{"id":12145,"date":"2023-11-18T22:43:55","date_gmt":"2023-11-18T22:43:55","guid":{"rendered":"https:\/\/mathscitech.org\/articles\/?p=12145"},"modified":"2026-04-18T22:51:31","modified_gmt":"2026-04-18T21:51:31","slug":"mathematics-prehistory-2","status":"publish","type":"post","link":"https:\/\/mathscitech.org\/articles\/mathematics-prehistory-2","title":{"rendered":"The Prehistoric Origins of Mathematics"},"content":{"rendered":"


\n3rd ed. Aug 2023 (expanded appendices). 2nd ed. Nov 2019 (revised to include advances in linguistics, genomics, interpretive theory, and Mesopotamian mathematics); 1st ed. (Dec 29, 2009)<\/em><\/p>\n

Part 1 in Ancient Mathematics series. (Part 2: The Mathematics of Uruk and Susa 3500-3000 BCE<\/a>, Part 3: Exploring Cuneiform Culture 8500-2500 BCE<\/a>)<\/em><\/p>\n

Abstract<\/strong>
\nHow far back in time can we trace mathematical understanding and mathematical practice? When did humans acquire the neurological circuitry for the cognitive and linguistic capabilities on which mathematics depends? Advances in multiple disciplines over the past 30 years have fundamentally changed what we know about our past and about the biological capacity for, and cultural impulses behind, cognitive precision (language, number sense, cultural transmission). Exploring these questions will take us on a journey across archaeology, Assyriology, artifact analysis (close reading theory), anthropology, genomics, linguistics, neurobiology, and animal cognition.<\/p>\n

\"\"<\/a>

The Anthropology and Archaeology of Conceptual Thought leading to the Birth of Mathematics<\/p><\/div>\n

<\/p>\n

We proceed backwards<\/em> in time from c.3000 BCE, at the dawn of written mathematics (archaic bookkeeping) in Sumeria (modern S.Iraq). From here we will move to the prehistoric evidence of practitioner geometry in the cultures of the late Neolithic as evidenced in the layout of permanent houses, granaries and temples (c.5000 BCE in Ubaid culture), and in geometric pottery designs (c.6000 BCE, Halaf and Samarran cultures). Further back, we see the appearance of plain tokens from c.7000-8000 BCE, the same plain tokens that we know were used for counting by herders and which were instrumental in the invention of writing for book-keeping purposes within the temple institutions running centralized economic control in the urban city-states. Looking beyond the Near East, in Paeleolithic\/Neolithic Europe and Britain there is evidence for monolithic monuments c.4000-7000 BCE oriented toward midwinter and midsummer solstice that suggest an awareness of the periodicity of the solar and lunar cycles, and the relation of the solar cycle to the seasons. <\/p>\n

Before 10,000 BCE marking the start of the warming period (holocene) that began after the retreat of the 4th glacial period, the density of artifactual evidence is insufficient to draw firm conclusions: there are fewer than 5 isolated finds of artifacts between 70,000-18,000 BCE (one find per 10,000 years), with contested interpretations. So we switch to indirect evidence (genetic, anthropologic, and linguistic) to establish the capability of symbolic thought in anatomically modern humans (H. Sapiens)<\/a> from c.200,000 BCE onwards. Here culture becomes a critical factor, under-scored by the fascinating example of the modern Piraha tribe in Brazil that have broken previous assumptions about the inevitability of symbolic thought in anatomically modern humans. The Piraha are the only known tribe\/people whose language and culture appear not to have progressed beyond an analog notion of magnitude similar to that of higher animals, skipping entirely the granular linguistic numeracy present in every other known language, primitive or modern. Why? It appears to be cultural: the Piraha reject the value of future planning and are completely non-materialistic. This leads to an interesting philosophical observation: quantitative mathematics initially develops within a culture that values planning and material control.<\/p>\n

From anatomically modern humans c.200,000, we jump backwards to c.2.4 million years ago and consider the capacity for conceptual thinking implicit in the tool-making capability of early hominids. We look at C.S. Peirce’s “semiotic model” (index, icon, symbol)<\/a> of conceptual and linguistic development, and conclude that bladed tool-making (Lokalalei site evidence) required at least stage 2 or stage 3 conceptual development. Having gone back as far as we can with the capabilities of humans and hominids, we consider the origin of number sense in humans, animals, birds, and reptiles, and trace back the neurological circuitry supporting an analogue number sense to a latest common ancestor (LCA), a stem reptile that would have existed some 260 million years ago.<\/p>\n

A set of Appendices<\/a> provide additional color on: <\/p>\n

    \n
  1. Appendix 1<\/a>: Dialectic nature of arithmetic (and mathematics),\n
  2. Appendix 2<\/a>: Invention of writing and the advancement of book-keeping\n
  3. Appendix 3<\/a>: Birth of the universe up to the early period of life on earth (Pre-Cambrian Eon)\n
  4. Appendix 4<\/a>: Acceleration of living diversity (Cambrian explosion period) to the dawn of humanity (Pliocene epoch)<\/a>\n
  5. Appendix 5<\/a>: Paleolithic (Stone Age) Culture from Lomweki (3.3mya) to Shanidar (c.50kya)\n
  6. Appendix 6<\/a>: The Mesolithic, the last glacial maximum (26kya) to settled life 10,000 BCE)\n
  7. Appendix 7<\/a>: Timeline for domestication of animals.\n
  8. Appendix 7b: Foraging for Food – What the wild landscapes might have held for ancient humans (and still today)\n
  9. Appendix 8<\/a>: Near Eastern Cultural History: from pre-Pottery Neolithic (c7500 BCE) to city states Uruk period (c4000 BCE)\n<\/ol>\n

    A list of recommended readings, most of which can be downloaded freely, is provided in the Bibliography<\/a>.<\/p>\n

    \n

    This is a long paper (62pp) with many images and tables.
    \nYou may find it easier to     download the article as a PDF<\/a> for offline reading\/printing.<\/strong><\/em><\/span><\/p>\n\n

    <\/a><\/p>\n

    1. Evidence from the dawn of written mathematics (c.3000 BCE): accounting with clay tokens.<\/h3>\n

    By 3,200 BCE (5200 years ago) there is indubitable evidence for mathematical practice within the sophisticated cultural context of Neolithic Sumerian city states with a strong centralized control of production resources and economic activity through temple-statal administration. This led to the breakthrough advancement of proto-writing: scribes used the clay tokens typically kept in “bullae” (clay envelopes) to impress upon flattened clay tablets to create the earliest known system of accounting, or book-keeping. In this context, the token combined quantity and commodity, so correct interpretation required knowing the context of the transaction. Within 50 years, there was a further advance: pictographic signs that could specify commodity separately from quantity. Over the next 500 years, temple and state control of economic planning and supply chain management grew more extensive and more ambitious, developing syllabic writing, standardizing traditional metrologies (the same signs could still take on different values depending on metrological context), and improving arithmetic technologies (the emergence of the sexagesimal system, reciprocal tables, other aids to calculation\/computation\/solving problems).
    \n<\/em><\/p>\n

    The strongest archaeological evidence of mathematical practice dates to at least 3,200 BCE (5200 years ago) in ancient near eastern city-states. Archaeological finds in the past century have shown that geometrical clay tokens which appear to have been used for counting and measuring across the region, became established at this time in the Sumerian city-state of Uruk (southern Mesopotamia\/Iraq) as the standard administrative procedure for recording commercial transactions (archaic book-keeping). Similar finds have been made in Elamite Susa (Zagros mountains\/western Iran), a rival city-state to Uruk. [NissenDE\/1993<\/a>], [Friberg\/1984<\/a>].<\/p>\n

    <\/a>
    \n

    \"\"<\/a>

    Near East toward the end of the Ubaid period (4300 BCE), before the earliest discovery of accounting. Notice the many city-states that had arisen in the alluvial flood plains between the Tigris and Euphrates rivers. Uruk and Susa would rise as leading city-states in Sumer and Elam respectively. The blue markers identify what would be considered the three religious centres in the following millenia. (Source: Wikipedia<\/a>)<\/p><\/div><\/p>\n

    Key to this conclusion were the finds by Denise Schmandt-Besserat of tokens enclosed in clay “bullae”, or sealed clay envelopes, with matching token-impressions on the clay surface. ([Besserat\/1977<\/a>], [Oppenheim\/1959<\/a>]) The impressed indentations made by pushing the tokens into the wet clay are also the earliest examples of proto-writing. [Damerow\/1999w<\/a>]<\/p>\n

    \"Clay

    Clay tokens and bulla (envelope). Note the impressions of the tokens on the surface of the bulla (Source: Besserat\/1977<\/a>, SMS 1, p53)<\/p><\/div>\n

    \"Bulla

    Bulla with 48 tokens found at Nuzi site dated from 2000-1500 BCE after cuneiform writing was fully developed, with an inscription describing the meaning of the tokens inside. (Source: Oppenheim\/1959<\/a>)<\/p><\/div>\n

    \"Clay

    Clay tokens mapped to the pictographs and numerical values assigned to them. (Source: Besserat\/1977<\/a>]<\/p><\/div>\n

    The meaning of these impressions can be worked out from the historical sequence of clay tablets that initially record token-impressions only with no additional written context, to later the juxtaposition of number signs with additional signs indicating the commodity (e.g. 3 sheep), and finally the use of separate cuneiform<\/a> signs for number and for commodity. [Nissen\/1986<\/a>], [NissenDE\/1993<\/a>], [Robson\/2000<\/a>].<\/p>\n

    There is evidence of elaborate systems of metrology (measurement) that linked the tokens variously to different length, area, volume, weight, and time units, in nested factors of 2,3,6, and 10. The decipherment of these metrologies was based on painstaking studies of hundreds of archaic tablets with numbers matched to those in cuneiform tablets using the associated cuneiform symbols (see below on the number system). [Powell\/1971], [Nissen\/1986], [Nissen\/1993], [Englund\/2004] The situation with ancient Sumerian metrologies was similar to
    \n    customary measures in medieval Europe<\/a>, see also Mathematics of Uruk and Susa<\/a>)<\/p>\n

    What must be remembered is that the mathematical and metrological understanding that is captured in the earliest tablets in 3200 BCE was pre-existing<\/em> and hence pre-dated 3200 BCE, before writing. It was the technology of book-keeping through writing that was the invention of the time (see the next section).<\/strong> <\/p>\n

    \"\"<\/a>

    Early metrology (counting & measurement) used separate systems depending on the commodity being measured. (Source: Nissen\/1993<\/a>, pp.28-29, Englund\/2004<\/a>, pp.32-33)<\/p><\/div>\n

    \"\"<\/a>

    SZE system for measuring grain capacity in units of 1 sila3 (bowl). Notice the fractions measured in shekels with 180 shekels in a gin, and 360 1\/2 shekels.<\/p><\/div>\n

    How far back did the use of plain tokens for counting and measurement go?
    \nWe have clear evidence for their use c.3200 BCE for administrative purposes associated with temple management of Ubaid period economy controlling surpluses and labor. Before 3,200 BCE, while plain tokens are find in many sites, they are without sufficient context (e.g. the bullae with imprints) to conclude definitively that they were used for counting and measurement. Thus we can provide only a date range for the start of plain token use for counting and measurement, from 8,000 BCE to 3,200 BCE (see [    Niemi\/2016<\/a>: 33-34], and [Bennison\/2018: 20-22]).<\/p>\n

    Archaic Tablet Texts<\/strong>
    \nFrom 3,200 BCE onwards, there is increasing archaeological record of clay tablets [    Friberg\/1984<\/a>]. The anthropological and sociological work of Nissen, Damerow, Englund, Hoyrup, Robson, and several others, have led since the 1980s to an understanding of how the temple economy evolved the scribal-statal system built around written accounting practice [Hoyrup\/1991<\/a>]. Mathematically, this proceeded over 1000 years (i.e. from 3200 BCE to 2300 BCE) in various stages: (1) an initial stage in which quantity and commodity were combined in a single token\/impression; (2) adding pictographic symbols for representing the commodity for which the number provided the quantity; (3) logographic cuneiform (writing with the wedge-end of a reed) in which the symbol represented the full word or idea, giving no indication of the pronounciation; (4) syllabic representation of spoken language using the same cuneiform symbols enabling Sumerian and Akkadian scribes to record concepts and literary ideas as well as numerical transactions. [Nissen\/1986<\/a>], [Robson\/2008<\/a>]<\/p>\n

    \"The

    The complexity of administrative tablets and notation evolved from quantity\/commodity combined (commodity implicit) to quantity and commodity indicated with separate signs. (Source: Nissen\/1986<\/a>)<\/p><\/div>\n

    \"The

    The evolution of writing from pictographs\/logographs to cuneiform, as the type of stylus changed. (Source: Nissen\/1986<\/a>)<\/p><\/div>\n

    \"Style

    Style of writing evolved from pictographic to cuneiform as the stylus changed. (Source: Nissen\/1986<\/a>)<\/p><\/div>\n

    \"(left)

    (left) simple clay tablet with token impressions for quantity (commodity is presumably implicitly known by context). The shape of the token impressions led to the early metrological signs. (middle & right) complex numerical tablets showing quantity and commodity separately. (Source: Damerow\/1999w<\/a>)<\/p><\/div>\n

    <\/a>
    \nSumerian written and spoken numerals<\/strong><\/p>\n

    The cuneiform representation of the Sumerian\/Akkadian\/Babylonian number system is ADDITIVE, and in this way has the same sense of cumbersomeness common to all additive systems (e.g. Roman numerals). This is because of the shortage of symbols. The Sumerians had two symbols: a symbol for 1 (vertical wedge) and a symbol for 10 (horizontal wedge nail end only). All numbers up to 60 (their base) were written by accumulations of these symbols. The digits 1 to 9 were expressed by writing the requisite number of 1’s, either consecutively or bundled together in groups of three with one group on top of another. Similarly the ‘digits’ 10, 20, …, 50 were expressed by the requisite number of 10’s, etc. The symbol for 60 was the same symbol as for 1, interpreted by context or, in a far-reaching innovation, BY POSITION. Thus the number 147 would be represented by two symbols of 60, two of 10, and seven of 1. The digit 0 was not used and presumed to be understood from the context, although in the later Babylonian period it was denoted by a wedge. [Roy, 2003]<\/p>\n

    \"\"<\/a>

    Cuneiform representation of Sumerian numbers<\/p><\/div>\n

    How were the numbers spoken? (A * next to an s renders it as ‘sh’ in pronounciation). 1-10 were: Dis*, min, es*, limmu, ia, as*, imin, ussu, ilimmu, u. Then higher number formation in a similar structure as with many languages that take 10 as base. u-dis*, u-min, etc.
    \nThe spoken numbers show a fascinating linguistic pattern, with some trace of base 5 (the numbers 6 through 9 are named as 5+1,…,5+4, though not consistently), strongly base 10 structure (the numbers 11-19 are 10+1,…,10+9), some trace of base 20 (40 is nimin, or 2×20), and mixing (50 is 2×20+10). Finally the sexagesimal unit 60 is reached (ges) and the pattern repeats.
    \nThere is an ambiguity in the verbalizing of numbers higher than 60 (gesh). Is gesh-u 70 (=60+10) or 600 (=60×10)? Apparently both, with the amount resolved in context. Compare the linguistc structure of languages from Inuit (base 20), western languages (nominally base 10 but with various inconsistencies in formulation) [Gullberg, 1997, 7-60], and east Asian languages (base 10 with clean structure). [Takasughi, 1996]<\/p>\n

    <\/a>
    \n

    \"\"<\/a>

    Sumerian number words and counting patterns<\/p><\/div><\/p>\n

    The deciphering of cuneiform languages: Sumerian, Akkadian, Babylonian, and Old Persian<\/h4>\n

    How do we know what these signs mean? Our understanding of cuneiform is relatively recent, with pioneering deciphering work beginning from 1838 (CE) onwards [Friberg\/1984<\/a>]. But our understanding of Mesopotamian mathematics is burdened by how we came to this understanding. Before 1850, the major investigators were Edward Hincks<\/a>, Henry Rawlinson, and Jules Oppert, who collectively deciphered the language working off the tri-lingual Behistun Inscription (see section below). From the 1850s onward, the focus was on deciphering specific tablet finds. The first set of tablets discovered pre-1945 were very different from the ones from 1945-1951 (Susa tablets). The investigators of the first set of tablets were either philologists (language studyiers, Abraham Sachs, Goetze) or mathematicians (Neugebauer, Thureau-Dangin) but rarely both. Neugebauer was the closest to both, and he, as well as others, fell victim to using modern symbolism, and interpreting Babylonian discussions in an modern, even pre-modern context. By the time the next set of data had come in, there had been several decades and multiple secondary sources telling the story that Babylonian mathematics was the antecedant of the Greek mathematics, and that Babylon was the origin of the stream that fed the rest of Western Mathematics. Thus, in terms of what is known about Babylonian mathematics, one should disregard all that was thought to be known up to the 80s, and certainly the received knowledge that is in textbooks and histories even through the 2010s. The new knowledge began in the 1970s from a revised look by Marvin Powell and Joran Friberg at existing works, and by Denise Schmandt-Besserat at archaeological findings. All three showed that Mesopotamian mathematics needed to be seen in historical development. [Hoyrup\/1991b: 27]. The new knowledge from the 1980s is not only from new sources, corrected transliterations, and better translation\/interpretation (e.g. Hoyrup’s conformal translations), but also new information about cultural context (scribal roles, social\/economic\/political\/military trends, use of clay tokens, and location of finds — i.e. exercise books, teacher training (e.g. Susa tablets), scrapheap (exams\/discards), problem sets with abbreviated solutions for instructors. These insights were contributed by Hans Nissen, Robert Englund, Peter Damerow, Jens Hoyrup, Joran Friberg, and Marvin Powell, who, amongst others, were part of the Berlin Workshop on Concept Development in Babylonian Mathematics. [Hoyrup\/1991b]<\/p>\n

    \"Earliest

    Earliest attestations of writing, beginning with proto-cuneiform c.3200 BCE (Source: Damerow\/1999<\/a>)<\/p><\/div>\n

    Behistun: from Herodotus to Old Persian to Babylonian Akkadian<\/strong>
    \nThe trilingual     Behistun Inscription<\/a> is to cuneiform writing what the rosetta stone (discovered 1799) is for the understanding of Egyptian hieroglyphics. The Inscription was engraved into the face of sheer cliffs near the ancient crossroads of Behistun (Kermanshah province of Iran) by the Achaemenid (Persian) king Darius the Great (Darius I) in 550 BCE. Its message proclaiming his conquest of all the lands and his right to rule, was intended for the entire Near East, and so was written in the three major cuneiform languages of his day: Old Persian (his own language), Elamite (the language of Susa), and Babylonian Akkadian (the semetic language understood across Assyria and Mesopotamia). All three languages had died even by 400 BCE and over the millenia fanciful suggestions were put forth as to what the inscription signified. Additional background on the Behistun Inscription (2017)<\/a><\/p>\n

    In 1802, Grotefend had deciphered ten of the 37 symbols of Old Persion. Sir Henry Rawlinson started in 1835 using Grotefend’s efforts. He found the first part of the Inscription contained the same list of Persian kings as given in Herodotus (400 BCE) but in their Persian forms. By 1838 Rawlinson had succeeded, in part due to the fact that Old Persian used an efficient syllabic representation of 37 characters. In 1844 and 1847 he studied the Babylonian section. Edwin Norris, a colleague, completed the study of the Elamite section by 1855. By 1855 [Rawlinson\/1855<\/a>] and Norris with a few others (Hincks 1854) had deciphered all three cuneiform sections: Old Persian (37 characters), Elamite (131 characters) and Babylonian (500 characters, more than 10x the number for syllabic Old Persian). The decipherment of Akkadian and Sumerian (Cathcart, 2011 paper<\/a>)<\/p>\n

    \"Behistun

    Trilingual Behistun Inscription carved into sheer cliffs in the Kermanshah province of Iran, engraved c.550 BCE by the Achaemenid king Darius the Great.<\/p><\/div>\n

    \"The

    The trilingual inscription at Behistun commissioned by Darius the Great in 550 BCE with messages written in the three cuneiform languages of the time: Old Persian, Elamite, and Babylonian (Source: LW King & RC Thompson, 1907<\/a><\/p><\/div>\n

    From Old Persian to Babylonian to Akkadian to Sumerian<\/strong>
    \nBehistun provided the key to Babylonian through     Old Persian<\/a>, which is accessible through Middle and Modern Persian (Farsi). Babylonian (and Assyrian) Akkadian are derivative dialects of an older Semetic language Akkadian<\/a>. Their decipherment was completed by Hincks, Rawlinson, and Oppert in the mid 1800s, and from 1956 through 2011, the 26-volume Akkadian dictionary was compiled by University of Chicago (freely available online). To get to Sumerian<\/a> we needed to rely on Sumero-Akkadian bilingual texts, and fortunately there are many, primarily the sign lists written by the early scribes that lived during the time of the transition from Sumerian city states to the Akkadian Empire, after Sargon unified the Sumerian city states under his rule. History of Akkadian (2003)<\/a><\/p>\n

    Deciphering proto-cuneiform (pictographs) from Sumerian cuneiform<\/h4>\n

    Deciphering an unknown but syllabic written language is hard. Deciphering the meaning of pictographs is harder still. To get to the meaning of the proto-literate writings it has taken the efforts of the Berlin group, a cross-disciplinary group of researchers who have used computer aided digitization of dozens of fragments to complete the work begun in 1936 by Adam Falkenstein who first published the Archaic Texts of Uruk (ATU).<\/p>\n

    Nissen 1986<\/a> (p.317) explains: even in 1936 it was recognized that a few texts were lists. Later more lists were discovered. And it was noticed that these proto-cuneiform lists corresponded word for word, position for position with the same lists almost 600 years later which were written in Sumerian cuneiform which by this time we did understand.<\/p>\n

    \"Deciphering

    Deciphering pictographs. Here: NINDA (bread, ration) and GU7 (to eat, distribute ration) (Source: Damerow\/1999<\/a>)<\/p><\/div>\n

    \"Interpreting

    Interpreting pictographs in early accounting texts. This tablet, formerly from the Erlenmayer Collection appears to have been part of the administrative archive of a production unit concerned with the distribution of beer and the ingredients used in beer brewing (unprocessed grain emmer and barley, malt, coarse-ground barley groats). The primary administrative activity in archaic Mesopotamia was of grain storage and distribution, and these are by far have the greatest number of accounts in Uruk. (Englund\/2001<\/a>,p.3) (Source: Tablet MSVO 3, 02<\/a> (3 columns). Interpretation<\/a>. Publictions: Nissen\/1993<\/a> frontspiece, p.42, Englund\/1998<\/a>)<\/p><\/div>\n

    Let us now leave 3,200 BCE, the dawn of writing and move back further into the previous period, when the mathematics was developed that the scribes of Uruk and Susa would later capture.<\/p>\n

    <\/a><\/p>\n

    2. Mathematical practice in the transition between Neolithic to Chalcolithic (Ubaid period): evidence from 8,000 BCE<\/h3>\n

    Tokens of the kind associated with the start of tablet accounting are found in Neolithic settlements across the Near East dating back through 8000-7000 BCE. Between 6,000 BCE and 4,000 BCE (8000 to 6,000 years ago) there is evidence of (1) painted pottery showing elaborate designs using sophisticated symmetries, and (2) layout of prestige buildings (eg temples, shrines) showing architectural competency in geometric design (rectangles with proper corners suggesting knowledge of 3:4:5 or 5:12:13 ratios) and use of moulded bricks (standardized dimensions per site, but not across time).<\/em><\/p>\n

    Anthropological context<\/strong>
    \nThe formation of settled society occurred from 12,000 to 10,000 BCE, with evidence for the deliberate cultivation of crops occurring c.9,000 BCE. This coincides with the time after the last ice-age receded from the Near East (c.12,000 BCE). Early Neolithic settlements were small, without walls, whose residents combined cultivation of crops and management of domestic livestock (primarily sheep and goats) within a largely egalitarian social structure. [Charvat\/2002]<\/p>\n

    \"The

    The neolithic began during the warming period (holecene, c.10,000 BCE) that occurred after the retreat of the 4th glacial ice age ca 12,000 years ago.<\/p><\/div>\n

    By c.6000 BCE, we see a clear shift into the transitional Neolithic-Chalcolithic Ubaid period culture, with larger settlement sizes, semi-permanent dwellings, further specialization in crafts, and emerging evidence of hierarchical social status. [Charvat\/2002<\/a>] The results were quite remarkable and are part of the documented acceleration in Neolithic cultural sophistication. [Charvat\/2002] (See Appendix 6<\/a> for more details on what life was like then.)<\/p>\n

    This was a time of practitioner level mathematical knowledge, what Hoyrup describes as sub-scientific, learned “on the job”, in terms of procedures, in apprenticeship arangements. [Hoyrup\/1988], [Hoyrup\/1989], [Hoyrup\/1994]. Evidence for sub-scientific, practitioner level mathematical understanding can be found in the artifacts of Neolithic life: designs in pottery showing geometric regularity and the exploration of geometric patterns; building layouts showing an understanding of form, symmetry, composition; an understanding of seasonal regularity and calendarized activities: migration, planting, harvesting, all of which required reasonable mastery of the solar calendar (without which seasonal regularity is impossible); and number, which is required in cooperative behaviour: equitable distribution gains from hunt or harvest, planning for the retention of sufficient seed for sowing next season’s crop, and trade\/exchange across increasingly longer distances. All of these have socio-anthropological-archaeological evidence in the period between 8000-4000 BCE. We may thus pull backwards the date of the development of mathematical understanding to this period from c.9,000 to 6,000 BCE, i.e. from the period of the deliberate cultivation of crops and management of small livestock (sheep, goas) to the period of sophisticated Neolithic practitioner technology within larger permanent settlements with longer distance trade and hierarchical organization.<\/p>\n

    Let’s look at each:<\/p>\n

    (1) painted pottery dating from 6,000-4,000 BCE show designs that use complex mathematical symmetries, and rotational frieze patterns, providing evidence for strong geometrical stylisation [RobsonSelin\/2000<\/a>].<\/p>\n

    Hassuna culture: painted and applique designs on pottery from the Yarimtepe I site (in Iran):<\/p>\n

    \"Painted

    Painted pottery from Hassuna culture Yarimtepe I site dated c6500-5000 BCE. (Source: Charvat\/2002<\/a> p.23)<\/p><\/div>\n

    Samarran culture: Pottery from Samarra from 6000 BCE-4000 BCE show confidence in geometrical form:<\/p>\n

    \"Samarran

    Samarran pottery, 6000 BCE-4000 BCE (Source: Charvat\/2002<\/a>, p.35))<\/p><\/div>\n

    In the photo below of the Samarra Bowl (c.4000 BCE), we see:<\/p>\n

    “Four stylised herons catch fish in their mouths while eight fish circle round them. An outer band of stepped lines moves outwards, countering the swirling effect of the animal figures.” [RobsonSelin\/2000<\/a>]<\/p><\/blockquote>\n

    \"Samarra

    Samarra Bowl (Pergamon Museum, Berlin), 4,000 BCE.
    Painted pottery during the Ubaid period showing strong geometric stylisation (Source:     Wikipedia<\/a>, RobsonSelin\/2000<\/a>). Note Robson’s image appears reversed from the Wikipedia .photo, making the flow clockwise instead of counter-clockwise<\/p><\/div>\n

    (2) analysis of nine successive temple layouts<\/a> at Eridu (first Sumerian city<\/a> mentioned in the King List) from Temples XVII c.5750 BCE through Temple VI, and comparison to other Ubaid period sites (6,500 BCE – 3800 BCE) show an architectural discipline in which prestige and communical buildings began to be laid out with increasing sophistication resulting in the use of modules with dimensions suggesting the use of a standardized length measure (Ubaid cubit of 0.72cm) as well as knowledge of side ratios of right triangles (3:4:5, and 5:12:13). [Kubba\/1990<\/a>], [Forest\/1991<\/a>] makes similar findings at Tell Hammam et-Turkman, Soudipour\/2007<\/a>.<\/p>\n

    \"Temple

    Temple layers at Eridu, built one on top of another over the course of 3000 years (5570 BCE through Ur III dynasty 2000 BCE) Source: Soudipour\/2007<\/a>, p.100<\/p><\/div>\n

    \"Nine

    Nine floorplans from Eridu temples c.5570 BCE onward, showing increasing architectural sophistication until Temple VI with a distinctly mathematical layout (see Kubba 1990). Source: Soudipour\/2007<\/a>, p.101-102<\/p><\/div>\n

    In addition to geometry, there appears to be some significant master builder experience involved even in the choice of orientation of the temple layout to maximze sunlight. From the earliest temple site (Temple XVII), all the buildings have a fixed orientation with corners at compass points N,E,S,W, creating a northwest-southeast axis.<\/p>\n

    “The fact that buildings were built in such a way that their corners were made to face the central axis indicates an excellent knowledge of climatic matters. When the corners of the building face the north-south axis, the four sides of the building receive maximum sunlight, the northeastern and south eastern wall receive the sunlight until midday and the northwestern and the southwestern walls receive the sunlight after midday. Thus, all four sides of the building receive sunlight daily\u201d (Youkana, 1997: 63)”<\/p><\/blockquote>\n

    Why have complex temple building activities not yielded written mathematical evidence?<\/strong>
    \nEssentially, it is due to what could be considered guilds in the early society. The guild of master builders had their own domain knowledge. The guild of temple administrators (SANGA) and chief administrators of a city (EN) had their own domain knowledge, essentially that of a quartermaster (senior individual supervising stores and distributing supplies and equipment) crossed with bookkeeper\/accountant (recording of financial transations, including purchases, sales, receipts, payments). The invention of written mathematics was in the guild of the quatermaster\/bookkeeper in the context of running an increasingly complex temple economy. The builders had no such practical pressure\/problem for which written mathematics was needed.
    \n.
    \nJens Hoyrup explains: “Temple building must have involved a fair measure of practical geometrical knowledge, but evidence from later times suggests that this knowledge was the posession of master builders and did not communicate with the mathematics of the literate managers.” (    Hoyrup\/2011<\/a> p.4)<\/p>\n

    Sophisticated naive geometrical knowledge and its associated geometrical algebra appears more or less fully clothed in scribal mathematics at the end of the Ur III period in the northern periphery of the Ur III empire (Eshnunna, c.2030 BCE) leading eventually to the sophisticated Old Babylonian mathematics that Neugebauer was able to decipher. [Hoyrup\/1985], [Hoyrup\/1990], [Hoyrup\/1993], [Hoyrup\/1996], [Hoyrup\/2002]. The traditional practitioner, guild-level knowledge of geometry understood by the master builders and field surveyors (rope stretchers) appears to have remained unchanged through to Islamic mathematics when al-Khwarizmi documented it as the science of the al-jebr guild, in his masterwork “Algebra”, through which this knowledge made its way to medieval Europe, persisting still in its naive geometrical form until the 1500s (Pacioli), finally dying in the work of Viete who transformed algebra into a symbolical instead of rhetorical discipline in the 1590s. [Hoyrup\/1994] When was it discovered? Again, we can give no firm date, but we can give a range, from 6,000 BCE during the extensive temple building and agricultural organization of larger settlements during the Ubaid period, through to 2,030 BCE when it first appears in the written record.<\/p>\n

    The interaction between culture and mathematical development<\/h4>\n

    By the end of the Ubaid and at the start of the Uruk period, settlements were for the first time able to generate significant food surpluses through centralized management of labor resources directed to building large-scale irrigation systems to improve food output. The resulting economic and social changes, the transition from settlements to city-states, the rise of an established urban elite, and the shift to a redestributive, centrally controlled temple-economy drove the use of tokens as accounting devices, as we have seen above. [Charvat\/2002<\/a>], [Niemi\/2016<\/a>]<\/p>\n

    In this and the previous period, what we have seen is that mathematical practice has arisen within a social context. It has been developed (invented?) and perfected within that social context for an application with a social purpose (accounting, recording of commercial transactions, state management of labor and food surpluses, design of prestige buildings, or the manufacture of status goods). Through its use, mathematics then affects and in many cases transforms the culture it arises within, and shifts it in new directions, which in turn affect the direction of further mathematical practice. See [Hoyrup\/1991<\/a>] and [Robson\/2008<\/a>] for examples of how the culture of the scribal schools varied from Uruk to Akkad to Ur to Hammurabi<\/a>‘s Babylon to the fall of Babylon, a period covering 1500 years.<\/p>\n

    By this time, the essential concepts for mathematics are present in the archaeological evidence: counting (using tokens for tracking quantities of sheep, goats, male, female, kids), keeping time (migration of animals being hunted, transhumance behaviour, right times for sowing and reaping), shape and symmetry in craft (symmetry present in the bifacial working of Mode 2 stone tools in which both sides are worked to improve the quality of the edge and to produce a blade, symmetry and shape in reed weaving, symmetry and shape in pottery, and in the decorative patterns that adorn it), practical matters of building and measuring (lengths of poles, doorways, size of sleeping areas, circular mud dwellings). Progress into settled life has been through the desire to exert control over subsistence security and to improve the material quality of life. In this context, experience has been gained on how to do things efficiently, and on the underlying methods for this control. Sufficient agricultural surplus allow practitioners to specialize and refine their craft and develop the technologies they use. All these practitioner level knowledge are attested to in the archaeological records of the Neolithic Near East. <\/p>\n

    <\/a><\/p>\n

    3. Exploring tthe Paleolith: limitations of direct archaeological evidence, and a look at controversial claims dating mathematical practice to 70,000 BCE.<\/h3>\n

    Before 10,000 BCE, there are a few isolated finds with controversial mathematical or calendrical interpretations, but nothing convincing. For example, we exclude the Ishango, Lebombo, and Wolf bones, and exclude also the engraved ochre from Blombos Cave. The argument for their mathematical nature (Marschak) is based on close reading of their markings and association with tallies, prime number groupings, or calendric tabulation. But the notches on the bones (for example) could have non-mathematical hypotheses, e.g. to improve their grip for use as a tool or weapon. These finds do not pass Newton’s test (1713) against speculation: “Hypothesis non fingo”<\/a> meaning “[I am certain if] I need feign no hypotheses!” [Walsh\/2010].<\/em><\/p>\n

    Can we find direct evidence of mathematical practice in the Paleolithic before humanity became settled?<\/p>\n

    There are two problems with older archaeological evidence: the first problem is that many materials that may have been part of mathematical practice are bio-degradable and would not have survived (e.g. markings on sand with a stick, tallies on wood). Those that could survive (pebbles, bones) lack any cultural context to confirm mathematical usage. For example: notches on bones could suggest tallying, arithmetic, an understanding of prime numbers, or pre-historic calendar cf. [Marshack\/1971]. But they could equally well be explained by non-mathematical intent, e.g. to improve the grip of the object used as a hammer or club [Elkins\/1996].<\/p>\n

    The second problem is that the paeleolithic finds are isolated geographically (Ishango bone in Uganda, Lebombo bone in Swaziland, and Wolf bone in the Czech Republic) and in time (dated between 18,000 and 35,000 BCE). There is little to no archaeological context of the finds that would suggest mathematical intention, which therefore relies entirely on interpretation of marks which remain tenuous and highly controversial (cf. Claim [Marschack\/1971] and rebuttal [Elkins\/1996]; Claim [Huylebrouke\/1996], and rebuttal [Keller\/2010]). <\/p>\n

    Similar problems beset the interpretation of an engraved red ochre lump from S.Africa dated to c.70,000 BCE. Suggestions of geometric decoration are hard to conclude without repetition or other context. They could also have been attempts at cleaning the point of a blade, or use as a cutting board, or scrapings to release coloured powder from the ochre for pigment dye.<\/p>\n

    \"Notched

    Notched bone (Ishango) and engraved red ochre (Blombos cave). There is no context that indicates whether the markings have meaning.<\/p><\/div>\n

    These are the circumstances surrounding all paleolithic artifacts discovered so far to which a mathematical culture has been ascribed. We simply do not have enough archaeological context on why or what they were carved for in order to interpret them. Unfortunately, speculative interpretations have made their way into news media and non-specialist literature covering ethno-mathematics and, regrettably, even into textbooks on mathematics history. The interpretations have ranged from lunar calendars and fertility tallies, to multiplication tables and prime number lists. As an example, textbook historian David Burton follows Marshack and represents a current enthusiastic popularization when he writes of the Ishango bone: “It had been used for reckoning time “in sequences of numbers that agree with the number of days included in successive phases of the moon.” [Burton\/1982] [Burton\/1982], [Huyle\/1996], [PletserHuyle\/1999].<\/p>\n

    More critical recent scholarship has drawn important cautions: [Elkins\/1996] takes apart Marshack’s microscopic readings of notched bone and highlights the repeated unjustified leaps in going from evidence to conclusion. [Keller\/2010] summarizes:<\/p>\n

    “The siren song of mathematical illusion is never far away when it comes to prehistoric artifacts. A notch may be nothing more than a mark [unless] one is obsessed with arithmetic [in which case the sign joins] the common denominator of all the ethnographic artifacts of this kind [showing] item by item [bijective] symmetry between objects and signs. Faced with the raw artifacts of prehistory, it is impossible to know … whether the markings are decorative or not, and if they are not, whether we are dealing with an artificial memory system.” (Keller\/2010)<\/p><\/blockquote>\n

    <\/a><\/p>\n

    4. Re-examining the Paleolithic for indirect evidence of symbolical capability in humans from 315,000 years ago<\/h3>\n

    We have seen that all the conceptual precursors for mathematics are directly present in the archaeological record by 6,000 BCE (end of Section 2 above). How far back can we trace them? We have seen in Section 3 that we don’t have high enough artifact density to categorically assert their presence. So we turn now to consider the indirect evidence which we do have to support the claim that capability for mathematical thinking<\/a> (number, shape, time, change) were present in the Paleolithic culture<\/a> of anatomically modern humans, i.e. from 315kya.<\/em><\/p>\n

    As early as 230kya, the archaeological record (Omo 1 site in the Ethiopian side of the Rift Valley)<\/a> shows changes in human species as anatomically modern humans (H. sapiens) diverged from Homo Erectus. New finds in Morocco (Jebel Irhoud site), push this date back to 315kya<\/a> (though some contest whether the latest finds are H. Sapiens). Evidence of complex behaviour (ritual burial of the dead, cooperative hunting, the controlled use of fire, language capability) suggests the capacity<\/em> for symbolical thinking that would be a prerequisite for any sort of mathematical practice (counting, bijection, keeping a tally, measuring, symmetry, or abstract artistic design).<\/em><\/p>\n

    In light of the previous section, we do not currently have direct conclusive evidence of mathematical practice from before 10,000 BCE. But if we modify the question to inquire when humanity developed the symbolic capability to support numeracy, then we can go back further to 315,000 years ago (note this is the last 9% of human presence on earth, which stretches back 3.5 million years).<\/p>\n

    Archaeological evidence shows that intelligence, communication, and social living stretch back to 315,000 years ago (Middle Pleistocene), when humans had already evolved into what is essentially their modern form, Homo Sapiens, and were using speech, tools, fire for warmth and cooking, were hunting large adult animals, and had diversified into all of the major races. By the time of the fourth glacial advance 100,000 years ago (Upper Pleistocene), anatomically modern humans (H. Sapiens) dressed and sewed skins, were able to live beyond the frost line, had a culture of arts and crafts and a ceremonial society that buried the dead and showed solicitude to the aged and maimed. (See Appendix 4<\/a> for details of life in the paleolithic to the start of the neolithic.) Presumably, then, there would already have been utility in comparing, for example, the number of men in a hostile encampment with those in the home group, and in communicating this numerical information for group action. Similarly, a builder or toolmaker needing material for a particular purpose would have needed to specify dimensions, even if roughly. An elder needing to know how long a hunting party had been absent before setting off to investigate would have needed to mark time.<\/p>\n

    Until early in the current century, the prevailing opinion was that humankind developed symbolic capability between 50,000 year ago and 315,000, coincident with the emergence of anatomically modern humans (H. Sapiens). This was based on:
    \n(1) the discovery of the earliest human art (cave paintings, jewelry\/decorative power),
    \n(2) anthropological evidence of ritual burial of the dead,
    \n(3) anthropological evidence of cooperative hunting which presumes the ability to communicate intentionally and with precision,
    \n(4) the practice of language, indicated by earliest presence of the human version of the FOXP2 gene which regulates learning and complex speech, combined with the assumption that (a) the ability to speak implies that speech and language occurred, and (b) that any language, no matter how primitive, must be symbolic and include at least a rudimentary number concept (e.g. one-two-many, or even one-many). The correctness of this last assumption was justified by the evidence of all known primitive languages encountered before the 1970s) (cf. [Conant\/1897], [Smith\/Ginsburg\/1937] and [Gullberg\/1997])<\/p>\n

    These views have changed in the past 20 years following extensive analysis and study of the Piraha people of the Brazilian Amazon discovered by Western sociologists and anthropologists in the 1970s, whose language surprisingly has no numerical concepts at all [Piraha\/2006]. The Piraha (both the people and their language) provide observational evidence that there can exist a state of being in which symbolic capability is present but numerical capability in language and culture does not result. [C.Everett\/2016] [DL.Everett\/2018]<\/p>\n

    Language and the Number Concept<\/h4>\n

    Speech had previously been viewed as a proxy indicator of numeracy, since before the Piraha every language and culture previously known had numeric concepts. [Conant\/1896]: “We know of no language in which the suggestion of number does not appear, and we must admit that the words which give expression to the number sense would be among the early words to be formed in any language.” The unusual language and culture of the Piraha people has no numbers, not even the “one”-“two”-“many” pattern found in other primitive languages. They become the first known counter-example, in the process changing our view of what language is and how it may have evolved. [SG\/1937], [Piraha\/2006], [Piraha\/2007], [Gordon\/2004], [FEFG\/2008], [EM\/2012] <\/p>\n

    Studies of the Piraha suggest that numerical capability appears to require three things: (1) the capability for symbolic thought (e.g. grasping the notion of bijection, which underpins discrete comparison); (2) a mechanism to keep the count (e.g. fingers, marks\/notches, pebbles, or linguistic counting words), and, taken for granted before the Piraha, and, most importantly perhaps, (3) a culture that assigns value<\/em> to planning, forethought, and material acquisition, all of which are supported by numeracy. The Piraha culture rejects planning, forethought, and is non-materialistic to the extreme, resulting in placing no value for number in their culture. As a result, not only have they not developed any mechanisms for counting, but they actively resist the learning and retention of these mechanisms when they are introduced to them, despite being able learners of other things [Gordon\/2004], [Frank, DL.Everett\/2008], [C.Everett\/2016].<\/p>\n

    This places the development of mathematical practice within cultural context once the fundamental neurological ability for symbolical thought exists. While one may indeed grasp the notion of bijection, without a mechanism to keep a precise tally one cannot actually count, only match. How the tally itself is made is less important and can take many forms: visually by using fingers of the hands or creating marks or notches, physically by collecting pebbles or other tokens or calculi, or linguistically using by words and\/or signs. But without valuing the act\/outcome of counting\/accounting\/planning, the Piraha example shows that humans essentially fall back on what appears to be a biologically innate analogue number sense that is also present in animals, birds,1<\/a><\/sup> and even some reptiles, but which decreases in precision as magnitudes get larger. This is why animal counting degrades quickly beyond four or five. [Gordon\/2004] [Everett\/2012], [Dehaene\/1997] Note the 2024 results in carrion crows<\/a> (May 2024, Scientific American).<\/p>\n

    \"Experiments

    Experiments on Piraha numeracy (Source: Gordon\/2004); (right) Piraha homeland (Source: Der Spiegal\/2006)<\/p><\/div>\n

    The evidence for speech<\/h4>\n

    What evidence exists for speech? Genomic investigations into speech defects have identified the FOXP2 gene<\/a> as a critical link to and enabling factor of speech control. Absence leads to non-viability, reduction leads to significant vocal disability. While the FOXP2 gene is expressed in birds, mice, primates, and humans, the human variation is different from all the others. The modern functioning version has been present in humans between 120,000-260,000 years ago, either the last common ancestor of Neanderthals and Homo sapiens, or specific to Homo sapiens. And so we form the basis of the argument: the capability for complex vocalization means the ability to realize speech and language. From language comes symbolism. Within a culture that valued planning, control, and materialism, the number concept can develop. All of the pieces for were therefore in place by 230,000 years ago (FOXP2 gene by 260,000 years ago, evidence of social\/cooperative living by 315,000 years ago). (Origin of speech<\/a>) <\/p>\n

    <\/a><\/p>\n

    5. Paleo-anthropological evidence from 2.3 million years ago and the semiotic model of human conceptual development<\/h3>\n

    The earliest known stone tool fossils are from the Lomekwi3 in W. Turkana (Kenya) dating back 3.3 mya. These stone tools are mostly single strike flakes taken from large cores that were rotated before striking. Their dating makes them interesting as there were no Homo species at the time, and it is suspected that perhaps Kenyanthropus are the knappers. [Harmand\/2015] However, the most interesting find is the next one, artifacts from Lokalalei, Kenya (in the same Turkana region) from 2.3mya (1 million years later) which show the emergence of sophisticated stone knapping techniques among the early hominids there. [Delagnes,Roche\/2005] By now it could be Homo Erectus, or continue to be Kenyanthropus or AAustralopithecus. If these were the simple split-stone variety (one strike, one split, use the edges that result), it would not be a surprise since the simpler Oldowan stone tool culture dates already from 2.6mya. What is surprising is that these stone tools from Lokalalei were made using the complex multi-strike techniques for forming blades from a carefully selected blank flint core using a sequence of strikes to create a razor sharp tapered edge. This technique requires considerable experience with how stone shatters as well as advance planning requiring to visualize how the sequence will work to create the tapered edges.<\/p>\n

    Except for the Lokalalei site, such tools are only found in the fossil record from 1.7mya onward (Acheulean culture), some 600,000 years later. <\/p>\n

    Looking at C.S Peirce’s semiotic model for conceptual and linguistic development (see below), we have in the Lokalalei stone knapping process two indications of early hominids having reached Stage 3 symbolic behaviour: the considerable planning requirements to shape the blade, and the cultural transmission (teaching) of the technique. This provides a terminus ante quem<\/a> (latest date) of 2.3 million years ago for abstract symbolic thought.
    \n<\/em><\/p>\n

    <\/a>Using C.S. Peirce’s semiotic progression (index, icon, symbol) for evaluating linguistic and conceptual development.<\/strong><\/p>\n

    C.S. Peirce’s semiotic model posits that conceptual and linguistic development pass through 3 stages: physical\/index, associated icon, and abstract symbolic (cf. [Everett\/2017]). This makes it a useful model for empirically situating any given activity and placing it within the 3 sequential states. <\/p>\n

    \"Semiotic

    Semiotic Progression model of language acquisition, following C.S. Peirce (Source: Everett DL, 2016, p.18)<\/p><\/div>\n

    Index conditioning (Stage 1) is the ability of creatures with a nervous system to perceive an “index” (physical stimulus) and produce an appropriate response (e.g. recoil from a hiss, be wary of yellow and black insects, recognize footprints or smells). This capability gives intelligent animals and humans have the ability to recognize and respond to sounds (bell, word, clap, sound of water suggesting presence of water) or visual cues (hand signal, position of ears, baring of teeth, etc.) or any of the other senses. Both the perception of stimulus and the pathways for response are biological and neurological. This is what allows a variety of animals, birds, and reptiles to possess a number sense and to perceive shape, time, and change (the cognitive precursors of mathematics). Memory, adaptation, and trained learning are forms of index conditioning. <\/p>\n

    Icon communication (Stage 2) involves the intentional use of signs (“icons”) chosen because of their close association with the intended physical meaning (e.g. smoke for fire, a figurine for motherhood, a stick drawing for a person, or a footprint or smell for the creature that caused it). Stage 2 is the understanding and use of “icons” which are associations intentionally chosen to represent physical phenomena (e.g. picture of cow, picture of fire, emojis, etc.). The majority of animals have not been found to be able to reach stage 2, with the exception of some primates, but even when they do show the ability to understand icons, they do not show the ability to take these learnings back to their communication with each other<\/a>.<\/p>\n

    Symbolic communication (Stage 3) involves the intentional selection of arbitrary signs whose meaning is established by cultural convention (e.g. male\/female signs, traffic light colors, arbitrary gestures, tallies, arithmetic signs, numerals, etc.). Stage 3 is the use of abstract symbols, i.e. signs that are purely arbitrary and require establishment by cultural convention in order to interpret. Examples include symbols such as $ or \u00a3 or traffic light green for go\/red for stop, a heart sign for love, sign language, alphabet, logograms, words\/names, and NUMBERS. 2<\/a><\/sup><\/p>\n

    Over the past 20 years, the fieldwork approach to linguistics has challenged the Chomskian theory of language acquisition in early humans. The importance of the Piraha to theories about prehistoric language development and numeracy is that they provide field evidence that anatomically modern humans have not reached Stage 3 in the semiotic progress, remaining at Stage 2, apparently by cultural choice. <\/p>\n

    Inferring human capability from the fossil record<\/strong><\/p>\n

    The making of effective bladed stone tools for cutting and scraping requires the ability to think abstractly and to conceptualize and foresee the consequence of a certain way of striking the stone to create a certain kind of fracture. When done expertly, the resulting blades are sharper than razors, sharper than surgical knives. Indeed, in the modern era of medicine, before super thin metal blades could be produced, surgical knives were indeed made from expertly knapped flint.<\/p>\n

    Stone tool-making among early humans is considered to be an aspect of transmitted culture based on the ability to consistently produce such artefacts through broad geographic regions and through time. The first of these cultures, the Oldowan c.2.6 mya produced simple split stones, but already this was enough to show that communal sharing of knowledge had developed in early hominins to enable this method of production to continue unbroken for the next 900,000 years (till 1.7mya). At this point, it was replaced by the improved Acheulean tool-making method of the late Homo Erectus period (1.7 mya). Evidence from [Delagnes,Roche\/2005] have shown that similar exceptional stone knapping capability to produce bladed tools were present at the Lokalalei site 2.3 million years ago. This kind of complex stone knapping was highly dangerous as evidence from modern lithic workers attempting to reconstruct the old ways have found out<\/a> (sharp fragments flying off at high velocity, with the makers wearing no gloves, no shoes, no protective eyewear, no trousers).<\/p>\n

    The archaeological data and evidence of sophisticated bladed stone tool creation at Lokalalei provides two arguments for humankind reaching Stage 3 symbolic capability at least by 2.3 million years ago, almost 14x earlier than the earliest Chomskian estimates: the capability itself, and the ability to transmit that knowledge. <\/p>\n

    \"Stone

    Stone Tool Cultures of early Hominids: Oldowan (2.6mya) and Acheulean (1.6mya) (Source: EverettDL\/2016)<\/p><\/div>\n

    <\/a><\/p>\n

    6. Culture transmission and the cognitive precursors for mathematics in animals: back to 260 million years ago<\/h3>\n

    How early does culture transmission in animals manifest itself? When does the analog perception of number appear in animals? Around 260 million years ago, neurological and biological evolution had progressed to the last common ancestor of birds and mammals, a reptile that shared the brain circuitry of both and which underpins the index\/response mechanism that gives an analog number sense (small numbers) to animals, birds, and monitor lizards (reptiles).<\/em><\/p>\n

    Culture transmission and iconic or symbolic association are necessary conditions for mathematical understanding. Cultural transmission in order to teach the understanding and use of complex ideas, tools, or technology. Iconic\/symbolic capability to be able to recognize and work with quantity, form, and the perception of change. <\/p>\n

    The presence of culture (socially transferred knowledge) has been observed in chimpanzees (Oct 2009 study<\/a>) who share tools and teach tool use (Oct 2016 study<\/a>).<\/p>\n

    <\/a>
    \nAdult members of the Piraha tribe use what appears to be a biologically innate analogue number sense that is also present in animals, birds, and even some reptiles. The precision of this number sense decreases as magnitudes get larger, and explains why animal counting accuracy degrades quickly beyond four or five, cf. [Everett\/2012], [Dehaene\/1997]. <\/p>\n

    Do number sense (but perhaps not measurement or counting per se) and the perception of shape and change (but perhaps not their description or communication) occur outside human species? Investigations have found evidence of number sense in animals (birds, dogs, monkeys, dolphins). Perception of the passage of time, the ability to distinguish one from many (in particular, quantities other than two), and the ability to distinguish shapes from each other, have all been documented in various animals. [Koehler\/1950] <\/p>\n

    “A man was anxious to shoot a crow. To deceive this suspicious bird, the plan was hit upon of sending two men to the watchhouse, one of whom passed on, while the other remained; but the crow counted and kept her distance. The next day three went, and again she perceived that only two retired. In fine, it was found necessary to send five or six men to the watch house to put her out in her calculation. The crow, thinking that this number of men had passed by, lost no time in returning.’ From this he inferred that crows could count up to four.” John Lubbock, _Nature_, Vol. XXXIII. p. 45., from [Conant\/1896<\/a>].<\/p><\/blockquote>\n

    “A nightingale which was said to count up to three. Every day he gave it three mealworms, one at a time. When it had finished one it returned for another, but after the third it knew that the feast was over….” Lichtenberg, _Nature_, Vol. XXXIII. p. 45., from [Conant\/1896<\/a>].<\/p><\/blockquote>\n

    “Dinah, my spaniel, … was overlooking half a dozen of her new-born puppies, which had been removed two or three times from her, and her anxiety was excessive, as she tried to find out if they were all present, or if any were still missing. She kept puzzling and running her eyes over them backwards and forwards, but could not satisfy herself. She evidently had a vague notion of counting, but the figure was too large for her brain.” Galton, _Nature_, Vol. XXXIII. p. 45, from [Conant\/1896<\/a>].<\/p><\/blockquote>\n

    The semiotic perspective highlights that perception and response to indexes (unintentional physical associations) is common to all living things that can think (sense their environment and choose a response). <\/p>\n

    If we ask when this analogue number sense may have developed in animals, this takes us back much further. Primates go back to 13 million years ago, birds to 150 million years ago, mammals to 220 million years ago, taking us back to the last common ancestor of birds and mammals having the same brain structure, which would have been a stem reptile c.260m years ago<\/a>. A research paper in 2024 on crows that count<\/a> concludes a 320 million date for a common ancestor, and presents evidence that this common ancestor would not have had the capability that the crows have, providing a limitation on the level of counting sophistication in the evolutionary progression.<\/p>\n

    <\/a>
    \nDating the capability for mathematical cognition then becomes a question of the timeline of intelligent, perceptive life itself. The intelligent tree-dwelling primates of 13 million years ago likely had the mental capacity for cognition of the precursors of mathematics. Are animals able to progress from the lowest rung (indexes) of Peirce’s 3-step evolution to the next rung (icons)? Experiments have shown that animals can proceed from icon to correct decision (this is learning and cataloguing new indexes e.g. Pavlov’s dog, trained monkey, crow), the challenge not yet demonstrated (as far as I am aware) is of an animal taking the stimulus, and picking the right descriptive icon, i.e. classification. Similarly, I am not aware of a non-human animal intentionally adopting a completely arbitrary symbol or sign, whose interpreted meanings need to be established as part of a cultural convention, unless we take animal language to be such an example.<\/p>\n

    Can we pin an upper time limit to the existence of a brain capable of perceiving number, shape, change, time and responding? Studies have shown that an analog number sense (recognition of numbers smaller than 6 in an analog\/imperfect way) exists in mammals (dogs, monkeys), birds (parrots, crows) and even reptiles (monitor lizards, [Pianka\/King 2004, Murphy 2019]). Neurological studies have established that in mammals, it is the mammalian neocortex (evolved 220 mya) that is the seat of complex cognitive functions such as sensory perception, spatial reasoning, learning and memory, decision making, motor control, and conceptual thinking. In birds, it is the DVR (dorsal vernicular ridge) that provides neocortical-like functioning. Both the neocortex and the DVR have been found to develop out of the same region in the embryonic brain. This points back to the neurological circuitry of a common ancestor of mammals, birds, and monitor lizards, i.e. a stem reptile (amniote)<\/a> existing some 260 million years ago.<\/p>\n

    \"Timeline

    Timeline of Life on Earth. First birds at 150mya. Reptiles as the last common ancestor of birds and animals c.260mya (Source: Wikipedia)<\/p><\/div>\n

    <\/a><\/p>\n

    7. Conclusions<\/h3>\n

    How far back do we have evidence for mathematical practice? What about the cognitive, social, and cultural aspects needed for its cognitive precursors?<\/p>\n

    We have seen in this article that:<\/p>\n

    Direct evidence for mathematical knowledge exists from c.6,000 BCE or 8,000 years ago. <\/p>\n

    Humans developed the capability for abstract thought around 2.3 million ago based on ability to create bladed stone tools requiring multiple precise strikes to a flint core, Lokalalei site evidence. The application of bladed stone tools drove innovation through to 315,000 years ago, by which point the last of the major evolutionary changes leading to anatomically modern humans was complete.<\/p>\n

    The intrinsic ability to perceive number, size, shape, time, and change trace back beyond humans themselves and into mammals and birds, back some 260 million years ago, to the last common ancestor of mammals and birds. <\/p>\n

    If we tell the story in the correct order, it looks like this:<\/p>\n

      \n
    1. c.220m years ago the mammalian neocortex, and by 150m years ago the dorsal vernicular ridge (DVR) in birds had evolved, and these are the neurological seats of cognitive recognition the index\/response mechanism that underlies the analog number sense (for small numbers) that is documented in mammals and birds. The number sense documented in monitor lizards (reptiles) would push the date further back to a reptilean common ancestor of mammals and birds, a stem reptile, c.260m years ago, sharing the neurological circuitry common to both<\/a>. In birds, their brains developed further developing what appears to be magnetoreceptors in bird retinas that are sensitive enough to transmit changes in orientation vs. earth’s magnetic field<\/a> directly to the brain, essentially working like a neurally integrated compass, and allowing long distance migration<\/li>\n
    2. The extinction event for land dinosaurs which occurred at 66 million years ago (mya) touched off a rapid cooling off period in global temperatures, present in geologic evidence. The disappearance of the dinosaurs led to the proliferation of mammals<\/a> into the ecological niches vacated by the dinosaurs. Mammals it turns out, had existed since c. 200 mya<\/a>, but had remained small, mostly nocturnal, and either tree-dwelling or burrowing, to avoid competition with the dominant dinosaurs. From 34 mya to 23 mya the Earth transitioned from a tropical world to modern ecosystems. From 23 mya to 2.6 mya the cooling continued. Primates were already living in trees by 13 million years ago, and hominids had branched off between 7.5 to 5.6 mya. At 2.6 mya, the four ice ages began (Pleistocene period) with the last glacial retreat occurring around 12,000 years ago (12 kya) and the inter-glacial warming period (holocene) beginning 10 kya. [Coon\/1996]. See Timeline (PDF) of Early Human Life, from 55mya to 5kya (Tom Conklin, 2009)<\/a>\n
    3. Around 2.3m years ago, we see in the fossil record (Lokalalei) the first appearance of technologically complex stone knapping tool-making to create blades by early human species, that provide evidence of the progression of human-kind from index\/response (shared with animals) to iconic\/symbolic thinking.<\/li>\n
    4. As early as 315,000 years ago, we see in the archaeological record (Omo 1<\/a>), changes in human species as anatomically modern humans emerge, and we have evidence of complex symbolic behaviour, which in principle could support numeracy.<\/li>\n
    5. Around 6,000 BCE (8000 years ago) there is evidence of elaborate pottery with mathematical designs, disciplined building layouts showing the use of a standardized length measure and an understanding of principles of geometry including the application of right triangles.<\/li>\n
    6. By 3,200 BCE (5200 years ago) there is indubitable evidence for mathematical practice in clay tokens and bullae (“envelopes”) in Mesopotamian city states within a centralized temple economy and a scribal-statal context. This is the earliest known system of metrology (counting and measuring), of writing, and of book-keeping (accounting)<\/li>\n<\/ol>\n

      But of course, we continue to uncover more about our prehistoric past, so the story of the prehistoric origins of mathematics is undoubtedly not yet complete.<\/p>\n\n

      Jump to Bibliography<\/a><\/p>\n

      Download article as PDF.<\/a><\/strong><\/em><\/span><\/p>\n

      \n

      Appendices have been MOVED to a separate page<\/a> – click here to continue<\/strong><\/p>\n

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      <\/div>\n
        \n
      1. How do birds migrate long distances? The answer appears to be magnetoreceptors in bird retinas that are sensitive enough to transmit changes in orientation vs. earth’s magnetic field<\/a> directly to the brain, essentially working like a neurally integrated compass. ↩<\/a><\/span><\/li>\n
      2. if we think of language as symbolic, the question is at which stage animals that communicate with each other, do so. ↩<\/a><\/span><\/li>\n<\/ol>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

        3rd ed. Aug 2023 (expanded appendices). 2nd ed. Nov 2019 (revised to include advances in linguistics, genomics, interpretive theory, and Mesopotamian mathematics); 1st ed. (Dec 29, 2009)<\/p>\n

        Part 1 in Ancient Mathematics series. (Part 2: The Mathematics of Uruk and Susa 3500-3000 BCE, Part 3: Exploring Cuneiform Culture 8500-2500 BCE)<\/p>\n

        Abstract How far back [Read More…]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","footnotes":""},"categories":[3],"tags":[],"coauthors":[112],"class_list":["post-12145","post","type-post","status-publish","format-standard","hentry","category-general","odd"],"views":95252,"_links":{"self":[{"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/posts\/12145","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/comments?post=12145"}],"version-history":[{"count":3,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/posts\/12145\/revisions"}],"predecessor-version":[{"id":12150,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/posts\/12145\/revisions\/12150"}],"wp:attachment":[{"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/media?parent=12145"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/categories?post=12145"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/tags?post=12145"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/coauthors?post=12145"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}