subitize

=Subitize=

toc Read here about people's and animal's ability to recognize numbers at a glance. This is helpful in scale studies when we consider the nature of our perception, and why we divide the world into different scales.

=Definition= People share with animals the ability to tell instantly at a glance whether there are one, two, three or up to four objects present to the sensory fabric. The ability is named subitizing, from Latin subitus for “sudden.” The ability applies both to visual arrays and arrays in memory, as for example when we ascertain the number of drum beats or light blinks in a sequence, or feel bumps pass beneath our fingers. More than four objects cannot be as quickly discerned. Distinguishing whether there are seven or eight, or thirteen or fourteen objects requires extra time and extra cognitive operation. To decide, a person will group the objects into smaller, subitizable groups and count them. These results have been well established in human perception studies since the 1950’s. Nunez reviews the literature and reports four day-old babies can discriminate between collections of two and three items, and under sometimes three items from four. The ability is not restricted to visual discrimination: these babies can also discriminate between sounds of two or three syllables. Infants four and a half months old “can tell” that one plus one is two and that two minus one is one. At around six months infants “can tell” that two plus one is three and that three minus one is two. And at about seven months, babies can recognize the numerical equivalence between arrays of objects and drumbeats of the same number (Lakoff p. 15ff). Many animals show similar abilities--primates, raccoons, rats, parrots and pigeons--though primates alone are proven to be also capable of comprehending written numbers as symbols for the subitized amounts. (Lakoff p.23).

=Logarithmic Nature of Natural Number Sense=

Dehaene wrote the following callout in Annual Review of Neuroscience 2009: Several recent psychological studies in remote human cultures confirm that the sense of approximate numerosity is a universal domain of human competence (Gordon 2004, Pica et al. 2004, Frank et al. 2008, Butterworth et al. 2008). In the Amazon, for instance, the Mundurucu and the Piraha people have very little access to education. Furthermore, their langauges have reduced sets of number words, five and two number words, respectively, which are not used for counting and may be similar to the approximate English words “dozen” or “handful.” Yet adults and children of these cultures pass tests of approximate number perception, matching, comparison, and simple arithmetic (Gordon 2004, Pica et al. 2004). A vigorous debate concerns the limits of these native numerical abilities. On the one hand, some who adhere to Whorf’s hypothesis believe the lack of a developed langauge for number drastically affects numerical cognition and prevents any understanding of exact arithmetic (Gordon 2004). On the other hand, a recent study of Australian children concludes that a reduced number vocabulary does not prevent the emergence of “the same numerical concepts as a comparable group of English-speaking children” (Butterworth et al. 2008), including a thorough understanding of large exact numbers. Studies of the Mundurucu, however, show clear cross-cultural differences; unlikeWestern controls, the Mundurucu appear to think of number as an analog, an approximate and logarithmically compressed continuum (Pica et al. 2004, Dehaene et al. 2008), exactly as expected from the neural representation of numerosity observed in human and nonhuman primates (Nieder & Miller 2003, Nieder & Merten 2007, Piazza et al. 2004). Education in counting and measurement, more than spoken langauge, may play a critical role in bringing about concepts of exact number, a conceptual change whose neural underpinnings probably involve the left intraparietal sulcus.

=Preferred Number= In industrial design, preferred numbers (also called preferred values) are standard guidelines for choosing exact product dimensions within a given set of constraints. Product developers must choose numerous lengths, distances, diameters, volumes, and other characteristic quantities. While all of these choices are constrained by considerations of functionality, usability, compatibility, safety or cost, there usually remains considerable leeway in the exact choice for many dimensions. Preferred numbers serve two purposes:
 * Using them increases the probability of compatibility between objects designed at different times by different people. In other words, it is one tactic among many in standardization, whether within a company or within an industry, and it is usually desirable in industrial contexts.
 * They are chosen such that when a product is manufactured in many different sizes, these will end up roughly equally spaced on a logarithmic scale. They therefore help to minimize the number of different sizes that need to be manufactured or kept in stock.

In electronics, international standard IEC 60063 defines another preferred number series for resistors, capacitors, inductors and zener diodes. It works similarly to the Renard series, except that it subdivides the interval from 1 to 10 into 6, 12, 24, etc. steps. These subdivisions ensure that when some arbitrary value is replaced with the nearest preferred number, the maximum relative error will be on the order of 20%, 10%, 5%, etc.

Preferred Number E12 tolerances in decade 1-12. Public domain.

=Links and Citations=

Lakoff 2000: Where Mathematics Comes From by George Lakoff and Rafael E. Nunez, Basic Books, 2000.