A matHUMANtical history of mathematics: towards more scientific literacy

Prehistorical Mathematics?

Blombos Ochre

Dating as early as 70,000 years ago, this ochre engraving from Blombos Cave suggests ancient humans may have been thinking about ways to tile the plane. While other interpretations are certainly possible, it looks like an ancient human attempted to fill the space between the upper and lower lines with copies of a single triangle. Below is my version of the engraving. What other tilings are possible? Submit your solutions!

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Between the time of that engraving and roughly 6000 BCE, there are too few artifacts to trace any sort of intellectual path from the rudiments we see to the record of mathematical knowledge left by the Babylonians. We are left to forge a path ourselves, from scratching stone to generating triples of whole numbers for the construction of right triangles. It wasn't until my fourth year as a math major, during a math history course, that I learned of the Babylonian knowledge of the theorem attributed to Pythagoras, but at least two millenia before his time. The facts of pre-Hellenic mathematics were never offered up for study by any of my math teachers prior to that. What are the missing links? Submit your findings!


A Brief History of Numbers

“God made natural numbers; all else is the work of man“, said Kronecker. I suppose what he meant is that God had no concern for measuring or accounting for what he created as He has no accountability. Natural numbers are an epiphenomena of that which has been brought into existence. All other mathematics is man's attempt to understand and harness Creation. The natural numbers are the sets upon which numerals and numeration systems have been invented to account for and discuss those sets we find important and interesting. In that sense, we are speaking of the natural numbers as a set of sets. However, we will avoid the tedium of that for now and begin with the Old Babylonian approach.

The ancient peoples of the Fertile Crescent invented the following scheme. For counting, ^ was used for one and > was used for ten, up to fifty-nine distinct digits that were used in a positional system much like the decimal system we use today.

The decimal system is actually just the most commonly used case of a more general numeration system called radix. Less common, but no less important, are binary, octal, and hexadecimal. In fact, the Old Babylonian system is an historically notable example of radix numeration which we now call sexagesimal, for it was the first of its kind on record.

There is an account of the problem of taxing shepherds in medieval England which suggests a binary system of accounting was used by administrators to the king. As the story goes, medieval English shepherds were illiterate and innumerate, and thus were required by the king's accountants to do perform an elaborate process which allowed the accountants to get an accurate count of the sheep they held. The idea was to make successive divisions by two and record the remainders from right to left. This would of course result in series of 1s and 0s. For example, if the shepherd had 123 sheep, the accountant would record the series of equations; 123 = 2 ⋅ 61 + 1, 61 = 2 ⋅ 30 + 1, 30 = 2 ⋅ 15 + 0, 15 = 2 ⋅ 7 + 1, 7 = 2 ⋅ 3 + 1, 3 = 2 ⋅ 1 + 1, and 1 = 2 ⋅ 0 + 1 as 1111011. This is not to be mistaken for the number we call by the decimal name one million, one hundred and eleven thousand, and eleven.

We can also do this in another way. Observe that 26 < 123 < 27. Thus, 123 = 26 + 59. If we repeat this process with each remainder after extracting the largest power of 2, then we see that 123 = 26 + 25 + 24 + 23 + 21 + 20. We can rewrite this as 123 = 1 ⋅ 26 + 1 ⋅ 25 + 1 ⋅ 24 + 1 ⋅ 23 + 0 ⋅ 22 + 1 ⋅ 21 + 1 ⋅ 20 to see the digits in order from left to right. We call 26 + 25 + 24 + 23 + 21 + 20 the 2-adic (binary) representation of the number which we write as 123 in decimal (10-adic) and its radix notation is 11110112.

Radix, which is the string of digits resulting from the b-adic representation of numbers, should be taught to elementary school children. The first reason is that there is no understanding without comparison -- in the current curriculum, only decimal notation is taught; there is nothing to compare it to. The second reason is that it provides a deeper understanding of the concept of number as distinct from name, numeral, and representation. The third reason is that it reveals the nature of binary, octal, and hexadecimal which is important for computer science.

Let's make this more formal. For a given number x, if there are natural numbers b and d0, ... , dn, not all zero, such that x = dnbn + ... + d0b0, then we call the sum a b-adic representation of x.

Now that we have defined a numeration for natural numbers, we can begin to define its arithmetic. We may define the addition of natural numbers a and b as the mapping +: N x N -> N defined by +(x,y) = |X U Y| where |X| = x and |Y| = y. However, this will prove difficult to extend to Q. Thus, we will define addition in the geometric sense of laying magnitudes end to end. More formally, for any natural numbers x and y, we define the mapping +: N x N -> N by +(x,y) or x + y denoting the length of the segment formed by laying x and y copies of a unit length end to end.


Pythagorean Triples

Egyptian ropestretchers knew that tying 13 equally spaced knots in a rope would make a handy device for approximating right angles. By fixing an end knot, stretching out five spaces, bringing the end knots together, then stretching three or four spaces, the ropestretchers got an approximation of a right angle as good as their knot spacing. Find some rope and try this.

On grid paper, if you measure three equal spaces horizontally and four more such spaces vertically from either end of the three, then the segment connecting the loose ends will measure pretty close to five such spaces. We call this a 3-4-5 right triangle. The Babylonians recorded many other triples of numbers that, when taken as the lengths of a triangle, will always form a right angle. Try to discover some of them by yourself.

How the Egyptians and Babylonians discovered such triples is an intriguing question. Consider the ratio of the areas of the two squares below.

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We see that the ratio of the areas is 1:2. Thus, the ratio of the sides is 1:1:√2. Is making more right triangles as simple as extending the observation that the sides are in the ratio 1:1:√(1 + 1)? No, because 3:4:√(3 + 4) ≠ 3:4:5. However, if we take the squares of the sides, we see that it works for both: 1:1:(1 + 1) = 1:1:2 and 9:16:(9 + 16) = 9:16:25. Could this be the key? Is every triple of perfect squares associated with a right triangle? Were they using something akin to tangrams to test triples? What was the key observation which led to the generating algorithm?

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Most likely, the ancients were finding these triples by trial and error. It certainly explains the interest in gnomons. Finding a Pythagorean triple is the same as finding a perfect square whose gnomon is also a perfect square. However, they did demonstrate an algorithm for generating Pythagorean triples, so they moved beyond trial and error at some point.

A chronological list of important publications in mathematics

  1. Moscow Mathematical Papyrus (1850 BCE)
  2. Rhind Mathematical Papyrus by Ahmes (1620 BCE)
  3. Baudhayana Sulba Sutra by Baudhayana (8th century BCE)
  4. Elements by Euclid (300 BCE)
  5. Archimedes Palimpsest by Archimedes of Syracuse (212 BCE)
  6. The Sand Reckoner by Archimedes of Syracuse (212 BCE)
  7. The Nine Chapters on the Mathematical Art from the 10th–2nd century BCE
  8. The Conics by Apollonius of Perga (190 BCE)
  9. Haidao Suanjing by Liu Hui (220-280 CE)
  10. Surya Siddhanta by Unknown Author (400 CE)
  11. Sunzi Suanjing by Sunzi (5th century CE)
  12. Aryabhatiya by Aryabhata (499 CE)
  13. Jigu Suanjing by Wang Xiaotong (626 CE)
  14. Brāhmasphuṭasiddhānta by Brahmagupta (628 CE)
  15. Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa'l-muqābala by Muhammad ibn Mūsā al-Khwārizmī (820 CE)
  16. Līlāvatī, Siddhānta Shiromani and Bijaganita by Bhāskara II
  17. Yigu yanduan by Liu Yi (12th century)
  18. Mathematical Treatise in Nine Sections by Qin Jiushao (1247)
  19. Ceyuan haijing by Li Zhi (1248)
  20. Jade Mirror of the Four Unknowns by Zhu Shijie (1303)
  21. Yuktibhāṣā by Jyeshtadeva (1501)
  22. Ars Magna by Gerolamo Cardano (1545)
  23. La Géométrie by René Descartes (1637)
  24. Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illi calculi genus by Gottfried Leibniz (1684)
  25. Philosophiae Naturalis Principia Mathematica by Isaac Newton (1687)
  26. Method of Fluxions by Isaac Newton (1736)
  27. Solutio problematis ad geometriam situs pertinentis by Leonhard Euler (1741)
  28. De fractionibus continuis dissertatio by Leonhard Euler (1744) [Translation]
  29. Introductio in analysin infinitorum by Leonhard Euler (1748)
  30. Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum by Leonhard Euler (1755)
  31. Recherches sur la courbure des surfaces by Leonhard Euler (1760)
  32. Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies by Joseph Louis Lagrange (1761)
  33. Vollständige Anleitung zur Algebra by Leonhard Euler (1770)
  34. Réflexions sur la résolution algébrique des équations by Joseph Louis Lagrange (1770)
  35. Recherches d'Arithmétique by Joseph Louis Lagrange (1775)
  36. Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse by Carl Friedrich Gauss (1799)
  37. Disquisitiones Arithmeticae by Carl Friedrich Gauss (1801)
  38. Mémoire sur la propagation de la chaleur dans les corps solides by Joseph Fourier (1807)
  39. Disquisitiones generales circa superficies curvas by Carl Friedrich Gauss (1827)
  40. Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données by Peter Gustav Lejeune Dirichlet (1829)
  41. "Beweis des Satzes, daß jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält" by Peter Gustav Lejeune Dirichlet (1837)
  42. Articles Publiés par Galois dans les Annales de Mathématiques, published by Journal de Mathematiques pures et Appliquées, II (1846)
  43. Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse by Bernhard Riemann (1851)
  44. Über die Hypothesen, welche der Geometrie zu Grunde Liegen by Bernhard Riemann (1854)
  45. The Laws of Thought by George Boole (1854)
  46. Theorie der Abelschen Functionen by Bernhard Riemann (1857)
  47. "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" by Bernhard Riemann (1859)
  48. Vorlesungen über Zahlentheorie by Peter Gustav Lejeune Dirichlet and Richard Dedekind (1863)
  49. Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe by Bernhard Riemann (1867)
  50. Traité des substitutions et des équations algébriques by Camille Jordan (1870)
  51. "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" by Georg Cantor (1874)
  52. Begriffsschrift by Gottlob Frege (1879)
  53. Leçons sur la théorie génerale des surfaces et les applications géométriques du calcul infinitésimal by Gaston Darboux (1887 - 1890)
  54. Theorie der Transformationsgruppen by Sophus Lie and Friedrich Engel (1888–1893)
  55. Formulario mathematico by Giuseppe Peano (1895)
  56. Analysis situs by Henri Poincaré (1895, 1899–1905)
  57. Zahlbericht by David Hilbert (1897)
  58. Grundlagen der Geometrie by David Hilbert (1899)
  59. Intégrale, longueur, aire by Henri Lebesgue (1901)
  60. Principia Mathematica by Bertrand Russell and Alfred North Whitehead (1910–1913)
  61. Grundzüge der Mengenlehre by Felix Hausdorff (1914)
  62. "Zur Theorie der Gesellschaftsspiele" by John von Neumann (1928)
  63. "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I" (On Formally Undecidable Propositions of Principia Mathematica and Related Systems) by Kurt Gödel (1931)
  64. Théorie des opérations linéaires by Stefan Banach (1932)
  65. "The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory" by Kurt Gödel (1938)
  66. Systems of Logic Based on Ordinals - Alan Turing's PhD thesis (1938)
  67. "Математические методы организации и планирования производства" by Leonid Kantorovich (1939)
  68. Theory of Games and Economic Behavior by Oskar Morgenstern, John von Neumann (1944)
  69. "General Theory of Natural Equivalences" by Samuel Eilenberg and Saunders Mac Lane (1945)
  70. L'anneau d'homologie d'une représentation, Structure de l'anneau d'homologie d'une représentation by Jean Leray (1946)
  71. Regular Polytopes by H.S.M. Coxeter (1947)
  72. Fourier Analysis in Number Fields and Hecke's Zeta-Functions by John Tate (1950)
  73. "Equilibrium Points in N-person Games" by Nash, John F. (January 1950)
  74. Quelques propriétés globales des variétés differentiables by René Thom (1954)
  75. Faisceaux Algébriques Cohérents by Jean-Pierre Serre Publication data: Annals of Mathematics, 1955
  76. Produits Tensoriels Topologiques et Espaces Nucléaires by Grothendieck, Alexander (1955)
  77. Géométrie Algébrique et Géométrie Analytique by Jean-Pierre Serre (1956)
  78. Homological Algebra by Henri Cartan and Samuel Eilenberg (1956)
  79. Sur Quelques Points d'Algèbre Homologique (the Tohoku paper) by Alexander Grothendieck (1957)
  80. Le théorème de Riemann–Roch, d'après A. Grothendieck by Armand Borel and Jean-Pierre Serre (1958)
  81. Solvability of groups of odd order by Walter Feit and John Thompson (1960)
  82. "On the evolution of random graphs" by Paul Erdős and Alfréd Rényi (1960)
  83. "Decomposition Principle for Linear Programs" by George Dantzig and P. Wolfe in Operations Research 8:101–111, 1960.
  84. "Network Flows and General Matchings" by L. R. Ford, Jr. & D. R. Fulkerson by Flows in Networks. Prentice-Hall, 1962.
  85. "The Independence of the Continuum Hypothesis" by Paul J. Cohen (1963, 1964)
  86. On convergence and growth of partial sums of Fourier series by Lennart Carleson (1966)
  87. Éléments de géométrie algébrique by Alexander Grothendieck (1960–1967)
  88. How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension by Benoît Mandelbrot (1967)
  89. Séminaire de géométrie algébrique by Alexander Grothendieck (1960–1969)
  90. "Automorphic Forms on GL(2)" by Hervé Jacquet and Robert Langlands (1970)
  91. "How Good is the Simplex Algorithm?" by Victor Klee and George J. Minty (1972)
  92. Categories for the Working Mathematician by Saunders Mac Lane (1971, second edition 1998)
  93. Topological Vector Spaces by Grothendieck, Alexander (1973)
  94. "La conjecture de Weil. I." by Pierre Deligne (1974)
  95. "On sets of integers containing no k elements in arithmetic progression" by Endre Szemerédi (1975)
  96. On Numbers and Games by John Horton Conway (1976)
  97. "Полиномиальный алгоритм в линейном программировании" by Khachiyan, Leonid Genrikhovich (1979)
  98. Winning Ways for your Mathematical Plays by Elwyn Berlekamp, John Conway and Richard K. Guy (1982)
  99. "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" by Gerd Faltings (1983)
  100. Sur certains espaces vectoriels topologiques by Bourbaki, Nicolas (1987)
  101. "Modular Elliptic Curves and Fermat's Last Theorem" by Andrew Wiles (1995)
  102. The geometry and cohomology of some simple Shimura varieties by Michael Harris and Richard Taylor (2001)
  103. "Le lemme fondamental pour les algèbres de Lie" by Ngô Bảo Châu (2008)
  104. Higher Topos Theory by Jacob Lurie (2010)
  105. "Perfectoid space" by Peter Scholze (2012)