.
Theorem 2: At least one object might have the “God-like” property.
Definition 2: A property is an “essential property” of an object if, and only if, every property that object has must be implied by the essential property, in every possible situation.
Assumption 4: Any property in P must be within P in every situation.
Theorem 3: If an object has the “God-like” property, that property must be an essential property.
Definition 3: An object has the “Anselmian God” property if, and only if, all its essential properties imply that, in every possible situation, an object exists with that essential property.
Assumption 5: The “Anselmian God” property is in P.
Theorem 4: There must exist an object with the “God-like” property in every situation.

I think this is enough examples to begin pulling out some common threads. To begin with, Ontological proofs ignore concepts grounded in reality and instead focus on using pure logic. The only potential exception is “existence,” depending on whether or not you think ideas have an existence outside or independent of the universe. From there, they start rattling off attributes and properties of god, and through some carefully-constructed logic come to the conclusion “god exists” by proving something god-like has the attribute of “existence.”

[22] This is the Latinized version of his name; in the Middle East, he was known as Abū ʿAlī al-Ḥusayn ibn ʿAbd Allāh ibn Sīnā, which was sensibly shortened down to Ibn Sīnā.

[23] Points 6 through 9 match up with the section of Proslogion that I quoted earlier, so you can decide how accurate my attempts at translation were for yourself.

[24] The second version is the rephrasing of Anselm’s argument I quoted earlier.

[25] My best translation from philosopher-ese is “it follows from and is proven from prior knowledge that…”