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.

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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.

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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.

Fractions

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.

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Transcoding

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 (McConey 1992)

Bibliography

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

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What Counts

Numerical Abilities

As well as being able to roughly compare continuous quantities, humans and animals of various sorts share a method of recognising small numbers of objects or sequences of events that is independent of language.  See Subitising and  Counting Ants

Both animals and pre-verbal children can judge proportions and numbers of things, sounds, time intervals, smells etc. (Reznikova and Ryabko 2011)

In the natural world the ability to perceive quantities is helpful in many situations, for example, in keeping track of predators or selecting the best foraging grounds.

Jackfruit_tree_in_Gujarat

Counting

In child and animal studies, the following 5 principles are widely accepted as defining the process of 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.

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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

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

3. The cardinal principle (cardinality). The label that is applied to the final item represents the absolute quantity of the set. In children, it seems likely that the cardinal principle presupposes the one-to-one principle and the stable-order principle and, therefore, should develop after the child has some experience in selecting distinct tags and applying those tags in a set.

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4. The abstraction principle (property indifference). Counting can be applied to heterogeneous items. In experiments with children, a child should be able to count such different items as toys of different kinds, colour or shape and to demonstrate skills of counting even actions or sounds. There are indications that many 2 or 3 year old children  can count mixed sets of objects. 

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5. The order irrelevance: the order in which objects are counted is irrelevant.

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Arithmetic in Young Children

In humans numerical ability can be demonstrated in 2 to 3 month old children.

After a period of habituation young pre-verbal children spend less time looking at a familiar scene and more time looking at an unexpected or unfamiliar one. So  an ‘expectancy violation technique’ can be used to assess a child’s understanding of a situation or problem.

The idea is that if infants can keep track of the number of toys being placed behind a screen, they will look longer if the removal of the screen reveals an outcome that violates their expectations. Using this technique very young children can be shown to be capable of  simple small number arithmetic (Wynn 1990)

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So with addition, if  1 doll is initially on the stage and another doll is visibly put onto the closed stage, children expect there to be 2 dolls on the stage when it is opened. This is shown by a lack of surprise. However, when a doll is visibly added and then secretly removed, children are surprised that only 1 doll is on the stage when it is opened, and show extra attention to this outcome.

Similarly with subtraction, when 2 dolls are initially on the stage and 1 doll is visibly removed, children show no surprise when it is opened, but show surprise if 1 doll is visibly removed and then secretly put back so that 2 dolls are on the stage when it is opened.

From Subitising to Counting

Lakoff and Núñez suggest that subitising is the a-priori foundation upon which all other mathematical  abilities are built

Gelman & Gallistel’s 5 Principles describe, in rather set theoretic terms, what is entailed in counting but they do not adequately describe the process of acquiring this capability. This is particularly so with the first two principles; one-to-one correspondence and the stable order (ordinality) principle.

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In particular it is not clear how an innate subitising ability, that is not available to conscious scrutiny, can be used to help tag objects. It might be imagined that the tags could be assigned in order by subitising the size of growing groups of objects, as illustrated above. But this pre-supposes that numerical tags of some sort are available.

It is suggested that children, obviously without any innate knowledge of number words, must learn the number words of their language and map them onto their own innate ordered list of number tags.(Gelman & Gallistel 1978)

Wynn suggests that the necessary tagging is not possible without at least some number words having been learnt.

“In order to understand the counting system-that is, to know how counting encodes numerosity–children must know the meanings of (some of) the number words. They must also know, at least implicitly, that each word’s position in the number word list relates directly to its meaning-the farther along a word occurs in the list, the greater the numerosity it refers to. Without this knowledge, though children might understand the meaning of a given number word, they would not understand how counting determines which number word applies to any given collection of counted entities. Thus children’s developing knowledge of the meanings of the number words is a central part of their understanding of the counting system”. (Wynn 1990)

Counting is thus a culturally supported linguistic activity.

Bibliography

Cantlon, J.F. & Brannon, E.M. (2007). How much does number matter to a monkey (Macaca mulatta)?  J. Exp. Psychol. Anim. Behav. Process. 33: 32-41

Fuson, K. C. (1988). Children’s counting and concepts of number. Springer-Verlag, New York.

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

Koehler, O. (1956). Thinking without words. — In: Proceedings of the 14th International Congress of Zoology, Copenhagen, pp. 75-88.

Lakoff, G. & Núñez, R. E. (2000) Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being Basic Books

Markman, E. M. (1989). Categorization and naming in children. MIT Press, Cambridge, MA:        .

Reznikova, Z. and Ryabko, B. 2011.  Numerical competence in animals, with an insight from ants Behaviour 148, 405-434

Wynn, K. (1990) Children’s Understanding of Counting. Cognition 36 155-193

 

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Counting Ants

This is not about how to count ants but how ants count.

The post follows research by Zhanna Reznikova and Boris Ryabko that investigates the numerical capacities of ants using ideas from Information Theory such as Shannon entropy and Kolmogorov complexity.  (Reznikova & Ryabko 2012)

Ants

Some species of red wood ant, that live in colonies of approximately 800 – 2000 individuals,  have a highly specialised social structure that includes having stable foraging teams of 5 to 9 ants. These teams are lead by a scout ant whose function is to find food sources, the location of which is then communicated to the other members of the foraging team.

A scouting ant contacting members of its team. Photo by Nail Bikbaev.

A scouting ant contacting members of its
team. Photo by Nail Bikbaev.

Experimental Procedures

In the experiments the ants in the foraging teams are individually identified with coloured dots. Sugar syrup is is placed  in one of the small reservoirs on the terminal leaves of floating mazes of various designs. All the other terminal locations have water in their reservoirs.

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A scout ant is placed at a randomly selected location of the food and then allowed to return through the maze to its foraging team in the colony. The maze is then replaced by a new but identical maze but with the food source replaced with water. This prevents the foragers simply following any scent trail left by the returning scout or being directly attracted by the smell of the sugar syrup.

The scout ant sometimes needs up to four trips before he contacts his foraging team. Once he has contacted them, he is given time to communicate the location of the food to his foragers, and is then temporally isolated. The foraging team then have to find the location on their own using the scout’s instructions. After they find the correct location, food is given to them. No ants are harmed in these experiments although during the experiments the ants could only obtain food from the maze and only once every 2 to 3 days.

The lack of food on the replacement maze(s) means that the last lap of the foragers’ search cannot be guided by the sight or smell of the food.

With maze type A in the diagram above, the communicated message might be something like “walk forward distance X then turn left and food is in front of you”. With maze B, the message might be “after turning right walk forward distance Y then turn left and the food is in front of you”. For maze C this might be “after turning left walk forward distance Z then turn left and the food is in front of you”

Alternatively the distance and direction could be given as an absolute bearing plus a distance, so the message would always be “walk distance X along bearing A from the sun or magnetic north”. This might be interpreted as trying to keep as near as possible to the bearing whilst consuming the travel distance.

/Users/grahamshawcross/Documents/blog_drafts/animal counting/Ant

In the experiments 2 things were measured:-

1. how long the scouts took to communicate the location of the food source to the other members of their foraging team. This was measured from first antenae contact to at least 2 foragers leaving to recover the food.

2. the success of the foragers in finding the location of the food source.

Information Content

In rational communications systems the length of a message is a measure of its information content.

The information content of messages like “turn right, walk forward distance X, turn left etc” depends on the number of turns, more turns result in longer messages.

With the bearing and distance method “walk distance X along bearing A”, the message length and information content remain constant for all target positions, unless longer distances require longer messages such as would be the case in tapping out the number of steps to be taken.

So what is required is a maze design in which the experimenters know the amount of information that has to be transmitted. Such a design is the binary tree maze where the subject only has to repeatedly decide whether to turn left or right.

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The simplest binary tree maze, with 1 fork and 2 leaves, is the Y-shaped maze . This has 1 fork representing 1 binary choice, turn left or right. This corresponds to 1 bit of information which the scout ant has to transmit to the foragers. In the experiments the number of forks was increased incrementally from 2 to 6. So the number of turns required to choose the correct path was equal to the number of bits of information that had to be transmitted.

Artificial ant nest and a binary tree maze placed in a bath with water. Photo by Nail Bikbaev.

Artificial ant nest and a binary tree maze
placed in a bath with water. Photo by Nail Bikbaev.

Results

335 scout ants and their foraging teams took part in all the experiments with the binary tree mazes, and each scout took part in ten or more trials.

338 trials were carried out with mazes with 2, 3, 4, 5 and 6 forks.

The scout ants took progressively longer to communicate paths in deeper mazes  (with more turns) that is they transmitted more information.

In simple terms, if t is the time taken to transmit the required information then

           t = ai + b

Where i is the number of forks (the depth of the maze)

a is the amount of time required to transmit 1 bit of information

and b is an introduced constant used to represent extra information that might be transmitted such as the signal “food”.

The rate of information transmission a derived from the above equation was approximately 1 minute per bit in three ant species.

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Complexity

In the 4 bit binary tree maze diagram above, the highlighted path to the food source is represented by the coded leaf string [RLRR] meaning start [, turn right [R, then left [RL, then right [RLR and right again [RLRR and finish [RLRR].

All the possible routes through a 4 bit deep maze can be represented by the 16 combinations of its possible end leaf codes. The question arises “are any of these routes less complex than some of the others?”. The authors attempt to investigate this by seeing if any of the coded end strings can be compressed. They call this Kolmogorov complexity but it is perhaps easiest to understand in terms of run-length encoding where repeated values are replaced, wherever possible, by a count of the values plus the value.

[LLLL] –>[4L]          [LLLR]–>[3LR]       [LLRL]–>[2LRL]     [LLRR]–>[2L2R]    

[LRLL]–>[LR2L]     [LRLR]–>2[LR]       [LRRL]–>[L2RL]    [LRRR]–>[L3R]

[RLLL]–>[R3L]        [RLLR]–>[R2LR]    [RLRL]–>2[RL]       [RLRR]–>[RL2R]    

[RRLL]–>[2R2L]     [RRLR]–>[2RLR]    [RRRL]–>[3RL]      [RRRR]–>[4R]

So for instance turn right 4 times [RRRR]–>[4R] is a less complicated route than right, left, right and right again, [RLRR]–>[RL2R]. The first example [RRRR]–>[4R] shows a compression ratio of 50%, from 4 to 2 characters. On the other hand the second example [RLRR]–>[RL2R] shows 0% compression. This is because the original and compressed strings both have 4 characters.

In total in a 4 bit maze there are 2 routes with 50% compression, 6 with 25% compression (4 to 3 characters) and 8 with 0% giving a total compression ratio for the maze of ((2 x 2) + (6 x 1)) / (16 x 4) = 15.6%.

There are 8 combinations of possible routes through a 3 bit deep maze

[LLL]–>[3L]             [LLR]–>[2LR]           [LRL]                         [LRR]–>[L2R]

[RLL]–>[R2L]          [RLR]                           [RRL]–>[2RL]        [RRR]–>[3R]

[LLL]–>[3L] and [RRR]–>[3R] both represent 33% compression from 3 to 2 characters with a total compression for the maze of (2 x 1) / (8 x 3) = 8.3%.

Finally their are 4 possible routes through a 2 bit maze.

[LL]–>[2L]               [LR]                              [RL]                             [RR]–>[2R]

No compression is possible with this maze because with [LL]–>[2L] the original and compressed strings both have 2 characters as does [RR]–>[2R]. In general more compression is possible with longer strings.

Complexity Experiments

A number of experiments were carried out with the express purpose of seeing if scout ants recognised some routes as being less complex than others. This was done by using selected paths that were thought to be more or less complex (I have added the Run-Length Compression columns to the authors’ data below).

Compression.xlsx

The authors judged the routes in the lower shaded portion of the table as being more complex than the unshaded part. These results are represented in the graph below.

/Users/grahamshawcross/Documents/blog_drafts/animal counting/ComThe authors claim that where the maze depth is 6, and most compression is possible, the communication time for the simpler, more repetitive (more compressible) routes is significantly less than the communication time for the more complex (less compressible) routes.

Arithmetic

In assessing the ability of ants to do arithmetic another piece of information theory was used. This says that in any reasonable communication system the frequency of use of a message is inversely correlated to its length.

The informal pattern is quite simple: the more frequently a message is used in a language, the shorter is the word or the phrase coding it. Professional slang, abbreviations, etc. can serve as examples. This phenomenon is manifested in all known human languages as well as in technical systems of information transmission. (Reznikova and Ryabko, 2011)

AntMaze30

With this in mind, the following series of experiments were carried out where the statistical distribution of the location of the food source was deliberately manipulated. Using the set-up illustrated above, all the experiments were undertaken in 3 stages:-

Stage 1. where the location of the food source was selected randomly with an equal chance of being in any particular location. The chance is 1 in 30 or 3.33%.

Stage 2. in which the statistical location of the food source was manipulated. In some experiments locations 7 and 14 were favoured and locations 10 and 20 in others. In both cases the 2 favoured locations had a 30% chance of being selected and the remaining locations a (100% – (30% x 2)) divided by 28 or a 1.43% chance. In other experiments one location, number 15, had a 50% chance of being selected and the others a (100% – 50%) divided by 29 or a 1.73% chance.

Stage 3. in which the location was again randomly selected exactly as in Stage 1.

  ArithmeticGraphAfter Ryabko and Reznikova, 2009

Stage 1 results were consistent with the binary maze experiments described earlier with a near linear relationship between the number of the branch with the food i and the amount of time t needed to transmit the necessary information and  t = ai +b. Stage 1 results are indicated by the black dots in the graph.

In Stage 2, by design, the supposed messages the food is on branch 7 (or 10) and the food is on branch 14 (or 20) was transmitted many more times than the food is on any other branch. In fact more than 40 times as often. With the one favoured selection on branch 15 the message the food is on branch 15 is transmitted a little less than 30 times as often as the food is on any other branch. In the graph the favoured branch indices are 10 and 20.

In Stage 3 the results are different to those in Stage 1. Times are much shorter and there is no linear relation between time and branch number. There is also a reduction in time around the favoured position(s) of Stage 2. Stage 3 results are indicated by pink squares in the graph.

In the first stage of  the experiments for example the ants took 70 to 82 seconds to transmit the information that the syrup was on branch number 11 and only 8 to 12 seconds for when it was on branch number 1. At the third stage, it only took 5 to 15 seconds to transmit the information that branch number 11, which was nearest to the favoured branch number 10, had the syrup on it.

The authors suggest that this means the ants  have changed their mode of presenting the data about the number of the branch containing the food. They suggest that the information is transmitted in two parts; firstly information about the index number of the nearest favoured location and secondly the offset which has to be added to or subtracted from this number.

The number of the favoured location has to be communicated because the scout ant has no other way of marking it. This is because the maze is replaced as soon as the scout makes contact with his scouts in the nest. The fact that the favoured index and offset are communicated in some way is vouched for by the forager ants’ remarkable success in finding the correct location.

Statistical analysis (Ryabko & Reznikova, 2009) supports the hypothesis that at the third stage of the experiment the transmission time is shorter when the branch is near a favoured branch.

The authors’ interpretation is that at this stage of the experiment the ants used simple additions and subtractions, achieving economy in a manner reminiscent of the human numerical system. When using numerical systems, people unconsciously have to perform simple arithmetical operations, for example, 13 = 10 + 3. They suggest that this is particularly obvious with Roman numerals, for example, VII = V + II and IV = V – I.

Discussion

As  illustrated here with ants, mazes are useful in experimental situations where verbal communication is impossible or undesirable. Mazes can be designed to require precise amounts of information to be communicated.

Information Theory provides a number of useful paradigms for the investigation of numerical capabilities. In particular that information content and message length are positively correlated and that frequency of use and message length are inversely correlated.

In all the experiments extreme care was taken to ensure that the foraging ants could only find the location of the food by receiving information  communicated directly to them by the scout ant.

It is not clear in the binary maze experiments or Stage 1 of the arithmetic experiments if distance alone, or in combination with a bearing, could account for the increased length of the messages transmitted. In the binary maze experiments, where distance to every leaf was the same, it seems likely that sequences of left and right  turns were being communicated but in Stage 1 of the arithmetic experiment distance alone might be sufficient, for instance by recalling the number of steps to be taken.

This is similar to the somewhat controversial bee-waggle dance where distance is communicated by the length of the dance or in some accounts the number of cycles performed. (Frisch 1968) (Gould 1979) In bees and ants sound also seems to be important in the recruitment process.

waggle

Kolmogorov complexity or run-length encoding is an interesting way of investigating complexity. Is the degree of compression possible a good measure of the complexity of an image?

The complexity experiments show that ants are able to modify their communications to take advantage of repetitions in the messages. This is particularly so when the binary maze depth is 5 or 6 and more compression is possible.

In the arithmetic experiments the authors suggest that ants are able to perform addition and subtraction with small numbers and have numerical capacities that are approximately equivalent to those of 2 year old children, rhesus monkeys and chimpanzees.

Bibliography

Frisch, K. Von, 1968. The role of dances in recruiting bees to familiar sites. Anim. Behav., 16: 531-533.

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

Gould, J., 1976. The Dance-Language Controversy. The Quarterly Review of Biology, Vol. 51, No. 2 (Jun., 1976), pp. 211-244

Shannon, C.E., 1948.  A mathematical theory of communication. Bell Sys. Tech. J.  27, 379-423, 623-656.

Reznikova, Z. and Ryabko, B. 2011.  Numerical competence in animals, with an insight from ants. Behaviour 148, 405-434

Reznikova, Z. and Ryabko, B. 2012.  Ants and Bits.  IEEE Information Theory Society Newsletter March 2012

Ryabko, B.and  Reznikova, Z. 2009. The Use of Ideas of Information Theory for Studying “Language” and Intelligence in Ants. Entropy 11, 836-853; doi:10.3390/e1104083

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Cafetières, Disorder, Chaos and Anarchy

At breakfast this morning my wife spilt the coffee because she hadn’t aligned the strainer in the lid of a Bodum cafetière with its pouring sprout.

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I suggest this is a stupid design, and “why aren’t the strainer holes all round the edge of the lid, and then it would always pour properly”.

At which my anatomy professor brother-in-law mischievously explodes that this “would bring disorder, chaos and anarchy into the world”.

My wife and her sister agree that there is more order in the current design and that it in some way it teaches you how it works. This is because with an approximate 5 to 1 chance of failure you just have to get it right and that having to think about the objects you are using is in some way good for you and society in general.

I disagree, I think the world is better ordered if you don’t have to think about how things work.

Incidentally, here on Skye cuckoos are signing loudly right outside the house.

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The Bodum cafetière also has a strange unperforated symmetrical bulge opposite the strainer section. Is this a safety measure, so that you can actually prevent it from pouring altogether?

A haptic improvement would be to put a finger or thumb shaped indentation, or even a mark, in the lid above the strainer so that it could be more easily aligned with the spout.

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William Tutte’s Hidden Past

If William Tutte is remembered at all by architects, it is for his contribution to solving the problem of Squaring the Square . (Tutte 1958) A solution using Graph Theory and Kirchhoff’s Laws for electrical flow in wires that was subsequently used in Philip Steadman’s The Automatic Generation of Minimum Standard House Plans. (Steadman 1970)

112x112squaredsquare

This last, ultimately failed enterprise is explained in some detail here.

Bill Tutte died in 2002 and his obituaries, like this from the Guardian, reveal that he also had a wider influence on world events.

Continue reading

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Missing Pullover Found

My long lost 1970s pullover, slip-over, or perhaps more properly 70s tank-top, has turned up at the back of a cupboard. Last year we had turned the house upside down looking for it.

P1020204 copy

The Missing Pullover

The design is apparently based on the colour theory of Interaction of Colour, (Albers 1963)  and illustrates the first of  The Twelve Fold Ways from Stanley’s Enumerative Combinatorics, (Stanley 1986 and 1997). The first way being n-tuples of x things with enumeration formula x to the power n.

Here there are 2 objects, an inside and an outside, and 4 colours giving 24 = 16 different combinations. Ignoring the 4 same-on-same combinations gives the 12 unique combinations numbered below.

pullover_one

Using a spreadsheet type program each of these 12 combinations is then associated with a random number function.

pullover_three

The combinations are then sorted on their associated random numbers and this is repeated as often as necessary with newly generated random numbers.

pullover_two

Repeatedly applied selections

This gives an even mixture because every combination is used before it is used again. This ensures that there are equal numbers of each combination, and therefore that equal numbers of balls of wool are required.

pullover_four

Knitted Sample

An equivalent procedure would be to put, say cardboard samples, representing each of the 12 combinations in a bag and drawing them out blindfold one-by-one until none are left, then putting all the cardboard samples back in the bag and repeating the procedure.

An even mixture would not be guaranteed if each sample was drawn out blindfold and then immediately put back in the bag before making another selection, such a method would just statistically tend towards an even distribution.

Architectural applications of this techniques to follow.

Unfortunately, as perhaps the observant will have noticed, the long lost pullover was not made in accordance with the knitted sample or the procedure above, but appears to just randomly list all the 2 colour combinations of 3 colours. I think then that it had better go back in the cupboard.

Bibliography

Albers, J., 1963. Interaction of Colour, Yale University Press.

Knuth, D.E., 2005. The Art of Computer Programming, Volume 4 Fascicle 2, Generating All Tuples and Permutations .Addison-Wesley

Stanley, R.P., 1986. Enumerative Combinatorics (Volume 1),Wadsworth & Brook.

Stanley, R.P., 1997. Enumerative Combinatorics (Volume 2), Cambridge University Press.

 

Posted in Aesthetics, Architecture, Design, Enumeration, Knitting | Tagged , , , , , , | 1 Comment

Spatial Representation of Number

Francis Galton

“…this peculiarity is found so far as my observations have extended, in about 1 out of every 30 adult males or 15 females. It consists in the sudden and automatic appearance of a vivid and invariable “Form” in the mental field of view, whenever a numeral is thought of, and in which each numeral has its own definitive place. This Form may consist of a mere line of any shape, of a peculiarly arranged row or rows of figures, or of a shaded space.” (Galton 1880)

VisualisedNumerals1

VisualisedNumerals2The term numeral is now more usually used to refer to the digits of a say  base 10 number system, with numerals 0 1 2 3 4 5 6 7 8 9. In modern parlance the visualisations above are therefore of numbers rather than numerals. They also seem to be surprisingly biased towards representing the verbal -ty number words, twenty, thirty etc. (B, S1, H also -teen word numbers, thirteen, fourteen etc. R, EN, NL, RN and ED) see earlier post about verbal number words.

Colour and brightness play a part in some representations (C, GE, GS, EN, MT, TEW) and lines shown in the diagrams sometimes do not appear in the actual mental representation (WHP, MT). The number 12 also seems to be well represented (B, WS, C, GS,MT, GH, THW, PGE, LMH, NL, RN,CH) probably because of its importance in clock counting. There is also a left-to-right and / or bottom-to-top preference in representing increasingly large numbers, but this is by no means universal.

In summary this effect is automatic and involuntary, located in a fixed internal visual space, is idiosyncratic, emerges in childhood, is stable over time but is far from universal. Because of this the effect is usually thought to be a specific form of synesthesia. (Hubbard et al 2005)

SNARC Effect

The Spatio-Numerical Association of Response Codes (or SNARC) effect refers to the fact that when subjects, doing some task, are presented with Arabic numbers in the range 1-9, they respond more quickly to small numbers (1-4) with the left hand and more quickly to large numbers (6-9) with the right hand. (Dehaene et al 1993)

According to Dehaene and his associates, the effect works for single (experiment 1) or two digit numbers (experiment 2) and is not affected by the subjects being left or right handed (experiment 5) or crossing hands (experiment 6) but is reduced for subjects with a right-to-left reading habit (experiment 7). It is absent for letters (experiment 4) but present for number words (experiments 8 and 9).

The effect seems to be a relative one, because when the presented number range is limited to the numbers 1 to 5, people respond more quickly with the right hand to the numbers 4 and 5, the  right-hand end of the putative number line. But when the range presented is 4 to 9 people respond more quickly with the left hand to the same numbers 4 and 5, that have now become the left-hand end of the number line.

From these experiments Dehaene and his collaborators suggested that:-

“…the representation of number magnitude is automatically accessed during parity judgment of Arabic digits. This representation may be linked to a mental number line […], because it bears a natural and seemingly irrepressible correspondence with the natural left – right coordinates of external space.”

As with subitising the response is automatic but learnt, see Subitising.

But unlike subitising which is identifiable in children as young as 3 months old, the SNARC effect only appears in children’s responses when they are at least 7 years or by some accounts 9 years old.

The task in Dehaene’s two digit experiment (experiment 2) was a parity (odd or even) judgement and it has been pointed out that the parity of 2 digit numbers can be assessed entirely from the last digit, so there is some doubt that the effect extends to two digit numbers.

The effect is also observed with eye movement, responses by looking to the left being faster when recognising small numbers and faster to the right with large numbers.

A similar effect occurs when people are asked to generate numerals randomly. If their head is turned to the left they tend to generate more lower numbers and more higher numbers when their head is turned to the right.

It has also been observed that when the distance between thumb and forefinger is varied; the greater the distance being held the more likely a random numeral is to be larger than expected. (Andres et al. 2004)

Priming

The SNARC effect requires no conscious effort and in fact takes place entirely unconsciously a fact that is emphasised by the fact that a priming effect has also been observed. So if a numeral is displayed that is unrelated to the task being undertaken, then the numeral displayed can affect whether the right or left hand is used preferentially.

Most authors assume that the parity (odd or even) judgement, used in most of the reported experiments, is just such a task; but the subjects are making a judgment about the displayed number and therefore paying attention to it. I think to demonstrate a true priming effect the task being carried out should not draw attention to the displayed numeral or use it in any way.

Reading Direction

The effect also seems to be related to the cultural background of subjects, particularly their habitual reading direction. The graph below shows the difference between right-hand and left-hand reaction times for a group of Canadian students with left-to-right reading habits and displays a normal SNARC effect.

canadians

After Shaki S., Fischer M., & Petrusic W. (2009)

However, monolingual Arabic speaking Palestinians who are therefore right-to-left readers exhibit an inverse SNARC effect.

palestinians

After Shaki S., Fischer M., & Petrusic W. (2009)

And the effect almost disappears with monolingual Hebrew speaking Israelis, who read text from right-to-left but read Arabic numbers from left-to-right, .

israelis

After Shaki S., Fischer M., & Petrusic W. (2009)

This shows that the SNARC effect is susceptible to habitual reading direction, but there is also evidence that it is an effect that is fairly easy to un-train. For instance if subjects are asked to think about numbers as if they are on a clock face, where the numbers increase in an approximate right-to-left direction, then the effect is not observed. (Bächtold et al 1998)

Incidentally, the graphs above all display a particular cultural preference, which might be relevant to the SNARC effect itself. The abscissa (x axis) is shown with values that increase from left to right. And the secondary ordinates (y axis) show values that increase from bottom to top, exhibiting the lower-to-higher small-to-large number metaphor.

Finger Counting

Apparently in all human cultures children use finger counting before being taught arithmetic. (Butterworth 1999)

FingerCounting

In a study of Scottish students, 83% right-handed, 10% left-handed and 7% ambidextrous, 66% started finger counting on their left hand (left starters) and 34% on their right hand (right starters), a significant difference. Of the left starters 92% started counting on the thumb, 3% on the index finger and 4% on the pinkie or little finger. With the right starters 80% started counting on the thumb. So in total 61% (92% of 66%) are left starters who start to count with the left hand thumb and exhibit the counting pattern illustrated above, that is a pattern that corresponds to a left-to-right number line representation. (Fischer 2008) Fischer himself suggests that there are studies that disagree with these findings. (Satoa and Lalaina 2008)

I think there may be some problems with the survey method used to collect this data; in particular the fact that the questionnaire included the diagram above (without the added numerals) showing the hands palm-up, perhaps unconsciously suggesting to respondents a left-right number line.

Statistically right starters have a small and significantly weaker SNARC effect than left starters who have a normal SNARC effect. This at least indicates that that finger counting habits  exert an influence on numerical cognition. See also Five Finger Exercises

 Sex Difference

It is reported that the SNARC effect is weaker in females than in males as shown in the graph below. (Bull, Cleland and Mitchell 2013) It is hypothesised that this may be due to the fact that the inferior parietal lobe, identified by fMRI studies as being involved in numerical cognition, is 25% bigger in men than in women, or in jest that men have a greater arm span than women.

SexDifferences

from Bull, Cleland and Mitchell 2013

Button Labels

Interesting things start to happen if labels “left” and “right” are put on the left-hand and right-hand buttons (congruently) or in Stroop like fashion put on the wrong buttons (incongruently). The arrangements are shown in the physical account on the right hand side of the diagram below, where the expected preference for the centrally displayed target number is indicated by the bold buttons. (van Dijck & Fias 2011)

/Users/grahamshawcross/Documents/blog_drafts/SNARC-effect/Incong

After van Dijck & Fias (2011)

With a size comparison task and instructions if  the target number is less than 5 press the  button labelled “left” and if the target number is greater than 5 press the button labelled “right” or vice-versa, the following results are obtained.

/Users/grahamshawcross/Documents/blog_drafts/SNARC-effect/Incong

After van Dijck & Fias (2011)

The authors believe that subjects are responding to the conceptual account on the right in the table above, where the incongruent preferences are reversed. They suggest that the SNARC effect derives from congruency between conceptual categories and not from congruency between a position on a mental line and left to right responses. They also suggest that the conceptual account listed above, maps to a more general set of left-to-right categories such as:-

/Users/grahamshawcross/Documents/blog_drafts/SNARC-effect/Incong

After Proctor & Cho 2006

This has lead some observers to believe that the SNARC effect is as much a verbal phenomena as a numerical one.

Discussion

All authors seem to agree that numbers are at least to some extent represented spatially and  that the SNARC effect is in some way a real effect.

Sequence-space synaesthetes, like Galton’s subjects, do not appear to have unusually strong SNARC responses, probably indicating that they are using a separate mental mechanism.

The SNARC effect is essentially statistical, that is not experienced by everyone or experienced in varying degrees by others. For instance cultural differences, the weaker effect in the 34% of right starting finger counters and the weaker effect in women.

There is an emerging tendency to see the SNARC effect as a verbal rather than a specifically numerical effect.

The embodied cognition paradigm suggests that all of our knowledge is represented together with the sensory and motor activity that was present during its acquisition. So it should be expected that an abstract ability such as numerical cognition inherits the functional properties of more basic perceptual and motor process such as subitisation,  finger counting, number lines and other relational metaphors. (Lakoff & Núñez 2000)

Speculation

It would be good to see if the SNARC effect could be used to prime some desired action such as a preferential turn to the left if a small number is displayed in the field of vision or to the right if a large number is displayed. But first it would be necessary to establish that a true priming effect can be established. That is to have the display of small or large numbers influence peoples’ actions without attention having been drawn to the displayed number.

For instance the famous effect of having a poster with eyes rather than flowers over a British university staff common room honesty box increasing the amount of money being put into the box. (Kaheman 2011) (Bateman, Nettles and Roberts 2006)

eyes_chart_1

If such a priming effect could be demonstrated for numbers, then it would be fairly simple to build a maze like computer game which users could walk through and only have to decide to turn left or right at each T junction where a randomly selected number was displayed. A game that could obviously be instrumented to automatically collect data on the effect.

Finally it would be interesting to see if the SNARC effect could be exported into the real world, for instance 7 George Square, which from this position has symmetrical left and right stairways leading into the building.

PyscholgyDepartment

Bibliography

Andres, M. , Davare, M. , Pesenti, M. , Olivier, E. & Seron, X. (2004) Number magnitude and grip aperture interaction. Neuroreport 15, 2773–2777 .

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

Bateson M., Nettle, D., Roberts, G. (2006) Cues of Being Watched Enhance Cooperation in a Real World Setting Biology Letters 2 412‐14

Bull, R., Cleland, A.A., & Mitchell, T. (2013) Sex Differences in the Spatial Representation of Number Journal of Experimental Psychology: General 142(1) 181-192

Butterworth B. (1999) The Mathematical Brain Macmillan, London

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

van Dijck J. P. & Fias W. (2011) A working memory account for spatial-numerical associations Cognition 119(1) 114-119

Fischer M. H. (2008) Finger counting habits modulate spatial-numerical associations cortex 44 386-392

Fischer M. H. (2011) The spatial mapping of numbers – its origin and flexibility in Language and Action in Cognitive Neurosciences, eds Coello Y., Bartolo A., editors. London, Psychology Press

Fischer M. H., Castel A. D., Dodd M. D., & Pratt J. (2003). Perceiving numbers causes spatial shifts of attention. Nature Neuroscience, 6, 555–556.

Galton F. (1880) Visualised Numerals Nature Vol. 21 252-6, 494-5

Galton, F. (1881) The Visions of Sane Persons Popular Science Monthly Volume 19 August 1881

Hubbard E. M, Manuela Piazza, M., Pinel, P. & Dehaene, S. (2005) Interactions between number and space in parietal cortex Nature Reviews Neuroscience 6, 435-448 (June 2005)

Kahneman D. (2011) Thinking Fast, Thinking Slow. Allen Lane. p 57

Lakoff G. & Núñez R. E. (2000) Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being Basic Books

Proctor R. W. & Cho Y. S. (2006) Polarity correspondence: A general principle for performance at speeded binary classification tasks. Psychological Bulletin 132(3) 416-442

Satoa M. and Lalaina M. (2008) On the relationship between handedness and hand-digit mapping in finger counting cortex 44 393-399

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

Wood G., Nuerk H. C., Willmes K., Fischer M. H. (2008). On the cognitive link between space and number: a meta-analysis of the SNARC effect. Psychol. Sci. Q. 50, 489–525.

Zebian S. (2005) Linkages between number concepts, spatial thinking and directionality of writing: The SNARC effect and the reverse SNARC effect in English and Arabic monliterates, biliterates and illiterate Arabic speakers. Journal of Cognition and Culture 8(1–2) 165-190

Posted in Architecture, Brain Physiology, Embodiment, Enumeration | Tagged , , , | 1 Comment