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.

Location

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

/Users/grahamshawcross/Documents/blog_drafts/seriation/ColourSor

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)

Origins

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 “

Acknowledgement

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

Bibliography

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 | http://www.nature.com 138-139

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

Seriation

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.

IMG_1581

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. (Dictionary.com 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)

SeriationWordUsage

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.

/Users/grahamshawcross/Documents/blog_drafts/seriation/SquirrelM

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 Charmers 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.

/Users/grahamshawcross/Documents/blog_drafts/seriation/SquirrelM

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.

Pointing

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.

/Users/grahamshawcross/Documents/blog_drafts/seriation/SquirrelM

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)
   repeat 
     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.

/Users/grahamshawcross/Documents/blog_drafts/seriation/ColourSor

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.

/Users/grahamshawcross/Documents/blog_drafts/seriation/ColourSor

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.

/Users/grahamshawcross/Documents/blog_drafts/seriation/ColourSor

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.

chist9

Indecks

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.

Rassenamt

Discussion

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

Dictionary.com (2015). Dictionary.com Unabridged. Random House, Inc. http://dictionary.reference.com/browse/seriation (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

FourRabits

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.

dalmatian_fig2

Algorithm

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)

Captcha

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.

Colour

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.

arcimboldo_1600-1200

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.

oho_1

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

AccidentalFace

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.

Discussion

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.

mullerlyer-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.

basic_refraction

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.

Conclusion

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.

Bibliography

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: http://discovery.ucl.ac.uk/1329981/.

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.

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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.

/Users/grahamshawcross/Documents/blog_drafts/subitising/Subitisi

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.

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

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 (McCloskey 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

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

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.

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

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.

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

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. 

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

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

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

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)

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

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.

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

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.

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

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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