![]() Washington, DC: Mathematical Association of America. Surfaces and Essences: Analogy as the Fuel and Fire of Thinking. G ödel, Escher, Bach: An Eternal Golden Braid. What Is Mathematics Really? Oxford: Oxford University Press. American Scientist, /article/gausss-day-of-reckoning. A Famous Story About the Boy Wonder of Mathematics Has Taken on a Life of Its Own. The Solution of the Four-Color-Map Problem. (2019) The Proof of Goldbach’s Conjecture on Prime Numbers. Ingenious Mathematical Problems and Methods. Monatshefte für Mathematik und Physik 38: 173–189. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, Teil I. The Golden Ticket: P, NP, and the Search for the Impossible. Journal of the London Mathematical Society 9: 282–288.įortnow, L. New York: Columbia University Press.Įrdös, P. (2009) Logicomix: An Epic Search for Truth. Washington: Joseph Henry Press.ĭoxiadis, A. Prime Obsession: Bernhard Riemann and His Greatest Unsolved Problem in Mathematics. Pi (π) in Nature, Art, and Culture: Geometry as a Hermeneutic Science. New York: Dover.ĭahl, O.-J., Dijkstra, E. Princeton: Princeton University Press.Ĭostello, M. In Pursuit of the Traveling Salesman Problem. American Journal of Mathematics 58: 345–363.Ĭook, W. An Unsolvable Problem of Elementary Number Theory. New Haven: Yale University Press.Ĭhurch, A. The Good, the True, and the Beautiful: A Neuronal Approach. Historia Mathematica 4: 397–404.Ĭhangeux, P. Gauss and the Eight Queens Problem: A Study in Miniature of the Propagation of Historical Error. Boston: Little, Brown, and Co.Ĭampbell, P. Trends in Cognitive Sciences 9: 322–328.īoyer, C. Critical Thinking: An Introduction to Logic and Scientific Method. Princeton: Princeton University Press.īlack, M. New York: Basic Books.īenjamin, A., Chartrand, G., and Zhang, P. The King of Infinite Space: Euclid and His Elements. Ithaca: Cornell University Press.īerlinski, D. ![]() Oxford: Oxford University Press.īergin, T. ![]() The Moment of Proof: Mathematical Epiphanies. Featured Reviews in Mathematical Reviews 1997–1999: With Selected Reviews of Classic Books and Papers from 1940–1969. Jacquette (ed.), Philosophy of Mathematics, 193-208. The Mathematical Intelligencer 8: 10-20.Īppel, K. Paradox: The Nine Greatest Enigmas in Physics. Fermat’s Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. ![]() This chapter deals with this faculty of poetic logic, as it manifests itself in problem-solving, discoveries, inventions, conjectures, and proofs. In other words, the graph is the end product of poetic logic, with Euler’s ingegno leading him to devise something new that enfolded something significant. This episode in mathematical history (among many others) brings out how the fantasia is much more than the brain’s ability to generate spontaneous mental imagery from perceptual input-it is a form of insight thinking that interprets the input and then sparks an abduction (a flash of insight), which led Euler to convert the insight into a graph (a model), which highlighted the structural features of the original map, removing extraneous information from it. Reconstructing his proof allowed us to see how Euler was guided by poetic logic-whereby he initially used his fantasia to envision the geographical map in an image schematic (outline) way, converting his inner vision into a diagrammatic model, via his ingegno, from which he could then use logical reasoning to establish why the network was impossible to traverse as such and, as a result, what this implied more generally. Euler’s demonstration of the impossibility of traversing the Könisgberg network without having to double back on one of its paths (previous chapter) made it possible to flesh out a hidden mathematical principle of connected networks, which laid the foundation for graph theory and topology. ![]()
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