Prodigious calculators and the Maya long count

Prodigious calculators and the Maya long count Circulation /Draft Roger M. Blench Prodigious calculators and the Maya long count Roger Blench McDonald Institute for Archaeological Research University of Cambridge SECTIONS Introduction ..................................................................................................................................................... 1 Very large numbers in Ancient India............................................................................................................ 2 Large numbers ≠ Mathematics ...................................................................................................................... 3 Prodigious calculators..................................................................................................................................... 3 Savant skills, acquired or innate?.................................................................................................................. 4 Funes the memorious ...................................................................................................................................... 5 Do we know everything already? Plato’s Meno ............................................................................................ 5 Where does this leave us? ............................................................................................................................... 6 References ........................................................................................................................................................ 6 PHOTOS Photo 1. The Buddha at Tushita, depicted at Borobudur, Java ......................................................................... 2 Photo 2. Wall panel at Chiapas de Corzo.......................................................................................................... 2 Photo 3. The Popol Vuh, Manuscript, 1701 ...................................................................................................... 3 Photo 4. Jedediah Buxton.................................................................................................................................. 4 Photo 5. Dialogie between Plato and Philosophy ............................................................................................. 5 Introduction Numeration systems vary widely across traditional societies in the world. From reported societies in the Amazon with no numbers at all, to widespread spread numeration systems which count only to three, to the elaborate numerology of Ancient India which conceived of numbers larger than the atoms in the universe and the ‘long count’ of the Maya, systems very wildly. These latter are the most surprising, as there seems to be no obvious value in constructing them. The interpretation of the evolution of number systems is broadly functionalist. We devise the number systems we need. If there is no obvious requirement in the Amazon or New Guinea to count items in any detail, number systems hardly evolve. Once trade systems develop, numeration systems become more elaborate to manage exchanges, since counting goods is necessary to ascertain their value. Once large machines, big social groups and long distances need to be measured, very large numbers are required. There is clearly a problem of interpreting the higher number systems of traditional societies. What is the possible use of numbers larger than can be experienced? What is the need for a system which can assign meaning to billions or trillions except in recent physical sciences. Clearly such numbers are more part of philosophical systems or intellectual games than a practical development. The question then becomes, why such systems are elaborated? This chapter argues that part of the answer may lie in the phenomenon of savant calculators who somehow infiltrate their private fascinations into a cultural nexus. Records of savants who have exceptional powers of recall of each day of their lives, and individuals who can calculate very large numbers extremely rapidly only begin in the 18th century. Many of these were individuals with otherwise no scientific background. Presumably though, the existence of such individuals was not confined the short period in which they have been recorded. Hence it is probable that elaborated number systems may be the product of exceptional individuals who flourished in past societies where there are no records. Prodigious calculators and the Maya long count Circulation /Draft Roger M. Blench Very large numbers in Ancient India Numbers beyond any possible use were a subject of considerable speculation in Ancient India. There are Sanskrit names in Vedic literature for the powers of 10 up to a trillion and even 1062. As early as the 3rd or 2nd Century BC, Jain mathematicians recognized five different types of infinities: infinite in one or two directions, in area, infinite everywhere and perpetually infinite. The Lalitavistara Sutra (lit. ‘The Play in Full’) is a text from the 3rd century AD which recounts a contest at Tushita including writing, arithmetic, wrestling and archery (Foucaux 1892). Episodes of the Lalitavistara Sutra are represented on the friezes at Borobudur, the great 9th century Buddhist monument in Central Java and Photo 1 shows one scene of the Buddha at Tushita. Photo 1. The Buddha at Tushita, depicted at Borobudur, Java Source: CC The Buddha was pitted against the great mathematician Arjuna and demonstrated his numerical skills by citing the names of the powers of ten up to one tallakshana, (1053). He then explains that this is part of a series of counting systems that can be expanded geometrically. The last number at which he arrived after going through nine successive counting systems was 10421. Given that there are an estimated 1080 atoms in the whole universe, this is as close to infinity as any in the ancient world came. The text also describes a series of iterations in decreasing size, to demonstrate the size of an atom. The Mayan Long Count The Mayans, the dominant civilization in Mesoamerica for more than 2000 years, devised two calendrical systems, the Long and Short Count, based on a vigesimal Photo 2. Wall panel at (i.e. base 20) system (Edmonson 1976, 1988). The Maya situated the start date of Chiapas de Corzo their calendar on August 11, 3114 BC1. Mayan mythology recounts that on this day, Raised-up-Sky-Lord allowed three stones to be set by deitiess at LyingDown-Sky, First-Three-Stone-Place, which centred the cosmos and allowed the sky to be raised, revealing the sun (Freidel et al. 1993). The first record of the Mayan calendar is on a wall panel at Chiapas de Corzo in Mexico, which rose to prominence around 600 BC. Photo 2 shows Stela 2 (actually a wall panel), showing a date of 7.16.3.2.13, or December 36 BC, which is the earliest Mesoamerican Long Count calendar date yet found. However, Mayan numeration far exceeded the few thousand years required for their actual calendar. Its scale and the units by which it is reckoned are shown in Table 1, illustrating the vigesimal system. These are very large numbers, not quite on the Source: CC scale of the Indian numbers cited above, but well beyond any practical needs. 1 Archaeologists date the formation of Mayan civilisation, the Preclassic Period, from 2000 BC onwards. Prodigious calculators and the Maya long count Circulation /Draft Roger M. Blench Table 1. Mayan Long Count units Long Count Unit Period 1 Kʼin 1 Winal 20 Kʼin 1 Tun 18 Winal 1 Kʼatun 20 Tun 1 Bʼakʼtun 20 Kʼatun 1 Piktun 20 Bʼakʼtun 1 Kalabtun 20 Piktun 1 Kʼinchiltun 20 Kalabtun 1 Alautun 20 Kʼinchiltun Source: Adapted from CC source Days 1 20 360 7,200 144,000 2,880,000 57,600,000 1,152,000,000 23,040,000,000 Solar Years 1 20 394 7,885 157,704 3,154,071 63,081,429 The Maya calendar has inevitably been the subject of considerable New Photo 3. The Popol Vuh, Age speculation. In 2012 it was calculated that one of the 5126 year cycles Manuscript, 1701 of the Mesoamerican calendar would end on 21 December. It was widely held that this might herald the end of the world, or at the very least, major events including the transformation of humanity to a new age of spirituality. Maya specialists protested in vain that no such prophecy was part of the calendrical tradition. However, the concept of prior failed ages, and that we are living in a ‘fourth world’, can be found in the Popol Vuh (Photo 3) giving some credence to this idea. Evidently, the world did not end and indeed there is a conspicuous lack of spiritual transformation, but it is unlikely to be the fault of the Mayan calendar. Large numbers ≠ Mathematics There is no direct connection between the concepts of very large numbers and the sophistication of mathematical systems. As the usual histories of mathematics show, the Ancient Egyptians, the Babylonians, the Greeks and Source: CC the Chinese all had elaborate mathematical treatises. However, they did not develop numerical systems reaching such very large numbers. Hence the Indian and Mayan mathematicians embedded such numbers in their culture for other reasons. Prodigious calculators In the annals of neuroscience, there are rare individuals who have a grasp of very large numbers, first called ‘prodigious calculators’. Many, but not all of these, are on the Austistic Spectrum. These capacities of autistic individuals seem to be innate and were noted long before autism was identified as a specific disorder. There have been many reports of startling numerical skills, particularly calendar skills, such that they can rapidly calculate the day of the week for any date thousands of years in the past and future. Perhaps in some ancient cultures, such individuals would have influenced the priestly hierarchy to enshrine concepts of very large numbers in religious discourse. In other words, because numbers could be calculated, they should be calculated. One of the first calculators in Europe to be documented was Jedediah Buxton (1707–1772) who was born in a virtually innumerate household in Derbyshire (Photo 4). He was able to carry out enormously complex calculations, apparently rather slowly, such that the result might appear months after the task was set. In 1754 he walked to London to be tested by the Royal Society, who pronounced his abilities genuine. The caption to the engraving of his portrait reads; Jedediah Buxton, A poor Day Labourer: born at Elmton in Derbyshire: who without being able to write or cast Accounts in the Ordinary method: perform'd the longest Calculations and solv'd the most difficult Problems in Arithmetics, by the strength of his Memory; – neither Noise, nor Conversation cou'd interrupt him: he would either go on with his Calculations all the time or leave off in the midst and resume them again even though it should be Years afterwards. Prodigious calculators and the Maya long count Circulation /Draft Roger M. Blench Exactly where Jedediah Buxton would have been placed on the autism spectrum is now impossible to establish, as there have been impressive Photo 4. Jedediah Buxton calculators with more standard social skills. Nonetheless, the way his calculations seemed to continue in the background is very characteristic of this type of autism and far from the competitive world of metnal arithmetic. Prime numbers are notoriously impossible to predict by any algorithm, despite centuries of research. It is therefore all the more remarkable that some autistic savants seem to be able to find extremely large prime numbers by unknown methods. Oliver Sachs (1985), in a study of autistic twins with prodigious calculating skills, demonstrated that they were able to make proposals for large prime numbers extremely rapidly, and indeed used this skill as a sort of competitive game. Some of the numbers they gave were impossible to test at the time of the research but have been confirmed by subsequent tests. Even now it is unclear what mental mechanism they could have used, but the suspicion must be that they Source: New York Public systematically checked numbers individually, in a lowl-level long-running Library metnal process. Savant skills, acquired or innate? If there are individuals with these special aptitudes, are these acquired or innate? In other words, do we all have these capacities just waiting to be awakened, or are they specific to individuals with particular brain configurations? Case studies of the sudden development of exceptional abilities, known as Acquired Savant Syndrome (ASS), suggests strongly they are innate (Treffert 2009). ASS is applied to the unexpected appearance of remarkable new slills, typically in music, art or mathematics, in ordinary people. These can be the consequence of a head injury, stroke or other problems in the central nervous system. However, there are also plenty of examples where ASS appears with no prior neurological episode or following a chance external shock. In his book Musicophilia, Oliver Sacks (2009), records the case of Tony Cicoria, a surgeon who was struck by lightning in 1994. He had no prior interest in classical music, bur rapidly became an obsessive and skilled pianist following his recovery. A comparable striking case of acquired mathematical ability is the case of Jason Padgett, who became a talented mathematician after being struck on the head with an iron bar in 2002 (Padgett & Seaberg 2014). According to his narrative, following the attack Mr.Padgett first experienced symptoms of OCD and PTSD, and then began to develop visualisation of mathermatical theories he was unable to name. A chance meeting enabled him to begin to apply standard terminology to his understanding. One of the earliest cases to be exhaustively investigated was Solomon Shereshevsky, a stage memory-artist (mnemonist). The Russian neuropsychologist Alexander Luria worked with Shereshevsky for three decades and describes his skills in his book, The Mind of a Mnemonist (Luria 1987). Apart from his mnemonist skills, Shereshevsky was also a synaesthete, and could control bodily functions, such as his pulse, through an act of will. As with other individuals with overwhelming recall, he found the abundance of memory confusing, and sometimes resorted to writing things down on paper and burning it. The ashes of the paper then symbolised his desire to forget them. These cases suggest that these capacities are present in all human beings, but are only realised in exceptional circumstances, or among autistic individuals for whom the advantages are counterbalanced by their difficulties in other areas. There is probably a good reason they are suppressed in most people. Too much recall tends to obscure or defocus attention on current life. Jill Champion, the mnemonist described in Price & Davis (2008), can describe what she was doing on every day of her life in considerable detail. However, she characterises the ability as a burden, since she is constantly oppressed by visions of the past in parallel to experiencing everyday life. The technical term for this condition is hyperthymesia (Baron-Cohen et al. 2009), now known as highly superior autobiographical memory (HSAM) (LePort et al. 2016)), the ability to recall one's past day-by-day, Prodigious calculators and the Maya long count Circulation /Draft Roger M. Blench has been confirmed to exist by some neuroscientists (Parker et al. 2006). Despite these examples, some researchers have questioned the existence of a truly eidetic (photographic) memory because recall only contains personally relevant autobiographical information. Marvin Minsky (1998: 153) dismissed sucb claims as ‘unfounded myths’ fit only for stage magicians. In some ways this was a curious throwaway line since the most exhaustively studied mnemonist, Solomon Shereshevsky (see above) was a stage performer, but his gifts were evidently real. Intutively, it might seem that forgetting is simply a process of erosion and degradation, that memories gradually fade. However, recent research, admittedly with fruit flies, suggesting that forgetting is an active process, in some ways the mirror image of forming memories (Berry et al. 2012). The neurotransmitter dopamine is important to remembering, but apparently it also actively regulates the forgetting of new memories. In creating new memories, the dCA1 receptor was activated, whereas forgetting was initiated by the activation of the DAMB receptor. This argues that forgetting is as crucial a process as recall in synthesising the tangled data of experience into a useable strategy. These cases remain rare and their explanation remains opaque. It may be that some savants operate by directly accessing low-level, less-processed information stored in all human brains. We take in far more than we can remember and process and for obvious reasons, our brains are highly selective, only bringing summaries of the incoming data to our attention. Individuals are highly variable in what they ‘notice’. If you consciously observe everything you see, it will stifle action and creative proceses. Sherlock Holmes, ticking off Watson as usual, famously observed ‘You see, but you do not observe’ in A Scandal in Bohemia (1891). In fact there is an extremely good reason for this. If we access brain activities that are not normally available to conscious awareness we miss out on the illogic of normal social interaction. Funes the memorious Documentation of savants is relatively recent, but we must suppose they existed before the syndrome was recognised. One of the most striking literary representations of savant syndrome is a short story by Borges (1899–1986), Funes the Memorious first published in La Nación in June 1942. It later appeared in the 1944 anthology Ficciones. The narrator, a version of Borges, encounters an illiterate farm boy with a prodigious memory, who can recall via a system of ‘enumeration’. This is described as a system of associating individual items with objects so that he can recall them individually, but which gives him no ability to generalise. Funes claims to have invented a system which gives every numeral up to at least 24,000 its own arbitrary name. Borges left no record of whether this was based on a personal encounter, but subsequent cases in the psychological literature are so similar that it may well have been (Quiroga 2010). Funes resembles an autistic savant, in that he has acquired an extraordinary ability, memory, without the obvious need for study or practice (Verberne 1976). The story raises the unresolved question of how much unfulfilled potential the human brain truly contains. Do we know everything already? Plato’s Meno The neurological evidence suggests that all these capacities are Photo 5. Dialogie between Plato latent. In principle, we can all develop savant skills, become and Philosophy prodigious calculators and memory marvels. If we do not, it is because most of these skills, impressive as they seem, impede everyday life. The argument for innate knowledge which just has to be recalled first appears in one of one of Plato’s dialogues, the Meno (probably written 385 BC although dated in the text in 402 BC). Woods (2022) represents a modern translation and Photo 5 a Dialogue between Plato and Philosophy, from a Rhineland manuscript, ca. 1230 AD. The Meno is framed as a dialogue between Meno, a young man visting from Thessaly and student of a prominent Sophist, Gorgias, and Socrates (Day 1994). Meno begins by asking Socrates Source: CC whether virtue is taught, acquired by practice, or comes by nature. Part of Socrates’ answer is to invoke the Prodigious calculators and the Maya long count Circulation /Draft Roger M. Blench theory of knowledge as recollection, anamnesis. The basis of his theory is that souls are immortal and know all things in a disembodied state, hence they only have to recollect them. To demonstrate the concept, Socrates makes use of one of Meno’s slaves to solve a problem in geometry by questioning him, prompting the slave to recall his subconscious knowledge. Despite a successful demonstration, at the end of the dialogue, Socrates’ argument seems to collapse as he attributes virtue to divine inspiration rather than the recollection of a buried disposition common to all human beings. The supernatural substructure of the Socrates’ arguments will scarcely appeal to the modern reader, especially as Plato makes him argue to opposite case in another dialogue, Protagoras. Nonetheless, the sense that everything is all somehow lying there, waiting to be recalled is too common an experience to be wholly denied. Most people, I assume, do have the strange experience of recalling events or scenes from many years ago, sometimes in apparent great detail. Our brains in principle have the storage capacity to keep this material until we die. But the Meno is rather dishonest because the slave solves a geometric problem, a skill rather different than simple recall. Where does this leave us? The data on world numeration systems establishes that although there is a very general correlation between their elaboration and their practical function, in some cultures the concepts of very large numbers developed with no use except for their role in philosophy and expansive calendrical calculations. Although the records are relatively recent, there are records of prodigious calculators, individuals with remarkable memory recall who are able to perform mathematical operations beyond the skills of ordinary individuals. Sometimes these capacities develop following brain injuries, but in other examples it appears to be innate. This is particularly striking in those who may even be illiterate or who are on the autism spectrum. It is really unclear what the evolutionary value of such skills might be; it has been suggested it is possible to connect to usually inaccessible lower-level processing in the brain. What then happens in cultures with a fascination for large numbers is that an individual with these capacities enters into an influential position, perhaps as a high priest. His own fascination with these elaborate calculations becomes reified in the culture, enshrined in sacred texts. The next generation of priests, having been shown the way are able to persist with these speculations, irrespective of their own mathematical skills. References Baron-Cohen, S, Ashwin, E, Ashwin, C, Tavassoli, T, Chakrabarti, B. 2009. Talent in autism: hypersystemizing, hyper-attention to detail and sensory hypersensitivity. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences. 364 (1522): 1377– 83. doi:10.1098/rstb.2008.0337. Berry, Jacob A., Isaac Cervantes-Sandoval, Eric P. 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