"The human mind possesses a basic probabilistic knowledge." So say Fontanari et al. in a newly published paper in PNAS ("Probabilistic cognition in two indigenous Mayan groups"). They asked whether formal schooling was a necessary foundation for a sense of chance by comparing two unschooled Mayan groups with Mayan schoolchildren and a control, and determined that no formal education is required for making "correct probabilistic evaluations."
This paper hit the popular news media. "We are all natural bookmakers," said New Scientist. And the senior author, Vittorio Girotto, said,
"We wanted to show that this sense of chance exists, that it is universal, and that you do not need to be trained to evaluate uncertainty," says Girotto. "We have good evidence now that the human mind does possess this ability."
Researchers have also reported that infants have a sense of "intuitive physics," seemingly being born with the ability to understand gravity (that is, by 2 months of age, they expect an object to fall -- really, who understands gravity?), and to expect that an object doesn't cease to exist when hidden from view.
And, studies (e.g., here and here) suggest that by 5 or 6 months, infants have a sense of numbers. But then, so, apparently, do non-human primates, such as tamarins. When two objects were hidden behind a screen, tamarins expected to see two objects when the screen was lifted; when there were three, the animals looked at the objects longer than when they were presented with the expected number, suggesting surprise or confusion. But then, dogs are good at playing Frisbee because they understand physics, too -- what does up there, comes down here.
And, even crows understand water displacement, knowing that if they raise the water level in a small beaker, they'll be able to pluck out a piece of floating food.
And look at how bats, and even dragonflies track their in-flight prey.
There seem to be several things going on when things like the probability study in unschooled Mayans make the news. To those of us who are schooled, probability, mathematics, physics -- or even grammar -- can seem like rather esoteric subjects that take years of training to master, or to even vaguely understand (though really, who understands probability?). Traditional schooling has divided the world we know into disciplines that have names and bodies of knowledge that must be mastered.
But, in large part, formal education is giving names to things we already knew. We have already internalized grammar as infants, we have a grasp of essential physical or mathematical principles, and it seems some basic understanding of chance as well. Essentially we're formalizing our description of the world we know from experience, but clearly we -- and dogs and tamarins, and crows and dragonflies and many other animals -- know that world before we know words or equations or models or principles that describe it. And indeed, most animals never get to that stage. I think we all can do this not because we have an innate sense of physics, or grammar, but because our brains have evolved to be able to recognize some kind of order, and to make generalizations from what we experience. It's apparently important to survival, because so many organisms have evolved the same ability.
In this context, we should keep in mind that mathematics is just an elegant way of describing relationships and really exists only because over the millennia humans did in fact realize that relationships had regularity. The long-known fact to western science, for example, that the Mayans had very sophisticated calendars shows that the recent news story is no surprise at all -- indeed, it would be very surprising were it not so. How things work in the brain is, however, a different order of question.
Holly elegantly suggests it simply comes down to pattern recognition. Frisbees follow predictable arcs, objects don't disappear inexplicably, if there are 5 yellow tokens and only 1 red, the chance of choosing a yellow one is higher than the chance of choosing a red one. I'm happy with that.
In 1931 Kurt Godel shocked the mathematical world. Math is the ultimate sanctuary for those who believe that some ultimately Platonic sense there is absolute, universal, unexceptionable--and understandable truth. The facts of geometry and mathematics are cosmically true (the Pythagorean theorem, the sum of angles =180 degrees, etc., 2+2=4, and the fact that the derivative of x-squared is 2x, etc). The way we do math may be a human or cultural convention, but the facts are not. Our cultural convention of how we do it, or whatever it is, may for cultural or historical reasons simply overlook many equivalent truths, but it is working with at least some set of ultimate truths. But how is it that Pythagorean theorem is true about right triangles, but there are no actual, perfect right triangles in the world??
2+2=4
At least, at the turn of the 20th century, for those studying such ultimate truths, it was widely thought that the principles of logic and logical reasoning were essentially the same as the principles of mathematics. Not only were both equally true but logical reasoning could be expressed in the same kinds of terms as those of mathematics. This would make the world a certain place, in a sense. Truth is truth. Truth is internally consistent. And truth is discoverable!
There were some problems. For example, what do we do with or about the notion of 'infinity'? Nineteenth century mathematicians, notably Cantor, showed that there are even different levels of infinity. The whole numbers are one level. You can match even numbers up one for one with odd numbers. But you can't match either up like that for the numbers between just 0 and 1. That's because there are far more of the latter: if we match 0, 1, and 2 with 0, 0.1, and 0.2 it might seem fine, but then how do we match up 0.001, 0.002, 0.00001, and so on?
And then there is the little problem of what 'randomness' means. For example, I recently learned that the digits in the value of pi (relating radius to circumference in a circle) are randomly distributed. Take any sequence, like 7623116, and you'll find it, on average every 10 million 7-digit sequencs in pi. Or look at Stephan Wolfram's 'cellular automata'; these are simple rules for transitions in a string of (say) black or white boxes if each box produces a 'descendant' box and the rule says what color it is based on the array of colors in the current generation. The resulting black/white pattern, determined by a simple rule, is, Wolfram says, indistinguishable from random: no pattern can be found along the string at any given time. Yet 'random' seems such an obvious concept...until you think about them too much, and then they become quite disturbing.
And I have not mentioned the very problematic notion of
probability. We know how a cause can lead to an effect--well, we think
we know that--but it is totally unclear how a cause can only lead to the
future probabilistically. Usually, the probabilistic nature of such
results is attributed to measurement error, poor theoretical
understanding, and so on, in a cosmos that, were we to know everything,
would be purely and rigidly law-like. But how could something cause something else 'with 27% probability'? If you think about it, it is not at all clear what that means, in terms of actual causation, beyond errors and sampling effects.
And then there is 'chaos' theory. Even in a purely deterministic, rule-bound process of cause and effect, where there is no uncertainty or probability involved, unless you have 100% measurement accuracy of things at some given time, you cannot predict with any accuracy what things will be like over the future. Your predictive power, even with perfectly true theory, is zilch. But how can you know what the underlying reason is?
Such things fly in the face of the views of the cosmos as a law-like place where certainty rules. We want a knowable universe. We spend our puny lives as scientists trying to understand it, and assuming it at least exists, even if it's hard to understand!
What we have learned
To general chagrin, what mathematician Kurt Godel showed halfway
through the last century, was that even if it were true that the
mathematical realm was all-of-perfection, an unknown fraction of it was
unknowable. That is, things that are true can't be proved and, worse, you
could never know whether something you thought were true and were trying
to prove it, was in reality untrue or just unprovably true.
Given all of this, it is surprising that in so many cases what we have learned is that the universe may not be law-like in the way we'd thought, true probability may or may not exist, not all things can be shown to be true even if they are true, and (in quantum mechanics and relativitiy and gravity at least) there are phenomena that seem truly to be unlike any of the above concepts or, as physicists often say, just are not consistent with 'common sense'....even if they're true.
Even physics and chemistry, not to mention biology and psychology or economics, are often swimming in uncertainty and claims of knowledge that, no matter how confidently asserted, simply don't hold up.
These various incarnations of indeterminacy, like probability, can shake our faith in the idea of a knowable universe whose causal nature we can pin down tight. In ordinary sciences, and sometimes even in our daily lives, we don't know exactly how we should be viewing, much less approaching, causation. What we end up doing is designing studies or experiments that we know how to design, using methods we know how to use, and crossing our fingers. Whether we're being ostriches to our peril, or whether it doesn't matter and we should just carry on regardless, is unclear.
To the young and thoughtful, these provide things to think about, both in the practical sense of actually moving the science forward more than a millimeter at a time.....and in terms of our ultimate hope to understand life as it really is, not just in a statistical analysis.
Remember in 4th grade when your teacher tried valiantly to teach your whole class to play the recorder? Learning the fingering was hard enough, but then you had to learn note values and how to count them, and halves of them, and quarters of them, and rests, and how to keep the beat and so forth. This is one of the most frustrating aspects of learning music for just about any beginner. But, by the time students actually enjoy playing their instruments, the whole issue of counting notes, and time between notes and all of that, has become second nature. They've internalized something that the rest of us never really did.
Our son-in-law Niocla Barbieri, an Italian professional double bass player who plays baroque and classical music, put it this way when I asked him about the experience of keeping time in an ensemble, including about the rubato, an expressive stretching of the beat. I quote him at length because it's such a beautiful and evocative description.
Staying together in synch is not usually a conscious thing and I would say that the more the ensemble we play with is good, or close, or “harmonious”, the less you have to think about that and the less it stays conscious… Or the less you have to want it, I would say, because it happens by “itself”… I wouldn’t say it is something automatic, either, because it is very far from any idea of being something stuck or rigid, but the feeling is more something towards a fluid idea of a continuous chain of perceiving and reacting, detecting and responding, more like a very relaxed dialogue…
Sometimes, from the top of a bridge you overlook down into the water of the river and you see the flowing of it, the general flowing, you perceive a whole fluid movement of a big mass… But some other times you watch better and you can notice several small currents and streams that seem to have independent “will” from the main one… They seem to slow down and then accelerate and then move sideways, as if a part of the water is going to move away from the rest… That’s just an impression, because all the big mass is still traveling as one...
This is the flowing of the time, in orchestra and this is the rubato, I would say… Of course in a good group, where nice things happens without effort and with a sort of natural relax (“sprezzatura", we used to call it in the Italian baroque era), like the one of the flowing of a river...
A new paper just published in the Journal of the Royal Society ("Optimal feedback correction in string quartet synchronization, Wing et al.) takes a look at how professional musicians correct lapses in synchronicity. Sensorimotor synchronization happens in many organisms (fireflies that synchronize their pulsing, e.g.), so synchronization is of long-standing interest for a variety of reasons. Musicians do sometimes have lapses -- indeed, sometimes it's intentional -- but they also are adept at getting back in synch. How? What have they internalized?
Wing et al. analyzed two different professional string quartets playing the fourth movement of Haydn's quartet Op. 74 no. 1.
The question was how the members of each quartet responded to expressive, unrehearsed variations in timing. Generally, members of a string quartet follow the lead of the first violinist, although with more or less strict adherence to this rule, depending on interpersonal dynamics and the philosophy of the group and so on. So, correcting timing may be a matter of getting back in synch with the first violinist, or it may be more fluid than that. In any case, the authors propose a feedback mechanism to correct timing that gets off, a linear phase correction.
Time series analysis of successive tone onset asynchronies [that is, musicians who aren't in time with each other at the start of a tone] was used to estimate correction gains for all pairs of players. On average, both quartets exhibited near-optimal gain. However, individual gains revealed contrasting patterns of adjustment between some pairs of players. In one quartet, the first violinist exhibited less adjustment to the others compared with their adjustment to her. In the second quartet, the levels of correction by the first violinist matched those exhibited by the others. These correction patterns may be seen as reflecting contrasting strategies of first-violin-led autocracy versus democracy. The time series approach we propose affords a sensitive method for investigating subtle contrasts in music ensemble synchronization.
If musicians always played the music in front of them exactly as scored, it could be dull. And, they aren't automatons -- they hear a piece in their own particular way and want to express what it means to them and they have plenty of freedom to do this, within limits. We, the audience, give them that freedom, and also adjust whatever internal metronomes we listen to music with (even those of us who failed 4th grade music have developed a sense of timing) to go with the flow of the music we're listening to -- within limits. This is the rubato. But there are limits, and apparently they are internalized.
According to Wing et al., musicians use linear phase correction to regain synchronicity with either other musicians or with the tick of a metronome. That is, they know when their count is off, because it has been set previously, by the relationship between note values and time between notes, and they are able to tell when they're off, and adjust to get back into the beat. Musicians learn their skill by spending tens of thousands of hours counting notes; that they can internalize it quickly, and correct it when it's off is no surprise.
It is often said that musicians are good mathematicians and vice versa. The earliest formal musical theory was by the Pythagorean school of mathematics, in ancient Greece. They figured out the mathematics of basic harmonies. But many musicians didn't take, or hardly squeaked by, in mathematics. So are they doing implicit time-series math in their heads implicitly? How?
And try this on for size:
This is an excerpt of an ensemble playing John Adams' "Shaker Loops", with its fiendishly minimalist, relentlessly repetitive and syncopated measures (with endless slight deviations) that goes on for around 25 minutes. We heard an ensemble play this a few years ago at Ithaca College, and the violist who was a friend told us the stress and horrors of trying to stay in synch. But they did, beautifully, and in that instance (unlike the YouTube) they had no conductor!
In this general vein, what explains this curious phenomenon that I've observed numerous times, after decades of knitting? Just last night I was casting on stitches to make a scarf. The pattern called for 85 stitches. There are too many distractions for me to be able to count stitches as I cast on, so I just take time out to count them a few times as I go along. Remarkably -- at least I think so -- last night when I stopped to count I had cast on exactly 85 stitches. But I've had this happen when the pattern called for 285 stitches too. No phase correcting there, do I have an internal counter that turns itself on as I start to cast on, and then alerts me when I've met the target number? If so, it seems like a rather frivolous way to spend brain cells though, even if useful.
I remember once being at my daughter's youth orchestra rehearsal when one of the violin teachers told me to watch the conductor when he stopped to talk to the players. "He'll pick up the beat right where he left off," she told me, and indeed he did. After years of conducting, he had developed an internal metronome that kept on ticking even when he wasn't waving his baton.
And then there's the internal alarm clock that always goes off 2 minutes before the alarm we'd set. I don't remember the last time I've heard an alarm -- except when I couldn't figure out how to turn the bloody thing off on my phone.
So, this mathematical explanation for musicians correcting themselves when they get off the beat. It might well be a good description of what happens, but it's not an explanation of what they are doing or feeling. To us, the deeper question is why we're able to do all this counting and time keeping anyway.
It's said that infants can count. Or at least have numerical awareness. As do non-human primates, and even dogs. Crows can count, parrots can count to six, and even have a concept of zero, according to at least one source. Apparently even frogs can count.
So the ability to count must have evolved long before humans. But why? Or better put, what kind of ability is it really? It's easy to imagine adaptive scenarios (the duck had to be able to count her ducklings, to shepherd any stragglers away from predators, the wolf had to know its pack was intact, and so forth), but like most such stories, impossible to test them.
But maybe it's not an adaptation at all, really. Maybe it's just one of the many ways we take in information about our surroundings, a by-product of there being more than one of any of us, or of any food, or of anything else in our environment. Maybe recognizing that there are two lions over the crest of the hill is exactly the same as noting their color or even just that they are there. It's just another observation; when we turn it into a number, that's when it becomes higher math.
Many pre-agricultural cultural groups are reported only to have numbers one, two, and many in their language, which may be consistent with the idea that formal 'math' and counting are recent cultural add-ons. It is interesting that math and music were seen by evolution's co-discoverer Alfred Wallace as attributes that could not have evolved because, after all (he argued), our primitive ancestors didn't need them and so could not have been selected for them. He used this as a reason to invoke the existence of God and human exceptionalism.
We wouldn't go that far, and instead are grateful for music, which, if it is but an evolutionary spandrel, is one of the more beautiful things we do with our ability to count.
