Diagramas euclidianos y justificación matemática
Palabras clave:
Filosofía de las prácticas matemáticas, Prueba, Euclides, Razonamiento diagramático
Resumen
Este trabajo presenta un panorama histórico de los problemas investigados en tres vertientes de la filosofía de las matemáticas: tradicional, maverick y conciliatoria. En la segunda y tercera secciones, el enfoque está en enseñar (1) como el uso de los diagramas en matemáticas, y más específicamente en la geometría euclidiana, fue fuertemente criticado por autores alineados con la primera escuela y (2) el impacto de tales críticas en las re-evaluaciones y reivindicación de la legitimidad del uso de los diagramas en Euclides por autores alineados con las últimas dos escuelas.
PALABRAS CLAVE: Filosofía de las prácticas matemáticas; prueba; Euclides; razonamiento diagramático.
Referencias
Aspray, William e Kitcher, Philip (1988). History and philosophy of modern mathematics. Minneapolis: University of Minnesota Press.
Ayer, Alfred J. (1936). Language, Truth and Logic. London: Dover
Benacerraf, Paul (1965). «What Numbers Could Not Be». In Benacerraf, P. & Putnam, H. (eds), Philosophy of Mathematics: Selected readings, (1983), Cambridge: Cambridge University Press, 2nd edition, pp. 272–294.
Benacerraf, Paul (1973). «Mathematical Truth». In Benacerraf, P. & Putnam, H. (eds), Philosophy of Mathematics: Selected readings, (1983), Cambridge: Cambridge University Press, 2nd edition, pp. 403–420.
Bourbaki, Nicolas (1949), «Foundations of mathematics for the working mathematician». The Journal of Symbolic Logic 14(1): 1–8.
Bourbaki, Nicolas (1950). «The architecture of mathematics». The American Mathematical Monthly 57: 221–232.
Bourbaki, Nicolas (1968). Theory of sets. Reading: Addison-Wesley.
Bourgess, John P. e Rosen, Gideon (1997). A subject with no object: strategies for nominalistic interpretation of Mathematics. Oxford: Claredon Press.
Brown, James (1999), Philosophy of Mathematics: an introduction to the world of profs and pictures. London: Routledge.
Carter, Jessica (2019). «Philosophy of mathematical practice – motivations, themes and prospects». Philosophia Mathematica (III) 27: 1-32.
Cellucci, Carlo (2012). «Top-down and bottom-up philosophy of mathematics». Found Sci 18: 93-103.
Cellucci, Carlo (2002). Filosofia e matemática. Bari: Laterza.
Colyvan, Mark (2001). The indispensability of Mathematics. Oxford: Oxford University Press.
Corfield, David (2003). Towards a philosophy of real mathematics. Cambridge: Cambridge University Press.
Curry, Haskell (1958). Outlines of a Formalist Philosophy of Mathematics. Amsterdam: North-Holland.
Dal Magro, Tamires e García-Perez, Manuel J. (2019). «On Euclidean diagrams and geometrical knowledge». Theoria. An International Journal for Theory, History and Foundations of Science 34(2): 222-276.
Davis, Philip e Hersh, Reuben (1980), The mathematical experience. Basel: Birkhäuser.
Dedekind, Richard (1963). «Continuity and the Irrational Numbers». In W. Beman, trans., Essays on the Theory of Numbers. New York: Dover Publications, pp. 1-27.
De Paz, María e Ferreirós, José (2018). «From Basic Cognition to Mathematical Practice». Special Issue, (eds), Theoria. An International Journal for Theory, History and Foundations of Science 33 (2): 267-269.
Dieudonné, Jean (1961). «New thinking in school mathematics. The Royaumont Seminar November 23–December 3». Paris: Organisation for European Economic Co-Operation, pp. 31–46
Dieudonné, Jean (1969). Foundations of modern analysis. New York: Academic Press.
Ferreirós, José (2016). Mathematical knowledge and the interplay of practices. Princeton University Press.
Ferreirós, José (2008). «The Crisis in the Foundations of Mathematics». In Gowers, T (ed.), Princeton Companion to Mathematics. Princeton: Princeton University Press, pp. 1-14
Ferreirós, José e Lassalle Casanave, Abel (eds) (2016). El árbol de los números: cognición, lógica y prática matemática. Sevilla: Editorial Universidad de Sevilla.
Ferreirós, José e Gray, Jeremy, eds. (2006). The architeture of modern Mathematics: essays in history and philosophy. Oxford University Press.
Ferreirós, José e García-Pérez, Manuel J. (2018). «¿“Natural” y “Euclidiana”? Reflexiones sobre la geometría práctica y sus raíces cognitivas». Theoria. An International Journal for Theory, History and Foundations of Science 33 (2): 325-344.
Field, Hartry (1989). Realism, Mathematics and Modality. Oxford: Blackwell.
Giaquinto, Marcus (2015). «The epistemology of visual thinking in mathematics». The Stanford Encyclopedia of Philosophy.
Available at http://plato.stanford.edu/archives/win2015/entries/epistemology-visualthinking/
Giaquinto, Marcus (2011). «Crossing curves: a limit to the use of diagrams in proofs». Philosophia Mathematica (III) 19 (3): 281-307.
Giaquinto, Marcus (2008). «Visualizing in mathematics». In P. Mancosu (ed.), The philosophy of mathematical practice. New York: Oxford University Press, pp. 22-42.
Giaquinto, Marcus (2007). Visual thinking in mathematics. Oxford: Oxford University Press.
Giardino, Valeria (2017). «Diagrammatic reasoning in mathematics». In Magnani & Bertolotti (eds), Springer Handbook of Model-Based Science. London-NewYork: Springer, pp. 499-522.
Giardino, Valeria (2016). «¿Dónde situar los fundamentos cognitivos de las matemáticas? ». En J. Ferreirós & A. Lassalle Casanave (Eds.), El árbol de los números: cognición, lógica y práctica matemática. Sevilla: Editorial Universidad de Sevilla, pp. 23-50.
Giardino, Valeria (2013). «A practice-based approach to diagrams». In M. Aminrouche & S. Shin (eds), Visual reasoning with diagrams. Basel: Birkhäuser, pp 135-151.
Giardino, Valeria (2010). «Intuition and Visualization in Mathematical Problem Solving». Topoi 29 (1): 29-39.
Gillies, Donald (1992). Revolutions in mathematics. Oxford: Oxford University Press.
Hallett, Michael e Ulrich, Majer (2004). David Hilbert's lectures on the foundations of geometry, 1891–1902. Berlin, Heidelberg, New York: Springer-Verlag.
Hale, Bob e Wright, Crispin (2001). The reason’s proper study: essays towards a neo-Fregean Philosophy of Mathematics. Oxford: Oxford University Press.
Hammer, Eric (1995). «Reasoning with sentences and diagrams». Notre Dame Journal of Formal Logic 35(1): 73-87.
Hahn, Hans (1933). «The crisis of intuition» in Hans Hahn, Empiricism, Logic and Mathematics. London: D. Reidel Publishing Company
Hersh, Reuben (1979). «Some proposals for reviving the philosophy of mathematics». Advances in Mathematics 31: 31–50.
Kitcher, Philip (1984). The nature of mathematical knowledge. Oxford: Oxford University Press.
Klein, Felix (2004). Elementary mathematics from an advanced standpoint: Geometry. Mineola: Dover.
Kline, Morris (1980). Mathematics: the loss of certainty. Oxford: Oxford University Press.
Krieger, Martin (2003). Doing mathematics: convention, subject, calculation, analogy. Singapore: World Scientific Publishing.
Lakatos, Imre (1978). A lógica do descobrimento matemático: provas e refutações. Rio de Janeiro: Zahar.
Lassalle Casanave, Abel (2013). «Diagramas en pruebas geométricas por reductio ad absurdum». In O. M. Esquisabel & F. T. Sautter (eds), Conocimiento simbólico y conocimiento gráfico. Buenos Aires: Centro de Estudios Filosóficos Eugenio Pucciare.
Lassalle Casanave, Abel (2019). Por construção de conceitos: em torno da filosofia kantiana da matemática. Rio de Janeiro: PUC-Rio.
Lassalle Casanave, Abel e Panza, Marco (2015). «Pruebas entinemáticas y pruebas canónicas en la geometría plana de Euclides». Revista Latinoamericana de Filosofía 41/2: 147-170.
Linnebo, Øysten (2017). Philosophy of Mathematics. Princeton: Princeton University Press.
Macbeth, Danielle (2014). Realizing reason: a narrative of truth and knowing. Oxford: Oxford University Press.
Macbeth, Danielle (2010). «Diagrammatic reasoning in Euclid's Elements». In B. Van Kerkhove, J. De Vuyst, and J. P. Van Bendegem (eds.), Philosophical perspectives on mathematical practice. London: College Publications, pp. 235-267.
Maddy, Penelope (1990). Realism in Mathematics. Oxford: Clarendon Press.
Maddy, Penelope (1997). Naturalism in Mathematics. Oxford: Clarendon Press.
Mancosu, Paolo, ed. (2008). The philosophy of mathematical practice. New York: Oxford University Press.
Mancosu, Paolo (2005). «Visualization in logic and mathematics». In P. Mancosu, K. F. Jørgensen and S. A. Pedersen, eds., Visualization, explanation and reasoning styles in mathematics. Dordrecht: Springer, pp. 13-30
Mancosu, Paolo, Jorgensen, Klaus F. e Pedersen, Stig A., eds. (2005). Visualization, explanation and reasoning styles in mathematics. Dordrecht: Springer.
Manders, Kenneth (2008a). «Diagram-based geometric practice». In P. Mancosu (ed.), The philosophy of mathematical practice. New York: Oxford University Press, pp. 65-79.
Manders, Kenneth (2008b). «The euclidean diagram». In P. Mancosu (ed.), The philosophy of mathematical practice. New York: Oxford University Press, pp. 80-133.
Mumma, John (2010). «Proofs, pictures and Euclid». Synthese, 175 (2): 255-287.
Norman, Jesse (2003). Visual reasoning in Euclid's geometry: an epistemology of diagrams. Philosophy PhD. University College London.
Overmann, Karenleigh (2013). «Material Scaffolds in Numbers and Time». Cambridge Archaeological Journal 23(1): 19–39.
Palmer, Stephen (1990). «Modern theories of gestalt perception». Mind and Language, 54: 289–323.
Palmer, Stephen (1999). Vision Science: Photons to Phenomenology. Cambridge, Mass.: MIT Press.
Panza, Marco (2012). «The twofold role of diagrams in Euclid's plane geometry». Synthese 186: 55-102.
Reck, Erich e Price, Michael (2000). «Structures and structuralism in contemporary Philosophy of Mathematics». Synthese 125: 341-383.
Resnik, Michael (1997). Mathematics as a science of patterns. Oxford: Claredon Press.
Robinson, Abraham (1969). «From a Formalist's Point of View». Dialectica 23: 45-49.
Russell, Bertrand (1901). «Mathematics and the Metaphysicians». In B. Russell, Mysticism and Logic and Other Essays. London: George Allen and Unwin LTD, pp. 74-96.
Russell, Bertrand (1919). Introduction to Mathematical Philosophy. Lodon: Routledge.
Secco, Gisele D. (2016). «Computadores nas práticas matemáticas: um exercício de micro história». O que nos faz pensar (25): 105-122.
Secco, Gisele D. e Pereira Luiz Carlos (2017). «Proofs Versus Experiments: Wittgensteinian Themes Surrounding the Four-Color Theorem». In: Marcos Silva. (Org.), How Colours Matter to Philosophy. Springer International Publishing (vol 388), pp. 289-307.
Seoane, José (2006). «Representar y demonstrar: observaciones preliminares sobre diagramas». Representaciones 2 (2): 105-126.
Shapiro, Stewart (2000). Thinking about Mathematics. Oxford: Oxford University Press.
Shapiro, Stewart (1997). Philosophy of Mathematics: structure and ontology. Oxford: Oxford University Press.
Shin, Sun-Joo (1994). The logical status of diagrams. Cambridge: Cambridge University Press.
Shin, Sun-Joo, Lemon, Oliver e Mumma, John (2014). “Diagrams». The Stanford Enciclopedy of Philosophy. Disponível em: http://plato.stanford.edu/entries/diagrams/.
Spelke, Elisabeth e Lee, Sang-ah (2012). «Core systems of geometry in animal minds». Philosophical Transactions of the Royal Society B: Biological Sciences 367 (1603): 2784-2793.
Tappenden, Jamie (2006). «The Riemannian background to Frege’s philosophy». In Ferreirós and Gray, The architeture of modern Mathematics: essays in history and philosophy. Oxford University Press, pp. 97–132.
Tymoczko, Thomas (1998 [1986]). New directions in the philosophy of mathematics. Revised and extended version. Princenton University Press.
Wilhelmus, Eva (2007). «Formalizability and Knowledge Ascriptions in Mathematical Practice». ILLC research report and technical notes series: Logic, philosophy and linguistics series. University of Amsterdam Institute for Logic, Language and Computation.
Disponível em: https://academic.oup.com/philmat/article-abstract/27/1/1/5316111
Van Bendegem, Jean P. (2014), «The impact of the Philosophy of Mathematical Practice to the Philosophy of Mathematics». In Soler, L., Zwart, S., Lynch, M., & Israel-Jost, V. (eds.), Science after the Practice Turn in the Philosophy, History, and Social Studies of Science. New York-London: Routledge, pp. 215-226.
Ayer, Alfred J. (1936). Language, Truth and Logic. London: Dover
Benacerraf, Paul (1965). «What Numbers Could Not Be». In Benacerraf, P. & Putnam, H. (eds), Philosophy of Mathematics: Selected readings, (1983), Cambridge: Cambridge University Press, 2nd edition, pp. 272–294.
Benacerraf, Paul (1973). «Mathematical Truth». In Benacerraf, P. & Putnam, H. (eds), Philosophy of Mathematics: Selected readings, (1983), Cambridge: Cambridge University Press, 2nd edition, pp. 403–420.
Bourbaki, Nicolas (1949), «Foundations of mathematics for the working mathematician». The Journal of Symbolic Logic 14(1): 1–8.
Bourbaki, Nicolas (1950). «The architecture of mathematics». The American Mathematical Monthly 57: 221–232.
Bourbaki, Nicolas (1968). Theory of sets. Reading: Addison-Wesley.
Bourgess, John P. e Rosen, Gideon (1997). A subject with no object: strategies for nominalistic interpretation of Mathematics. Oxford: Claredon Press.
Brown, James (1999), Philosophy of Mathematics: an introduction to the world of profs and pictures. London: Routledge.
Carter, Jessica (2019). «Philosophy of mathematical practice – motivations, themes and prospects». Philosophia Mathematica (III) 27: 1-32.
Cellucci, Carlo (2012). «Top-down and bottom-up philosophy of mathematics». Found Sci 18: 93-103.
Cellucci, Carlo (2002). Filosofia e matemática. Bari: Laterza.
Colyvan, Mark (2001). The indispensability of Mathematics. Oxford: Oxford University Press.
Corfield, David (2003). Towards a philosophy of real mathematics. Cambridge: Cambridge University Press.
Curry, Haskell (1958). Outlines of a Formalist Philosophy of Mathematics. Amsterdam: North-Holland.
Dal Magro, Tamires e García-Perez, Manuel J. (2019). «On Euclidean diagrams and geometrical knowledge». Theoria. An International Journal for Theory, History and Foundations of Science 34(2): 222-276.
Davis, Philip e Hersh, Reuben (1980), The mathematical experience. Basel: Birkhäuser.
Dedekind, Richard (1963). «Continuity and the Irrational Numbers». In W. Beman, trans., Essays on the Theory of Numbers. New York: Dover Publications, pp. 1-27.
De Paz, María e Ferreirós, José (2018). «From Basic Cognition to Mathematical Practice». Special Issue, (eds), Theoria. An International Journal for Theory, History and Foundations of Science 33 (2): 267-269.
Dieudonné, Jean (1961). «New thinking in school mathematics. The Royaumont Seminar November 23–December 3». Paris: Organisation for European Economic Co-Operation, pp. 31–46
Dieudonné, Jean (1969). Foundations of modern analysis. New York: Academic Press.
Ferreirós, José (2016). Mathematical knowledge and the interplay of practices. Princeton University Press.
Ferreirós, José (2008). «The Crisis in the Foundations of Mathematics». In Gowers, T (ed.), Princeton Companion to Mathematics. Princeton: Princeton University Press, pp. 1-14
Ferreirós, José e Lassalle Casanave, Abel (eds) (2016). El árbol de los números: cognición, lógica y prática matemática. Sevilla: Editorial Universidad de Sevilla.
Ferreirós, José e Gray, Jeremy, eds. (2006). The architeture of modern Mathematics: essays in history and philosophy. Oxford University Press.
Ferreirós, José e García-Pérez, Manuel J. (2018). «¿“Natural” y “Euclidiana”? Reflexiones sobre la geometría práctica y sus raíces cognitivas». Theoria. An International Journal for Theory, History and Foundations of Science 33 (2): 325-344.
Field, Hartry (1989). Realism, Mathematics and Modality. Oxford: Blackwell.
Giaquinto, Marcus (2015). «The epistemology of visual thinking in mathematics». The Stanford Encyclopedia of Philosophy.
Available at http://plato.stanford.edu/archives/win2015/entries/epistemology-visualthinking/
Giaquinto, Marcus (2011). «Crossing curves: a limit to the use of diagrams in proofs». Philosophia Mathematica (III) 19 (3): 281-307.
Giaquinto, Marcus (2008). «Visualizing in mathematics». In P. Mancosu (ed.), The philosophy of mathematical practice. New York: Oxford University Press, pp. 22-42.
Giaquinto, Marcus (2007). Visual thinking in mathematics. Oxford: Oxford University Press.
Giardino, Valeria (2017). «Diagrammatic reasoning in mathematics». In Magnani & Bertolotti (eds), Springer Handbook of Model-Based Science. London-NewYork: Springer, pp. 499-522.
Giardino, Valeria (2016). «¿Dónde situar los fundamentos cognitivos de las matemáticas? ». En J. Ferreirós & A. Lassalle Casanave (Eds.), El árbol de los números: cognición, lógica y práctica matemática. Sevilla: Editorial Universidad de Sevilla, pp. 23-50.
Giardino, Valeria (2013). «A practice-based approach to diagrams». In M. Aminrouche & S. Shin (eds), Visual reasoning with diagrams. Basel: Birkhäuser, pp 135-151.
Giardino, Valeria (2010). «Intuition and Visualization in Mathematical Problem Solving». Topoi 29 (1): 29-39.
Gillies, Donald (1992). Revolutions in mathematics. Oxford: Oxford University Press.
Hallett, Michael e Ulrich, Majer (2004). David Hilbert's lectures on the foundations of geometry, 1891–1902. Berlin, Heidelberg, New York: Springer-Verlag.
Hale, Bob e Wright, Crispin (2001). The reason’s proper study: essays towards a neo-Fregean Philosophy of Mathematics. Oxford: Oxford University Press.
Hammer, Eric (1995). «Reasoning with sentences and diagrams». Notre Dame Journal of Formal Logic 35(1): 73-87.
Hahn, Hans (1933). «The crisis of intuition» in Hans Hahn, Empiricism, Logic and Mathematics. London: D. Reidel Publishing Company
Hersh, Reuben (1979). «Some proposals for reviving the philosophy of mathematics». Advances in Mathematics 31: 31–50.
Kitcher, Philip (1984). The nature of mathematical knowledge. Oxford: Oxford University Press.
Klein, Felix (2004). Elementary mathematics from an advanced standpoint: Geometry. Mineola: Dover.
Kline, Morris (1980). Mathematics: the loss of certainty. Oxford: Oxford University Press.
Krieger, Martin (2003). Doing mathematics: convention, subject, calculation, analogy. Singapore: World Scientific Publishing.
Lakatos, Imre (1978). A lógica do descobrimento matemático: provas e refutações. Rio de Janeiro: Zahar.
Lassalle Casanave, Abel (2013). «Diagramas en pruebas geométricas por reductio ad absurdum». In O. M. Esquisabel & F. T. Sautter (eds), Conocimiento simbólico y conocimiento gráfico. Buenos Aires: Centro de Estudios Filosóficos Eugenio Pucciare.
Lassalle Casanave, Abel (2019). Por construção de conceitos: em torno da filosofia kantiana da matemática. Rio de Janeiro: PUC-Rio.
Lassalle Casanave, Abel e Panza, Marco (2015). «Pruebas entinemáticas y pruebas canónicas en la geometría plana de Euclides». Revista Latinoamericana de Filosofía 41/2: 147-170.
Linnebo, Øysten (2017). Philosophy of Mathematics. Princeton: Princeton University Press.
Macbeth, Danielle (2014). Realizing reason: a narrative of truth and knowing. Oxford: Oxford University Press.
Macbeth, Danielle (2010). «Diagrammatic reasoning in Euclid's Elements». In B. Van Kerkhove, J. De Vuyst, and J. P. Van Bendegem (eds.), Philosophical perspectives on mathematical practice. London: College Publications, pp. 235-267.
Maddy, Penelope (1990). Realism in Mathematics. Oxford: Clarendon Press.
Maddy, Penelope (1997). Naturalism in Mathematics. Oxford: Clarendon Press.
Mancosu, Paolo, ed. (2008). The philosophy of mathematical practice. New York: Oxford University Press.
Mancosu, Paolo (2005). «Visualization in logic and mathematics». In P. Mancosu, K. F. Jørgensen and S. A. Pedersen, eds., Visualization, explanation and reasoning styles in mathematics. Dordrecht: Springer, pp. 13-30
Mancosu, Paolo, Jorgensen, Klaus F. e Pedersen, Stig A., eds. (2005). Visualization, explanation and reasoning styles in mathematics. Dordrecht: Springer.
Manders, Kenneth (2008a). «Diagram-based geometric practice». In P. Mancosu (ed.), The philosophy of mathematical practice. New York: Oxford University Press, pp. 65-79.
Manders, Kenneth (2008b). «The euclidean diagram». In P. Mancosu (ed.), The philosophy of mathematical practice. New York: Oxford University Press, pp. 80-133.
Mumma, John (2010). «Proofs, pictures and Euclid». Synthese, 175 (2): 255-287.
Norman, Jesse (2003). Visual reasoning in Euclid's geometry: an epistemology of diagrams. Philosophy PhD. University College London.
Overmann, Karenleigh (2013). «Material Scaffolds in Numbers and Time». Cambridge Archaeological Journal 23(1): 19–39.
Palmer, Stephen (1990). «Modern theories of gestalt perception». Mind and Language, 54: 289–323.
Palmer, Stephen (1999). Vision Science: Photons to Phenomenology. Cambridge, Mass.: MIT Press.
Panza, Marco (2012). «The twofold role of diagrams in Euclid's plane geometry». Synthese 186: 55-102.
Reck, Erich e Price, Michael (2000). «Structures and structuralism in contemporary Philosophy of Mathematics». Synthese 125: 341-383.
Resnik, Michael (1997). Mathematics as a science of patterns. Oxford: Claredon Press.
Robinson, Abraham (1969). «From a Formalist's Point of View». Dialectica 23: 45-49.
Russell, Bertrand (1901). «Mathematics and the Metaphysicians». In B. Russell, Mysticism and Logic and Other Essays. London: George Allen and Unwin LTD, pp. 74-96.
Russell, Bertrand (1919). Introduction to Mathematical Philosophy. Lodon: Routledge.
Secco, Gisele D. (2016). «Computadores nas práticas matemáticas: um exercício de micro história». O que nos faz pensar (25): 105-122.
Secco, Gisele D. e Pereira Luiz Carlos (2017). «Proofs Versus Experiments: Wittgensteinian Themes Surrounding the Four-Color Theorem». In: Marcos Silva. (Org.), How Colours Matter to Philosophy. Springer International Publishing (vol 388), pp. 289-307.
Seoane, José (2006). «Representar y demonstrar: observaciones preliminares sobre diagramas». Representaciones 2 (2): 105-126.
Shapiro, Stewart (2000). Thinking about Mathematics. Oxford: Oxford University Press.
Shapiro, Stewart (1997). Philosophy of Mathematics: structure and ontology. Oxford: Oxford University Press.
Shin, Sun-Joo (1994). The logical status of diagrams. Cambridge: Cambridge University Press.
Shin, Sun-Joo, Lemon, Oliver e Mumma, John (2014). “Diagrams». The Stanford Enciclopedy of Philosophy. Disponível em: http://plato.stanford.edu/entries/diagrams/.
Spelke, Elisabeth e Lee, Sang-ah (2012). «Core systems of geometry in animal minds». Philosophical Transactions of the Royal Society B: Biological Sciences 367 (1603): 2784-2793.
Tappenden, Jamie (2006). «The Riemannian background to Frege’s philosophy». In Ferreirós and Gray, The architeture of modern Mathematics: essays in history and philosophy. Oxford University Press, pp. 97–132.
Tymoczko, Thomas (1998 [1986]). New directions in the philosophy of mathematics. Revised and extended version. Princenton University Press.
Wilhelmus, Eva (2007). «Formalizability and Knowledge Ascriptions in Mathematical Practice». ILLC research report and technical notes series: Logic, philosophy and linguistics series. University of Amsterdam Institute for Logic, Language and Computation.
Disponível em: https://academic.oup.com/philmat/article-abstract/27/1/1/5316111
Van Bendegem, Jean P. (2014), «The impact of the Philosophy of Mathematical Practice to the Philosophy of Mathematics». In Soler, L., Zwart, S., Lynch, M., & Israel-Jost, V. (eds.), Science after the Practice Turn in the Philosophy, History, and Social Studies of Science. New York-London: Routledge, pp. 215-226.
Publicado
2020-09-30
Cómo citar
[1]
Dal Magro, T. 2020. Diagramas euclidianos y justificación matemática. Disputatio. 9, 14 (sep. 2020), 73-102. DOI:https://doi.org/10.5281/zenodo.4603480.
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