Abstract
The search for a synthesis between formalism and constructivism, and meditation on Gödel incompleteness, leads in a natural way to conceive mathematics as dynamic and plural, that is the result of a human achievement, rather than static and unique, that is given truth. This foundational attitude, called dynamic constructivism, has been adopted in the actual development of topology and revealed some deep structures that had remained hidden under other views. After motivations for and a brief introduction to dynamic constructivism, an overview is given of the changes it induces in the practice of mathematics and in its foundation, and of the new results it allows to obtain.
Some of the content of this paper was anticipated in my talks at Continuity, Computability, Constructivity (CCC 2017), Nancy (France), 29 June 2017 and at Incontro di Logica Matematica, AILA, Padova (Italy), 25 August 2017. I thank my wife Silvia for support and advice. I thank the editors, in particular my corresponding editor Deniz Sarikaya, for patience and empathy.
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Notes
- 1.
The chapters by Priest and by Wagner, as well as the general introduction of this volume, contain analogous overviews.
- 2.
Assuming classical logic and completeness of the more concrete system, conservativity of the higher one coincides with its consistency.
- 3.
See the chapters by Friend, Priest, and Wagner in this volume for other views on pluralism.
- 4.
Note that it is not a mere matter of words; if the classical meaning of the word set is considered by now fixed, our sets might be called proper, or small, or effective sets.
- 5.
Beware that limit points in this sense require an impredicative definition, and thus we cannot use them to define closure in terms of interior.
- 6.
The notation \(\mathrel \between \) is now rapidly spreading in the community of constructive mathematicians.
- 7.
The case in which I depends on a, as in Coquand et al. (2003), is only apparently more general.
- 8.
Note that Kleene’s construction proving that BI +  AC! +  CT ⊢⊥, which is commonly understood as showing that effectivity is incompatible with choice sequences and Bar Induction, is not a problem here because of the absence of AC!.
- 9.
We do not consider here some fine distinctions between the three notions of cHa, frames or locales.
References
Asperti, A., Maietti, M. E., Sacerdoti Coen, C., Sambin, G., & Valentini, S. (2011). Formalization of formal topology by means of the interactive theorem prover Matita. In J. H. Davenport, W. M. Farmer, J. Urban, & F. Rabe (Eds.), Intelligent computer mathematics (Lecture notes in artificial intelligence, Vol. 6824, pp. 278–280). Berlin/Heidelberg: Springer.
Battilotti, G., & Sambin, G. (2006). Pretopologies and a uniform presentation of sup-lattices, quantales and frames. Annals of Pure and Applied Logic, 137, 30–61.
Bishop, E. (1967). Foundations of constructive analysis. Toronto: McGraw-Hill.
Ciraulo, F., & Sambin, G. (2010). The overlap algebra of regular opens. Journal of Pure and Applied Algebra, 214, 1988–1995.
Ciraulo, F., & Sambin, G. (2019). Reducibility, a constructive dual of spatiality. Journal of Logic and Analysis, 11, 1–26.
Coquand, T., Sambin, G., Smith, J., & Valentini, S. (2003). Inductively generated formal topologies. Annals of Pure and Applied Logic, 104, 71–106.
Dieudonné, J. (1970). The work of Nicholas Bourbaki. American Mathematical Monthly, 77, 134–145.
Enriques, F. (1906). Problemi della scienza, Zanichelli. English translation: Problems of Science (introd. J. Royce; trans. K. Royce). Chicago: Open Court, 1914.
Fourman, M. P., & Grayson, R. J. (1982). Formal spaces. In A. S. Troelstra & D. van Dalen (Eds.), The L.E.J. Brouwer centenary symposium (Studies in logic and the foundations of mathematics, pp. 107–122). Amsterdam: North-Holland.
Johnstone, P. T. (1982). Stone spaces. Cambridge/New York: Cambridge University Press.
Kuhn, T. S. (1962). The structure of scientific revolutions (2nd enl. ed., 1970). Chicago: University of Chicago Press.
Maietti, M. E. (2009). A minimalist two-level foundation for constructive mathematics. Annals of Pure and Applied Logic, 160, 319–354.
Maietti, M. E., & Maschio, S. (2014). An extensional Kleene realizability model for the minimalist foundation. In 20th International Conference on Types for Proofs and Programs, TYPES 2014 (pp. 162–186).
Maietti, M. E., & Sambin, G. (2005). Toward a minimalist foundation for constructive mathematics. In L. Crosilla & P. Schuster (Eds.), From sets and types to topology and analysis. Towards practicable foundations for constructive mathematics (Oxford logic guides, Vol. 48, pp. 91–114). Oxford/New York: Oxford University Press.
Martin-Löf, P., & Sambin, G. (2020, to appear). Generating positivity by coinduction. In Sambin (2020, to appear).
Martino, E., & Giaretta, P. (1981). Brouwer, Dummett, and the bar theorem. In S. Bernini (Ed.), Atti del Congresso Nazionale di Logica, Montecatini Terme, 1–5 ottobre 1979 (pp. 541–558). Naples: Bibliopolis.
Sacerdoti Coen, C., & Tassi, E. (2011). Formalizing overlap algebras in Matita. Mathematical Structures in Computer Science, 21, 1–31.
Sambin, G. (1987). Intuitionistic formal spaces – a first communication. In D. Skordev (Ed.), Mathematical logic and its applications (Vol. 305, pp. 187–204). New York: Plenum Press.
Sambin, G. (1991). Per una dinamica nei fondamenti. In G. Corsi & G. Sambin (Eds.), Nuovi problemi della logica e della filosofia della scienza (pp. 163–210). Bologna: CLUEB.
Sambin, G. (2002). Steps towards a dynamic constructivism. In P. Gärdenfors, J. Wolenski, & K. Kijania-Placek (Eds.), In the scope of logic, methodology and philosophy of science (Volume One of the XI International Congress of Logic, Methodology and Philosophy of Science, Cracow, Aug 1999, pp. 261–284). Amsterdam: Elsevier.
Sambin, G. (2008). Two applications of dynamic constructivism: Brouwer’s continuity principle and choice sequences in formal topology. In M. van Atten, P. Boldini, M. Bourdeau, & G. Heinzmann (Eds.), One hundred years of intuitionism (1907–2007) (The Cerisy Conference, pp. 301–315). Basel: Birkhäuser.
Sambin, G. (2011). A minimalist foundation at work. In D. DeVidi, M. Hallett, & P. Clark (Eds.), Logic, mathematics, philosophy, vintage enthusiasms. Essays in honour of John L. Bell (The Western Ontario series in philosophy of science, Vol. 75, pp. 69–96). Dordrecht: Springer.
Sambin, G. (2012). Real and ideal in constructive mathematics. In P. Dybjer, S. Lindström, E. Palmgren, & G. Sundholm (Eds.), Epistemology versus ontology, essays on the philosophy and foundations of mathematics in honour of Per Martin-Löf (Logic, epistemology and the unity of science, Vol. 27, pp. 69–85). Dordrecht: Springer.
Sambin, G. (2015). Matematica costruttiva. In H. Hosni, G. Lolli, & C. Toffalori (Eds.), Le direzioni della ricerca logica in Italia (Edizioni della Normale, pp. 255–282). Springer.
Sambin, G. (2017). C for constructivism. Beyond clichés. Lett. Mat. Int., 5, 1–5. https://doi.org/10.1007/s40329-017-0169-1
Sambin, G. (2020, to appear). Positive topology and the basic picture. New structures emerging from constructive mathematics (Oxford Logic Guides). Oxford/New York: Oxford University Press.
Sambin, G., Battilotti, G., & Faggian, C. (2000). Basic logic: Reflection, symmetry, visibility. The Journal of Symbolic Logic, 65, 979–1013.
Weyl, H. (1918). Das Kontinuum. Kritische Untersuchungen ĂĽber die Grundlagen der Analysis, Veit, Leipzig. English trans.: Pollard, S., & Bole, T. (1987). The continuum. A critical examination of the foundation of analysis. Philadelphia: Th. Jefferson University Press.
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Sambin, G. (2019). Dynamics in Foundations: What Does It Mean in the Practice of Mathematics?. In: Centrone, S., Kant, D., Sarikaya, D. (eds) Reflections on the Foundations of Mathematics. Synthese Library, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-030-15655-8_21
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