Steven Pinker -- How the Mind Works - Hampshire High Italian ...

Good Ideas 339Human adults use several mental representations of quantity. One isanalogue—a sense of "how much"—which can be translated into mentalimages such as an image of a number line. But we also assign numberwords to quantities and use the words and the concepts to measure, tocount more accurately, and to count, add, and subtract larger numbers.All cultures have words for numbers, though sometimes only "one,""two," and "many." Before you snicker, remember that the concept ofnumber has nothing to do with the size of a number vocabulary. Whetheror not people know words for big numbers (like "four" or "quintillion"),they can know that if two sets are the same, and you add 1 to one ofthem, that set is now larger. That is true whether the sets have four itemsor a quintillion items. People know that they can compare the size of twosets by pairing off their members and checking for leftovers; even mathematiciansare forced to that technique when they make strange claimsabout the relative sizes of infinite sets. Cultures without words for bignumbers often use tricks like holding up fingers, pointing to parts of thebody in sequence, or grabbing or lining up objects in twos and threes.Children as young as two enjoy counting, lining up sets, and otheractivities guided by a sense of number. Preschoolers count small sets,even when they have to mix kinds of objects, or have to mix objects,actions, and sounds. Before they really get the hang of counting andmeasuring, they appreciate much of its logic. For example, they will tryto distribute a hot dog equitably by cutting it up and giving everyone twopieces (though the pieces may be of different sizes), and they yell at acounting puppet who misses an item or counts it twice, though their owncounting is riddled with the same kinds of errors.Formal mathematics is an extension of our mathematical intuitions.Arithmetic obviously grew out of our sense of number, and geometry outof our sense of shape and space. The eminent mathematician SaundersMac Lane speculated that basic human activities were the inspiration forevery branch of mathematics:Counting-» arithmetic and number theoryMeasuring—> real numbers, calculus, analysisShaping—> geometry, topologyForming (as in architecture) —> symmetry, group theoryEstimating—> probability, measure theory, statisticsMoving—> mechanics, calculus, dynamicsCalculating—> algebra, numerical analysisProving—> logic