End Notes

INTRODUCTION

I. Kathleen Freeman, Ancilla to the Pre-Socratic Philosophers (Cambridge, MA: Harvard University Press, 1966), p. 74.

2. Plato, The Dialogues of Plato, trans. B. Jowett (New York: Random House, 1937) The Republic, VII, 525.

CHAPTER 1

I. Karl Menninger, Number Words and Number Symbols (New York: Dover Publications, 1969), p. 33.

2. Erich Harth, Windows on the Mind: Reflections on the Physical Basis of Consciousness (New York: William Morrow and Company, 1982), p. 101.

3. Paul Glees, The Human Brain (Cambridge: Cambridge University Press, 1988), p. 37. 4. Mathematical Disabilities (Gerard Deloche and Xavier Seron, eds.) (Hillsdale, New

Jersey: Lawrence Erlbaum Associates, 1987). 5. Paul D. MacLean, The Triune Brain in Evolution (New York: Plenum Press, 1990),

p. 549; Deloche and Seron, p. 140.

CHAPTER 2

1. Roger Lewin, Bones of Contention: Controversies in the Search for Human Origins (New York: Simon and Schuster, 1987), p. 108.

281

282 END NOTES

2. David Lambert, The Field Guide to Early Man (New York: Facts on File, 1987), pp. 98-105.

3. Ibid., p. 106. 4. David Eugene Smith, History of Mathematics (New York: Dover Publications, 1951),

p.6. 5. Paul D. MacLean, The Triune Brain in Evolution (New York: Plenum Press, 1990),

p. 555. 6. Richard E. Leakey, Origins (New York: E.P. Dutton, 1977), p. 205. 7. Karl Menninger, Number Words and Number Symbols, p. 35. 8. Graham F1egg, Numbers Through the Ages (London: MacMillan Educations LTD,

1989), p. 7. 9. Graham F1egg, Numbers: Their History and Meaning (New York: Schocken Books,

1983), p. 19. 10. Flegg, Numbers Through the Ages, p. 9. II. Flegg, Numbers: Their History and Meaning, p. 24. 12. Menninger, p. II. 13. F1egg, Numbers: Their History and Meaning, p. II. 14. Menninger, p. 32. IS. Leakey, p. 162. 16. F1egg, Numbers Through the Ages, p. 37. 17. Ibid., p. 11.

CHAPTER 3

1. Graham F1egg, Numbers: Their History and Meaning, p. 7. 2. H. Kalmus, ''l\nimals as Mathematicians," Nature 202 (June 20, 1964), p. 1156. 3. Levi Leonard Conant, "Counting," in The World of Mathematics. Vol. I (James R.

Newman, ed.) (New York: Simon and Schuster, 1956), p. 433. 4. Donald R. Griffin, Animal Thinking (Cambridge, MA: Harvard University Press,

1984), p. 204. 5. 0. Koehler, "The Ability of Birds to 'Count'," in The World of Mathematics. Vol. I

(James R. Newman, ed.) (New York: Simon and Schuster, 1956), p. 491. 6. Conant, p. 434. 7. Guy Woodruff and David Premack, "Primate Mathematical Concepts in the Chim­

panzee: Proportionality and Numerosity," Nature 293 (October IS, 1981), p. 568-570. 8. Phone conversation with Kenneth S. Norris, retired professor of natural history at the

University of California-Santa Cruz, Nov. 19, 1992. 9. Menninger, Number Words and Number Symbols, p. II.

10. John McLeish, Number (New York: Fawcett Columbine, 1991), p. 7. II. David Caldwell and Melba Caldwell, The World of the Bottle-Nosed Dolphin (New

York: J. B. Lippincott Co., 1972), p. 17.

END NOTES 283

12. Carl Sagan, Mind in the Waters (Joan McIntyre, ed.) (New York: Charles Scribner's Sons, 1974), p. 88.

CHAPTER 4

I. David Eugene Smith, p. 37. 2. Denise Schmandt-Besserat, Before Writing, Vol. I: From Counting to Cuneiform

(Austin, TX: University of Texas Press, 1992), p. 7. 3. Ibid., p. 6. 4. Ibid., p. 190. 5. Mortimer Chambers, Raymond Grew, David Herlihy, Theodore Rabb, and Isser

Woloch, The Western Experience: To 17/5 (New York: Alfred A. Knopf, 1987), p. 7. 6. Schmandt-Besserat, p. 114. 7. Ibid., p. 199. 8. Carl B. Boyer, A History of Mathematics (New York: John Wiley and Sons, 1968),

p.33. 9. H. L. Resnikoff and R. 0. Wells, Jr., Mathematics in Civilization (New York: Dover

Publications, 1973), p. 76. 10. David Eugene Smith, p. 43. II. Boyer, p. 22. 12. David Eugene Smith, p. 43. 13. Boyer, p. 12. 14. Morris Kline, Mathematical Thought from Ancient to Modern Times, Vol. / (New York:

Oxford University Press, 1972), p. 16. 15. Lucas !3unt, Phillip Jones, and Jack Bedient, The Historical Roots of Elementary

Mathematics (New York: Dover Publications, 1976), p. 37.

CHAPTER 5

I. McLeish, p. 53. 2. David Eugene Smith, p. 23. 3. Menninger, p. 452. 4. Boyer, p. 220. 5. McLeish, p. 70. 6. Ibid., p. 24. 7. Stuart 1. Fiedel, Prehistory of the Americas (Cambridge: Cambridge University Press,

1987), p. 282. 8. Ibid., p. 281. 9. The Codex Dresdensis in Dresden, the Codex Tro-Cortesianus in Madrid, and the

Codex Peresianus in Paris.

284 END NOTES

10. Bunt et al., p. 226. II. Thomas Crump, The Anthropology of Numbers (New York: Cambridge University

Press, 1990), p. 46. 12. Jacques Soustelle, Mexico (New York: World Publishing Company, 1967), p. 125. 13. Fiedel, p. 335.

CHAPTER 6

I. Chambers et al., p. 40. 2. Menninger, p. 272. 3. Ibid., p. 299. 4. Kline, p. 28. 5. David Eugene Smith, p. 64. 6. The two different positions are illustrated by Smith, p. 71; and Boyer, p. 52. 7. Michael Moffatt, The Ages of Mathematics: Vol. /, The Origins (New York: Doubleday

and Company, 1977), p. 96. 8. Boyer, p. 60. 9. Bunt et al., p. 83.

10. Aristotle, The Basic Works of Aristotle, trans. J. Annas (Richard McKeon, ed.) (New York: Random House, 1941); The Metaphysics, 986a, lines 1-3 and 15-18, Oxford University Press.

II. Ibid., 1090a, lines 20-25. 12. Two different visual proofs come from Stuart Hollingdale, Makers of Mathematics

(London: Penguin Books, 1989), p. 39; and Eric Temple Bell, Mathematics: Queen and Servant of Science (New York: McGraw-Hili, 1951), p. 190.

13. Kline, p. 33. 14. Moffatt, p. 92. 15. Bunt et al., p. 86.

CHAPTER 7

I. McLeish, p. 115. 2. Kline, p. 184. 3. David Eugene Smith, p. 157. 4. Menninger, p. 399. 5. McLeish, p. 122. 6. Kline, p. 184. 7. Bunt et al., p. 226. 8. Ibid., p. 227. 9. Menninger, p. 425.

10. Ibid., p. 432. II. Ibid., p. 400. 12. Hollingdale, p. 109.

END NarES 285

13. Jane Muir, Of Men and Numbers (New York: Dodd, Mead, and Company, 1961), p. 235.

CHAPTER 8

I. Freeman, p. 14. 2. Ibid., p. 19. 3. Aristotle, The Basic Works of Aristotle, Physics, Book III, 204b, lines 2-9. 4. Freeman, p. 75. 5. Aristotle, Physics, Book /II, 206b, lines 31-32. 6. Ibid., 204b, lines 6-8. 7. Ibid., 206a, line 26; 206b, line 13. 8. Ibid., 239b, lines 14-18. 9. Plato, The Dialogues of Plato, trans. B. Jowett (New York: Random House, 1937),

Timeaus, lines 25, 52. 10. Rudy Rucker, Infinity and the Mind (New York: Bantam Books, 1982), p. 3. II. Thomas Hobbes, Leviathan: Parts I and II (New York: Bobbs-Merrill Company, 1958),

p.36. 12. Thomas Hobbes, "Selections from the De Corpore," in Philosophers Speakfor Them­

selves: From Descartes to Locke (T. V. Smith and Marjorie Grene, eds.) (Chicago: University of Chicago Press, 1957), p. 144.

13. Rene Descartes, "Meditations on First Philosophy," in Philosophers Speak for Them-selves: From Descartes to Locke, p. 78.

14. Hollingdale, p. 359. 15. Rucker, p. 88. 16. Euclid, Elements, Book III (New York: Dover Publications, 1956), Sec. 14.

CHAPTER 9

I. A well-ordered set is a simply ordered set such that every subset contains a first element. See Zermelo's axiom of choice.

2. Sir Thomas Heath, A History of Greek Mathematics, Vol I (Oxford, England: The Clarendon Press, 1960), p. 385.

3. The English translation of this work can be found in Richard Dedekind, Essays on the Theory of Numbers (La Salle, IL: Open Court Publishing Company, 1948).

4. Ibid., p. 6. 5. Ibid., p. 12.

286 END NarES

6. Ibid., p. 13. 7. Ibid., p. IS. 8. Boyer, p. 307. 9. Ibid., p. 348.

10. Richard Preston, "Profiles: The Mountains of Pi," The New Yorker (March 2, 1992), p.36.

CHAPTER 10

1. Hollingdale, p. 275. 2. Boyer, p. 361. 3. Muir, p. 217. 4. Quoted in Sherman K. Stein, Mathematics: The Man-Made University, (New York:

W. H. Freeman and Company, 1963), p. 252. 5. Ibid., p. 253. 6. Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite

(Princeton, NJ: Princeton University Press, 1979), p. 50. 7. Leo Zippin, Uses of Infinity (Washington, D.C.: The Mathematical Association of

America, 1962), p. 56.

CHAPTER 11

1. Kline, p. 143. 2. Ibid., p. 253. 3. Eric Temple Bell, Men of Mathematics (New York: Simon and Schuster, 1965), p. 35. 4. Hollingdale, p. 126. 5. Bell, Men of Mathematics, p. 43. 6. Muir, p. 172. 7. Dauben, p. 54. 8. Ibid., p. 55. 9. Hollingdale, p. 337.

CHAPTER 12

1. E. Kamke, Theory of Sets (New York: Dover Publications, 1950), p. 47. 2. Rucker, pp. 48-50. 3. Dauben, p. 232.

END NOTES 287

4. Rucker, p. 276. 5. Ibid., pp. 281-285. 6. Muir, p. 237. 7. Dauben, p. 285. 8. Ibid., p. 243.

CHAPTER 13

I. Steven B. Smith, The Great Mental Calculators (New York: Columbia University Press, 1983).

2. Darold A. Treffert, Extraordinary People (New York: Harper & Row Publishers, 1989). 3. Steven B. Smith, p. 97. 4. Ibid., p. 289. 5. Ibid., p. 245. 6. Oliver Sacks, The Man Who Mistook His Wife for a Hat (New York: Harper Perennial,

1985), p. 203. 7. Treffert, p. 41. 8. W. W. Rouse Ball, "Calculating Prodigies," in The World of Mathematics, Vol. 1,

p.467. 9. Steven B. Smith, p. xv.

10. Treffert, p. 220. II. Ibid., p. 222. 12. A. E. Ingham, The Distribution of Prime Numbers (Cambridge: Cambridge University

Press, 1990), p. 3.

CHAPTER 14

1. Bertrand Russell, The Problems of Philosophy (London: Oxford University Press, 1959), p. 12.

2. George Berkeley, "Three Dialogues Between Hylas and Philonous," in Philosophers Speak for Themselves: Berkeley, Hume, and Kant, pp. 1-95.

3. Russell, p. 98. 4. Jerry P. King, The Art of Mathematics (New York: Plenum Press, 1992), p. 29. 5. Ibid., p. 43. 6. Ibid., p. 139. 7. W. H. Werkmeister, A Philosophy of Science (Lincoln, NE: University of Nebraska

Press, 1940), p. 141.

288 END NOTES

CHAPTER 15

I. Boyer, p. 645. 2. E. 1. Borowski and 1. M. Borwein, The HarperCollins Dictionary of Mathematics (New

York: HarperCollins Publishers, 1991), p. 589. 3. Ibid., p. 49. 4. King, p. 6. 5. Roger T. Stevens, Fractal: Programming in Turbo Pascal (Redwood City, CA: M&T

Publishing, 1990). 6. James Gleick, Chaos: Making a New Science (New York: Viking Penguin, 1987),

p.217. 7. Ibid., p. 222. 8. Ibid., p. 239. 9. Bell, Men of Mathematics, p. 57.

10. Ibid., p. 71. II. Michael D. Lemonick, "Fini to Fermat's Last Theorem," Time (5 July 1993), p. 47.

Glossary

abacus: an ancient calculating device consisting of beads strung on rods mounted in a wooden frame. Calculations are performed by moving the beads along the rods. Absolute Infinite: entity that is identified with a collection of all infinities, sometimes associated with God and designated as n (capital omega). A concept that cannot be understood rationally but only mystically. absolute value: a positive value of a number regardless of the number's original sign. Alexandrian number system: the Greek numbering system based on twenty-seven different numerals and used predominately after 100 B.C.

Also called the Ionic system. algebraic number: a number that is a solution to a polynomial whose coefficients are all rational numbers. analytic geometry: the geometry where positions are represented by number coordinates and algebraic methods apply. Attic number system: an early, written Greek number system based on six primary numerals and used until approximately 100 B.C. Also called the Heroqianic system. body-counting: an extension of finger-counting where different body parts represent natural numbers. Brahmi numerals: Hindu numerals for the numbers 1 through 9, which were used as early as the third century B.C. These symbols were to eventually lead to the modern Hindu-Arabic numerals used today. cardinal number: a number specifying how many elements are in a set.

289

290 GLOSSARY

Cartesian coordinate system: the use of two perpendicular number lines to identify every point in the plane with two real numbers. closed numbers: a set of numbers is closed for an operation if every application of that operation on numbers from the set yields another number in the set. cluster point: another name for a limit, especially when more than one limit is involved. complex number: a number of the form (a,b) ora + bi where both a and b are real numbers and i is v=i. Real numbers and imaginary numbers are both subsets of the complex numbers. composite number: a natural number that can be evenly divided by more numbers that 1 and itself. concrete counting: when the objects being counted are mapped one-to­one with counting tokens or symbols. continued fraction: a number consisting of an integer plus a fraction such that the denominator of the fraction is also an integer plus a fraction of the same kind. Every irrational number can be represented as an infinite continued fraction. constructivism: the belief that mathematical objects do not exist indepen­dently of human minds. continuum hypothesis: the hypothesis that there exists a transfinite num­ber between Xo and Xl' The question is undecidable. convergence: state of an infinite sequence of numbers or number series that approaches a limit. coordinates: the two real numbers associated with each point in the plane in analytic geometry. countable set: an infinite set that can be put into a one-to-one mapping with the natural numbers. Also called "enumerable." counting: finding the cardinal number of a set. counting board: an ancient counting device consisting of a marked board and counting sticks or pebbles. dense: the property that between every two numbers there exists another number. The rational, irrational, real, and complex numbers are all dense. denumerable set: a set that can be mapped one-to-one with the natural numbers. See countable set.

GLOSSARY 291

digit: anyone of the ten numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 of the Hindu-Arabic number system. divergence: state of an infinite sequence of numbers or number series that has no bound or limit. element: one particular item or member of a set. Euclidean geometry: the geometry developed by Euclid and satisfying the parallel postulate that says: Given a line and a point not on the line, one and only one other line can pass through the point that is parallel to the original line. exponent: a number written as a superscript to a second (base) number and indicating the power to which the second number is to be raised. extendable cardinal number: the largest cardinal number known at present. factorial: the product of all the natural numbers less than and equal to a specific natural number and indicated with the symbol "!" placed after the specific natural number. For example, 5! = 1·2·3·4·5 = 120. false position: a method of solving equations using a guess substituted into the equation to generate a proportional adjustment for an improved guess. Fibonacci sequence: the number sequence beginning with 1 where each succeeding term is the sum of the previous two terms. The first seven terms of the sequence are 1, 1, 2, 3, 5, 8, 13, ... finger-counting: using fingers to map onto a set of objects to be counted. five-counting (5-counting): an early number system with a base of five. fractal: a complex geometric curve that does not change into simple forms under magnification and generally has a dimension that is between whole numbers. Gobar numerals: the numerals used by the Arabs during the ninth century A.D. and transcribed from the earlier Indian Brahmi numerals. golden mean or golden ratio: the ratio of (1 + Vs) to 2 or the fraction (1 + Vs)I2. Discovered by the Greeks and found in numerous mathe­matical relationships. hieratic writing: early Egyptian writing used by scribes to conduct daily record-keeping. hieroglyphics: early Egyptian writing used for formal occasions and on monuments. Hindu-Arabic number system: the dominant number system used today

292 GLOSSARY

based on the ten unique symbols 0, 1, 2, 3,4, 5, 6, 7, 8, and 9, and using a position-value system. hypercomplex number: a number generated by extending the concept of number to dimensions beyond the two-dimensional complex numbers. See quaternions. hyperinaccessible transfinite number: a transfinite number that is so inaccessible that its inaccessibility cannot be defined beginning with smaller transfinite numbers. See inaccessible transfinite number. hypotenuse: in a triangle, the side opposite the right angle. imaginary number: a number on the vertical axis of the complex number plane; a number in the form ai where a is a real number and i is v=I. inaccessible transfinite number: a transfinite number that cannot be defined in terms of smaller transfinite numbers. incommensurable: when two magnitudes cannot be expressed as the ratio of two whole numbers. indeterminate equations: equations with an infinite number of solutions, for example, the equation x + y = 7. Such equations are useful in the study of numerous physical systems. indirect method of proof: a method of proof discovered by various ancient societies where one assumes the opposite of what is to be proven and then shows a contradiction. Also called reductio ad absurdum. infinite: without bounds, unbounded, not finite. For a set, able to be mapped onto a proper subset of itself, for example, the natural number map onto the squares of natural numbers. integer: one of the set of numbers consisting of the positive and negative natural numbers plus zero. intuitionism: the belief that mathematics should not deal with infinite sets and that only those proofs involving finite steps or constructions should be admitted. irrational number: a number on the real number line that cannot be expressed as the ratio of two whole numbers. limit: the sequence of terms AI' A 2, A 3, ... An' ... has a limit L if for any positive value E there exists a number N such that for all n > N, the absolute value of L - An < E.

linear equation: an equation where all the unknowns have exponents equal to 1.

GLOSSARY

Liouville numbers: transcendental numbers of the following form:

a/WI! + ai102! + ai103! + ...

where the a's are integers in the range of 0 to 9.

293

logarithm: the exponent to which a number called a base must be raised to obtain a given number. The logarithm of n with the base of a is written as logan. For example, 10glOlOO = 2, or 102 = 100. logistic: the ancient Greek art of calculation, considered a trade rather than an intellectual pursuit. method of exhaustion: a technique developed by the Greeks to find the area within curved geometric shapes. The technique finds successively better approximations by computing the areas of rectangles and triangles within the curve. neo-2-counting: an early form of counting using number words equiva­lent to "one" and "two" plus conjunctions of these two symbols and involving addition, subtraction, and multiplication. non-Euclidean geometry: any geometry based on substituting a different postulate for Euclid's parallel postulate. See Euclidean geometry. non positional number system: any number system where the position of a specific numeral does not help determine the numeral's value. Hence, the numerals may be written in any order without changing the number's value. normal number: a number whose decimal expansion contains an equal proportion of all ten digits from 0 through 9. An absolutely normal number is one whose decimal expansion under all bases contains an equal proportion of each digit. number field: any set of real or complex numbers such that the sum, difference, product, and quotient (except for zero) is another number in the set. Therefore a number field is closed under the four operations of arithmetic. number line: an infinite line where each point is associated with one and only one real number. number-number: the decimal number formed by writing each natural number in succession, for example, 0.12345678910111213 .. . number sequence: a set of numbers: Ai' A 2 , A3 , ••• An' ... forms a sequence of numbers if the numbers are well ordered, that is, if the subscripts are in the order of the natural numbers.

294 GLOSSARY

number series: a collective sum of a set of numbers. number theory: the formal study of the natural numbers and their relationships. numeral: the written symbol representing a number or representing a digit of a number. numerical analysis: the use of computers and computing algorithms to approximate the solutions to complex problems. one-to-one mapping: assignment of exactly one and only one element from a set (e.g., number words) to each element in a second set (e.g., fingers). ontology: the study of the meaning of existence. ordinal number: a number that specifies the relative position of an element in a set. origin: the point on the number line associated with the number zero, or the point in the complex plane where the two axes intersect. paradox: an argument that derives self-contradictory conclusions by valid deduction from intuitively acceptable premises. perfect number: a natural number that is the sum of its divisors. Six is the first perfect number since 6 = I + 2 + 3, and 28 is the second since 28 = I + 2 + 4 + 7 + 14. polynomial equation: an equation of one or more unknowns raised to powers and mUltiplied by numbers called coefficients. A polynomial with one unknown, x, has the general form aoxn + alxn- I ... + an-Ix + an = O. positional number system: a number system that combines the value of a numeral with its position within the number to give its final value. The modern Hindu-Arabic number system is a positional system. prime number: a natural number that can only be evenly divided by itself and the number 1. projective geometry: that branch of mathematics concerned with those properties of geometrical figures that do not change when the figures are projected onto a different space. proof a sequence of logical steps that establishes the truth of a conclusion based on a set of axioms. The first fully articulated proofs were developed by the Greeks to prove certain geometric relationships.

GLOSSARY 295

Pythagorean numbers: any set of three natural numbers that satisfy the Pythagorean theorem. For example, the triplet 3, 4, and 5 are Pythag­orean numbers since 32 + 42 = 52.

Pythagorean theorem: the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the adjacent sides, or a2 + b2 = c2 where c is the length of the hypotenuse and a and b are the lengths of the adjacent sides. quaternions: complex numbers of the form a + bi + cj + dk where a, b, c, and d are real numbers and i, j, and k are hypercomplex numbers such that i2 = P = k2 = ijk = -1. quipu: a collection of knotted strings used by the Incas of the New World to record possessions and transactions. rational number: any number that can be expressed as the ratio of two nonzero integers. real number: the numbers associated with all the points on the number line; the union of the algebraic and transcendental numbers. rhetorical algebra: an ancient form of algebra where problems were stated in ordinary language without precise symbolism. root: a number that satisfies an equation; that is, when substituted into the equation for an unknown, both sides of the equal sign are equal in value. Russian peasant method: a method originally used by the early Egyptians to multiply numbers by successively doubling one number and then adding the appropriate multiples; adopted later by Central Europeans. set: a collection of items or elements. sexagesimal number system: a number system based on sixty, as opposed to our Hindu-Arabic system, which is based on ten. simply ordered: a set of numbers is simply ordered when the following two conditions hold for any three numbers of the set, x, y, and z: (l) x = y or x < y or y < x and (2) if x < y and y < z then x < z. The real numbers are simply ordered while the complex numbers are not. simultaneous linear equations: two or more equations where each equa­tion contains one or more unknowns and every unknown is raised to an exponent of 1. square number: a natural number that is the square of another natural

296 GLOSSARY

number. For example, 4 and 9 are square numbers because they are the squares of 2 and 3. stick-counting: a form of counting that does not require the use of language. Counting objects, such as sticks or pebbles, are mapped one­to-one onto the set of objects being counted. subitizing: the immediate awareness of the manyness of a set of objects. symbolic algebra: algebra employing well-defined symbols rather than ordinary language. syncopated algebra: algebra that is midway between rhetorical algebra and symbolic algebra; the use of a mix of symbols and ordinary words to make algebraic statements. tag number: a number used in place of a name. tally stick: a stick, usually split lengthwise, marked with notches to record financial transactions. When the two parts are placed together, the notches match. theorem: a statement or formula that is deduced from a set of axioms and! or other theorems. token: in counting, a small clay figurine used in western Asia between 8500 and 3000 B.C. for recording the cardinal number of sets of objects. transcendental number: a real number that is not the root or solution to any polynomial with rational coefficients. A real number that is not algebraic. transfinite number: the cardinal or ordinal number of an infinite set. trigonometry: the study of the relationships between the lengths of the sides of a triangle and the measures of its interior angles. two-counting (2-counting): an early form of counting using number words equivalent to "one" and "two" plus additive conjunctions of these two symbols. uncountable: an infinite set whose elements cannot be mapped one-to­one with the natural numbers. undecidable: within a formal mathematical or logical system, a statement that cannot be proved or disproved based on the axioms used in the system. unit fraction: a fraction of the form lin; a fraction whose numerator is 1. vigesimal number system: a number system based on twenty, as opposed to the Hindu-Arabic system, which is based on ten.

GLOSSARY 297

well-ordered: a set is well-ordered if for every subset, including the set itself, there is a first element. The empty set, 0, is considered well­ordered. zero: the cardinal number of the empty set; the symbol that has no value but is used for a placeholder in a positional number system; the numeralO.

Bibliography

Aristotle, The Basic Works of Aristotle (Richard McKeon, ed.). New York: Random House, 1941.

Bell, Eric Temple, The Magic of Numbers. New York: Dover Publications, 1974. Bell, Eric Temple, Men of Mathematics. New York: Simon & Schuster, 1965. Bell, Eric Temple, Mathematics: Queen and Servant of Science. New York: McGraw-Hili

Book Company, 1951. Borowski, E. 1., and Borwein, 1. M., The HarperCollins Dictionary of Mathematics. New

York: HarperCollins Publishers, 1991. Boyer, Carl B., A History of Mathematics. New York: lohn Wiley and Sons, 1968. British Museum (Natural History), Man's Place in Evolution. London: Cambridge Univer­

sity Press (undated). Bunt, Lucas; lones, Phillip; and Bedient, lack, The Historical Roots of Elementary Mathe­

matics. New York: Dover Publications, 1976. Burgess, Robert E, Secret Languages of the Sea. New York: Dodd, Mead, and Company, 1981. Caldwell, David, and Caldwell, Melba, The World of the Bottle-Nosed Dolphin. New York:

1. B. Lippincott Company, 1972. Calvin, William H., The Ascent of Mind: Ice Age Climates and the Evolution of Intelligence.

New York: Bantam Books, 1990. Chambers, Mortimer; Grew, Raymond; Herlihy, David; Rabb, Theodore; and Woloch,

isser, The Western Experience: to 1715. New York: Alfred A. Knopf, 1987. Crump, Thomas, The Anthropology of Numbers. New York: Cambridge University Press,

1990. Dauben, loseph Warren, Georg Cantor: His Mathematics and Philosophy of the I'!finite.

Princeton, New lersey: Princeton University Press, 1979. Dedekind, Richard, Essays on the Theory of Numbers. La Salle, Illinois: Open Court

Publishing Company, 1948. Deloche, G~rard, and Seron, Xavier, eds., Mathematical Disabilities: A Cognitive Neuro­

psychological Perspective. London: Lawrence Erlbaum Associates, 1987.

299

300 BIBLIOGRAPHY

de Villiers, Peter A., and de Villiers, Jill G., Early Language. Cambridge: Harvard University Press, 1979.

Euclid, Elements. New York: Dover Pub1icatiC''1s, 1956. Fiede1, Stuart 1., Prehistory of the Americas. New York: Cambridge University Press, 1987. Fiegg, Graham, Numbers: Their History and Meaning. New York: Schocken Books, 1983. Fiegg, Graham, Numbers Through the Ages. London: MacMillan Education LTD, 1989. Freeman, Kathleen, Ancilla to the Pre-Socratic Philosophers. Cambridge: Harvard Univer-

sity Press, 1966. Gillings, Richard, Mathematics in the Time of the Pharoahs. New York: Dover Publica-

tions, 1972. GIess, Paul, The Human Brain. New York: Cambridge University Press, 1988. Gleick, James, Chaos: Making a New Science. New York: Viking Penguin, 1987. Griffin, Donald R., Animal Thinking. Cambridge: Harvard University Press, 1984. Hadamard, Jacques, The Psychology of Invention in the Mathematical Field. New York:

Dover Publications, 1945. Harth, Erich, Windows on the Mind: Reflections on the Physical Basis of Consciousness.

New York: William Morrow and Company, 1982. Heath, Sir Thomas, A History of Greek Mathematics. London: Oxford University Press,

1921. Hobbes, Thomas, Leviathan: Part I and fl. New York: Bobbs-Merrill Company, 1958. Hollingdale, Stuart, Makers of Mathematics. London: Penguin Books, 1989. Huntley, H. E., The Divine Proportion. New York: Dover Publications, 1970. Ingham, A. E., The Distribution of Prime Numbers. New York: Cambridge University

Press, 1990. James, Glenn, and James, Robert, eds., Mathematics Dictionary. New York: D. Van

Nostrand Company, 1959. Jaynes, Julian, The Origin of Consciousness in the Breakdown <1 the Bicameral Mind.

Boston: Houghton-Mifflin Company, 1976. Kalmus, H., "Animals as Mathematicians," Nature 202 (June 20, 1964.) Kamke, E., Theory of Sets. New York: Dover Publications, 1950. King, Jerry P, The Art of Mathematics. New York: Plenum Press, 1992. Kline, Morris, Mathematics: A Cultural Approach. Reading, MA: Addison-Wesley Pub­

lishing Company, 1962. Kline, Morris, Mathematical Thoughtfrom Ancient to Modern Times, Vol. 1. New York:

Oxford University Press, 1972. Knopp, Konrad, Infinite Sequences and Series. New York: Dover Publications, 1956. Kroner, Stephan, The Philosophy of Mathematics. New York: Dover Publications, 1968. Kosslyn, Stephen M., and Koenig, Oliver, Wet Mind: The New Cognitive Neuroscience.

New York: The Free Press, 1992. Lambert, David, The Field Guide to Early Man. New York: Facts on File Publications,

1987. Leakey, Richard E., Origins, New York: E. P Dutton, 1977. Lemonick, Michael D., "Fini to Fermat's Last Theorem," Time, 5 July 1993, p. 47.

BIBLIOGRAPHY 301

Lewin, Roger, Bones of Contention: Controversies in the Search for Humnn Origins. New York: Simon and Schuster, 1987.

Lumsden, Charles 1., and Wilson, Edward 0., Promethean Fire: Reflections on the Origin of the Mind. Cambridge: Harvard University Press, 1983.

MacLean, Paul D., The Triune Brain in Evolution. New York: Plenum Press, 1990. McLeish, John, Number. New York: Fawcett Columbine, 1991. Menninger, Karl, Number Words and Number Symbols: A Cultural History of Numbers.

New York: Dover Publications, 1969. Moffatt, Michael, The Ages of Mathemntics, Vol. I, The Origins. New York: Doubleday &

Company, 1977. Muir, Jane, Of Men and Numbers, New York: Dodd, Mead, & Company, 1961. Newman, James, ed., The World of Mathematics. New York: Simon and Schuster, 1956. Newsom, Carroll, Mathemntical Discourses. Englewood Cliffs, NJ: Prentice-Hall, 1964. Niven, Ivan, Numbers: Rational and Irrational. Washington, D.C.: The Mathematical

Association of America, 1961. Ottoson, David, Duality and Unity of the Brain. New York: Plenum Press, 1987. Pappas, Theoni, The Joy of Mathemntics. San Carlos, CA: World Wide PublishingiTetra,

1989. Peter, Rozsa, Playing with Infinity. New York: Dover Publications, 1961. Plato, The Dialogues of Plato, trans. B. Jowett. New York: Random House, 1937. Preston, Richard, "Profiles, The Mountains of Pi," The New Yorker (March 2, 1992). Resnikoff, H. L., and Wells, R. 0., Jr., Mathematics in Civilization. New York: Dover

Publications, 1984. Robins, Gay, and Shute, Charles, The Rhind Mathemntical Papyrus. New York: Dover

Publications, 1987. Rucker, Rudy, Infinity and the Mind. New York: Bantam Books, 1982. Russell, Bertrand, The Problems of Philosophy. London: Oxford University Press, 1959. Sacks, Oliver, The Man Who Mistook His Wife for a Hat. New York: Harper Perennial, 1985. Sagan, Carl, Mind in the Waters, ed. Joan McIntyre. New York: Charles Scribner's Sons, 1974. Schmandt-Besserat, Denise, Before Writing, Vol. I: From Counting to Cuneiform. Austin,

TX: University of Texas Press, 1992. Schmandt-Besserat, Denise, "The Earliest Precursor of Writing," Scientific American

(June 1978). Smith, David Eugene, History of Mathematics, Vol. I. New York: Dover Publications, 1951. Smith, Steven B., The Great Mental Calculators. New York: Columbia University Press,

1983. Smith, T. v., and Grene, Marjorie, eds., Philosophers Speak for Themselves: From Des­

cartes to Locke. Chicago: University of Chicago Press, 1957. Soustelle, Jacques, Mexico. New York: World Publishing Company, 1967. Stein, Sherman K., Mathematics: The Man-Made Universe. San Francisco: W. H. Freeman

and Company, 1963. Treffert, Darold A., Extraordinary People: Understanding Idiots Savants. New York:

Harper & Row Publishers, 1989.

302 BIBLIOGRAPHY

Werkmeister, W. H., A Philosophy of Science. Lincoln, NE: University of Nebraska Press, 1940.

Whitehead, Alfred North, and Russell, Bertrand, Principia Mathematica. Cambridge, England: Cambridge University Press, 1950.

Winick, Charles, Dictionary of Anthropology. Totowa, NJ: Littlefield, Adams, and Com­pany, 1970.

Woodruff, Guy, and Premack, David, "Primitive Mathematical Concepts in the Chim­panzee: Proportionality and Numerosity," Nature 293 (October 15, 1981).

Zaslavsky, Claudia, Africa Counts. New York: Lawrence Hill Books, 1973. Zippin, Leo, Uses of Infinity. Washington, D.C.: The Mathematical Association of

America, 1962.

Index

Abacists, 131 Abacus, 84, 129-130 Absolon, Karl, 33 Absolute Infinite, 226-232, 254 Akousmatikoi, 105 aI-Khowarizmi, Mohammad ibn Musa, 129 Algebra

Babylonian, 64 Chinese, 82 Egyptian, 71 Greek, 107 rhetorical, 74-75, 126, 167-168 symbolic, 74, 126, 167-168 syncopation, 74

Algorithmicists, 131 AI'Mansur, Caliph, 129 Analytic geometry, 119, 210-212 Anaximander of Miletus, 101, 138 Apollonius, 120 Arabs, 129 Archimedes, 103, 120, 168, 180 Argand, Jean Robert, 217 Aristotle, 103, Ill, 119, 138-139, 141,

145-148, 230, 253 Arithmetic-, 107 Aryabhata, 125 Australopithecines, 21 Aztecs, 91

303

Babylonia, 32, 51, 56, 60, 62-66, 72, 96, 101, 127, 174

Berkeley, Bishop, 251 Berry, G.G., 266 Bhiiskara, 126 Bidder, George Parker, 239 Binet, Alfred, 236 Bolzano, Bernhard, 200 Bolzano-Weierstrass theorem, 200 Brahmagupta, 125, 133, 167 Brahmans, 124 Brain

cerebellum, 12 corpus callosum, 13 cortex, 13, 45 gray matter, 13 human, 11-15 neocortex, 12, 20, 45 neuron, 12 prefrontal cortex, 15 size in animals, 44-46 stem, 12

Buddha, 124 Bullae, 57-58 Buxton, Jedediah, 235

Caesar, Julius, 86 Cahokia, 85

304 INDEX

Camayoc, 92 Cantor, Georg, 194-205, 217-218, 221-

222, 223-226, 229-232, 254 Cantor's theorem, 224 Cantor's paradox, 265 Ch'ang Ts'ang, K'iu-ch'ang Suan-shu, 82 Chasquis, 92 Chimpanzees, 19-21, 40-41, 43, 44, 46 China, 65, 77-84, 127 Chou-per, 82 Chuquet, Nicolas, 131 Clever Hans, 38-39 Cohen, Paul, 204 Colburn, Zerah, 238-239 Computers, 233-234, 241, 245-246,

269-277 Consciousness, 11, 249-253, 257 Constructivism, 248, 251, 259-260; see

also Realism Continuum hypothesis, 204-205, 230 Coordinates, 211 Counting

boards, 84, 100, 129-130 body-,27 concrete, 58 early evolution of, 19-26 finger-, 26 five-, 30, 34, 59 in other species, 37-48 learning, 15-17 location in brain, 11-15 neo-2, 30, 34 stick-, 10-11,27,38,43 ten-, 31-32, 59 two-, 29, 34, 59

Cuzco, 91-92

Dase, Johann Martin Zacharias, 235, 237 da Vinci, Leonardo, 156 Decad (ten-ness), 141 Decimal point, 64, 174 Dedekind, Richard, 169-173, 180, 195,

197-198, 218, 224

Descartes, Rene, 119, 148-149, 208-212, 278

Devi, Shakuntala, 235 Diophantus, 74, 119-120, 126, 208,

278 Dolphins, 37, 40, 43, 45-48 Dualism, 251, 257 Dubois, Eugene, 23

e, 183-188, 203, 219 Eberstark, Hans, 236 Egypt, 36, 62, 64, 66, 67-73, 96, 101 Eleatic School, 142 Element, 6 Envelopes, 57-58 Equations

Babylonian, 64 Chinese, 82 Egyptian, 71 exponential, 181-182, 188 indeterminate, 82 logarithmic, 181-182, 188-189 polynomial, 168, 181, 186, 196-197,

207-208, 216, 272 in symbolic algebra, 74 trigonometric, 182, 189

Euclid, 47, 97, 113, 118, 119-120, 151, 165, 204, 263

Eudoxus, 151, 165-167 Euler, Leonhard, 159, 183, 186-187,219,

245 Exhaustion, method of, 149-151; see also

Limit

Fatou, Pierre, 277 Fermat, Pierre de, 277-279 Fertile Crescent, 50-51, 53, 56, 85 Fibonacci, 130-131, 155; see also

Leonardo of Pisa Finkelstein, Salo, 236 Fractals, 270-277 Fraenke1, Adolf, 205 Fuh-hi,78

INDEX 305

Fundamental Theorem of Algebra, 168-169, 2l3-217

Fundamental Theorem of Arithmetic, 237

Galilei, Galileo, 190, 262 Gauss, Carl Friedrich, 149, 168-169,

170, 196, 212-217, 245-246 Gelfond, Aleksander Osipovich, 188 Gerstmann syndrome, 14 Girard, Albert, l32, 208 God, 147-149, 226, 230, 251, 253-254 Giidel, Kurt, 149, 267 Golden Mean (Golden Ratio), 155, 180 Greece, 62, 64, 95-120, 127, l36, 140 Gutenberg, Johannes, 268

Hamilton, William Rowan, 219-222 Hermite, Charles, 187 Hippasus of Metapontum, 117 Hobbes, Thomas, 148 Homo erectus, 22, 33, 35 Homo habilis, 21 Homo sapiens, 25 Homo sapiens sapiens, 25, 36, 49 Hubbard, John, 274 Hunter-gatherers, 32-36, 50, 73, 84, 86

I-Ching, 81-82 Idealism, 248-249, 251, 256-258; see

also Platonism Idiot savants, 240-245 Inaccessibility, 227-228 Inaudi, Jacques, 236 Inca, 91-93 Incommensurable, 115-119, 165-167 Incompleteness theorem, 267 India, 124-129, 167 Infinity, 122-123, l35-159, 167, 191,226 Intuitionism, 230-231

Julia, Gaston, 277

Kant, Immanuel, 251

Khayyam,Omar, 167 Klein, Wim, 236 Kronecker, Leopold, l32, 194, 229-231

Landa, Diego de, 86 Language

and animals, 37 brain area for, 14 evolution of, 29

Leonardo of Pisa, l30-l31, 155; see also Fibonacci

Limit, 149-159; see also Exhaustion, method of

Lindemann, Ferdinand, 187-188 Liouville, Joseph, 187 Logistic, 107

MahiivIra, 126 Mandelbrot, Benoit, 277 Manyness

brain area for, 14-15 of sets, 8-11, 38, 73, 225, 227, 259-

260 Mapping, 8-11, 24, 38, 43-44, 189-194,

197-203, 218, 222 Materialism, 251 Mathematical proof

Babylonian, 66 based on the ratios of magnitudes,

165-167 Chinese, 83 complex and real numbers have the

same cardinality, 218 by computers, 269 countability of algebraic numbers,

196-197 countability of rational numbers, 191-193 deductive method, 103 Egyptian, 73 false position, 71, 82 Fermat's Last Theorem, 277·-279 incommensurability of the square root

of 2, 117-118

306

Mathematical proof (Cont.) indirect method, 118, 140, 199; see

also Reductio ad absurdum Pythagorean theorem, 113-115 Pythagorean, 112 transcendental numbers are

uncountable, 199-202 Mathematikoi, 105 Maya, 81, 85-91, 127 Miletus, 100 Mohenjo Daro, 124 Moscow papyrus, 68 Muhammad, the Prophet, 129

Napier, John, 174 Native Americans, 84-85 Neumann, John von, 245 Newton, Sir Isaac, 168, 272 Number

abstraction of, 28, 58, 81, 102, 260-261

algebraic, 186-189, 196-197,203 as attribute, 31-32 cardinal, 6, 27, 38, 189-194, 201,

203, 222-228 closure of, 75-76, 164-165, 173,216,

219 complex, 207-219, 272 complex conjugates, 217 decimal, 173-178 fractions, 35, 59, 61-63, 68-73, 99,

108, 123-124, 180, 203 imaginary, 210; see also Complex

number infinite, 65, 82, 223-232 irrational, 66, 71, 76, 119, 161-180,

183 Liouville, 187 natural, 6, 135, 140, 146-147, 203 negative, 83, 125-126, 131-133, 161 normal, 177-178 number number, 177 ontology of, 247-262

INDEX

Number (Cont.) ordinal, 7, 27, 225, 227 prime and composite, 110, 235, 237-

242 quaternions, 219-222 rational, 133, 161, 192-194 tag, 7 transcendental, 181-206 transfinite, 223-232 unnamed,39 whole, 6 zero, 63-64, 76, 81, 87, 91, 92, 122-

124, 125-126, 131-133 Number field, 219 Number line, 121-124, 133, 199-201 Number sequence, 152-157, 183,200-

201 Number series, 156-159, 178-180, 183-185 Number theory, 101, 107, 109-111 Numerals

Brahmi,128 East Arabic, 129 Gobar, 128-129 Hindu-Arabic, 81, 91, 99, 127-133 West Arabic, 129

Numerology, 109

Ordering of complex and rational numbers, 218-

219 of natural numbers, 7, 170-171 simply ordered, 163-164 well-ordered, 231-232

Parmenides, 142, 262 Peano, Giuseppe, 264 Pellos, Francesco, 174 Pherecydes of Syros, 138 Philolaus of Tarentum, 3, 141 Phoenicians, 96 Pi

Babylonian, 83 calculation, 178-179

INDEX

Pi (Cant.) Chinese, 83 decimal expansion, 177-178, 236 and e, 219 Egyptian, 71 Indian, 125 and limits, 159 as transcendental, 186-188

Pizarro, Francisco, 91 Plato, 3, 97, 105, 118, 138, 141, 144-145,

165, 230, 253 Platonism, 248-249, 251; see also

Idealism Pope Sylvester II, 130 Proportion, Eudoxus' theory of, 165-167 Prime Number theorem, 245-246 Ptolemy of Alexandria, 126 Pythagoras, 64, 101, 104-108, 138, 230 Pythagorean theorem,

Babylonians, 64-66 Chinese, 65, 82 Egyptians, 71 and the Golden Mean, 155 Pythagoreans, 112-115

Pythagoreans, 105, 107-120, 121, 133, 138, 141-142,144-145,156,161,245,253

Quipus, 53-54, 85, 92-93

Ramanujan, Srinivasa, 129, 245 Realism, 248, 250-251; see also

Constructivism Rechenbucher, 131 Reductio ad absurdum, 118, 140, 199; see

also Mathematical proof Reflection principle, 227-228 Rhind papyrus, 68 Russell, Bertrand, 249, 257-258, 264-265 Russell's Paradox, 265

Sacks, Oliver, 240-241 Sacrobosco, Johannes de, 130 Schmandt-Besserat, Denise, 53-56, 58

Schneider, Theodor, 188 Set

axiom of powers, 224 countable, 190-194 definition, 6 empty, 224 uncountable, 197-203, 225-229

Shi" Huang-ti, 78, 86 Shu-king, 81 Siddhiintas, 125 Socrates, 3 Stifel, Michael, 132 Subitizing, 41-43 Sulvasutras, 125 Sumer, 32, 51, 56-63, 66, 67, 96 Siirya Siddhiinta, 125

Tally sticks, 52-53 Thales, 100-103 Theaetetus, 118 Theophilus, 86 Thomson's Lamp paradox, 266 Tokens, 53-59

Undecidability, 204-205 Universals, 249-251, 253, 257-258

Variihamihira, 125 Viete, Fran<;ois, 174

Weierstrass, Karl, 194, 200 Wessel, Caspar, 217 Whales, 37, 44-48, 256 Whitehead, Alfred North, 264 Widman, Johann, 132 Wiles, Andrew, 278-279 Won-wang, 81

Yau, Emperor, 81

Zeno of Elea, 142-144, 147-148, 151, 154, 158, 262

Zermelo, Ernst, 205, 232

307