Threesology Research Journal
Artificial Intelligence and 3sology (56K)
Page 29

Note: the contents of this page as well as those which precede and follow, must be read as a continuation and/or overlap in order that the continuity about a relationship to/with the typical dichotomous assignment of Artificial Intelligence (such as the usage of zeros and ones used in computer programming) as well as the dichotomous arrangement of the idea that one could possibly talk seriously about peace from a different perspective... will not be lost (such as war being frequently used to describe an absence of peace and vice-versa). However, if your mind is prone to being distracted by timed or untimed commercialization (such as that seen in various types of American-based television, radio, news media and magazine publishing... not to mention the average classroom which carries over into the everyday workplace), you may be unable to sustain prolonged exposures to divergent ideas about a singular topic without becoming confused, unless the information is provided in a very simplistic manner.

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Artificial Intelligence and 3sology Introduction
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Because binary computations relies on a type of conservation of number, though not explicitly in the quantitative sense of calculation beyond an analogical usage of pairing, it is of need, as previously mentioned, that the topic of Subitizing needs to be taken account of:


Subitizing is the rapid, accurate, and confident judgments of numbers performed for small numbers of items. The term was coined in 1949 by E.L. Kaufman et al.,[1] and is derived from the Latin adjective subitus (meaning "sudden") and captures a feeling of immediately knowing how many items lie within the visual scene, when the number of items present falls within the subitizing range.[1] Number judgments for larger set-sizes are referred to either as estimating if insufficient time is available for observers to accurately count all the items present, or counting otherwise.

The accuracy, speed, and confidence with which observers make judgments of the number of items are critically dependent on the number of elements to be enumerated. Judgments made for displays composed of around one to four items are rapid,[2] accurate[3] and confident.[4] However, as the number of items to be enumerated increases beyond this amount, judgments are made with decreasing accuracy and confidence.[1] In addition, response times rise in a dramatic fashion, with an extra 250–350 ms added for each additional item within the display beyond about four.[5]

While the increase in response time for each additional element within a display is 250–350 ms per item outside the subitizing range, there is still a significant, albeit smaller, increase of 40–100 ms per item within the subitizing range.[2] A similar pattern of reaction times is found in young children, although with steeper slopes for both the subitizing range and the enumeration range.[6] This suggests there is no span of apprehension as such, if this is defined as the number of items which can be immediately apprehended by cognitive processes, since there is an extra cost associated with each additional item enumerated. However, the relative difference in costs associated with enumerating items within the subitizing range are small, whether measured in terms of accuracy, confidence, or speed of response. Furthermore, the values of all measures appear to differ markedly inside and outside the subitizing range.[1] So, while there may be no span of apprehension, there appear to be real differences in the ways in which a small number of elements is processed by the visual system (i.e., approximately less than four items), compared with larger numbers of elements (i.e., approximately more than four items).

A 2006 study demonstrated that subitizing and counting are not restricted to visual perception, but also extend to tactile perception, when observers had to name the number of stimulated fingertips.[7] A 2008 study also demonstrated subitizing and counting in auditory perception.[8] Even though the existence of subitizing in tactile perception has been questioned,[9] this effect has been replicated many times and can be therefore considered as robust.[10][11][12] The subitizing effect has also been obtained in tactile perception with congenitally blind adults.[13] Together, these findings support the idea that subitizing is a general perceptual mechanism extending to auditory and tactile processing.

Enumerating afterimages

As the derivation of the term "subitizing" suggests, the feeling associated with making a number judgment within the subitizing range is one of immediately being aware of the displayed elements.[3] When the number of objects presented exceeds the subitizing range, this feeling is lost, and observers commonly report an impression of shifting their viewpoint around the display, until all the elements presented have been counted.[1] The ability of observers to count the number of items within a display can be limited, either by the rapid presentation and subsequent masking of items,[14] or by requiring observers to respond quickly.[1] Both procedures have little, if any, effect on enumeration within the subitizing range. These techniques may restrict the ability of observers to count items by limiting the degree to which observers can shift their "zone of attention"[15] successively to different elements within the display.

Atkinson, Campbell, and Francis[16] demonstrated that visual afterimages could be employed in order to achieve similar results. Using a flashgun to illuminate a line of white disks, they were able to generate intense afterimages in dark-adapted observers. Observers were required to verbally report how many disks had been presented, both at 10 s and at 60 s after the flashgun exposure. Observers reported being able to see all the disks presented for at least 10 s, and being able to perceive at least some of the disks after 60 s. Despite a long period of time to enumerate the number of disks presented when the number of disks presented fell outside the subitizing range (i.e., 5–12 disks), observers made consistent enumeration errors in both the 10 s and 60 s conditions. In contrast, no errors occurred within the subitizing range (i.e., 1–4 disks), in either the 10 s or 60 s conditions.[17]

Brain structures involved in subitizing and counting

The work on the enumeration of afterimages[16][17] supports the view that different cognitive processes operate for the enumeration of elements inside and outside the subitizing range, and as such raises the possibility that subitizing and counting involve different brain circuits. However, functional imaging research has been interpreted both to support different[18] and shared processes.[19]

Balint's syndrome

Clinical evidence supporting the view that subitizing and counting may involve functionally and anatomically distinct brain areas comes from patients with simultanagnosia, one of the key components of Balint's syndrome.[20] Patients with this disorder suffer from an inability to perceive visual scenes properly, being unable to localize objects in space, either by looking at the objects, pointing to them, or by verbally reporting their position.[20] Despite these dramatic symptoms, such patients are able to correctly recognize individual objects.[21] Crucially, people with simultanagnosia are unable to enumerate objects outside the subitizing range, either failing to count certain objects, or alternatively counting the same object several times.[22]

However, people with simultanagnosia have no difficulty enumerating objects within the subitizing range.[23] The disorder is associated with bilateral damage to the parietal lobe, an area of the brain linked with spatial shifts of attention.[18] These neuropsychological results are consistent with the view that the process of counting, but not that of subitizing, requires active shifts of attention. However, recent research has questioned this conclusion by finding that attention also affects subitizing.[24]

Imaging enumeration

A further source of research upon the neural processes of subitizing compared to counting comes from positron emission tomography (PET) research upon normal observers. Such research compares the brain activity associated with enumeration processes inside (i.e., 1–4 items) for subitizing, and outside (i.e., 5–8 items) for counting.[18][19]

Such research finds that within the subitizing and counting range activation occurs bilaterally in the occipital extrastriate cortex and superior parietal lobe/intraparietal sulcus. This has been interpreted as evidence that shared processes are involved.[19] However, the existence of further activations during counting in the right inferior frontal regions, and the anterior cingulate have been interpreted as suggesting the existence of distinct processes during counting related to the activation of regions involved in the shifting of attention.[18]

Educational applications

Historically, many systems have attempted to use subitizing to identify full or partial quantities. In the twentieth century, mathematics educators started to adopt some of these systems, as reviewed in examples below, but often switched to more abstract color-coding to represent quantities up to ten.

Aleister Crowley advocated subitizing in 1913 in Liber Batrachophrenoboocosmomachia, published in The Equinox. In the nineties, babies three weeks old were shown to differentiate between 1–3 objects, that is, to subitize.[22] A more recent meta-study summarizing five different studies concluded that infants are born with an innate ability to differentiate quantities within a small range, which increases over time.[25] By the age of seven that ability increases to 4–7 objects. Some practitioners claim that with training, children are capable of subitizing 15+ objects correctly.[26]


The hypothesized use of yupana, an Inca counting system, placed up to five counters in connected trays for calculations.

In each place value, the Chinese abacus uses four or five beads to represent units, which are subitized, and one or two separate beads, which symbolize fives. This allows multi-digit operations such as carrying and borrowing to occur without subitizing beyond five.

European abacuses use ten beads in each register, but usually separate them into fives by color.

Twentieth century teaching tools

The idea of instant recognition of quantities has been adopted by several pedagogical systems, such as Montessori, Cuisenaire and Dienes. However, these systems only partially use subitizing, attempting to make all quantities from 1 to 10 instantly recognizable. To achieve it, they code quantities by color and length of rods or bead strings representing them. Recognizing such visual or tactile representations and associating quantities with them involves different mental operations from subitizing.

Other applications

One of the most basic applications is in digit grouping in large numbers, which allow one to tell the size at a glance, rather than having to count. For example, writing one million (1000000) as 1,000,000 (or 1.000.000 or 1000000) or one (short) billion (1000000000) as 1,000,000,000 (or other forms, such as 1,00,00,00,000 in India) makes it much easier to read. This is particularly important in accounting and finance, as an error of a single decimal digit changes the amount by a factor of ten. This is also found in computer programming languages for literal values; see Integer literal: Digit separators.

Dice, playing cards and other gaming devices traditionally split quantities into subitizable groups with recognizable patterns.

See also: Approximate number system


  1. Note=1 Kaufman, E.L., Lord, M.W., Reese, T.W., & Volkmann, J. (1949). "The discrimination of visual number". American Journal of Psychology (The American Journal of Psychology) 62

  2. Note=2 Saltzman, I.J., & Garner, W.R. (1948). "Reaction time as a measure of span of attention". The Journal of Psychology 25

  3. Note=3 Jevons, W.S. (1871). "The power of numerical discrimination". Nature 3 (67)

  4. Note=4 Taves, E.H. (1941). "Two mechanisms for the perception of visual numerousness". Archives of Psychology 37.

  5. Note=5 Trick, L.M., & Pylyshyn, Z.W. (1994). "Why are small and large numbers enumerated differently? A limited-capacity preattentive stage in vision". Psychological Review 101

  6. Note=6 Chi, M.T.H. & Klahr, D. (1975). "Span and rate of apprehension in children and adults". Journal of Experimental Child Psychology 19

  7. Note=7 Riggs, K.J., Ferrand, L., Lancelin, D., Fryziel, L., Dumur, G., & Simpson, A. (2006). "Subitizing in tactile perception". Psychological Science 17

  8. Note=8 Camos, V., & Tillmann, B. (2008). "Discontinuity in the enumeration of sequentially presented auditory and visual stimuli". Cognition 107

  9. Note=9 Gallace A., Tan H. Z., Spence C. (2008). "Can tactile stimuli be subitised? An unresolved controversy within the literature on numerosity judgments". Perception 37 (5): 782.

  10. Note=10 Plaisier, M.A., Bergmann Tiest, W. M., & Kappers, A. M. L. (2009). "One, two, three, many - Subitizing in active touch". Acta Psychologica 131

  11. Note=11 Plaisier, M.A., Bergmann Tiest, W. M., & Kappers, A. M. L. (2010). "Range dependent processing of visual numerosity: similarities across vision and haptics". Experimental Brain Research 204

  12. Note=12 Plaisier,M.A., & Smeets, J. B. J. (2011). "Haptic subitizing across the fingers". Attention, Perception, & Psychophysics 73

  13. Note=13 Ferrand, L., Riggs, K. J., & Castronovo, J. (2010). "Subitizing in congenitally blind adults". Psychonomic Bulletin & Review 17

  14. Note=14 Mandler, G., & Shebo, B.J. (1982). "Subitizing: An analysis of its component processes". Journal of Experimental Psychology: General 111

  15. Note=15 LaBerge, D., Carlson, R.L., Williams, J.K., & Bunney, B.G. (1997). "Shifting attention in visual space: Tests of moving-spotlight models versus an activity-distribution model". Journal of Experimental Psychology: Human Perception and Performance 23 (5)

  16. Note=16 Atkinson, J., Campbell, F.W., & Francis, M.R. (1976). "The magic number 4±0: A new look at visual numerosity judgements". Perception 5

  17. Note=17 Simon, T.J., & Vaishnavi, S. (1996). "Subitizing and counting depend on different attentional mechanisms: Evidence from visual enumeration in afterimages". Perception & Psychophysics 58

  18. Note=18 Corbetta, M., Shulman, G.L., Miezin, F.M., & Petersen, S.E. (1995). "Superior parietal cortex activation during spatial attention shifts and visual feature conjunction". Science 270 (5237)

  19. Note=19 Piazza, M; Mechelli, A; Butterworth, B; Price, CJ (2002). "Are subitizing and counting implemented as separate or functionally overlapping processes?". NeuroImage 15

  20. Note=20 Balint, R (1909). "Seelenlahmung des 'Schauens', optische Ataxie, raumliche Storung der Aufmerksamkeit". Monatschrift für Psychiatrie und Neurologie 25

  21. Note=21 Robertson, L., Treisman, A., Freidman-Hill, S., & Grabowecky, M. (1997). "The interaction of spatial and object pathways: Evidence from Balint's Syndrome". Journal of Cognitive Neuroscience 9

  22. Note=22 Dehaene, S (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press.

  23. Note=23 Dehaene, S., & Cohen, L. (1994). "Dissociable mechanisms of subitizing and counting: neuropsychological evidence from simultanagnosic patients". Journal of Experimental Psychology: Human Perception and Performance 20

  24. Note=24 Vetter, P; Butterworth, B; Bahrami, B (2008). Warrant, Eric, ed. "Modulating attentional load affects numerosity estimation: Evidence against a pre-attentive subitizing mechanism". PLOS ONE 3

  25. Note=25 Rouselle, L., & Noël, M. P. (2008). "The development of automatic numerosity processes in preschoolers: Evidence for numerosity-perceptual interference". Developmental Psychology 44

  26. Note=26
Source: (Wikipedia)

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Initial Posting: Saturday, 13-Feb-2016... 10:59 AM
Updated Posting: Monday, 24-June-2019...10:49 AM

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Herb O. Buckland