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Proofs Without Words - References - Acknowledgments - About the Authors

Tim Doyle (Whitman College), Lauren Kutler (Whitman College), Robin Miller (Whitman College), and Albert Schueller (Whitman College)


  1. Alsina, Claudi, and Roger B. Nelsen. Math Made Visual: Creating Images for Understanding Mathematics. Washington, DC: Mathematical Association of America, 2006.
  2. Alsina, Claudi, and Roger B. Nelsen. When Less Is More: Visualizing Basic Inequalities. Washington, D.C.: Mathematical Association of America, 2009.
  3. Boyer, Carl B., A History of Mathematics (2nd edition), John Wiley & Sons, Inc., 1991.
  4. Burge, T., "Frege on Knowing the Foundation," Mind, Vol. 107, No. 426 (1998), pp. 305-347.
  5. Burge, T., Truth Thought Reason: Essays on Frege, Oxford University Press, 2005.
  6. Byrne, Oliver, The first six books of the Elements of Euclid, in which coloured diagrams and symbols are used instead of letters for the greater ease of learners. London: W. Pickering, 1847.
  7. Cooke, Roger, "The Mathematics of the Hindus," The History of Mathematics: A Brief Course, Wiley-Interscience, pp. 213-215, 1997.
  8. Doyle, Timothy, "The Mathematical Content of Meno 82b-85b," Draft manuscript, presented at the Northwest Philosophy Conference, Pacific University, October 2013.
  9. Dummett, Michael, Frege: Philosophy of Mathematics, Harvard University Press, 1991.
  10.  Fredrickson, George, Dissections: Plane & Fancy, Cambridge University Press, 1997.
  11.  Frege, Gottlob. The Foundations of Arithmetic: A Logico-mathematical Enquiry into the Concept of Number. Evanston, IL: Northwestern UP, 1980.
  12. Fry, Alan L., "Sum of Cubes," Mathematics Magazine, Vol. 58, No. 1 (1985), p. 11.
  13. Gallant, Charles D., "A Truly Geometric Inequality," Mathematics Magazine, Vol. 50, No. 2 (1977), p. 98.
  14. Gerstein, Larry J., Introduction to Mathematical Structures and Proofs, Springer Publishing, 1996.
  15. Goodfriend, Jason, Gateway to Higher Mathematics, Jones and Bartlett Learning, 2005.
  16.  Gossett, Eric, Discrete Mathematics With Proof, John Wiley and Sons, 2009.
  17. Isaacs, Rufus, "Two Mathematical Papers without Words," Mathematics Magazine, Vol. 48, No. 4 (Sep., 1975), p. 198.
  18.  Katz, Victor J., A History of Mathematics, Harper-Collins, 1993.
  19. Kawasaki, Ken-ichiroh, "Viviani's Theorem," Mathematics Magazine, Vol. 78, No. 3 (2005), p. 213.
  20. Kung, Sidney H., "The Law of Cosines," Mathematics Magazine, Vol. 63, no. 5 (1990), p. 342.
  21.  Lakatos, Imre. Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge: Cambridge UP, 1976.
  22. Lucas, John F., Introduction to Abstract Mathematics (2nd edition), Ardley House Publishers, 1990.
  23. Mabry, Rick, "Proof Without Words: \((1/4) + (1/4)^2 + (1/4)^3 + \cdots = 1/3,"\) Mathematics Magazine, Vol. 72, no. 1 (1999), p. 63.
  24. Morash, Ronald P., Bridge to Abstract Mathematics: Mathematical Proof and Structures (2nd edition), McGraw Hill, 1991.
  25. "News and Letters," Mathematics Magazine, Vol. 49, No. 1 (Jan., 1976), pp. 49-52.
  26. Nelsen, Roger B., Interview with Robin Miller, March 23, 2012.
  27. Nelsen, Roger B., "Proof without Words: The Harmonic Mean - Geometric Mean - Arithmetic Mean - Root Mean Square Ineqality," Mathematics Magazine, Vol. 60, No. 3 (June, 1987), p. 158.
  28. Nelsen, Roger B. Proofs without Words: Exercises in Visual Thinking. Washington, D.C.: Mathematical Association of America, 1993.
  29. Nelsen, Roger B. Proofs without Words II: More Exercises in Visual Thinking. Washington, DC: Mathematical Association of America, 2000.
  30. Plato, Meno (2nd edition), G.M.A. Grube, translator, Hackett, 1976.
  31. Romaine, William, "(no title)," Mathematics Magazine, Vol. 61, No. 2 (1988), p. 113.
  32. Siu, Man-Keung, Personal email to Albert Schueller, June 10, 2014.
  33. Siu, Man-Keung, "Proof without Words: Sum of Squares," Mathematics Magazine, Vol. 57, No. 2 (1984), p. 92.
  34. Steen, Lynn A., Personal email to Robin Miller, April 20, 2012.
  35. Steiner, Mark, "Mathematical Proof" in Philosophy of Mathematics: An Anthology, ed. Dale Jacquette, Blackwell Publishers, 2002.
  36. Swetz, Frank J. and Katz, Victor J., "Mathematical Treasures - Oliver Byrne's Euclid," Loci: Convergence (January 2011).
  37. Wolf, Robert S., Proof, Logic, and Conjecture: The Mathematician's Toolbox, W.H. Freeman and Company, 1998.
  38. Wolf, Samuel, "(No title)," Mathematics Magazine, Vol. 62, no. 3 (1989), p. 190.
  39. Yates, Robert C., "The Trisection Problem," National Mathematics Magazine, Vol. 15, no. 6 (1941), pp. 278-293.


The authors wish to thank Moira Gresham, a member of the Department of Physics at Whitman College, and E. Sonny Elizondo, a member of the Department of Philosophy at UC Santa Barbara, for their thoughtful feedback on this paper.

About the Authors

Tim Doyle is originally from western Washington, and completed undergraduate work at Reed College and graduate work at UCLA, primarily in philosophy, but with lots of math thrown in for good measure. His research interests include, but are definitely not limited to ancient philosophy, philosophy of mathematics and logic, and early analytic philosophy. When he's not talking to students or buried in books, he moonlights as a winemaker.

Lauren Kutler is a Math for America Teaching Fellow currently teaching algebra and geometry at Malden High School (near Boston, MA). She double majored in mathematics and philosophy at Whitman College and then completed a Master of Arts in Teaching at Boston University. Lauren also enjoys exploring the outdoors, traveling, and cooking with a lot of vegetables.

Robin Miller grew up in Corvallis, Oregon, and attended Whitman College, where she double majored in Mathematics and Economics and was an active member of the college orchestra and choir. After working as a business analyst for two years, she decided to go back to school and study to be a web developer. She is now living in Portland, Oregon and working as a software engineer for a local tech startup. In her spare time, she enjoys cribbage, strategy board games, cooking, and spending time with her family.

Albert Schueller is originally from Philadelphia, PA. He majored in mathematics at Penn State University ('90) and earned a PhD in mathematics at the University of Kentucky ('96) in inverse eigenvalue problems. He has always enjoyed computing and programming. He installed Linux for the first time in 1994 using 120 1.44Mb floppy disks. He is an avid proponent of open-source software and open-source textbooks.

Tim Doyle (Whitman College), Lauren Kutler (Whitman College), Robin Miller (Whitman College), and Albert Schueller (Whitman College), "Proofs Without Words - References - Acknowledgments - About the Authors," Convergence (August 2014)