Abstract
The history of the continuous inclusion of mathematics in liberal education in the West, from ancient times through the modern period, is sketched in the first two sections of this chapter. Next, the heart of this essay (Sects. 3, 4, 5, 6, and 7) delineates the central role mathematics has played throughout the history of Western civilization: not just a tool for science and technology, mathematics continually illuminates, interacts with, and sometimes challenges fields like art, music, literature, and philosophy – subjects now universally considered to be liberal arts. Section 8 adds an international perspective to the contemporary liberal arts story by describing some instructive mathematical achievements from many cultures and societies. Finally, Sect. 9 addresses how contemporary mathematics teaching can use the history of mathematics viewed as a liberal art to enhance the appreciation and understanding of mathematics for all students.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In modern discourse about education, the term “liberal arts” has been defined and characterized in a variety of ways. It is of course built into the history of Western education. The influential Carnegie Foundation for the Advancement of Teaching has provided a list of contemporary “liberal arts” that goes beyond the traditional Western canon: English language and literature, foreign languages, letters, liberal and general studies, life sciences, mathematics, physical sciences, psychology, social sciences, visual and performing arts, area and ethnic studies, multi- and interdisciplinary studies, philosophy, and religion (Carnegie 1994, p. xx; Ferrall 2011, p. 9). In antiquity, as the next sections of this paper will describe, the list came to have seven items: arithmetic, geometry, astronomy, and music theory (the quadrivium), and grammar, rhetoric, and logic (the trivium) (Stahl 1977; Wagner 1983b). Those who compile such lists characterize liberal arts education as study undertaken for its own sake, as opposed to vocational education. As for the purpose of liberal arts education, it has been described as educating a free person, or as liberating the mind to pursue the truth, or as producing a cultivated person who can be both a good citizen and a leader of society. Liberal arts instruction has sometimes focused on the canonical texts of Western civilization as ways to build and reinforce the shared values of society. But such instruction has also been championed as suitable for the education of free individuals to be citizens in a democracy by developing the capacities for independent and critical thinking, logical analysis, effective communication, an understanding of the interrelations between different fields of learning, and imagination (Ferrall 2011; Kimball 1995; Nussbaum 2010; Sinaiko 1998). The present chapter recognizes and appreciates these disparate views. The Carnegie list is useful to illustrate the types of subjects that constitute a liberal arts education today, and the present essay shares the view that modern liberal education’s most important goal is to educate independent and thoughtful citizens.
- 2.
For Proclus, see Proclus (1970, p. 53). The term “mathematics” itself reveals a liberal arts origin; the Greek root “mathema” was first more general, connoting merely “something learned,” and the “mathematikoi” were the inner initiates of the Pythagorean school. For pre-Euclidean logically structured geometry, see Knorr (1975, esp. p. 7) and McKirahan (1992, pp. 16–18).
- 3.
- 4.
Distinguishing experimental philosophy from reasoning from arbitrary hypotheses, Newton wrote, “In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction” (Newton 1934, p. 547). Once Newton had his general principles, his Principia could take the logical structure familiar from Euclid’s Elements.
- 5.
Khwārizmī’s title can be translated as “the book of restoring and balancing,” where the Arabic “al-jabr” or “restoring” was interpreted as adding the same thing to both sides of an equation and “al-muqabala” or “balancing” the subtraction of the same quantity from both sides of an equation (Berggren 1986, p. 7). The sense of “al-jabr” as “restoring” remains in Spanish, where, for instance, in Don Quixote, Part II, Chap. XV, a bonesetter is an “algebrista” (Merzbach and Boyer 2011, p. 207).
- 6.
The point will be clearer with a fuller quotation: “With the feeling…that he was setting forth an important axiom, he wrote: Freedom is the freedom to say that two plus two make four. If that is granted, all else follows” (Orwell 1949, p. 81; his italics).
- 7.
- 8.
(Olson 2008, pp. 96–121; Richards 2002, pp. 11, 308–310). The Romantics would not admit that what epistemologists call “secondary qualities” like color, so constitutive of human experience, are mere epiphenomena reducible to “primary qualities” of matter in motion nor that greater understanding necessarily follows from mathematical description. As John Keats put it in criticizing “philosophy” (science) in his poem Lamia (part 2):
Philosophy will clip an angel’s wings,
Conquer all mysteries by rule and line,…
Unweave a rainbow.
- 9.
(Joseph 2011, p. 462). Sebokht, a Syrian bishop, was challenging the supposed universal superiority of Greek scientific thought by praising the superior methods of calculation using the base-10 place-value number system from India.
- 10.
There is now an extensive and reliable English-language literature on the mathematics of other cultures. See, for instance, Ascher (1998, 2002), Berggren (1986), (2007), Closs (1986), Dauben (2007), Gerdes (1999), Gillings (1972), Imhausen (2007), Katz (2000), (2007), Martzloff (1997), Plofker (2007), (2009), Robson (2007), (2008), Robson and Stedall (2009), Van Brummelen (2009), and Zaslavsky (1999).
- 11.
- 12.
The question is, if long or heavy syllables are two beats and short or light syllables are one beat, what is the number of different arrangements A(n) of long and short syllables for a line of n beats? For example, if there are two beats and if we use “S” for short and “L” for long, the arrangements are SS and L. If there are three beats, the arrangements are SSS, SL, and LS. If there are four beats, the arrangements are SSSS, SSL, and SLS (formed by placing an S in front of each of the arrangements for three beats), plus LSS and LL (formed by placing an L in front of each of the arrangements for two beats). Thus, A(4) = A(3) + A(2). Since A(2) = 2 and A(3) = 3 and since the method of forming A(n) from A(n-2) and A(n-1) must follow the same pattern, this gives the Fibonacci series (Singh 1985).
- 13.
Discussions about how this can and has been done, and how it has been assessed, may be consulted in Alternatives for Rebuilding Curricula Center (2003), Ball et al. (2005), Boaler and Staples (2008), Hill et al. (2005), and Tarr et al. (2008). A cross-cultural study involving Chinese and American teachers at the elementary-school level can be found in Ma (1999).
- 14.
Examples of books that might be suitable for such courses include Ascher (1998 and 2002), Frantz and Crannell (2011), Gerdes (1999), and Packel (1981). For details about the author’s courses, see Grabiner (2011). A superb online resource for liberal arts mathematics teaching is the Mathematical Association of America’s “magazine” of the history of mathematics and its uses in the classroom, Convergence (n.d.).
- 15.
On women in mathematics in general, see the online biographies maintained by Agnes Scott College (2012), the sourcebook Grinstein and Campbell (1987), and the Mathematical Association of America’s poster Women of Mathematics (MAA 2008). On important individual women in mathematics, see Arianrhod (2012), Brewer and Smith (1989), Dahan-Dalmédico (1991), Deakin (2007), Hagengruber (2012), Katz (2009, pp. 189–190, 616–617, 714–715, 787, 874, 896–898, 899), Koblitz (1983), Mazzotti (2007), Neeley (2001), Reid (1996), and Zinsser (2006).
References
Abbott, E. (1953/1884). Flatland: A Romance of Many Dimensions. New York: Dover.
Abbott, E. (2010/1884). Flatland. With Notes and Commentary by W. F. Lindgren & T. F. Banchoff. Washington, DC: Mathematical Association of America.
Agnes Scott College (2012). Biographies of Mathematicians. Online only. http://www.agnesscott.edu/lriddle/women/women.htm. Consulted 6 July 2012.
Alternatives for Rebuilding Curricula Center (2003). Tri-State Student Achievement Study. Lexington, MA. Available at http://www.comap.com/elementary/projects/arc/index.htm. Consulted 15 June 2012.
Arianrhod, R. (2012). Seduced by Logic: Émilie Du Châtelet, Mary Somerville and the Newtonian Revolution. New York: Oxford University Press.
Aristotle, De Anima. In (Aristotle 1941), pp. 535–603.
Aristotle, Metaphysics. In (Aristotle 1941), pp. 689–926.
Aristotle (1941). The Basic Works of Aristotle. Ed. R. McKeon. New York: Random House.
Ascher, M. (1998). Ethnomathematics: A Multicultural View of Mathematical Ideas. New York: Chapman & Hall/CRC.
Ascher, M. (2002). Mathematics Elsewhere: An Exploration of Ideas across Cultures. Princeton and Oxford: Princeton University Press..
Ashcraft, M. H. (2002). Math Anxiety: Personal, Educational, and Cognitive Consequences. Current Directions in Psychological Science, Vol. 11, No. 5 (October), 181–185.
Babbage, C. (1832). On the Economy of Machinery. London: Charles Knight.
Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade and how can we decide? American Educator, Fall, 14–46.
Barrett, W. (1958). Irrational Man: A Study in Existential Philosophy. New York: Doubleday.
Barzun, J. (1945). Teacher in America. London: Little Brown.
Baudrillard, J. (1994) Reversion of History. In CTheory.net, April 20, 1994. www.ctheory.net/articles.aspx?id = 54 Consulted 15 May 2012.
Becker, C. L. (1922). The Declaration of Independence: A Study in the History of Political Ideas. New York: Harcourt Brace.
Berggren, J. L. (1986). Episodes in the Mathematics of Medieval Islam. New York et al: Springer.
Berggren, J. L. (2007). Mathematics in Medieval Islam. In (Katz 2007), pp. 515 – 675.
Boaler, J. & Staples, M. (2008). Creating Mathematical Futures through an Equitable Teaching Approach: The Case of Railside School. Teacher’s College Record. 110 (3), 608–645.
Bonola, R. (1955). Non-Euclidean Geometry: A Critical and Historical Study of Its Development. With a Supplement containing “The Theory of Parallels” by Nicholas Lobachevski and “The Science of Absolute Space” by John Bolyai. Tr. H. S. Carslow. New York: Dover.
Boole, G. (1854). An Investigation of The Laws of Thought. London: Walton and Maberly. Reprint: New York: Dover, n.d.
Boyer, C. B. (1956). History of Analytic Geometry. New York: Scripta Mathematica.
Brewer, J. W., & Smith, M. K. (1989). Emmy Noether: A Tribute to Her Life and Work. New York: Marcel Dekker.
Burger, E., & Starbird, M. (2012). The Heart of Mathematics: An Invitation to Effective Thinking. 4th Edition. New York: Wiley.
Carnegie Foundation for the Advancement of Teaching. (1994). A Classification of Institutions of Higher Education. Princeton: Carnegie Foundation.
Cauchy, A.-L. (1892/1821). Cours d’analyse de l’Ecole Royale Polytechnique. lre Partie: Analyse algébrique [all published]. Paris: Imprimérie royale. Reprinted in A.-L. Cauchy, Oeuvres, series 2, vol. 3. Paris: Gauthier- Villars. English translation in (Cauchy 2009).
Cauchy, A.-L. (2009). Cauchy’s Cours d’analyse: An Annotated Translation. Tr. Bradley, R. E. & Sandifer, C. E. Dordrecht et al: Springer. English translation of (Cauchy 1892).
Cesareo, F. C. (1993). Quest for Identity: The Ideals of Jesuit Education in the Sixteenth Century. In (Chapple 1993b), pp. 17–33.
Chapple, C. (1993a). Introduction. In (Chapple 1993b), pp. 7–12.
Chapple, C. (Ed.). (1993b). The Jesuit Tradition In Education and Missions. Scranton, PA: University of Pennsylvania Press.
Chemla, K. (2012). Using documents from ancient China to teach mathematical proof. In (Hanna & de Villiers 2012), pp. 423–429.
Cicero, M. T. De oratore. In Two Volumes. Books I-II, Vol. 1; Book III, Vol. 2. Tr. H. Rackham. London: William Heinemann, and Cambridge, MA: Harvard University Press, 1942.
Cicero, M. T. De re publica; de legibus. Tr. C. W. Keyes. London: William Heinemann, and Cambridge, MA: Harvard University Press, 1928.
Clifford, W. K. (1956). The Postulates of the Science of Space. In (Newman 1956), Vol. I, pp. 552–567.
Closs, M. (1986). The Mathematical Notation of the Ancient Maya. In M. Closs, ed., Native American Mathematics. Austin: University of Texas Press, pp. 291–369.
Cohen, I. B. (1980). The Newtonian Revolution. Cambridge, UK: Cambridge University Press.
Cohen, I. B. (1995). Science and the Founding Fathers. New York: W. W. Norton.
COMAP (Consortium for Mathematics and Its Applications). (2013). For All Practical Purposes: Mathematical Literacy in Today’s World. Ninth Edition. San Francisco: W. H. Freeman.
Comte, A. (1830). Cours de philosophie positive. Vol. I. Paris: Bachelier.
Condorcet, Marquis de. (1976/1793). Sketch for a Historical Picture of the Progress of the Human Mind. Tr. J. Barraclough. In Baker, K. M., ed., Condorcet: Selected Writings. Indianapolis: Bobbs-Merrill.
Convergence (n. d.). (http://mathdl.maa.org/mathDL/46/). Consulted 24 June 2012.
Cruz, J. A. H. M. (1999). Education. Encyclopedia of the Renaissance, vol. 2. New York: Scribner’s, pp. 242–254.
Dahan-Dalmédico, A. (1991). Sophie Germain. Scientific American (265), 117–122.
D’Ambrosio, U. (1985). Ethnomathematics and Its Place in the History and Pedagogy of Mathematics. For the Learning of Mathematics, 5(1), 44–48.
Dauben, J. (2007). Chinese Mathematics. In (Katz 2007), pp. 187–385.
Deakin, M. (2007). Hypatia of Alexandria: Mathematician and Martyr. Amherst, NY: Prometheus Books.
Descartes, R. (1637a). Discourse on Method. Tr. L. J. Lafleur. New York: Liberal Arts Press, 1956.
Descartes, R. (1637b). La géométrie. Tr. D. E. Smith and M. L. Latham as The Geometry of René Descartes. New York: Dover, 1954.
Dickens, C. (1854). Hard Times. Norton Critical Edition. G. Ford and S. Monod (Eds.). New York and London: W. W. Norton, 1966.
Euclid, Elements. In (Heath 1925). On-line version: http://aleph0.clarku.edu/~djoyce/java/elements/toc.html. Consulted 2 July 2012.
Ferrall, V. E. (2011). Liberal Arts at the Brink. Cambridge, MA and London: Harvard University Press.
Fauvel, J,. & Gray, J. (1987). The History of Mathematics: A Reader. Houndmills and London: Macmillan.
Fauvel, J., Flood, R., & Wilson, R. (Eds.) (2003). Music and Mathematics: From Pythagoras to Fractals. Oxford: Oxford University Press.
Field, J. V. (1997). The Invention of Infinity: Mathematics and Art in the Renaissance. Oxford: Oxford University Press.
Field, J. V. (2003). Musical cosmology: Kepler and His Readers. In (Fauvel et al 2003), pp. 28–44.
Frantz, M., & Crannell, A. (2011). Viewpoints: Mathematical Perspective and Fractal Geometry in Art. Princeton: Princeton University Press.
Friedman, M. (1992). Kant and the Exact Sciences. Cambridge, MA: Harvard University Press.
Gaukroger, S. (1995). Descartes: An Intellectual Biography. Oxford: Oxford University Press.
Gerdes, P. (1999). Geometry from Africa: Mathematical and Educational Perspectives. Washington, DC: Mathematical Association of America.
Gillings, R. J. (1972). Mathematics in the Time of the Pharaohs. Cambridge, MA: M. I. T. Press.
Gillispie, C. C. (1960). The Edge of Objectivity. Princeton: Princeton University Press.
Grabiner, J. V. (1988). The Centrality of Mathematics in the History of Western Thought. Mathematics Magazine 61, 220–230. Reprinted in (Grabiner 2010), pp. 163–174.
Grabiner, J. V. (2010). A Historian Looks Back: The Calculus as Algebra and Selected Writings. Washington, DC: Mathematical Association of America.
Grabiner, J. V. (2011). How to Teach Your Own Liberal Arts Course. Journal of Humanistic Mathematics. Vol. I (1), 101–118. http://scholarship.claremont.edu/jhm/vol1/iss1/8. Consulted 7 July 2012.
Grant, H. (1999a). Mathematics and the Liberal Arts I. College Mathematics Journal 30, No. 2, 96–105.
Grant, H. (1999b). Mathematics and the Liberal Arts II. College Mathematics Journal 30, No. 3, 197–203.
Gray, J. (1989). Ideas of Space: Euclidean, Non-Euclidean and Relativistic. 2nd ed. Oxford: Clarendon Press.
Grendler, P. F. (1989). Schooling in Renaissance Italy: Literacy and Learning, 1300-1600. Baltimore and London: Johns Hopkins University Press.
Grendler, P. F. (2002). The Universities of the Italian Renaissance. Baltimore and London: Johns Hopkins University Press.
Grinstein, L. S., & Campbell, P. J. (Eds.) (1987). Women of Mathematics: A Biobibliographic Sourcebook. New York: Greenwood Press.
Hadid, Z. (2008). http://lan-haiyun.blogspot.com/2012/03/5-concepts-of-zaha-hadid.html. Consulted 14 May 2012.
Hagengruber, R. (Ed.) (2012). Emilie du Châtelet between Leibniz and Newton. Dordrecht and New York: Springer.
Hanna, G. & de Villiers, M. (Eds.) (2012). Proof and Proving in Mathematics Education. Dordrecht et al: Springer.
Hardy, G. H. (1967/1940). A Mathematician’s Apology. Foreword by C. P. Snow. Cambridge: Cambridge University Press.
Haskins, C. H. (1927). The Renaissance of the Twelfth Century. Cambridge MA: Harvard University Press.
Heath, T. L. (Ed.). (1925). The Thirteen Books of Euclid’s Elements, 3 vols. Cambridge, UK: Cambridge University Press. Reprinted New York: Dover, 1956.
Heath, T. L. (1949). Mathematics in Aristotle. Oxford: Oxford University Press.
Helmholtz, H. (1954/1863). On the sensations of tones. Tr. A. J. Ellis. New York: Dover.
Helmholtz, H. (1962/1870). On the origin and significance of geometrical axioms. Reprinted in H. von Helmholtz, Popular Scientific Lectures, ed. M. Kline. New York: Dover, pp. 223–249.
Henderson, L. (1983). The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Princeton: Princeton University Press.
Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal (42), 371–406.
Hobbes, T. (1939/1651). Leviathan, or the Matter, Form, and Power of a Commonwealth, Ecclesiastical and Civil. Reprinted in E. Burtt (Ed.), The English Philosophers from Bacon to Mill. New York: Modern Library.
Hoeflich, M. H. (1986). Law and Geometry: Legal Science from Leibniz to Langdell. American Journal of Legal History (30), 95–121.
Høyrup, J. (1994). In Measure, Number, and Weight: Studies in Mathematics and Culture. Albany, NY: State University of New York Press.
Horng, W. S. (2000). Euclid versus Liu Hui: A Pedagogical Reflection. In (Katz 2000), pp. 37–47.
Huffman, C. (2011). Archytas. The Stanford Encyclopedia of Philosophy (Fall 2011 Edition), ed. E. N. Zalta. http://plato.stanford.edu/archives/fall2011/entries/archytas/. Consulted 12 June 2012.
Hutcheson, F. (2004/1728). On computing the morality of actions. In F. Hutcheson, Inquiry into the Original of Our Ideas of Beauty and Virtue in Two Treatises, ed. Wolfgang Leidhold. Indianapolis: Liberty Fund.
Hyman, A. (1982). Charles Babbage: Pioneer of the Computer. Princeton: Princeton University Press.
Imhausen, A. (2007). Egyptian Mathematics. In (Katz 2007), pp. 7–56.
Jacobs, H. R. (2012). Mathematics: A Human Endeavor. 4th Edition. San Francisco: Freeman.
Jaeger, W. (1944). Paideia: The Ideals of Greek Culture. Tr. Gilbert Highet. 3 vols. Oxford: Oxford University Press. [Vol. 1, 1939, vol. 2, 1943, vol. 3, 1944]
Jeans, J. (1956). Mathematics of Music. In (Newman 1956), vol. 4, pp. 2278–2309.
Joseph, G. G. (2011). The Crest of the Peacock: Non-European Roots of Mathematics. 3d edition. Princeton and Oxford: Princeton University Press.
Kalman, L. (1986). Legal Realism at Yale, 1927-1960. Chapel Hill and London: University of North Carolina Press.
Kant, I. (1950/1783). Prolegomena to Any Future Metaphysics. Ed. L. W. Beck. New York: Liberal Arts Press.
Kant, I. (1961/1781). Critique of Pure Reason. Tr. F. M. Müller. New York: Macmillan.
Karp, T. C. (1983). Music. In (Wagner 1983), pp. 169–195.
Katz, V. J. (Ed). (2000). Using History to Teach Mathematics: An International Perspective. Washington, DC: Mathematical Association of America.
Katz, V. J. (Ed.) (2007). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton and Oxford: Princeton University Press.
Katz, V. J. (2009). A History of Mathematics: An Introduction. 3d edition. Boston et al: Addison-Wesley.
Kemp, M. (1990). The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat. New Haven: Yale University Press.
Kimball, B. (1995). Orators and Philosophers: A History of the Idea of Liberal Education. Expanded edition. New York: The College Board.
Klein, J. (1968). Greek Mathematical Thought and the Origins of Algebra. Tr. E. Brann. With an Appendix containing Vieta’s Introduction to the Analytic Art. Cambridge, MA: M. I. T. Press.
Kline, M. (1953). Mathematics in Western Culture. Oxford and New York: Oxford University Press.
Knorr, W. R. (1975). The Evolution of the Euclidean Elements. Dordrecht: D. Reidel.
Koblitz, A. H. (1983). A Convergence of Lives: Sofia Kovalevskaia, Scientist, Writer, Revolutionary. Boston: Birkhaüser.
LaPiana, W. P. (1994). Logic and Experience: The Origin of Modern American Legal Education. New York and Oxford: Oxford University Press.
Laplace, P.-S. (1951/1819). A Philosophical Essay on Probabilities. Tr. F. W. Truscott and F. L. Emory. New York: Dover.
Leibniz, G. W. (1951/1677). Preface to the General Science and Towards a Universal Characteristic. In P. P. Weiner (Ed.), Leibniz: Selections. New York: Scribner’s, pp. 12–25.
Leibniz, G. W. (1969/1684). A new method for maxima and minima…and a remarkable type of calculus for them. Tr. D. J. Struik. In (Struik 1969), pp. 271–280.
Leslie, D. (1964). Argument by contradiction in Pre-Buddhist Chinese Reasoning. Canberra: Australian National University.
Levinson, S. (1996). Frames of Reference and Molyneux’s Question: Crosslinguistic Evidence. In P. Bloom et al, Language and Space. Cambridge, MA: M. I. T. Press, pp. 109–170.
Levinson, S. (2003). Language and Mind: Let’s Get the Issues Straight. In D. Gertner and S. Goldin-Meadow (Eds.), Language in Mind: Advances in the Study of Language and Thought. Cambridge, MA: M. I. T. Press, pp. 25–46.
Lovejoy, A. (1936). The Great Chain of Being. Cambridge, MA: Harvard University Press.
Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. Mahwah, NJ: Erlbaum.
MAA (2008). Women of Mathematics Poster. Washington, D. C.: Mathematical Association of America. http://www.maa.org/pubs/posterW.pdf. Consulted 6 July 2012.
Maclaurin, C. (1714). De viribus mentium bonipetis. MS 3099.15.6, The Colin Campbell Collection, Edinburgh University Library. Translated in (Tweddle 2008).
Mahoney, M. (1973). The Mathematical Career of Pierre de Fermat, 1601-1655. Princeton: Princeton University Press.
Malthus, T. R. (1798). An Essay on the Principle of Population, as it Affects the Future Improvement of Society: with Remarks on the Speculations of Mr. Godwin, M. Condorcet, and other writers. London: J. Johnson.
Marr, A. (2011). Between Raphael and Galileo: Mutio Oddi and the Mathematical Culture of Late Renaissance Italy. Chicago: University of Chicago Press.
Marrou, H. I. (1956). A History of Education in Antiquity. Tr. G. Lamb. New York: Sheed and Ward. Mentor reprint, New York, 1964.
Martzloff, J.-C. (1997). A History of Chinese Mathematics. Tr. S. S. Wilson. Berlin: Springer.
Masi, M. (1983). Arithmetic. In (Wagner 1983), pp. 147–168.
Mazzotti, M. (2007). The World of Maria Gaetana Agnesi: Mathematician of God. Baltimore: Johns Hopkins Press.
McInerny, R. (1983). Beyond the Liberal Arts. In (Wagner 1983b), pp. 248–272.
McKirahan, R. D. Jr. (1992). Principles and Proofs: Aristotle’s Theory of Demonstrative Science. Princeton: Princeton University Press.
Merz, J. T. (1904). A History of European Thought in the Nineteenth Century. Vol. I. London: Blackwood. Reprinted New York: Dover, 1965.
Merzbach, U., & Boyer, C. B. (2011). A History of Mathematics. 3d edition. New York: Wiley.
Morrison, K. F. (1983). Incentives for Studying the Liberal Arts. In (Wagner 1983b), pp. 32–57.
Neeley, K. A. (2001). Mary Somerville: Science, Illumination, and the Female Mind. Cambridge: Cambridge University Press.
Newman, J. R. (Ed.). (1956). The World of Mathematics. 4 vols. New York: Simon and Schuster.
Newton, I. (1934/1687). Sir Isaac Newton’s Mathematical Principles of Natural Philosophy and His System of the World. Tr. A. Motte. Revised and edited by F. Cajori. Berkeley: University of California Press.
Nussbaum, M. (2010). Not for Profit: Why Democracy Needs the Humanities. Princeton: Princeton University Press.
Olson, R. (2008). Science and Scientism in Nineteenth-Century Europe. Urbana and Chicago: University of Illinois Press.
Ortega y Gasset, J. (1968/1961). The Historical Significance of the Theory of Einstein. In (Williams 1968), pp. 147–157.
Orwell, G. (1949). Nineteen Eighty-Four: A Novel. New York: Harcourt Brace.
Packel, E. (1981). The Mathematics of Games and Gambling. Washington, DC: Mathematical Association of America.
Pascal, B. (1931/1669). Pensées. Tr. W. F. Trotter. New York: E. P. Dutton.
Peirce, B. (1881). Linear associative algebra with notes and addenda by C. S. Peirce. American Journal of Mathematics (4), 97–229.
Plato. (1961). The Collected Dialogues of Plato, Including the Letters. Ed. E. Hamilton and H. Cairns. New York: Pantheon Books.
Plato. Republic. Tr. P. Shorey. In (Plato 1961), pp. 575–844.
Plofker, K. (2007). Mathematics in India. In (Katz 2007), pp. 385–514.
Plofker, K. (2009). Mathematics in India. Princeton and Oxford: Princeton University Press.
Poincaré, H. (1952/1905). Science and Hypothesis. New York: Dover.
Popkin, R. H. (1979). The History of Scepticism from Erasmus to Spinoza. Berkeley and Los Angeles: University of California Press.
Porter, T. (1986). The Rise of Statistical Thinking, 1820-1900. Princeton: Princeton University Press.
Proclus. (1970/5th century CE). A Commentary on the First Book of Euclid’s Elements. Tr. G. R. Morrow. Princeton: Princeton University Press.
Quetelet, A. (1828). Instructions populaires sur le calcul des probabilités. Brussels: Tarlier & Hayez.
Rabinovitch, N. L. (1973). Probability and Statistical Inference in Ancient and Medieval Jewish Literature. Toronto and Buffalo: University of Toronto Press.
Reid, C. (1996). Julia [Robinson]: A Life in Mathematics. Washington, DC: Mathematical Association of America.
Richards, J. L. (1988). Mathematical Visions: The Pursuit of Geometry in Victorian England. Boston: Academic Press.
Richards, R. (2002). The Romantic Conception of Life: Science and Philosophy in the Age of Goethe. Chicago and London: University of Chicago Press.
Robson, E. (2007). Mesopotamian Mathematics. In (Katz 2007), pp. 57–186.
Robson, E. (2008). Mathematics in Ancient Iraq: A Social History. Princeton and Oxford: Princeton University Press.
Robson, E., & Stedall, J. (Eds.) (2009). The Oxford Handbook of the History of Mathematics. Oxford and New York: Oxford University Press.
Rose, P. L. (1975). The Italian Renaissance of Mathematics: Studies on Humanists and Mathematicians from Petrarch to Galileo. Geneva: Librairie Droz.
Rudolph, F. (1962). The American College and University. New York: Vintage.
Seligman, J. (1978). The High Citadel: The Influence of Harvard Law School. Boston: Houghton Mifflin.
Sinaiko, H. L. (1998). Energizing the Classroom: The Structure of Teaching. In H. L. Sinaiko, Reclaiming the Canon: Essays on Philosophy, Poetry, and History. New Haven: Yale University Press, pp. 241–252.
Schiebinger, L. (1989). The Mind Has No Sex: Women in the Origins of Modern Science. Cambridge, MA: Harvard University Press.
Singh, P. (1985). The so-called Fibonacci numbers in ancient and medieval India. Historia Mathematica (12), 229–244.
Siu, M. K. (2012). Proof in the Western and Eastern Traditions: Implications for Mathematics Education. In (Hanna & de Villiers 2012), pp. 431–440.
Smith, A. (1974/1776). The Wealth of Nations. London: Penguin Books.
Spinoza, B. (1953/1675). Ethics Demonstrated in Geometrical Order. Ed. J. Gutmann. New York: Hafner.
Stahl, W. H. (1977). Martianus Capella and the Seven Liberal Arts. 2 vols. [vol. I 1971, vol. II 1977]. New York and London: Columbia University Press.
Steen, L. A. (1988). The Science of Patterns. Science 29 April 1988, 240, 611–616.
Struik, D. J. (1969). A Source Book in Mathematics. Cambridge, MA: Harvard University Press.
Taton, R. (Ed.). (1964). Enseignement et diffusion des sciences en France au XVIII e siècle. Paris: Hermann.
Tarr, J. E., Reys, R. E., Reys, B. J., Chavez, O., Shih, J., & Osterlind, S. J. (2008). The impact of middle-grades mathematics curricula and the classroom learning environment on student achievement. Journal for Research in Mathematics Education 39 (3), 247–280.
Tobias, S. (1993). Overcoming Math Anxiety. New York: W. W. Norton.
Tweddle, I. (2008). An early manuscript of MacLaurin’s: Mathematical modelling of the forces of good; some remarks on fluids. Edinburgh University Library, GB 237 Coll-38, MSS 3096-3102. Includes translation of (Maclaurin 1714).
Van Brummelen, G. (2009). The Mathematics of the Heavens and the Earth: The Early History of Trigonometry. Princeton and Oxford: Princeton University Press.
Voltaire, F. M. A. de. (1901a). Philosophical Dictionary. In The Works of Voltaire: A Contemporary Version, vol. VI. Tr. W. F. Fleming. New York: Dumont.
Voltaire, F. M. A. de. (1901b). “Morality.” In (Voltaire 1901a).
Voltaire, F. M. A. de. (1901c). “Sect.” In (Voltaire 1901a).
Wagner, D. L. (1983a). The Seven Liberal Arts and Classical Scholarship. In (Wagner 1983b), pp. 1–31.
Wagner, D. L. (1983b). The Seven Liberal Arts in the Middle Ages. Bloomington, IN: University of Indiana Press.
Wagner, M. (2006). The Geometries of Visible Space. Mahwah, NJ and London: Erlbaum.
Wardhaugh, B. (2009). Mathematics, music, and experiment in late seventeenth-century England. In (Robson & Stedall 2009), pp. 639–661.
Washburn, D. K. & Crowe, D. W. (1998). Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. Seattle: University of Washington Press.
Weil, A. (1949). Sur l’étude algébrique de certains types de lois de mariage. In C. Lévi-Strauss, Les structures élémentaires de la parenté. Paris: Presses Universitaires de France, pp. 278–287.
Weil, A. (1969). On the Algebraic Study of Certain Types of Marriage Laws. In C. Lévi-Strauss, The Elementary Structures of Kinship. Tr. J. H. Bell, J. R. von Sturmer, and R. Needham, ed. Boston: Beacon Press, pp. 221–229.
Weizenbaum, J. (1976). Computer Power and Human Reason: From Judgment to Calculation. San Francisco: Freeman.
Westman, R. S. (1975). The Melanchthon Circle, Rheticus, and the Wittenberg Interpretation of the Copernican Theory. Isis 66 (2), 165–193.
Williams, L. P. (Ed.) (1968). Relativity Theory: Its Origins and Impact on Modern Thought. New York et al: Wiley.
Wilson, S. (2003). California Dreaming: Reforming Mathematics Education. New Haven: Yale University Press.
Wollenberg, S. (2003). Music and Mathematics: An Overview. In (Fauvel et al 2003), pp. 1–9.
Zamyatin, E. (1952). We. Tr. M. Ginsburg. New York: Penguin Books.
Zaslavsky, C. (1994). Fear of Math: How to Get over It and Get on with Your Life. New Brunswick, NJ: Rutgers University Press.
Zaslavsky, C. (1999). Africa Counts: Number and Pattern in African Culture. 3d edition. Chicago: Lawrence Hill Books.
Zinsser, J. (2006). La Dame d’Esprit: A Biography of the Marquise du Châtelet. New York: Viking.
Acknowledgments
This essay is dedicated to the memory of Herman Sinaiko (1929–2011), peerless teacher and scholar of the liberal arts. I am grateful to the anonymous referees and to the editor, for their scholarly expertise, comments, and criticisms, which have materially improved this chapter. I also thank the Pitzer family, donors of the Flora Sanborn Pitzer professorship, for their generous support of my research.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Grabiner, J.V. (2014). The Role of Mathematics in Liberal Arts Education. In: Matthews, M. (eds) International Handbook of Research in History, Philosophy and Science Teaching. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7654-8_25
Download citation
DOI: https://doi.org/10.1007/978-94-007-7654-8_25
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-7653-1
Online ISBN: 978-94-007-7654-8
eBook Packages: Humanities, Social Sciences and LawEducation (R0)