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

340 I HOW THE MIND WORKSPuzzling —» combinatorics, number theoryGrouping —> set theory, combinatoricsMac Lane suggests that "mathematics starts from a variety of humanactivities, disentangles from them a number of notions which are genericand not arbitrary, then formalizes these notions and their manifold interrelations."The power of mathematics is that the formal rule systems canthen "codify deeper and nonobvious properties of the various originatinghuman activities." Everyone—even a blind toddler—instinctively knowsthat the path from A straight ahead to B and then right to C is longerthan the shortcut from A to C. Everyone also visualizes how a line candefine the edge of a square and how shapes can be abutted to form biggershapes. But it takes a mathematician to show that the square on thehypotenuse is equal to the sum of the squares on the other two sides, soone can calculate the savings of the shortcut without traversing it.To say that school mathematics comes out of intuitive mathematics isnot to say that it comes out easily. David Geary has suggested that naturalselection gave children some basic mathematical abilities: determiningthe quantity of small sets, understanding relations like "more than"and "less than" and the ordering of small numbers, adding and subtractingsmall sets, and using number words for simple counting, measurement,and arithmetic. But that's where it stopped. Children, he suggests,are not biologically designed to command large number words, large sets,the base-10 system, fractions, multicolumn addition and subtraction,carrying, borrowing, multiplication, division, radicals, and exponents.These skills develop slowly, unevenly, or not at all.On evolutionary grounds it would be surprising if children were mentallyequipped for school mathematics. These tools were inventedrecently in history and only in a few cultures, too late and too local tostamp the human genome. The mothers of these inventions were therecording and trading of farming surpluses in the first agricultural civilizations.Thanks to formal schooling and written language (itself arecent, noninstinctive invention), the inventions could accumulate overthe millennia, and simple mathematical operations could be assembledinto more and more complicated ones. Written symbols could serve as amedium of computation that surmounted the limitations of short-termmemory, just as silicon chips do today.How can people use their Stone Age minds to wield high-tech mathematicalinstruments? The first way is to set mental modules to work on