## Posts Tagged ‘counting’

### Numbers and mathematics are possible only because time flows

August 13, 2022

It is probably just a consequence of ageing that I am increasingly captivated (obsessed?) by the origin of things. And of these things, I find the origins of counting, numbers and mathematics (in that order) particularly fascinating. In that order because I am convinced that these developed within human cognition – and could only develop – in that order.  First counting, then numbers and then mathematics. The entire field of what is called number theory, which studies the patterns and relationships between numbers, exists because numbers are what they are. All the patterns and relationships discovered in the last c. 10,000 years all existed – were already there – as soon as the concept of numbers crystallised. Whereas counting and numbers were invented, all the wonders of the patterns and relationships that make up number theory were – and are still being – discovered. And what I find even more astonishing is that the entire edifice of numbers is built upon just one little foundation stone- the concept of identity which gives the concept of oneness.

#### Croutons in the soup of existence

The essence of identity lies in oneness. There can only be one of any thing once that thing has identity. Once a thing is a thing there is only one of it. Half that thing is no longer that thing. There can be many of such things but every other such thing is still something else.

Numbers are abstract and do not exist in the physical world. They are objects (“words”) within the invented language of mathematics to help us describe the physical world. They enable counting and measuring. The logical one or the philosophical one or the mathematical one all emerge from existence and identity. Neither logic nor philosophy nor mathematics can explain what one is, except that it is. Every explanation or definition attempted ends up being circular. It is what it is.

Given one (1), all other numbers follow.

#### Where numbers come from

Numbers start with one (1), and without a one (1) there can be no numbers. …… . Given the abstract concepts of identity (oneness, 1) and arithmetical addition (+), all natural numbers inevitably follow. With a 1 and with a +, and the concept of a set and a sum, all the natural numbers can be generated.

1 + 1 + 1 + 1 ……

…. Numbers, ultimately, rest on the concept of identity (oneness).

Equally fascinating are the questions that existence, time and causality are answers to. I am coming to the conclusion that the flow of time (whatever time is) does not emerge from existence but, in fact, enables existence.

#### Revising Genesis

…. What time is remains a mystery but the first act of creation is to set it flowing. Note that the flow of time does not need existence. To be, however, requires that time be flowing. Time itself, whatever it is, is a prerequisite for the flow of time and the flow of time is prerequisite for existence. ………. For even the concept of existence to be imaginable, it needs that the flow of time be ongoing. It needs to be present as a permanent moving backdrop. The potential for some particular kind of existence then appears, or is created, only when some particular rules of existence are defined and implemented. These rules of existence must therefore also be in place before the concept of things, whether abstract or material or otherwise, can be conjured up.

It is inevitable that my views have evolved and they may well evolve further but my current conclusion is that for mathematics to exist time needs to be flowing.

The bottom line:

1. All branches of mathematics, though abstract, are existentially dependent upon the concept of numbers.
2. Numbers depend on the concept of counting.
3. Counting derives from the concept of oneness (1).
4. Oneness depends upon the concept of a unique identity.
5. The existence of a unique identity requires a begin-time.
6. Beginnings require time to be flowing.
7. Existence is enabled by the flow of time

Therefore

### Mathematics started in prehistory with counting and the study of shapes

January 8, 2021

Mathematics today is classified into more than 40 different areas by journals for publication. I probably heard about the 3 R’s (Reading, Riting and Rithmetic) first at primary school level. At high school, which was 60 years ago, mathematics consisted of arithmetic, geometry, algebra, trigonometry, statistics and logic – in that order. I remember my first class where trigonometry was introduced as a “marriage of algebra and geometry”. Calculus was touched on as advanced algebra. Some numerical methods were taught as a part of statistics. If I take my own progression, it starts with arithmetic, moves on to geometry, algebra and trigonometry and only then to calculus and statistics, symbolic logic and computer science. It was a big deal when, at 10, I got my first “compass box” for geometry, and another big deal, at 13, with my first copy of trigonometric tables. At university in the 70s, Pure Mathematics was distinguished from Applied Engineering Mathematics and from Computing. In my worldview, Mathematics and Physics Departments were where the specialist, esoteric mysteries of such things as topology, number theory, quantum algebra, non-Euclidean geometry and combinatorics could be studied.

I don’t doubt that many animals can distinguish between more and less. It is sometimes claimed that some primates have the ability to count up to about 3. Perhaps they do, but except for in the studies reporting such abilities, they never actually do count. No animals apply counting, They don’t exhibit any explicit understanding of geometrical shapes or structures, though birds, bees, ants and gorillas seem to apply some structural principles, intuitively, when building their nests. Humans, as a species, are unique in not only imagining but also in applying mathematics. We couldn’t count when we left the trees. We had no tools then and we built no shelters. So how did it all begin?

Sometimes Arithmetic, Geometry and Algebra are considered the three core areas of mathematics. But I would contend that it must all start with counting and with shapes – which later developed into Arithmetic and Geometry. Algebra and its abstractions came much later. Counting and the study of shapes must lie at the heart of how prehistoric humans first came to mathematics. But I would also contend that counting and observing the relationship between shapes would have started separately and independently. They both require a certain level of cognition but they differ in that the study of shapes is based on observations of physical surroundings while counting requires invention of concepts in the abstract plane. They may have been contemporaneous but they must, I think, have originated separately.

No circle of standing stones would have been possible without some arithmetic (rather than merely counting) and some geometry. No pyramid, however simple, was ever built without both. No weight was dragged or rolled up an inclined plane without some understanding of both shapes and numbers. No water channel that was ever dug did not involve some arithmetic and some geometry. Already by the time of Sumer and Babylon, and certainly by the time of the Egyptians and the Harappans, the practical application of arithmetic and geometry and even trigonometry in trade, surveying, town planning, time-keeping and building were well established. The sophisticated management of water that we can now glimpse in the ancient civilizations needed both arithmetic and geometry. There is not much recorded history that is available before the Greeks. Arithmetic and Geometry were well established by the time we come to the Greeks who even conducted a vigorous discourse about the nobility (or divinity) of the one versus the other. Pythagoras is not happy with arithmetic since numbers cannot give him – exactly – the hypotenuse of a right triangle of sides of equal length (√2). Which he can so easily draw. Numbers could not exactly reflect all that he could achieve with a straight edge and a compass. The circle could not be squared. The circumference was irrational. The irrationality of the numbers needed to reflect geometrical shapes was, for the purists, vulgar and an abomination. But the application of geometry and arithmetic were common long, long before the Greeks. There is a great distance before counting becomes arithmetic and the study of shapes becomes geometry but the roots of mathematics lie there. That takes us back to well before the Neolithic (c. 12,000 years ago).

That geometry derives from the study of shapes and the patterns and relationships between shapes, given some threshold level of cognition, seems both obvious and inevitable. Shapes are real and ubiquitous. They can be seen in all aspects of the natural world and can be mimicked and constructed. The arc of the sun curving across the sky creates a shape. Shadows create shapes. Light creates straight lines as the elevation of the sun creates angles. Shapes can be observed. And constructed. A taut string to give a straight line and the calm surface of a pond to give a level plane. A string and a weight to give the vertical. A liquid level to give the horizontal. Sticks and shadows. A human turning around to observe the surroundings created a circle. Strings and compasses. Cave paintings from c. 30,000 years ago contain regular shapes. Circles and triangles and squares. Humans started not only observing, but also applying, the relationships between shapes a very long time ago.

Numbers are more mystical. They don’t exist in the physical world. But counting the days from new moon to new moon for a lunar month, or the days in a year, were also known at least 30,000 years ago. Ancient tally sticks to count to 29 testify to that. It would seem that the origins of arithmetic (and numbers) lie in our ancient prehistory and probably more than 50,000 years ago. Counting, the use of specific sounds as the representation of abstract numbers, and number systems are made possible only by first having a concept of identity which allows the definition of one. Dealing with identity and the nature of existence take us before and beyond the realms of philosophy or even theology and are in the metaphysical world. The metaphysics of existence remain mystical and mysterious and beyond human cognition, as much today as in prehistoric times. Nevertheless, it is the cognitive capability of having the concept of a unique identity which enables the concept of one. That one day is distinguishable from the next. That one person, one fruit, one animal or one thing is uniquely different to another. That unique things, similar or dissimilar, can be grouped to create a new identity. That one grouping (us) is distinguishable from another group (them). Numbers are not physically observable. They are all abstract concepts. Linguistically they are sometimes bad nouns and sometimes bad adjectives. The concept of one does not, by itself, lead automatically to a number system. That needs in addition a logic system and invention (a creation of something new which presupposes a certain cognitive capacity). It is by definition, and not by logic or reason or inevitability, that two is defined as one more than the identity represented by one, and three is defined as one more than two, and so on. Note that without the concept of identity and the uniqueness of things setting a constraint, a three does not have to be separated from a two by the same separation as from two to one. The inherent logic is not itself invented but emerges from the concept of identity and uniqueness. That 1 + 1 = 2 is a definition not a discovery. It assumes that addition is possible. It is also significant that nothingness is a much wider (and more mysterious and mystical) concept than the number zero. Zero derives, not from nothingness, but from the assumption of subtraction and then of being defined as one less than one. That in turn generalises to zero being any thing less than itself. Negative numbers emerge by extending that definition. The properties of zero are conferred by convention and by definition. Numbers and number systems are thus a matter of “invention by definition”, but constrained by the inherent logic which emerges from the concept of identity. The patterns and relationships between numbers have been the heady stuff of number theory and a matter of great wonder when they are discovered, but they are all consequent to the existence of the one, the invention of numerals and the subsequent definition that 1 + 1 = 2. Number theory exists only because the numbers are defined as they are. Whereas the concept of identity provides the basis for one and all integers, a further cognitive step is needed to imagine that the one is not indivisible and then to consider the infinite parts of one.

Mere counting is sometimes disparaged, but it is, of course, the most rudimentary form of a rigorous arithmetic with its commutative, associative and distributive laws.

The cognitive step of getting to count in the first place is a huge leap compared to the almost inevitable evolution of counting into numbers and then into an arithmetic with rigorous laws. We will never know when our ancestors began to count but it seems to me – in comparison with primates of today – that it must have come after a cognitive threshold had been achieved. Quite possibly with the control of fire and after the brain size of the species had undergone a step change. That takes us back to the time of homo erectus and perhaps around a million years ago.

Nearly all animals have shape recognition to some extent. Some primates can even recognise patterns in similar shapes. It is plausible that recognition of patterns and relationships between shapes only took off when our human ancestors began construction either of tools or of rudimentary dwellings. The earliest tools (after the use of clubs) were probably cutting edges and these are first seen around 1.8 million years ago. The simplest constructed shelters would have been lean-to structures of some kind. Construction of both tools and shelters lend themselves naturally to the observation of many geometrical shapes; rectangles, polygons, cones, triangles, similar triangles and the rules of proportion between similar shapes. Arches may also have first emerged with the earliest shelters. More sophisticated tools and very simple dwellings take us back to around 400,000 years ago and certainly to a time before anatomically modern humans had appeared (c. 200,000 years ago). Both rudimentary counting and a sense of shapes would have been present by then. It would have been much later that circles and properties of circles were observed and discovered. (Our earliest evidence of a wheel goes back some 8,000 years and is the application of a much older mathematics). Possibly the interest in the circle came after a greater interest in time keeping had emerged. Perhaps from the first “astronomical” observations of sunrise and sunset and the motion of the moon and the seasons. Certainly our ancestors were well-versed with circles and spheres and their intersections and relationships by the time they became potters (earlier than c. 30,000 years ago).

I suspect it was the blossoming of trade – rather than the growth of astronomy – which probably helped take counting to number systems and arithmetic. The combination of counting and shapes starts, I think, with the invention of tools and the construction of dwellings. By the time we come to the Neolithic and settlements and agriculture and fortified settlements, arithmetic and geometry and applied mathematics is an established reality. Counting could have started around a million years ago. The study of shapes may have started even earlier. But if we take the origin of “mathematics” to be when counting ability was first combined with a sense of shapes, then we certainly have to step back to at least 50,000 years ago.

### Why did we start to count?

October 12, 2020

Counting and the invention of numbers and the abstractions enabling mathematics are surely cognitive abilities. Counting itself involves an abstract ability. The simple act of raising two fingers to denote the number of stones or lions or stars implies first, the abstract ability to describe an observed quality and second, the desire to communicate that observation.

What led humans to counting and when?

Before an intelligence can turn to counting it must first have some concept of numbers. When and how did our ancient ancestors  first develop a concept of numbers and then start counting? ……..

It seems clear that many animals do distinguish – in a primitive and elementary way – between “more” and “less, and “few” and “many”,and “bigger” and “smaller”, and even manage to distinguish between simple number counts. They show a sophisticated use of hierarchy and precedence.

Some primates show some primitive abilities when tested by humans

…..  Rhesus monkeys appear to understand that 1 + 1 = 2. They also seem to understand that 2 + 1 = 3, 2 – 1 = 1, and 3 – 1 = 2—but fail, however, to understand that 2 + 2 = 4. ……

But even chimpanzees and monkeys rarely, if ever, use counts or counting in interactions among themselves. The abilities for language and counting are not necessarily connected genetically (though it is probable), but they are both certainly abilities which appear gradually as cognition increases. Mathematics is, of course, just another language for describing the world around us. Number systems, as all invented languages, need that a system and its rules be shared before any communication is feasible. It is very likely that the expressions of the abilities to count and to have language follow much the same timeline. The invention of specific sounds or gestures to signify words surely coincided with the invention of gestures or sounds to signify numbers. The step change in the size of brains along the evolutionary path of humans is very likely closely connected with the expressions of the language and the counting abilities.

The ability to have language surely preceded the invention of languages just as the ability to count preceded the expressions of counting and numbering. It is not implausible that the first member of a homo erectus descendant who used his fingers to indicate one of something, or four of something else, to one of his peers, made a far, far greater discovery – relatively – than Newton or Einstein ever did.

We must have started counting and using counts (using gestures) long before we invented words to represent counts. Of course, it is the desire to communicate which is the driving force which takes us from having abilities to expressions of those abilities. The “cooperation gene” goes back to before the development of bipedalism and before the split with chimpanzees or even gorillas (at least 9 million years ago).

The simple answer to the question “Why did we start to count?” is because we could conceive of a count, observed it and wished to communicate it. But this presupposes the ability to count. Just as with language, the ability and the expression of the ability, are a consequence of the rapid increase in brain size which happened between 3 m and 1 m years ago.

I am persuaded that that rapid change was due to the control of fire and the change to eating cooked food and especially cooked meat. The digestion of many nutrients becomes possible only with cooked food and is the most plausible driver for the rapid increase in brain size.

Raw Food not enough to feed big brains

………. our brains would still be the size of an ape’s if H. erectus hadn’t played with fire: “Gorillas are stuck with this limitation of how much they can eat in a day; orangutans are stuck there; H. erectus would be stuck there if they had not invented cooking,” she says. “The more I think about it, the more I bow to my kitchen. It’s the reason we are here.”

### Counting on fingers leads naturally also to base-60

August 19, 2017

We have forgotten what it was like to count on our fingers. We have forgotten that counting itself was a mystery long before the mysteries of manipulation of numbers and the magic of mathematics. Yet the use of base-60 lies deep in our psyches. We still use it for time measurement and for geographical and spatial measurements. Attempts to use decimals for time and angle measurement have all failed miserably. Sixty still occurs in ancient Chinese and Indian calendars.

Today the use of 60 still predominates for time, for navigation and geometry. But generally only for units already defined in antiquity. A base of 10 is used for units found to be necessary in more recent times. Subdivision of a second of time or a second of arc is always using the decimal system rather than by the duodecimal or the sexagesimal system.

Usually the origin of sexagesimal systems of counting are traced back to the Babylonians (c. 1,800 BCE) and even to the Sumerians (c. 3,000 – 2,500 BCE). But I suspect that it goes back much further and that base-60 long precedes the Babylonians and the Sumerians.

I observe that twelve and then sixty come naturally from three factors:

1. using fingers for counting,
2. maximising the count with only one hand free, and
3. the growth of trade and the need for counts of greater than 20

That five comes naturally from the fingers of one hand is self-evident. With only one free hand, a count to twelve using the thumb and the digits of the other four fingers is also self-evident. I saw my great grandmother, and my grandmother after her, regularly count to twelve using only one hand. Sixty comes naturally from a hand of five of a hand of twelve. Counting to five and twelve would have been well known to our hunter-gatherer ancestors. It seems very plausible that a hunter would look to maximise the count on a single hand. So, it is not necessary to look for the origins of base-60 in the skies or in the length of the year or the number of its divisors or the beginnings of geometry. If the origins of counting lie some 50,000 years ago, the use of twelve and then of sixty probably goes back some 20,000 years.

The origins of base 60

I like 60. Equilaterals. Hexagons. Easy to divide by almost anything. Simple integers for halves, quarters, thirds, fifths, sixths, tenths, 12ths, 15ths and 30ths. 3600. 60Hz. Proportions pleasing to the eye. Recurring patterns. Harmonic. Harmony.

The origins of the use of base 60 are lost in the ancient past. By the time the Sumerians used it about 2,500 years ago it was already well established and continued through the Babylonians. But the origin lies much earlier.

I speculate that counting – in any form more complex than “one, two, many….” – probably goes back around 50,000 years. I have little doubt that the fingers of one hand were the first counting aids that were ever used, and that the base 10 given by two hands came to dominate. Why then would the base 60 even come into being?

The answer, I think, still lies in one hand. Hunter-gatherers when required to count would prefer to use only one hand and they must – quite early on and quite often – have had the need for counting to numbers greater than five. And of course using the thumb as pointer one gets to 12 by reckoning up the 3 bones on each of the other 4 fingers.

a hand of 12 – image sweetscience

My great-grandmother used to count this way when checking the numbers of vegetables (onions, bananas, aubergines) bought by her maid at market. Counting up to 12 usually sufficed for this. When I was a little older, I remember my grandmother using both hands to check off bags of rice brought in from the fields – and of course with two hands she could get to 144. The counting of 12s most likely developed in parallel with counting in base 10 (5,10, 50, 100). The advantageous properties of 12 as a number were fortuitous rather than by intention. But certainly the advantages helped in the persistence of using 12 as a base. And so we still have a dozen (12) and a gross (12×12) and even a great gross (12x12x12) being used today. Possibly different groups of ancient man used one or other of the systems predominantly. But as groups met and mixed and warred or traded with each other the systems coalesced.

And then 60 becomes inevitable. Your hand of 5, with my hand of 12, gives the 60 which also persists into the present.  (There is one theory that 60 developed as 3 x 20, but I think finger counting and the 5 x 12 it leads to is far more compelling). But it is also fairly obvious that the use of 12 must be prevalent first before the 60 can appear. Though the use of 60 seconds and 60 minutes are all pervasive, it is worth noting that they can only come after each day and each night is divided into 12 hours.

While the use of base 10 and 12 probably came first with the need for counting generally and then for trade purposes (animals, skins, weapons, tools…..), the 12 and the 60 came together to dominate the measuring and reckoning of time. Twelve months to a year with 30 days to a month. Twelve hours to a day or a night and 60 parts to the hour and 60 parts to those minutes. There must have been a connection – in time as well as in the concepts of cycles – between the “invention” of the calendar and the geometrical properties of the circle. The number 12 has great significance in Hinduism, in Judaism, in Christianity and in Islam. The 12 Adityas, the 12 tribes of Israel, the 12 days of Christmas, the 12 Imams are just examples. My theory is that simple sun and moon-based religions gave way to more complex religions only after symbols and writing appeared and gave rise to symbolism. ………

If we had six fingers on each hand the decimal system would never have seen the light of day. A millisecond would then be 1/ 1728th of a second. It is a good thing we don’t have 7 fingers on each hand, or – even worse – one hand with 6 fingers and one with 7. Arithmetic with a tridecimal system of base 13 does not entice me. But if I was saddled with 13 digits on my hands I would probably think differently.

### Number theory was probably more dependent upon live goats than on raindrops

June 14, 2017

It used to be called arithmetic but it sounds so much more modern and scientific when it is called number theory. It is the branch of mathematics which deals with the integers and the relationships between them. Its origins (whether one wants to call it a discovery or an invention) lie with the invention of counting itself. It is from where all the various branches of mathematics derive. The origin of counting can be said to be with the naming of the integers, and is intimately tied to the development of language and of writing and perhaps goes back some 50,000 years (since the oldest known tally sticks date from some 30,000 years ago).

How and why did the naming of the integers come about?  Why were they found necessary (necessity being the cause of the invention)? Integers are whole numbers, indivisible, complete in themselves. Integers don’t recognise a continuum between themselves. There are no partials allowed here. They are separate and discrete and number theory could as well be called quantum counting.

Quite possibly the need came from counting their livestock or their prey. If arithmetic took off in the fertile crescent it well may have been the need for trading their live goats among themselves (integral goats for integral numbers of wives or beads or whatever else they traded) which generated the need for counting integers. Counting would have come about to fit their empirical observations. Live goats rather than carcasses, I think, because a carcass can be cut into bits and is not quite so dependent upon integers.  Quanta of live goat, however, would not permit fractions. It might have been that they needed integers to count living people (number of children, number of wives …..) where fractions of a person were not politically correct.

The rules of arithmetic – the logic – could only be discovered after the integers had been named and counting could go forth. The commutative, associative and distributive properties of integers inevitably followed. And the rest is history.

But I wonder how mathematics would have developed if the need had been to count raindrops.

After all:

2 goats + 2 goats = 4 goats, and it then follows that

2 short people + 2 short people = 4 short people.

But if instead counting had been inspired by counting raindrops, they would have observed that

2 little raindrops + 2 little raindrops = 1 big raindrop.

They might then have concluded that

2 short people + 2 short people = one tall person

and history would then have been very different.

### Counting was an invention

March 19, 2017

A new book is just out and it seems to be one I have to get. I am waiting to get hold of an electronic version.

Number concepts are a human invention―a tool, much like the wheel, developed and refined over millennia. Numbers allow us to grasp quantities precisely, but they are not innate. Recent research confirms that most specific quantities are not perceived in the absence of a number system. In fact, without the use of numbers, we cannot precisely grasp quantities greater than three; our minds can only estimate beyond this surprisingly minuscule limit.

Numbers fascinate me and especially how they came to be.

The earliest evidence we have of humans having counting ability are ancient tally sticks made of bone and dating up to 50,000 years ago. An ability to tally at least up to 55 is evident. One of the tally sticks may have been a form of lunar calendar. By this time apparently they had a well developed concept of time. And concepts of time lead immediately and inevitably to the identification of recurring time periods. By 50,000 years ago our ancestors counted days and months and probably years. Counting numbers of people would have been child’s play. They had clearly developed some sophistication not only in “numbering” by this time but had also progressed from sounds and gestures into speech.  They were well into the beginnings of language.

Marks on a tally stick tell us a great deal. The practice must have been developed in response to a need. Vocalisations – words – must have existed to describe the tally marks. These marks were inherently symbolic of something else. They are evidence of the ability to symbolise and to think in abstract terms. Perhaps they represented numbers of days or a count of cattle or of items of food or of number of people in the tribe. But their very existence suggests that the concept of ownership of property – by the individual or by the tribe – was already in place. Quite probably a system of trading with other tribes and protocols for such trade were also in place. At 50,000 years ago our ancestors were clearly on the threshold of using symbols not just on tally sticks or in cave paintings but in a general way and that would have been the start of developing a written language. …….

My time-line then becomes:

• 8 million YBP           Human Chimpanzee divergence
• 6 million YBP           Rudimentary counting among Archaic humans (1, 2, 3 many)
• 2 million YBP           Stone tools
• 600,000 YBP          Archaic Human – Neanderthal divergence
• 400,000 YBP          Physiological and genetic capability for speech?
• 150,000 YBP           Speech and counting develop together
• 50,000   YBP           Verbal language, counting, trading, calendars in place (tally sticks)
• 30,000   YBP           Beginnings of written language?
Clearly our counting is dominated by the base of 10 and our penchant for 12-based systems. The joints on the fingers of one hand allows us to count to 12 and that together with the five fingers of the other clearly led to our many 60-based counting systems.

I like 60. Equilaterals. Hexagons. Easy to divide by almost anything. Simple integers for halves, quarters, thirds, fifths, sixths, tenths, 12ths, 15ths and 30ths. 3600. 60Hz. Proportions pleasing to the eye. Recurring patterns. Harmonic. Harmony.

The origins of the use of base 60 are lost in the ancient past. By the time the Sumerians used it about 2,500 years ago it was already well established and continued through the Babylonians. But the origin lies much earlier. ……

Why then would the base 60 even come into being?

image sweet science

The answer, I think, still lies in one hand. Hunter-gatherers when required to count would prefer to use only one hand and they must – quite early on and quite often – have had the need for counting to numbers greater than five. And of course using the thumb as pointer one gets to 12 by reckoning up the 3 bones on each of the other 4 fingers.

My great-grandmother used to count this way when checking the numbers of vegetables (onions, bananas, aubergines) bought by her maid at market. Counting up to 12 usually sufficed for this. When I was a little older, I remember my grandmother using both hands to check off bags of rice brought in from the fields – and of course with two hands she could get to 144. The counting of 12s most likely developed in parallel with counting in base 10 (5,10, 50, 100). The advantageous properties of 12 as a number were fortuitous rather than by intention. But certainly the advantages helped in the persistence of using 12 as a base. And so we still have a dozen (12) and a gross (12×12) and even a great gross (12x12x12) being used today. Possibly different groups of ancient man used one or other of the systems predominantly. But as groups met and mixed and warred or traded with each other the systems coalesced.

If we had 4 bones on each finger we would be using 5 x 16 = 80 rather than 60.

### The origins of base 60

September 14, 2015

I like 60. Equilaterals. Hexagons. Easy to divide by almost anything. Simple integers for halves, quarters, thirds, fifths, sixths, tenths, 12ths, 15ths and 30ths. 3600. 60Hz. Proportions pleasing to the eye. Recurring patterns. Harmonic. Harmony.

The origins of the use of base 60 are lost in the ancient past. By the time the Sumerians used it about 2,500 years ago it was already well established and continued through the Babylonians. But the origin lies much earlier.

I speculate that counting – in any form more complex than “one, two, many….” – probably goes back around 50,000 years. I have little doubt that the fingers of one hand were the first counting aids that were ever used, and that the base 10 given by two hands came to dominate. Why then would the base 60 even come into being?

The answer, I think, still lies in one hand. Hunter-gatherers when required to count would prefer to use only one hand and they must – quite early on and quite often – have had the need for counting to numbers greater than five. And of course using the thumb as pointer one gets to 12 by reckoning up the 3 bones on each of the other 4 fingers.

a hand of 12 – image sweetscience

My great-grandmother used to count this way when checking the numbers of vegetables (onions, bananas, aubergines) bought by her maid at market. Counting up to 12 usually sufficed for this. When I was a little older, I remember my grandmother using both hands to check off bags of rice brought in from the fields – and of course with two hands she could get to 144. The counting of 12s most likely developed in parallel with counting in base 10 (5,10, 50, 100). The advantageous properties of 12 as a number were fortuitous rather than by intention. But certainly the advantages helped in the persistence of using 12 as a base. And so we still have a dozen (12) and a gross (12×12) and even a great gross (12x12x12) being used today. Possibly different groups of ancient man used one or other of the systems predominantly. But as groups met and mixed and warred or traded with each other the systems coalesced.

And then 60 becomes inevitable. Your hand of 5, with my hand of 12, gives the 60 which also persists into the present.  (There is one theory that 60 developed as 3 x 20, but I think finger counting and the 5 x 12 it leads to is far more compelling). But it is also fairly obvious that the use of 12 must be prevalent first before the 60 can appear. Though the use of 60 seconds and 60 minutes are all pervasive, it is worth noting that they can only come after each day and each night is divided into 12 hours.

While the use of base 10 and 12 probably came first with the need for counting generally and then for trade purposes (animals, skins, weapons, tools…..), the 12 and the 60 came together to dominate the measuring and reckoning of time. Twelve months to a year with 30 days to a month. Twelve hours to a day or a night and 60 parts to the hour and 60 parts to those minutes. There must have been a connection – in time as well as in the concepts of cycles – between the “invention” of the calendar and the geometrical properties of the circle. The number 12 has great significance in Hinduism, in Judaism, in Christianity and in Islam. The 12 Adityas, the 12 tribes of Israel, the 12 days of Christmas, the 12 Imams are just examples. My theory is that simple sun and moon-based religions gave way to more complex religions only after symbols and writing appeared and gave rise to symbolism.

Trying to construct a time-line is just speculation. But one nice thing about speculation is that the constraints of known facts are very loose and permit any story which fits. So I put the advent of numbers and counting at around 50,000 years ago first with base 10 and later with base 12. The combination of base 10 with base 12, I put at around 20,000 years ago when agricultural settlements were just beginning. The use of 60 must then coincide with the first structured, astronomical observations after the advent of writing and after the establishment of permanent, settlements. It is permanent settlements. I think, which allowed regular observations of cycles, which allowed specialisations and the development of symbols and religion and the privileged priesthood. That probably puts us at about 8 -10,000 years ago, as agriculture was also taking off, probably somewhere in the fertile crescent.

Wikipedia: The Egyptians since 2000 BC subdivided daytime and nighttime into twelve hours each, hence the seasonal variation of the length of their hours.

The Hellenistic astronomers Hipparchus (c. 150 BC) and Ptolemy (c. AD 150) subdivided the day into sixty parts (the sexagesimal system). They also used a mean hour(124 day); simple fractions of an hour (14, 23, etc.); and time-degrees (1360 day, equivalent to four modern minutes).

The Babylonians after 300 BC also subdivided the day using the sexagesimal system, and divided each subsequent subdivision by sixty: that is, by 160, by 160 of that, by 160of that, etc., to at least six places after the sexagesimal point – a precision equivalent to better than 2 microseconds. The Babylonians did not use the hour, but did use a double-hour lasting 120 modern minutes, a time-degree lasting four modern minutes, and a barleycorn lasting 313 modern seconds (the helek of the modern Hebrew calendar), but did not sexagesimally subdivide these smaller units of time. No sexagesimal unit of the day was ever used as an independent unit of time.

Today the use of 60 still predominates for time, for navigation and geometry. But generally only for units already defined in antiquity. A base of 10 is used for units found to be necessary in more recent times. Subdivision of a second of time or a second of arc is always using the decimal system rather than by the duodecimal or the sexagesimal system.

If we had six fingers on each hand the decimal system would never have seen the light of day. A millisecond would then be 1/ 1728th of a second. It is a good thing we don’t have 7 fingers on each hand, or – even worse – one hand with 6 fingers and one with 7. Arithmetic with a tridecimal system of base 13 does not entice me. But if I was saddled with 13 digits on my hands I would probably think differently.

### Numeracy and language

December 2, 2013

I tend towards considering mathematics a language rather than a science. In fact mathematics is more like a family of languages each with a rigorous grammar. I like this quote:

R. L. E. SchwarzenbergerThe Language of Geometry, in A Mathematical Spectrum Miscellany, Applied Probability Trust, 2000, p. 112:

My own attitude, which I share with many of my colleagues, is simply that mathematics is a language. Like English, or Latin, or Chinese, there are certain concepts for which mathematics is particularly well suited: it would be as foolish to attempt to write a love poem in the language of mathematics as to prove the Fundamental Theorem of Algebra using the English language.

Just as conventional languages enable culture and provide a tool for social communication, the various languages of mathematics, I think, enable science and provide a tool for scientific discourse. I take “science” here to be analaogous to a “culture”. To follow that thought then, just as science is embedded within a “larger” culture, so is mathematics embedded within conventional languages. This embedding shows up as the ability of a language to deal with numeracy and numerical concepts.

And that means then the value judgement of what is “primitive” when applied to language can depend upon the extent to which mathematics and therefore numeracy is embedded within that language.

According to a recent article by Mike Vuolo in Slate.com, Pirahã is among “only a few documented cases” of languages that almost completely lack of numbers. Dan Everett, a renowned expert in the Pirahã language, further claims that the lack of numeracy is just one of many linguistic deficiencies of this language, which he relates to gaps in the Pirahã culture. …..

The various types of number systems are considered in the WALS.info article on Numeral Bases, written by Bernard Comrie. Of the 196 languages in the sample, 88% can handle an infinite set of numerals. To do so, languages use some arithmetic base to construct numeral expressions. According to Comrie, “we live in a decimal world”: two thirds of the world’s languages use base 10 and such languages are spoken “in nearly every part of the world”. English, Russian, and Mandarin are three examples of such languages. …..

Around 20% of the world’s languages use either purely vigesimal (or base 20) or a hybrid vigesimal-decimal system. In a purely vigesimal system, the base is consistently 20, yielding the general formula for constructing numerals as x20 + y. For example, in Diola-Fogny, a Niger-Congo language spoken in Senegal, 51 is expressed as bukan ku-gaba di uɲɛn di b-əkɔn ‘two twenties and eleven’. Other languages with a purely vigesimal system include Arawak spoken in Suriname, Chukchi spoken in the Russian Far East, Yimas in Papua New Guinea, and Tamang in Nepal. In a hybrid vigesimal-decimal system, numbers up to 99 use base 20, but the system then shifts to being decimal for the expression of the hundreds, so that one ends up with expressions of the type x100 + y20 + z. A good example of such a system is Basque, where 256 is expressed as berr-eun eta berr-ogei-ta-hama-sei ‘two hundred and two-twenty-and-ten-six’. Other hybrid vigesimal-decimal systems are found in Abkhaz in the Caucasus, Burushaski in northern Pakistan, Fulfulde in West Africa, Jakaltek in Guatemala, and Greenlandic. In a few mostly decimal languages, moreover, a small proportion of the overall numerical system is vigesimal. In French, for example, numerals in the range 80-99 have a vigesimal structure: 97 is thus expressed as quatre-vingt-dix-sept ‘four-twenty-ten-seven’. Only five languages in the WALS sample use a base that is neither 10 nor 20. For instance, Ekari, a Trans-New Guinean language spoken in Indonesian Papua uses base of 60, as did the ancient Near Eastern language Sumerian, which has bequeathed to us our system of counting seconds and minutes. Besides Ekari, non-10-non-20-base languages include Embera Chami in Colombia, Ngiti in Democratic Republic of Congo, Supyire in Mali, and Tommo So in Mali. ……

Going back to the various types of counting, some languages use a restricted system that does not effectively go above around 20, and some languages are even more limited, as is the case in Pirahã. The WALS sample contains 20 such languages, all but one of which are spoken in either Australia, highland New Guinea, or Amazonia. The one such language found outside these areas is !Xóõ, a Khoisan language spoken in Botswana. …….

Counting monkey?

In some societies in the ancient past, numeracy did not contribute significantly to survival as probably with isolated tribes like the Pirahã. But in most human societies, numeracy was of significant benefit especially for cooperation between different bands of humans. I suspect that it was the need for social cooperation which fed the need for communication within a tribe and among tribes, which in turn was the spur to the development of language, perhaps over 100,000 years ago. What instigated the need to count is in the realm of speculation. The need for a calendar would only have developed with the development of agriculture. But the need for counting herds probably came earlier in a semi-nomadic phase. Even earlier than that would have come the need to trade with other hunter gatherer groups and that  probably gave rise to counting 50,000 years ago or even earlier. The tribes who learned to trade and developed the ability and concepts of trading were probably the tribes that had the best prospects of surviving while moving from one territory to another. It could be that the ability to trade was an indicator of how far a group could move.

And so I am inclined to think that numeracy in language became a critical factor which 30,000 to 50,000 years ago determined the groups which survived and prospered. It may well be that it is these tribes which developed numbers, and learned to count, and learned to trade that eventually populated most of the globe. It may be a little far-fetched but not impossible that numeracy in language may have been one of the features distinguishing Anatomically Modern Humans from Neanderthals. Even though the Neanderthals had larger brains and that we are all Neanderthals to some extent!

### Baboons can tell “more” from “less” – but that is still a long way from counting

May 4, 2013

Being able to distinguish between “more” and “less” is – most likely – a capability that is a pre-requisite for the evolutionary development of the ability to count which itself must lead to the invention of numbers. Recent experiments with baboons demonstrates that they have a clear ability to make quite complex more/less distinctions.

Allison M. Barnard, Kelly D. Hughes, Regina R. Gerhardt, Louis DiVincenti, Jenna M. Bovee and Jessica F. Cantlon.Inherently Analog Quantity Representations in Olive Baboons (Papio anubis)Frontiers in Comparative Psychology, 2013 DOI: 10.3389/fpsyg.2013.00253

… Now a new study with a troop of zoo baboons and lots of peanuts shows that a less obvious trait—the ability to understand numbers—also is shared by man and his primate cousins.

“The human capacity for complex symbolic math is clearly unique to our species,” says co-author Jessica Cantlon, assistant professor of brain and cognitive sciences at the University of Rochester. “But where did this numeric prowess come from? In this study we’ve shown that non-human primates also possess basic quantitative abilities. In fact, non-human primates can be as accurate at discriminating between different quantities as a human child.”

“This tells us that non-human primates have in common with humans a fundamental ability to make approximate quantity judgments,” says Cantlon. “Humans build on this talent by learning number words and developing a linguistic system of numbers, but in the absence of language and counting, complex math abilities do still exist.” ……

……… The baboons’ choices, conclude the authors, clearly relied on this latter “more than” or “less than” cognitive approach, known as the analog system. The baboons were able to consistently discriminate pairs with numbers larger than three as long as the relative difference between the peanuts in each cup was large. Research has shown that children who have not yet learned to count also depend on such comparisons to discriminate between number groups, as do human adults when they are required to quickly estimate quantity.
Studies with other animals, including birds, lemurs, chimpanzees, and even fish, have also revealed a similar ability to estimate relative quantity, but scientists have been wary of the findings because much of this research is limited to animals trained extensively in experimental procedures. The concern is that the results could reflect more about the experimenters than about the innate ability of the animals. ……..

……… To rule out such influence, the study relied on zoo baboons with no prior exposure to experimental procedures. Additionally, a control condition tested for human bias by using two experimenters—each blind to the contents of the other cup—and found that the choice patterns remained unchanged.

A final experiment tested two baboons over 130 more trials. The monkeys showed little improvement in their choice rate, indicating that learning did not play a significant role in understanding quantity.

“What’s surprising is that without any prior training, these animals have the ability to solve numerical problems,” says Cantlon. The results indicate that baboons not only use comparisons to understand numbers, but that these abilities occur naturally and in the wild, the authors conclude. …….