Jevons’es Data

In 1871 the early economist and logician William Stanley Jevons published an article in Nature “The Power of Numerical Discrimination” (Jevons 1871)

According to Jevons, Sir William Hamilton had clearly stated the problem:

“Assuming that the mind is not limited to the simultaneous consideration of a single object, a question arises, How many objects can it embrace at once? ……

If you throw a handful of marbles on the floor, you will find it difficult to view at once more than six, or seven at most, without confusion; but if you group them into twos, or threes, or fives , you can comprehend as many groups as units, because the mind considers these groups as units; views them as wholes, and throws their parts out of consideration.”

Using a simple experimental technique, Jevons set out to  answer Sir William Hamilton’s question.

“A round paper box 4 1/2″ in diameter, lined with white paper and with the edges cut down to be only 1/4″ high, was placed in the middle of a black tray. A quantity of uniform black beans was then obtained, and a number of them taken up casually were thrown towards the box so that a wholly uncertain number fell into it. At the very moment when the beads came to rest, their number was estimated without the least hesitation, and then recorded together with the real number obtained by deliberate counting”

The data Jevons collected this way is tabulated below (taken directly from his article).


The table gives the number of times each set size was estimated correctly or incorrectly. So six beans were correctly identified 120 times but were incorrectly identified as five beans 7 times and as seven beans 20 times. Jevons was not surprised that he never made a mistake with three and four beans but was surprised that in 5% of trials he made mistakes with five beans.

Jevons analysed the error rates as follows:-


He supposed the error rate proportional to the excess of the real number over 4½ and obeying the formula error = m x (n – 4½) where n is the correct number and m is some constant. He than calculated m for each of the values of :-


Seeing them sufficiently equal he took the average (0.116) and proposed the following empirical formula:-

error = 0.116 x (n – 4½) or approximately n / 9 – ½


Jevons concluded that for him the absolute limit of discrimination was 4, but recognised that the limit probably varied for other people and might perhaps be taken as 4½ if that made any sense.

His estimate corresponds fairly accurately to modern estimates of the subitising range for randomly located dots. See Subitising.

The data also corresponds to Weber’s Law with accuracy getting worse as set size increases. A post on Weber’s Law to follow.

The diagram below summarises the error data.


Jevons calculated his formula  using unsigned error values (red outlines above) rather than signed errors (solid red above).

He also recognised that his method of throwing beans lead to the skewed distribution shown above with many more sets of seven beads thrown (156) than sets of three (23) or fifteen (11). Something that could easily be avoided using modern presentation techniques. See Find Your Own Space.


Jevons, W.S., (1871) The Power of Numerical Discrimination Nature, Thursday February 9, 1871.

Posted in Architecture, Brain Physiology, Enumeration, Randomness | Tagged , , , | Leave a comment

Shooting Baboons: A Story

If a man with a gun goes to shoot baboons near the edge of a forest, the baboons will see him coming, hide in the forest and not come out until he is seen to go away.

If the first man hides and a second man with a gun joins him, and then one of them walks away, the baboons will stay hidden and not come out of the forest. They know that there is still a man hiding with a gun .

The same is true if two, three, four or possibly five men join the first man and the same, or a smaller, number of men go away. The baboons stay hidden, they know that at least one man is still hiding.

However if six men with guns join the first one and then six of them ostentatiously walk away, after a while the baboons will come out of the forest and can be shot by the man they have failed to accounted for.


When a second man joins the first man and then one of them walks away, the baboons can calculate that one man remains hidden, they can subtract 1 from 2 and get the right answer that there is 1 left.

The baboons can also get the right answer with 1 plus (2, 3, 4 and possibly 5) minus (2, 3, 4 or 5 respectively).

It is therefore thought that baboons can do small number arithmetic with numbers up to about six, so have a number system something like none, one, two, three, four, five, six and many (more than six).

But when six men join the first man and then six  men walk away the baboon’s number system lets them down. This is because one plus six results in many, and many minus many gives none, the wrong answer. So it seems to the baboons that all the hunters have gone away and it is safe to come out of the forest.

Small Number Arithmetic

The diagram below illustrates the problem with small number arithmetic, using a slightly more restricted number system consisting of none, one, two, three and many (more than 3).

/Users/grahamshawcross/Documents/blog_drafts/shooting baboons/Sm

The system works well for addition, exhibiting closure (each operation results in a unambiguous instance of the number system itself). So one plus two gives three, two plus three gives many and many plus many gives many etc.

As baboons can apparently find out to their cost, problems arise with subtraction. Subtracting anything except none (one, two, three or many) from many is always problematic. In particular taking many from many seems most likely to result in none but in reality could also result in one, two, three or many depending on the unknowable “real” values of many.

It seems that the baboons are applying the simple rule “same minus same always results in none”. So one minus one, two minus two, three minus three and many minus many all give a result of none.


There is no evidence of anyone hunting baboons like this. The story was probably just made up to illustrate the limitations of languages that do not have words or a number system that can represent all numerosities.

There are human languages that use small number arithmetic and have a word like many to represent large numerosoties. (Butterworth et al., 2008) Isolated hunter gatherer cultures seem to have little need for an elaborate number system in their languages but tend to acquire them quite quickly upon contact and trade with the outside world.

The story also suggests an experimental technique for establishing the upper numerosity discrimination limits of animals and pre-verbal children, using an ‘expectancy violation technique’. See What Counts and  Otto Koehler.

Unneccessary Expansion

The problem caused by a lack of necessary number words is superficially similar to the apparent order in which languages “acquire” colour words. Some languages only have two colour words, cold and warm, corresponding to monochrome, black and white. Others have black and white plus red. Yet others add yellow then green or green then yellow, then blue, brown and finally purple, pink, orange or grey. (Berlin and Kay, 1969)

/Users/grahamshawcross/Documents/blog_drafts/shooting baboons/Co

Colour Hierarchy Diagram (after Berlin and Kay 1969)

The diagram above works from left to right (following the arrows and plus signs). If a language has a particular colour word then it will also have all the colour words to the left of that word. So if a language has a word for blue, then it will also have words for yellow, green, red, black and white. The diagram also indicates that if a language has a word for say pink, then it may, or may not, have a word for purple, but it will have colour words for brown, blue etc.

The colour words in a language provide foci, or prototypes, for the colour experience but say nothing about the boundaries between these colour foci.

This sequence corresponds fairly closely to the order in which children acquire colour words and is therefore a cultural example of the discredited evolutionary theory of recapitulation. This is summarised in Ernst Haekel’s phrase “ontogeny recapitulates phylogeny”, suggesting that as embryos develop into adults, they go through stages that resemble the evolution of their species.


Thanks again to my brother-in-law Tony Payne who told me the baboon story, although he cannot remember where he first heard it. See also Beau Geste Hypothesis and Cafetières, Disorder, Chaos and Anarchy


Berlin, B. and Kay, P. (1969) Basic Color Terms: Their Universality and Evolution. Berkeley: University of California Press.

Butterworth, B., Reeve, R., Reynolds, F., & Lloyd, D. (2008). Numerical thought with and without words: Evidence from indigenous Australian children. Proceedings of the National Academy of Sciences of the USA, 105, 13179-13184.

Posted in Architecture, Brain Physiology, Camouflage, Enumeration | Tagged , , , | 2 Comments

Beau Geste Hypothesis

In the 1924 book Beau Geste, and the many film versions that followed it, the climax of the action takes place in the desert at Fort Zinderneuf where members of the French Foreign Legion are attempting to hold off an Arab  attack.

‘As each man fell, throughout that long and awful day, he had propped him up, wounded or dead, set the rifle in its place, fired it, and bluffed the Arabs that every wall and every embrasure and loophole of every wall was fully manned.’ (Wren 1924).

Still from the 1939 film

Still from the 1939 film

Learning Many Songs

Many species of birds have larger repertoires of songs than seems strictly necessary. In an attempt to explain this John Krebs suggested what he called the Beau Geste Hypothesis. (Krebs 1977)

He describes the use by an individual, usually male, bird of a variety of different songs as an attempt to increase the apparent density of occupation of a territory and so discourage the interest of rivals. Singing the same song from many locations within the territory would most likely be interpreted as a single bird moving about, but singing different songs from lots of locations would more likely be interpreted as coming from a number of different birds. In effect different songs add verisimilitude to the individuality of the singer.

Being a good Darwinian, Krebs suggests that having a large repertoire of songs has an evolutionary advantage, because it allows birds to defend larger territories than would otherwise be the case.


Kreb’s hypothesis is that a large repertoire of songs allows a bird to falsely indicate a higher density of territory occupation than there is in reality. I think it not unreasonable to map density to numbers of individuals in the territory and say that the territory being defended has a larger number of competitive occupiers than is actually the case.


Thanks to my brother-in-law Tony Payne whose PhD was in animal behaviour and was latterly Professor of Anatomy at Glasgow University who told me this story, see also Cafetières, Disorder, Chaos and Anarchy


Krebs, J. R. (1977) The significance of song repertoires: The Beau Geste hypothesis. Animal Behaviour 25: 475-478

Wren, P. C. (1924) Beau Geste. John Murray

Yasukawa, K., Searcy, W. A. (1985) Song repertoires and density assessment in red-winged blackbirds: further tests of the Beau Geste hypothesis. Behavioral Ecology and Sociobiology. Vol. 16, Issue 2, 171-175

Posted in Architecture, Camouflage, Enumeration, Illusions | Tagged , , , , | 2 Comments

Counting Cormorants

As a small child I have a vivid memory of a picture in a Wonder Book that showed cormorants  being used by Chinese fishermen. Each bird having a ring round its neck that prevented it eating the fish it caught.

Recently my eldest son Noah brought to my attention a 1979 paper by Pamela Egremont and Miriam Rothschild called ‘Calculating Cormorants’ which described cormorant fishing in China & Japan around 1975.

Fishing at dusk on the Li-Kiang River

Fishing at dusk on the Li-Kiang River

Sometimes a single wading fishermen with a large hat casting a shadow on the water fished with a single bird, but more often a narrow bamboo raft with a single pole was used with 2 or 3 cormorants. These were ringed and tied to the raft. Exactly as shown in K’iP’Ei’s finger painting of 1665.

Finger Painting by K'iP'Ei 17th Century

Finger Painting by K’iP’Ei 17th Century

Egremont and Rothschild report that

“After each cormorant had caught seven fish — and no bird was allowed to return unsuccessfully to its perch — the knots holding their neck bands were loosened and the birds were rewarded by being allowed to fish for themselves. The eighth fish was by long tradition the cormorant’s fish. The procedure must have been followed faithfully in this particular region for decades, for V. Wyndham-Quin* had made careful and more extensive observations of the same phenomenon in 1914. Once these birds have retrieved their tally of seven fish (or to put it more precisely, seven successful sallies have been completed) they stubbornly refuse to move again until their neck ring is loosened. They ignore an order to dive and even resist a rough push or knock, sitting glumly and motionless on their perches………

One is forced to conclude that these highly intelligent birds can count up to seven.”

A footnote added to the proof states

One of us (M.R.) has just visited China (May 1979) and looked in vain for cormorant fishing in progress. Unfortunately the river in Kweilin district is heavily polluted, a matter of great concern to the authorities, but until measures have been taken to improve the situation it is unlikely that cormorant fishing can be pursued in this matchless beauty spot.

Brief Discussion

Counting to seven as illustrated here could simply be a trained response and does not of itself provide any evidence of counting per se, for instance of the ability to count other numerosities.

There are however a number of YouTube videos showing presumably relatively contemporary fishing  using cormorants, for example.

Added 21 July. Lovely picture of cormorant fishing at Xingping, Yangshou county, China in today’s Guardian, and lots of pictures of Japanese cormorant fishing (Ukai) here, but no mention of 8th fish.

Extraneous Information

James VI of Scotland (I of England) for a period took “great delight” in fishing with trained cormorants. (Laufer 1931)

Pamela Egremont (1925-2013) as Lady Egremont was the beautiful chatelaine of Petworth House who charmed Macmillan, Thesiger and Stalin’s daughter

According to her obituary in the Telegraph

(Prime Minister Harold) Macmillan, often shy with women, became devoted to Pamela, who could be very sympathetic. The explorer Wilfred Thesiger found her to be the only member of the opposite sex with whom he felt completely comfortable, apart from his old nanny. Stalin’s daughter, the redoubtable Svetlana, who lived in England for some years, adored and confided in Pamela. Among her other close friends, some of whom she cared for as they became frail, were the writers Gavin Young and Patrick Leigh-Fermor.

Dame Miriam Rothschild (1925-2005) was a world authority on fleas, a practical farmer, animal rights supporter and environmentalist.

According to her obituary in the Guardian

Her interests, although centred on insects and other animals, reached in all directions. To her the moth, its delicate odour, the tiny nematode, the sexual organs of a flea, a Shakespeare sonnet, traditional crafts, great paintings, wild grasses, animals of the field, grandchildren, the place and chemistry of life, all shared the same beauty, the same fascination.

During the Second World War Pamela Egremont and Miriam Rothschild both worked at Bletchley Park.


*Valentine Wyndham Quin was Pamela Egremont’s father.


Egremont E. and Rothschild M. (1979) The calculating cormorants. Biological Journal of the Linnean Society Vol 12 Pages 181-186

Laufer B. (1931) The domestication of the Cormorant in China and Japan. Field Museum Natuarl History (Anthropological Series) 18(3):200-262

Posted in Architecture, Brain Physiology, Enumeration | Tagged , | 3 Comments

Otto Koehler

Numerical Competence in Animals

The German zoologist Otto Koehler (1889-1974) was the first scientist to convincingly demonstrate numerical competence in animals.

The first part of this post is based upon a panel from Counting on neurons: the neurobiology of numerical competence” (Nieder 2005)

Koehler established a number of experimental paradigms, including simultaneous or successive stimulus presentation and matching to sample and oddity matching procedures. (Koehler 1956)

He thought that animals had two numerical capabilities; a visuo-spatial one when the items to be counted were displayed all at once or simultaneously, and a temporal one when the items were displayed successively, one after the other.

He called the first capability “simultaneously seeing the number of items”, what might now be called subitising, and the second “successively acting upon the number of items”.

He tested the simultaneous capability using his match to sample experimental paradigm. A sample numerosity is indicated by ink blots, pebbles or lumps of plasticine. The task of the animal subject is to find one of two box lids that match the sample number, lift the lid off the box and find a food reward inside.

KoehlerCrowOne of Koehler’s animal subjects attempting a match to sample test

One way Koehler tested the second, sequential capability was by training birds to peck a certain number of grains from two piles of grain. For example, a bird trained on ‘five’ could eat all three grains from a small pile and two additional grains from a second, larger pile, before flying off, leaving the rest of the grains untouched. The animals also learned to combine both the simultaneous and the sequential task.

Clever Hans Effect

Clever Hans was a horse who could apparently do quite difficult arithmetic but who was shown in 1907 to be getting unintentional cues from his owner or other people watching him.

Koehler was aware of potential non-numerical cues that the birds might have relied on to solve the tasks, so he eliminated figural, positional and temporal cues to the subjects. To avoid giving the animals unconscious cues, the experimenter was out of sight of each animal throughout the sessions. An automatic spring-loaded device shooed the birds if they made errors. The experimental sessions were filmed and thoroughly analysed off-line. Most notably, Koehler introduced transfer tests in which the punishment contingency was removed. During such transfer tests, the birds successfully applied the learned numerosity discriminations to novel situations without feedback on their behaviour.

Over the years, Koehler and his students tested eight animal species in the numerical competence project and derived upper numerosity discrimination limits for each. 5 for pigeons, 6 for budgerigars and jackdaws, and 7 for ravens, african grey parrots, amazones, magpies, squirrels and humans. His work, which was published in many scientific articles, served as the basis for all the following investigations into non-verbal numerical competence and its neural foundation.

Thinking Without Words

Koehler thought that the numerical competence displayed by animals showed the non-verbal ancestral nature of some human and animal thought.

“When I want to nail and have no hammer, I go round and seek. I do not think in words, this newspaper, that glass would not do; I should better say I do not see them. But casually seeing my foot, I know at once that my shoe will be an excellent hammer. I already have its tip in my hand and strike the nail with its heel, before finding words, such as tool-character, Ersatz and the like. Most brainwaves, even in Man, may be wordless in the beginning. One has enjoyed already the invention, before one seeks the first word to describe it.”

“We owe our deepest thanks to language which, from animals, made us humans and opened the new plane of the mind, developing itself in continuous interaction, between thinking with words and without words. We should be extremely ungrateful, if we did not appreciate our thinking without words which we owe to our animal ancestors. It links the mind to the earth on which we stand. It is in all earthly matters, the touchstone on which to test words as to their validity.”


If one maps emotional and irrational to non-verbal, and intellectual and rational to verbal, this seems to be getting very close to the ideas about problem solving espoused in Design Methods. In particular the hypotheses of Synectics described there.

i) creative efficiency in people can be markedly increased if they understand the psychological processes by which they operate;

 ii) in creative process the emotional component is more important than the intellectual, the irrational more important than the rational;

iii) it is these emotional, irrational elements which can and must be understood in order to increase the probability of success in a problem-solving situation

Donald Schön is credited with developing with Gordon the importance of the hedonic response in the creative process, which they say can take two forms:

1. It is a pleasurable feeling, developed toward the successful conclusion of a period of problem solving concentration, that signals the conceptual presence of a major new viewpoint which promises to lead to a useful solution.

2. It is a pleasurable feeling which occurs in a minor way acting as a moment-to-moment evaluation of the course of the creative process itself.

Gordon goes on to state:

The PROCESS of producing either aesthetic or technical objects is accompanied by certain useful emotional responses, and that these responses must not be rejected as irrelevant, but must be schooled and liberated.


Gordon W.J.J. (1961) Synectics. Harper & Row. New York, Evanston and London

Koehler, O. (1957) Thinking Without Words. Proceedings of the14th International Congress of Zoology. Copenhagen

Niedler, A. (2005)  Counting on neurons: the neurobiology of numerical competence. Nature Reviews Neuroscience 6, 177-190 (March 2005) | doi:10.1038/nrn1626

Posted in Architecture, Brain Physiology, Design Methods, Enumeration, Logic | Tagged , , , , | 1 Comment

Primed Number Lines

As noted in the previous post, there is a right brain hemisphere dominance in attending to visuospatial and numerical information. (Rogers et al 2013) So when patients with a right parietal lesion and therefore a spatial deficit for left side stimuli, are asked to point to the mid point of a line they point to a position right of centre. Or if asked for a number halfway between 2 and 6 might reply 5. (Zorz1 et al 2002)

Subjects without this deficit, judge the centre of visually presented horizontal lines fairly accurately, only making small errors to the left for short lines and small errors to the right for larger lines (Brooks, Della Sala, & Darling, 2014)

Things get interesting when the line to be bisected is made of repeated arabic numerals or number words. If the line is made up of small numbers the perceived midpoint shifts to the left and if the line is made up of large numbers shifts to the right. (Calabria & Rossetti, 2005; Fischer, 2001).

/Users/grahamshawcross/Documents/blog_drafts/children's countingCare in avoiding potential methodological problems such as getting matching line lengths and equal image densities were explored by using french number words of the same length, similar density but different magnitudes; in the case below DEUX and NEUF. (Calabria and Rossetti 2005)


The left edges of the two lines are both defined by a solid vertical edge and in the top pair the right hand edge by two points. But in the canonical and mirror presentations the characters are shifted so that the lines both start and end with the same letter ‘E’.

Importantly subjects were requested to bisect the strings without any reference being made to the numbers making up the lines.


Although in the literature this effect is described as a bias, I think that it demonstrates a priming effect operating on the spatial representation of number. This is because whilst the attention of subjects is not drawn to the number words that form the lines, the words do in fact subliminally effect the decisions being made by the subjects.

The arguments of Calabria and Rossetti are more nuanced than I have presented, including the fact that drawing attention to the task irrelevant number elicits a stronger effect. In most related experiments this is done by asking if the number displayed is odd or even.

However a great deal of unconscious thinking is going on in identifying and extracting the number words from the lines, so one might expect that lines made up of number dots would work.



Brooks, J.L., Della Sala, S. & Darling, S. (2014) Representational Pseudoneglect: A Review. Neuropsychology review, vol 24, no. 2, pp. 148-165

Calabria, M., and Rossetti, Y. (2005) Interference between number processing and line bisection: a methodology. Neuropsychologia 43 779–78

Fischer, M. H. (2001). Number processing induces spatial performance biases. Neurology, 822–826.

Rogers, L. J., Vallortigara G. , R. J. Andrew R. J. (2013) Divided Brains Cambridge Univ. Press, New York.

Shaki S., Fischer M., & Petrusic W. (2009). Reading habits for both words and numbers contribute to the SNARC effect Psychonomic Bulletin & Review, 16 (2), 328-331

Zorzi M., Priftis K. and Umila C. (2002) Brain Damage: Neglect disrupts the mental number line. NATURE | VOL 417 | 9 MAY 2002 | 138-139

Posted in Architecture, Brain Physiology, Embodiment, Enumeration | Tagged , , , | Leave a comment

Even 3-day-old Chicks Do It

An interesting series of experiments has recently been reported in Science.  (Regani et al 2015). These strongly support the idea that many animals and humans represent numbers by a mental number line where smaller values are located on the left and larger values on the right. See Spatial Representation of Number

In the first experiment 3-day-old chicks, once familiarized with a target number 5, spontaneously associated a smaller number (2) with their left side and a larger number (8) with their right.

Experiment 1
Experiment 1

In the second experiment,  when the trained target number was 20, the smaller number (8) was associated with the left side, rather than right side as it was in the first experiment, and the larger number (32) was associated with the right side. Showing that the effect is relative to the trained number and importantly using numbers that are outside the normal subitising range (greater than 5).

Experiment 2

Experiment 2

In both experiments once trained the chicks were presented with pairs of the same number, thus 2 versus 2, 8 versus 8 and 32 versus 32.

In humans and animals there is a distance effect. Numerical comparisons become easier as the difference between the numbers increases. In both experiments the difference between the target number and the smaller and larger numbers was the same, (5 – 2) = (8 – 5) = 3 in experiment 1 and (20 – 8) = (32 – 20) = 12 in experiment 2.

There is also a size effect with comparisons between larger numbers becoming more difficult.

Mental Number Line Direction

This research strongly supports the idea that the direction of the default mental number line is from left to right. However there is evidence that this default direction is fairly easy to un-train, for instance by asking subjects to image the numbers as being on a clock face. (Bächtold et al 1998) Habitual reading direction can also effect the direction of the number line. (Shaki et al 2009) Notwithstanding this, it is clear that cultural preferences, like reading or finger counting direction, cannot in any way cause the effects being exhibited here.


With brain asymmetry, there is a right hemisphere dominance in attending to visuospatial and or numerical information (Rogers et al 2013)


So when patients with a right parietal lesion and therefore a spatial deficit for left-side stimuli, are asked to point to the mid-point of a line they point to a position right of centre. Or if asked for a number halfway between 2 and 6 might reply 5. (Zorz1 et al 2002)


Brain asymmetry, like that described above, is a common and ancient attribute of vertebrates so that:

“Interspecific similarities suggest a continuous and analogical nonverbal representation of numerical magnitude. This indicates that numerical competence did not emerge de novo in linguistic humans but was probably built on a precursor nonverbal number system “


Thanks to my biologist son Noah, for bringing this research to my attention and knowing that I would be interested in it.


Bächtold D, Baumüller M, Brugger P. (1998)  Stimulus-response compatibility in representational space Neuropsychologia 36(8):731-5

Dehaene S., Bossini S., & Giraux P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General 122: 371-396.

Regani R., Vallortigara G., Priftis K., & Regolin L. (2015) Number-space mapping in the newborn chick resembles humans’ mental number line Science. DOI:10.1126/science.aaa1379

Rogers, L. J., Vallortigara G. , R. J. Andrew R. J. (2013) Divided Brains Cambridge Univ. Press, New York.

Shaki S., Fischer M., & Petrusic W. (2009). Reading habits for both words and numbers contribute to the SNARC effect Psychonomic Bulletin & Review, 16 (2), 328-331

Zorzi M., Priftis K. and Umila C. (2002) Brain Damage: Neglect disrupts the mental number line. NATURE | VOL 417 | 9 MAY 2002 | 138-139

Posted in Architecture, Brain Physiology, Enumeration, Logic | Tagged , , , | Leave a comment


This is Archie, our then 3 year old grandson, after spontaneously reassembling a sawn up branch and being asked to pose with it for his delighted grandfather.


Archie’s First Artwork

Seriation is defined as “the forming of an orderly sequence” and was apparently first used in this sense in the 1650s. ( 2015)

The peak usage of the term appears to have been in 1977. Seriation was one of the key tasks used by Piaget to investigate the development of children’s thinking. (Inhelder & Piaget 1958)


Piaget studied seriation by presenting children with a disordered collection of sticks and asking them to arrange the sticks into an ordered series. (Inhelder & Piaget 1958) (Piaget 1964)

PiagetSeriationPiaget’s results illustrated above can be summarised as follows.

Stage I: children (aged up to about 4) make no attempt to line up the sticks or just move them about randomly.

Stage II: children (aged about 5) are unable to arrange a complete series, but can combine the sticks in local ordered pairs or triplets.

Stage III: children (aged about 6) can construct a complete series but only by a process of trial and error.

Stage IV: children (aged about 7)  can ‘operationally’ complete the series, that is select and place say the smallest first followed by the next smallest etc. They are also able to insert a rod taken at random from the pool into its correct place.

Later research has generally supported these conclusions, but with some qualifications. The key stages are now thought to be less sharply defined and the ages a little earlier than those stated above. The ability to seriate also seems to depend on the children’s motivation and their ability to understand and carry out verbal instructions.

Performance is improved if there is greater differentiation between adjacent sizes. The number of sticks also seems to be important, with younger children only being able to serialise smaller numbers of sticks and five or six being the maximum number  before Stage IV is reached.

Transitive Inference

Transitive inferences is an ability that has long been supposed to differentiate humans from animals.

Given the propositions R(a, b) and R(b, c) then R(a, c) can be inferred. This is true for symmetric relations like “equals”, “similar” etc. and asymmetric relations like “bigger than”, “smarter than” etc.

According to Inhelder and Piaget, transitive inference is a late developing ability and 8 year old children can fail to solve transitive problems based on size. It appears even later, at up to 9 or 11, for problems based on weight or volume etc. (Inhelder and Piaget 1958, 1964)

They suggest that the Stage IV ability to seriate operationally at around 7 years of age results from children beginning to understand relational reversibility and relativity.

Reversibility of relations means understanding that if I have a “brother” called John, then John also has a “brother” – me, and that if I am “taller than” John then he is also “shorter than” me.

The relativity of relations means understanding that relations are not absolute attributes of objects, so John can be “taller than” me and “shorter than” Paul.

Another form of transitive inference is given by constructions like if  “A is shorter than B” and “B is shorter than C”, “which is the shortest?” This type of reasoning would seem to be important in putting sets of objects into some sort of order.

There is a “spatial model” theory which suggests that transitive inferences are made on the basis of a mental spatial model constructed in the process of understanding the premises and this is easily understood diagrammatically.





In fact, unless one is a trained logician, it is difficult to understand how one can answer this type of question verbally in one’s head without constructing a visual spatial model  of some sort.

Relational Competency in Animals

Many species display what McGonigle and Chalmers call relational competency, that is similar to those of 5 year old children. (McGonigle & Chalmers, 1996)  See also  What Counts. and Counting Ants

/Users/grahamshawcross/Documents/blog_drafts/seriation/SquirrelMThey trained squirrel monkeys and five year old children to associate box size with colour, the smallest size with red, the next smallest size with green etc. so that when the monkeys were presented with a randomly arranged row of say green boxes they would ‘know’ into which box food had been placed,  the second smallest box, fourth from the left in the example above.

So the uniform colour of the presented boxes was a sort of instruction to select a particular size of box.


The results for each size of box is summarised above, with little difference between squirrel monkeys and 5 year old children. There are good results for the extreme sizes (white and red) and poorer results for the intermediate ones (blue, black and green), probably because these sizes are easier to confuse in a random presentation.

By changing the absolute size of the boxes but conserving their relative sizes it was shown that relative size was being encoded rather than absolute size.

They also found that it was difficult to successfully extend the experiment to more than 5 or 6 boxes, suggesting that this was because of the larger number of possible random arrangements. But given the data above it seems more likely that the problem was differentiating between increasingly similar sizes when they were randomly presented.

If this is the case, it is an example of Weber’s Law because the just discernible difference between adjacent box sizes is inevitably reduced when there are more boxes.


To reduce the amount of dexterity required to seriate a set of objects the task was modified to one of sequentially pointing, on a computer screen, to the box sizes in ascending or descending size order.

/Users/grahamshawcross/Documents/blog_drafts/seriation/SquirrelMFeedback was given via bleeps (positive) and buzzes (negative) but the boxes were not moved and remained in their original place on the screen as indicated above. A record was kept of the number of trials required to reach a criterion of getting 8 out 10 attempts correct.


7 year old children had little difficulty achieving the criterion with 5 or 7 items but 5 year old children had increasing difficulty. Again supporting the notion that children achieve Piaget’s Stage IV, operational seriation, at around 7 years of age when they are becoming capable of transitive inference.

Comparison of Procedures

Whilst the two procedures are superficially similar there is at least one significant difference. Moving sticks or boxes allows one-to-one comparisons to be made on the fly, for instance by moving a candidate box across the other boxes. Pointing in-order on the other hand offers no such opportunities.

Bubble Sort

When one-to-one comparisons can be made, a bubble sort seems to be a way of overcoming the difficulties, experienced by both young children and animals, in ordering collections with more than five or six items.

This is because it only requires the repeated comparison of two adjacent elements to see if they need to be swapped. It is also a procedure that can be carried out manually, by simply swapping the position of two adjacent objects in the collection, leaving the rest in place.

Importantly it also avoids the need for any form of transitive inference.

Donald Knuth, the father of the formal analysis of algorithms, said that the bubble sort “seems to have nothing to recommend it, except a catchy name”. (Knuth 1973) Bubble sorts, however, have been remarkably resistant to his scorn partly because they are very easy to understand and implement. They are also reasonably efficient, at least for partly ordered lists.

Pseudo Code

procedure bubbleSort( A : list of sortable items )
   n = length(A)
     swapped = false
     for i = 1 to n-1 inclusive do
       /* if pair meet swap criteria */
       if swap_criteria(A[i-1], A[i]) then
         /* swap them and remember being swapped */
         swap( A[i-1], A[i] )
         swapped = true
       end if
     end for
   until not swapped
end procedure

The application of this pseudo code, illustrated below for sorting a set of objects in descending order of height, uses repeated iterations. These are necessary to ensure that all possible swaps are carried out, but it might be argued that this is a circular method of creating seriation because it  requires a knowledge of number series.


Manual Version

On the other hand in a manual, non-computer, implementation the swaps can be carried out in any order, without explicit iterations. Just continue to swap objects until no more can be swapped. The procedure shown below relies only on the ability to compare the values of two adjacent squares to see if they need to be swapped and recognising when no more swaps are possible. It is essentially non teleological, one does not have to specify the end result, one simply gets it by carrying out the procedure.


The need for manual dexterity could be reduced by displaying the list on a computer screen and allowing the simian or child  subjects to indicate when a swap was required by pointing at, say the left-hand item of, a pair needing to be swapped.


If an allowable swap is indicated, the swap would actually be made on the screen. As before, success would be measured by comparing the number of correct choices with the number of incorrect choices.

Media Matters

Given that doing a bubble sort does not require a computer, I thought it might be instructive to investigate sorting before the advent of the computer. One of the few examples I could find was instructions on how to sort a hand of cards using what would now be called an insertion sort but might just be thought of as the natural, if slightly geeky, way of sorting a hand of cards.

Sorting Hand of CardsAn insertion sort, as illustrated above , shows why sorting and searching are usually treated as opposite sides of the same coin. (Knuth 1993) In order to sort the cards efficiently one needs to be able to search the cards one is assembling on the left in order to insert the next card into the correct position.

Like playing cards, index cards have the essential quality that the cards, like sticks and boxes, can be easily handled and for instance a new card easily pushed into the correct position in an already ordered set of cards. A dropped set of cards can also be reordered by following a procedure similar to that used for ordering the playing cards above.

CardIndexBoxEdge notched card systems are an example of a mechanical sorting and data retrieval system in use before the advent of computers. They use inverted indexing to allow cards with particular properties to be extracted with a knitting needle like tool.



Finally an IBM Hollerith card, an horrendous example from the SS Race Office, of the card that later became ubiquitous in data processing by computer.



Seriation looks as if it might provide an underlying mechanism for satisfying the first two, and particularly the second, of Gelman and Gallistel’s five principles that define counting. (Gelman & Gallistel 1978)

1. The one-to-one principle. Each item in a set (or event in a sequence) is given a unique tag, code or label so that there is a one-to-one correspondence between items and tags.

2. The stable-order principle (ordinality). The tags or labels must always be applied in the same order (e.g., 1, 2, 3, 4 and not 3, 2, 1, 4). This principle underlies the idea of ordinality: the label ‘3’ stands for a numerosity greater than the quantity called ‘2’ and less than the amount called ‘4’.

/Users/grahamshawcross/Documents/blog_drafts/children's counting

For a fuller explanation see  What Counts.

Unfortunately, unlike Subitising which can be demonstrated in 3 month old children, seriation only appears to be available to children aged 5 or above.

There is a real difference between pointing to objects in the correct order on a computer screen and seriating objects, sticks, boxes or nesting cups, that are ordered by physically moving the objects.

If side by side comparisons can be made, a manual bubble sort method would be an improved experimental paradigm. It allows very  small differences to be discriminated and enables the ordering of large sets of objects where, for instance, adjacent boxes may differ by very little in size.

A bubble sort is simple enough to be carried out manually by simians or small children.  An insertion sort, on the other hand, requires some knowledge of transitive inference so that cards can be inserted into the correct position in the sorted set of cards. This applies to playing cards and index cards.

It is interesting that cards as media are retained in early mechanical data management systems, like edge notch cards and Hollerith cards.

Bibliography (2015). Unabridged. Random House, Inc. (accessed: January 12, 2015).

Gelman, R. & Gallistel, C. (1978) The Child’s Understanding of Number. Harvard University Press, Cambridge M.A.

Inhelder, B. & Piaget, J. (1958) The Growth of Logical Thinking from Childhood to Adolescence. London: Routledge and Kegan Paul. 1958. Pp. xxvi + 35t. 32s.

Inhelder, B. and Piaget, J. (1964) The early growth of logic in the child, London: Routledge & Kegan Paul.

Knuth, D. (1973) The Art of Computer Programming, Volume 3.  Sorting and Searching. First Edition Addison-Wesley Reading Massachusetts

Mareschal, D & Shultz, T. R. (1999) Development of Children’s Seriation: A Connectionist Approach. Connection Science, Vol 11, No. 2 Pp 149-186

McGonigle, B. & Chalmers, M. (1996) The ontology of order. In L. Smith, ed. Critical Readings on Piaget. Routledge, pp. 279-311.

Piaget, J. (1965) The Child’s Conception of Number. Norton, New York.



Posted in Architecture, Classification, Enumeration, Logic | Tagged , , | 2 Comments

Emergent Images

This is a short post on emergent images, still or moving images where objects at first only appear with effort and concentration, but once recognised are very easy to see again even after several months or years. In effect once you have recognised the object you remember it forever.

Emergence refers to the unique human ability to aggregate information from seemingly meaningless pieces, and to perceive a whole that is meaningful. (Mitra et al 2009)

Try your luck with the image below


There are four objects hidden in the image above, with some supposed to be harder to see than others.

I have used this example rather than the more common one, due to R.C. James, because I assume many people have already seen this and would easily recognise the hidden object.



The first image above, of four rabbits, was produced algorithmically from a computer generated 3D model. The aim of the system is to produce emerging images (and videos) of varying difficulty that can resist current object finding bots and yet be able to be recognised by humans. (Mitra et al 2009)


Rather in the style of a CAPTCHA (a Completely Automated Public Turing test to tell Computers and Humans Apart). The emergence algorithm is summarised below.

Algorithm Schematic

Algorithm Schematic

  1. Generate an importance map for the object from the 3D model taking account of its surface geometry, lighting and view position.
  2. Splat the object using the importance map to determine splat centres.
  3. Break the silhouette boundaries, removing some boundaries and perturbing others.
  4. Use enriched splat texture from object to splat the background
  5. Add clutter using copy-peturb-paste.

The success of the system can be seen in this video. Although no objects can be detected in the individual frames, humans can easily track the moving objects when the frames are connected into a video, whilst bots singularly fail to detect and track them.


There seems to be no intrinsic reason why optical illusions of this sort should only use black and white images. As in this 1590 painting ‘The Market Gardener’ by Arcimboldo which when viewed one way up looks like a bowl of vegetables and a face the other way up.


This probably has more to do with our innate ability from an early age to recognise faces when even very approximate features are presented in an appropriate orientation.


This is an effect that Rex Whistler also takes advantage of with his inverted images and symmetrical books. (Whistler and Whistler 1946 & 1978)


Other accidental faces emerge unbidden. A ‘good’ selection is here. Again I think this is an artefact of innate facial recognition rather than an example of an emergent image.


The purpose of this post is to  differentiate emergence, which is in some ways a one-off event, from some other sorts of  illusion where the effect is on-going, automatic, involuntary and cannot be consciously ignored even when you know you are looking at an illusion.


brain_model_fig7This applies to the Müller-Lyer illusion above and the acoustic illusion described in a previous post here and the McGurk effect here.

Illusions like the Ames Window here and Ames Room here are also of this automatic, involuntary type.

These are usually called cognitive illusions because the effect contradicts our expectations. Two other types of illusion are generally recognised, literal illusions and physiological illusions.

Literal illusions can be thought of as the result of normal physical laws, for instance the way a stick can look bent in water.


Physiological illusions are usually the result of sensory overload. Aristotle’s waterfall effect and after images generally fall into this category.

twinkleThe twinkle effect is a perhaps more complicated example where dark spots appear apparantlty randomly within the white dots at the intersection points of a grey on black grid.


Emergent images are to some degree a memory effect, the mental effort of finding the hidden objects somehow ensures that it is easier to find or remember them on subsequent occasions.

Emergent images also support the Gestalt theory that the object only emerges when its component parts are exposed together to give an impression of the whole object.

Emergent images probably relate more to camouflage than other types of optical illusion but in some ways are its opposite. The aim is to create images where objects emerge with effort rather than remain hidden. In effect emergent images might be seen as camouflage that is designed to fail under intense scrutiny.


Gregory, R., 1997. Knowledge in perception and illusion. Phil. Trans. R. Soc. Lond. B 352: 1121–1128

McGurk H., MacDonald J., 1976. Hearing lips and seeing voices. Nature 264 (5588): 746–8

Mitra, N. et al., 2009. Emerging Images. Available at:

Webster C., Glasze G. and Frantz K., 2005. The Two Faces of Rex Whistler Guest Editorial, Perception volume 34, pages 639 – 644

Whistler R., Whistler L., 1946. ¡OHO! Certain Two-faced Individuals now exposed by the Bodley Head.

Whistler R., Whistler L., 1978. ¡AHA! London: Murray


Posted in Architecture, Audiology, Brain Physiology, Camouflage, Embodiment, Graphics Technology, Illusions, Objects | Tagged , , , , , , , | 2 Comments

Number Names and Words

Number Names

George Lakoff has pointed out that we do not normally distinguish numbers from what might be more properly be called number names. (Lakoff 1989)  The most common number naming systems adopt base-10 and use ten single-digit number names, for instance (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9). They then form other multi-digit number names (21,  1342 etc) from these basic number names, or numerals, using a positional representation. With Arabic numerals this starts with the rightmost numeral being the quantity of units and the next leftward numeral being the quantity of tens and so on.

But many other bases are possible, the next most common probably being base-2 or binary, with just two basic number names (0 and 1). The difference between numbers and number names can therefore perhaps best be understood by realising that the number name ‘3’ in base-10 (3 × 100) represents the same quantity or numerosity as the number name ’11’ in base-2 ((1 × 21 ) + (1 × 20)). That is the base used, as well as the digits or glyphs adopted, can change the way any particular quantity, numerosity or number is represented.

Cardinal numbers measure the size of  collections or sets and therefore include the number zero needed to represent the size of an empty collection or set. In English cardinal numbers are nouns.

Ordinal numbers represent position or rank in a sequential, spatial or temporal lists or order and therefore do not include zero, there is no zeroth element in a sequential list. In English ordinal numbers are adjectives.

Number Words

Number names in this sense are different and distinct from number words, the verbal version of numbers, the way numbers are spoken or transliterated, (one, two, three etc.). see Five Finger Exercises

In English, verbal numbers are organised as a hybrid series of additions and multiplications summarised, for the cardinal Arabic number 350172, by the graph below where the plus signs indicate addition and the X signs multiplication.


After Dehaene (1992) Varieties of Numerical Abilities Cognition, 44 1-42

So that ((((three is multiplied by a hundred) and added to fifty) which is then multiplied by a thousand) and added to ((one multiplied by a hundred) added to (seventy added to two)))

This system involves a combination of simple number words; one, two, three etc., some special multiplier words like hundred, thousand etc and the particularly English -ty words like sixty, seventy, eighty and ninety plus the slightly modified twenty, thirty, forty and fifty. And -teen words like thirteen, fourteen etc. plus the unique eleven and twelve.

With Arabic numerals the same cardinal number (350172) is represented positionally; starting with the rightmost numeral being the quantity of units (2) and the next leftward numeral being the quantity of tens (7) etc. Note that Arabic numbers are read, or more accurately generated, from right to left, perhaps betraying their origin.

Chinese number words follow a similar but somewhat simpler, more regular pattern.

Comparison of English and Chinese Number Words

ChineseNumberWordsSome of the extra complexity of English number words derive from spelling conventions rather than word sound, for instance eigh[]teen, fo[]rty and eigh[]ty. There is also some evidence of pronunciation slippage. Thus twelve and twenty to avoid the awkwardness of twoteen and twoty, thirteen and thirty to avoid threeteen and threety and fifteen and fifty to avoid fiveteen and fivety.

Ordinal Number Words

In English the initial verbal ordinal words are the unique firstsecond, and third, but typically ordinals have a th suffix added to the cardinal name for the number, so fourth, sixth, seventh, nineth, and tenth plus the slightly modified in spelling terms fif[]th, and eigh[]th. The multiple powers of ten have an ieth suffix replacing the y ending of the cardinal name, so twentieth from twenty, thirtieth from thirty, fortieth from forty etc. Again these are organised as a hybrid series of additions and multiplications.


In English the Arabic version of ordinals borrow their suffices from the end of their verbal equivalents, so we have 1st (from first), 2nd (from second) and 3rd (from third), followed by 4th .. 20th then 21st, 22nd and 23rd etc.

The first three English ordinals have interestingly varied etymologies.  First derives from the Old English fyr(e)st and Old Norse fyrsthaving the sense of furthest forward, and the German Fürst, a prince, that is furthest forward in rank. Second derives from the Latin sequi followsecundus following and second via Old French into Middle English. Third derives from Old English thridda via English thrid which was the most common spelling until the 16th century.

A Latinate ordinal system is also used to represent importance and precedence,  primary, secondary, tertiary, quaternary etc. which are rarely used beyond the first four. So primarysecondary and tertiary education. This system is also used to indicate a sequence of  dependent effects, thus secondary picketing.

In technical and academic practice Greek ordinals are also used as prefixes proto-, deutero-, trite- and  tetarto-, thus proto-renaissance, protagonist and deuterium.


When speaking of fractions a half is used for 1/2, a quarter for 1/4 and three quarters for 3/4 but a fourth is also used in music. Otherwise ordinals are used as in a third for 1/3, a fifth, a sixth etc. In the more general case a cardinal number is used for the numerator and an ordinal for the denominator, so 2/3 is two thirds and 19/32 is nineteen thirty seconds etc.

/Users/grahamshawcross/Documents/blog_drafts/children's counting


Literate English speakers have no problem reading, writing, comprehending or producing all these systems and transcoding between them even though there is evidence, through the study of patients with deficits in one or more of these capacities, of a neurological dissociation between the verbal and written systems (McCloskey 1992)


Dehaene, S. (1992) Varieties of Numerical Abilities Cognition, 44 1-42

Lakoff, G. (1987) Women, Fire, and Dangerous Things University of Chicago, Chicago Page 150

Lakoff, G. and Núñez, R. (2000) Where Mathematics Comes From Basic Books, New York

McCloskey, M. (1992) Cognitive mechanisms in numerical processing: Evidence from acquired dyscalculia, Cognition 44 107-157

Posted in Architecture, Classification, Enumeration, Logic | Tagged , , , | 1 Comment