Recently many news outlets around the world carried the startling news that archaeologists had found the world’s oldest bread—as witnessed by this headline for a story by Helen Briggs published on 17 July 2018 in the BBC News:
Prehistoric bake-off: Scientists discover oldest evidence of bread
NPR carried a similar story by Lina Zeldovich on July 24th:
14,000-Year-Old Piece Of Bread Rewrites The History Of Baking And Farming
These stories brought to mind a spoof I had written back in the mid 1960s when I was in graduate school studying anthropology. I won’t discuss my motivation, but some may pick up on what was the source of my inspiration.
“Without mathematics there is no art.”
A commentary by Uliana Solovieva
Look at these two rectangles. Which one appeals to you?
Most people would readily agree that the left choice is somehow better. A new field of Neuroaesthetics, pioneered by Semir Zeki, has been busy understanding the mechanics of our gut feeling. It seems that inside the complex system, we call our brain, exists an autonomous math calculator. It sends signals to the brain when the art we view matches with intrinsic “pleasing” criteria. Such system is guided by the reward function of the internal model of the brain (Kawabata 2004:1700). This theory seems to downgrade us to mechanic calculating robots, but it can give us a new framework to understanding aesthetics. What I am arguing is that unnoticeably, mathematics guide our aesthetic appreciation of art, and the process by which artists create art.
The first obvious source of artistic appreciation is in the defined proportions/symmetry we see all around us. The internal OCD filter is satisfied when harmony is achieved. This phenomenon can be looked at through the lens of the golden ratio, famously utilized in the Fibonacci sequence. Under close observation, the golden ratio is found in the realistic art of Leonardo Da Vinci’s Vitruvian Man as well as surrealistic Salvador Dali’s Melting Clocks. Despite the obvious differences, both works of art are widely acknowledged as innately pleasing. By closely probing at these masterworks, math’s explanatory power can be traced in art as the source of aesthetic appreciation.
In discussion of art, one controversial issue has been the validity of art as a subjective experience. Some may readily challenge my stance by insisting a commonly accepted notion that “art is in the eye of the beholder.” However, new evidence suggests the underlying universality of art appreciation is based on math – symmetry and ratios. Mario Livio, in his article “Why Math Works” argues, “The universe has regularities, known as symmetries, that let physicists describe it mathematically. And no one knows why.” (Livio 2011:83). Most of us readily accept math’s power to explain fundamental properties of the universe, while art is left in the wishy-washy grey area of the human experience. Could these two seemingly incomparable subjects be treated as one? Let’s do so by returning to the rectangles from (Figure 1), Leonardo Da Vinci’s Vitruvian Man (Figure 2), and Salvador Dali’s Melting Clocks (Figure 3).
The key to unlocking this puzzle lies in mathematics of the golden ratio. It seems we are naturally predisposed to appreciate a ratio closest to 1.618033988749895… (Dense 2013:39) which is represented by phi (φ), a never ending and never repeating number. It has boggled the minds of people ever since Pythagoras discovered it in secrecy (Livio 2008:41). Look at the first rectangle in Figure 1 with sides 1.62 (length) to 1 (width). You are looking at the world’s most beautiful rectangle, called the golden rectangle. It needs no beauty pageants to be voted best, rather, it needs our brain circuitry to become so. Without the brain evolution handed down to us, there would be no aestheticians, no art critics, and no mathematicians. When the brain recognizes the golden mean, it activates reward pathways and we unnoticeably feel good and unconsciously ascribe the dopamine surge to the gut feeling “I like!”
Da Vinci’s Vitruvian Man (Figure 2) is the picture-perfect of all the perfect perfections of a perfect man. Da Vinci is said to have been inspired by the writing of the architect Vitruvius who once wondered what would a perfectly proportional human look like. Vitruvius suggested human proportions as a symbiosis between each constituent part of the body (Livio 2008:134). Ask any artist how to “properly” draw a human figure and you’ll get math – “Draw 8 heads horizontally, that’s your proportionate height. Legs will start on the “5th head” and it is also where hands end….” my art instructor would repeat time and time again. It becomes evident that the body’s proportionate relationship of each part to the other generates realistic proportions that we call beautiful. Moreover, just as our beautiful rectangle, humans are also quantified as beautiful within the frame of reference in accord to symmetry. Once again, it is all the working of golden ratio. Da Vinci shamelessly utilized these seemingly magical principles into the intricate layouts of his own artwork.
To understand the highly encompassing impact of φ in aesthetics, we must understand where it came from. Patterns are generated from sequences, and the Fibonacci pattern goes like this
1, 1, 2, 3, 5, 8, 13, 21, 34, 55…
This sequence is life’s way of generating φ! Adding up the previous two numbers together forms each consecutive number; then, by dividing current number by previous, we get fractions that inch up closer and closer to φ but never quite reach it 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13… (Dense 2013:39).
With this pattern, a logarithmic spiral is generated in Figure 4, notice that parameters of each “box” making up the rectangle is in correspondence to the Fibonacci sequence numbers. The box in the center is 1:1, the next is 1:1, then 2:2, 3:3 and so on to create the good old golden rectangle.
With this groundwork, it becomes easy to see a connection between math in realistic art. However, what if I want to steer clear of obvious shapes and sizes, as in surrealism? That art style is known to disregard the boundaries of possibility and rely on the feelings, “pathos”, emotions of each viewer. Salvador Dali’s art demonstrates lack of required proportions, shadows, or concrete meaning that Da Vinci’s work exhibited. Did we finally hit the sweet spot of art containing no math backbone?
Not quite. Even though Dali, a master surrealist, seems unconcerned with the proper rules of symmetry and reality, in actuality tells us a hidden story of math through his Melting Clocks piece (Figure 3). Unintentional implicit use of the golden ratio in the creative process in Dali’s mind can be noticed by inserting a Fibonacci spiral over his masterpiece. It evidently lines up! Our eyes start from the brightest spot in the center and move to the left following the course of the tree and extending down to the clocks and completing the golden spiral in the anti-clock-wise fashion. Strangely once more, math seems to be underlying not only our aesthetic appreciation, but the way a painter paints. Could these algorithmic internalized calculations be present in all of us, secretly staging our actions since the day we can hold a paintbrush (pen/spatula/cup). Our brains are known to precisely calculate each movement when we take on even the simplest of tasks such as grabbing a cup of tea. Oh the intricacy of trajectories needed to move every muscle in perfect synchrony in order do it right. Could same or similar mechanisms lie in our understanding of art and the creation of art as a two-fold system, leading to the sense of aesthetic pleasure spiked with dopaminergic pathways and feel good vibes? It surely seems so.
Mario Livio’s remark of math having explanatory and predictive power stands clear to identify two opposite works of art – one realistic and one surrealistic – under the same umbrella of universal mathematical laws. These laws narrow the subjective ambiguity of human “liking” into a well-defined boundary. Dictated by the golden ratio φ in all things math – numbers, patterns and sequences act as the unconscious factors that choose what we do and do not like. This idea, can help you and any other folk looking to predict likability of a product. From artists to scientists, the mystery of the mystical aesthetic sense is no further than a math equation away,
as H.E. Huntley adds:
The description of this proportion as Golden or Divine is fitting perhaps because it is seen by many to open the door to a deeper understanding of beauty and spirituality in life. That’s an incredible role for one number to play, but then again this one number has played an incredible role in human history and the universe at large. (Huntley 1970:63)
Indeed, the golden ratio is referred to as a divine proportion found far beyond the properties of aesthetics: physicists find it in cosmic pulsars; botanists find it in growth patterns of leaves and biologists in the structure of DNA. Even though artworks of Dali and Da Vinci might not paint (no pun intended) the whole picture of how φ is found in every type of art. Nevertheless, they illustrate a microcosmic example of the role mathematics play in our internalized sense of beauty.
Livio, Mario. The golden ratio: The story of phi, the world’s most astonishing number. Broadway Books, 2008: 1-268.
Livio*, Mario. “Why math works.” Scientific American 305.2 (2011): 80-83.
Dence, Thomas. “SOME OLD AND NEW RESULTS FOR THE WORLD’S MOST FAMOUS SEQUENCE OF NUMBERS.” Journal of Applied Global Research 6.16 (2013): 38-43.
Kawabata, Hideaki, and Semir Zeki. “Neural correlates of beauty.” Journal of neurophysiology 91.4 (2004): 1699-1705.
Bielski, Carolyn. Beginning Art 1 Class. Lake Forest High School. Lake Forest, IL. September 2011. Keynote Address.
Huntley, H. E. The Divine Proportion: A Study in Mathematical Beauty. New York: Dover Publications, 1970. Print. 50-69.
Images found in:
“Leonardo’s Vitruvian Man.” The Vitruvian Man. Web. 28 Mar. 2016.
“15 Things You Didn’t Know About ‘The Persistence Of Memory'” Mental Floss. Web. 28 Mar. 2016.
“Lesson Plans Based on Movies & Film Clips!” Donald in Mathmagic Land. Web. 28 Mar. 2016.
WHAT A DIFFERENCE A DECADE CAN MAKE. Back in 2006, Angela Davis remarked during a keynote address at the University of Wyoming honoring Martin Luther King Jr.’s birthday: “We have been basically persuaded that we should not talk about racism.” Following the acquittal of George Zimmerman in the shooting death of African-American teen Trayvon Martin in 2013, the activist movement Black Lives Matter was born. Since then the issue of racism has been front and center in American politics. What remains elusive, however, is why racism however motivated finds such fertile ground in the human psyche.
The Earth is flat
Kyrie Irving, who plays basketball brilliantly for the Cleveland Cavaliers, made headlines in February 2017 for declaring boldly that the Earth is flat. He was perhaps pulling our collective leg. His stated rationale, however, has more than a little bit of good old common sense to back it up:
“For what I’ve known for as many years and what I’ve come to believe, what I’ve been taught, is that the Earth is round,” he continued. “But if you really think about it from a landscape of the way we travel, the way we move, and the fact that—can you really think of us rotating around the sun and all planets aligned, rotating in specific dates, being perpendicular with what’s going on with these planets?”
Crazy thinking? Maybe, but then what about this? A poll published two years ago by the U.S. National Science Foundation found that 26% of Americans don’t know that despite appearances to the contrary, the Sun does not go around the Earth. Perhaps more astonishing, when asked, 52% of Americans evidently don’t agree with the statement that humans evolved from earlier animal species.
You don’t have to be Bill Nye or Neil deGrasse Tyson to see that all these instances of scientific ignorance make perfect sense from a common sense point of view despite being wrong. Furthermore, it wasn’t all that long ago most people on Earth in point of fact were misinformed in precisely these ways: yes, of course, the Earth is flat; yes, it is obvious that the Sun goes around the Earth; and haven’t you heard? Humans were created in their present form by a special act of Divine Will.
. . . and races are real
There are no polls I know of to back up the claim. Even so, it seems likely many people today—maybe even most—would also say they can’t possibly be at all racist because, don’t you know?, they don’t look down upon people in other races (see the dictionary definition reprinted above).
This argument may be socially honorable, but if this is what many truly believe, then there are people who need to hear that just like the idea that the Earth is flat, so too, the notion that human beings come in different kinds that can be labeled as “races” is just plain scientifically wrong.
As the anthropologist Jonathan Marks at the University of North Carolina – Charlotte and many others, too, have been saying for years, human genetic variation around the globe is real. But the same cannot be said for the commonsense claim that the Earth is peopled by separate and distinct human races.
As Marks has observed on numerous occasions, when we try to divide people up into different races, it’s not that we’re reading natural patterns of variation and simply extracting this idea from nature. Instead,
what we’re doing is we’re deciding that certain patterns of variation are less important than others, and certain patterns of variation are more important than others. We decide that the difference between a Norwegian and an Italian is not significant and so we’ll place them in the same category. And we decide that the difference between a Persian and a Somali is important; and so we’ll place them in different categories.
Sinner heal thyself
It is probably true that most human geneticists nowadays recognize that human beings don’t come in kinds—that is, races aren’t real. It is more than unfortunate, therefore, that geneticists today generally still don’t seem to know how to talk about human biological variation from place to place and down through time without using words—the term “population,” for example—that all too easily can mislead others less knowledgeable into believing science still endorses the old commonsense idea that human races exist in the real world to be embraced or savaged depending on one’s personal and moral proclivities.
No wonder, therefore, that dictionary definitions of racism (such as the one at the top of this commentary) can still make it sound like there is nothing wrong with the idea of race provided we don’t use this notion as an excuse for prejudice, discrimination, or antagonism.
The way forward
Most of us don’t believe the Earth is flat. Yet most of us live and act as if it were because this commonsense idea is a seemingly trivial lie that mostly works just fine in everyday life. Similarly, most of us may feel comfortable using the word race for the same reason. Truth be told, however, most of us also know the consequences of doing so can be deadly. Is there a way forward?
Here are 5 recommendations. They have been written specifically with geneticists in mind. But you don’t have to be a professional geneticist to add them to your own personal stock of “best practices.”
Avoid whenever possible using facile concepts and terms such as ancestry, migration, and admixture when writing about human diversity.
Abandon using the outdated concept of a “population,” and replace it with the statistician’s term “sample.”
Stop writing about the “population structure” of this or that species, and instead report on their “genetic structure” as a species.
Develop comparative databases documenting the genetic structure of other species to demonstrate publicly and repeatedly until the truth finally sinks in that geographic variation doesn’t have to be “racial” to be real.
Create mathematical tools and network algorithms to use when mapping, analyzing, and reporting on the genetic structure of a species that unlike current methods (e.g., the popular computer program Structure) are non-categorical.
How many immigrants does it take to make a migration?
When presented with a sample comprising only 3+1 skulls, both scientific caution and parsimony suggest you should assume that colonists coming ashore back at the beginning of human history in Vanuatu and Tonga were probably more diverse, biologically speaking, than is witnessed by these four—at least until there is further evidence showing they did indeed come not just from a genetically homogeneous place of origin, but also a place where the inhabitants were as sui generis as they appear to be vis-à-vis others on earth (Skoglund et al. 2016: fig. 1b).
Logic such as this is well worth attending to. But in this instance, there is an equally logical way to get around the usual working assumption that people are likely to be more diverse than first appearances may suggest. Given how poorly specified are the two hypotheses under scrutiny here, it is anyone’s guess how big we are supposed to think the boats must have been that brought early colonists to Vanuatu and Tonga around three thousand years ago. Even granting they may have arrived in more than one canoe, it would be reasonable to assume those arriving were fairly few in numbers. If so, then there is no need to assume blindly that those who came ashore in Vanuatu or Tonga constituted a representative (random) sample of the real human genetic diversity among those back home in the places where they came from, wherever on earth that was (Terrell 1986).
Furthermore, this is not all that might be reasonably assumed when trying to pin down the who, what, where, why, and when behind these four skulls. The number of pioneering colonists arriving in canoes from elsewhere with them or before them may not only have been relatively few. They may also have been kin, i.e., biologically related to one another. If so, then possibly what makes these crania look sui generis in comparison with other people on earth, living and dead, may just be that we are seeing a “family resemblance” in these human remains (Terrell 1986; Walker and Hill 2014).
For journalists and others, the real mystery of these remains, of course, is where these pioneers or their immediate forebears sailed from when they launched their boats to start a new life elsewhere. What is now known or can be reasonably assumed, therefore, about places to the west where they may have sailed from?
Until the Holocene stabilization of sea levels in the southwestern Pacific around 8,000–6,000 years ago, it is likely that much of the northern coastline of New Guinea was steep and uninviting of human settlement (as much of it still is today) except perhaps where favorable local circumstances may have at least temporarily trapped sediment in sandbars, coastal lagoons, and small river deltas. Little is currently known archaeologically about this coastline, which runs east-west for roughly 1,500 miles (2,400 km), and which would logically have been the most likely route between Asia and the farther reaches of the Pacific (Golitko et al. 2016). The best guess at the moment is that few people lived along this coast for the first 35,000–45,000 years of human history in the Pacific (Terrell 2006). In effect, earth and sea conspired to isolate New Guinea, like a sleeping giant, from frequent contact with islanders elsewhere both to the east in what is now popularly called Melanesia, and to the west in Island Southeast Asia (for biological support for this inference, see: Matisoo-Smith 2016: 391).
Following the Holocene stabilization of sea levels, however, coastal areas in Southeast Asia and the Pacific began to develop into rich floodplains, river deltas, and lagoons. By the mid Holocene, it is probable that people in the island realms to the east and west of New Guinea began to deal with one another back-and-forth more often as coastal people began to travel with greater reach along this immense island’s lengthy northern coastline (Torrence and Swadling 2008).
West meets East
Contrary to the notion that there are only two hypotheses about the prehistoric human settlement of the more remote islands in the Pacific east of New Guinea, there are numerous variants not only of those two old ideas but of others, too (for a recent review, see: Matisoo‐Smith 2016). Here we will introduce only one plausible reconstruction (Terrell, in press).
Initial baseline assumptions
Archaeologists now think people have been living in Southeast Asia for 50,000 years or so, and perhaps for not quite so long in the islands just east of New Guinea as far as Bougainville in the Northern Solomons.
The gradual flooding of the Sunda paleocontinent in what is now Southeast Asia since the Last Glacial Maximum ~21,000 years ago created extensive coastal environments that were ecologically rich and productive (Sathiamurthy and Voris 2006; Hanebuth et al. 2011: fig. 2). Similar extensive flooding did not occur in the area east of New Guinea labeled as Parkinson’s Islands (after the early ethnologist Richard Parkinson) on the map above (Lavery et al. 2016).
Due to this environmental advantage, it is probable that there were far more people living in Southeast Asia 6,000 years ago than there were in Parkinson’s Islands.
By the mid Holocene—contrary to the prevailing assumption in historical linguistics that doesn’t take this ecological advantage into consideration—it is probable that languages classifiable as Austronesian were widely spoken throughout Wallacea and elsewhere in Southeast Asia even as far north as Taiwan. But not yet in Parkinson’s Islands which had been isolated from Asia by the island of New Guinea.
Throughout the Late Pleistocene and early Holocene, Wallacea and Parkinson’s Islands were both areas of the Pacific where the advantages of travel by sea rather than by land nurtured the use of canoes and the development of local navigational methods and skills.
Canoes equipped with outriggers and sails were invented in Southeast Asia at some point in the Late Pleistocene or early Holocene. Simple dugout canoes remained the predominant boat type used for travel among coastal communities in Parkinson’s Islands.
Modeling the relocation of immigrants from Wallacea
A small Austronesian-speaking hamlet or village community left home for some particular local reason or reasons from somewhere in Wallacea—or possibly on the north coast of New Guinea—and made landfall in the Bismarck Archipelago.
It is possible that wherever it was they came ashore, they arrived not as strangers but rather as old friends of some of the local people there in the Bismarcks (Terrell 2015).
Among these immigrants were individuals skilled at pottery-making, and also skilled in the arts and rituals of building and sailing outrigger canoes with sails. Both of these technologies were new to the Bismarcks region. Moreover, such skills may not have arrived at the same time if travel back-and-forth between communities in Wallacea, northern New Guinea, and the Bismarcks became routine at least for awhile.
The local people not only welcomed them, but often also acquired new ways of doing things—such as the art of pottery-making—from their immigrant neighbors, in some instances even their foreign language skills. The reverse may have also been true.
Time passed, generations came and went. For now unknown reasons, it eventually became fashionable, prestigious, or perhaps even necessary for some people in the Bismarcks to set sail for islands yet farther to the south and east in the Pacific, although how many people in how many communities were involved, how often they sailed away, and for how many years this voyaging away from home in the Bismarcks went on are now all unknown and perhaps unknowable.
Even so, considering the passage of the time between (a) the first arrival of immigrants from the west and (b) the departure of some people generations later to settle down in other (more remote) places to the southeast, there is no reason to insist that these two separate episodes of human resettlement were similarly inspired or motivated (Walker and Hill 2014).
Furthermore, given that both the voyaging technology and navigational skills required to colonize the more remote islands of the Pacific may have been available then only in some communities in the Bismarcks, it is not surprising that early settlers in Vanuatu, Tonga, and elsewhere had similar material culture traits (i.e., the so-called “Lapita cultural complex”).
Because the first immigrants who reached Vanuatu and Tonga were entering a vast and uninhabited part of the Pacific, it is probably not surprising that many nowadays have been seduced by the modern global distribution of Austronesian languages—from Madagascar to Rapa Nui (Easter Island) and from New Zealand to Taiwan—into thinking that such a vast geographic compass could only be the historical product of some kind of massive human migration that was singularly intentional and singularly premeditated from the very moment the first Austronesian-speaking immigrant stepped into the first canoe to sail from somewhere in island Southeast Asia or on the north coast of New Guinea to the Bismarcks 3,000 and more years ago. It is wise to remember, therefore, that appearances can be deceiving.
Furthermore, today we know nothing about marriage (or sexual) practices in the Pacific in the prehistoric past. Although it is stating the case too simply, we do know that the basic building block of human genetic relatedness is the gene. Anyone who knows about the birds and the bees knows that genes can travel far and wide through sexual intercourse even if the people carrying them may only get as far away from home during their time on earth as the next village or two down the road. Consequently, there is no a priori reason to assume that race = language = culture. Or that genes necessarily traveled the Pacific millennia ago as the exclusive and enduring “property” of a massive and self-contained ethnic or ethnolinguistic migration that was able to keep its collective act together over thousands of miles and for hundreds, even thousands, of years. As some anthropologists like to say it, we need models of Pacific prehistory to work with that are “on the ground,” not “pie in the sky.”
Although we have been talking here almost exclusively about the Pacific Islands, the issue at stake is a global one. It is not just worrisome to find that even scientists may sometimes be unaware of the intellectual racism hidden in the conviction that the story of our species is a tale about ancestry, ancient migrations, and admixture. Commonsense ideas like these can be more than misleading. They can lend credence to other notions and old prejudices that can be harmful and sometimes deadly.
We thank Ethan Cochrane, Mark Golitko, Tyrone Lavery, Lisa Matisoo-Smith, and Robin Torrence for assistance in the preparation of this 3-part commentary.
Bellwood, Peter. 2011. Holocene population history in the Pacific region as a model for worldwide food producer dispersals. Current Anthropology 52: S363–S378.
Gibbons, Ann. 1994. Genes point to a new identity for Pacific pioneers. Science 263: 32–33, p. 32.
Gibbons, Ann. 2001. The peopling of the Pacific. Science 291: 1735–1737.
Golitko, Mark, Ethan E. Cochrane, Esther M. Schechter, and Jason Kariwiga. 2016. Archaeological and Palaeoenviromental Investigations Near Aitape, Northern Papua New Guinea, 2014. Journal of Pacific Archaeology 7: 139–150.
Green, Roger C. 2003. The Lapita horizon and traditions – signature for one set of oceanic migrations. In C. Sand (ed.), Pacific Archaeology: Assessments and Prospects. Le Cahiers de l’Archéologie en Nouvelle-Calédonie 15. Nouméa: Service de Musées et du Patrimoine de Nouvelle-Calédonie, pp. 95-120.
Hanebuth, Till JJ, Harold K. Voris, Yusuke Yokoyama, Yoshiki Saito, and Jun’ichi Okuno. 2011. Formation and fate of sedimentary depocentres on Southeast Asia’s Sunda Shelf over the past sea-level cycle and biogeographic implications. Earth-Science Reviews 104: 92-110.
Lavery, Tyrone H., Andrew D. Olds, Jennifer M. Seddon, and Luke K‐P. Leung. 2016. The mammals of northern Melanesia: speciation, ecology, and biogeography.” Mammal Review 46: 60–76.
Matisoo-Smith, Elizabeth A. 2016. Human biology and population histories in the Pacific–Is there such thing as a Lapita people?. In: The Routledge Handbook of Bioarchaeology in Southeast Asia and the Pacific Islands, edited by M. Oxenham and H. Buckley, pp. 389–408. Routledge, London.
Sathiamurthy, E. V. H. K., and Harold K. Voris. 2006. Maps of Holocene sea level transgression and submerged lakes on the Sunda Shelf. The Natural History Journal of Chulalongkorn University, Supplement 2: 1-43.