Abstract
We discuss the idea of concrete mathematics inspired by Hilbert’s idea of finitistic mathematics as the part of mathematics not engaged into actual infinity. We explicate it as the part of mathematics based on \(\varDelta ^0_2\) arithmetical concepts. The explication is justified by equivalence of \(\varDelta ^0_2\) definability with algorithmic learnability (an epistemic argument) and with FM–representability (representability in finite models, an ontological argument).
We show that the essential part of classical mathematics can be interpreted in the concrete framework. We claim that current mathematics is a social game of proving theorems on some axiomatic set theoretic background. On the other hand, concrete mathematics is the reality on which our mathematical experience is based. This is what makes the game intersubjective. Nevertheless, this game is one of the most efficient methods of building our mathematical knowledge.
This work was funded by the Polish National Science Centre grant number 2013/11/B/HS1/04168.
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Notes
- 1.
The distinction between potential and actual infinity is due to Aristotle, see [1].
- 2.
Another source of inspiration would be Leopold Kronecker’s view on foundations of mathematics, see [17] and a few famous remarks elsewhere. Unfortunately Kronecker never gave any systematic presentation of his views on foundations. Nevertheless, they are coherent, and probably they influenced Hilbert’s idea.
- 3.
Contemporarily we know that the ideal part of mathematics is essentially undetermined. However, in 1926 Hilbert was not aware of this fact.
- 4.
The notion of a model was introduced later.
- 5.
By axiomatic theory we mean a theory with a finite presentation i.e. recursively axiomatizable theory.
- 6.
Here we not only get that the structure of \(\mathbf {M}\) i.e. the universe and the relations is computable in S but the satisfaction relation in \(\mathbf {M}\) is also computable in S.
- 7.
We need a slightly stronger version: every low consistent theory has a low model.
- 8.
Czarnecki requires concrete models to have both concrete structure and satisfaction relation.
- 9.
The book General Topology [15] published in 1955 by Kelley gives a presentation of topological concepts in set theoretical framework. It gives explicitly axioms of set theory assumed. Later on Chang and Keisler in their Model Theory [3] give an equivalent set of axioms as the declared background of the theory. In both cases it was so called Kelley–Morse set theory, shortly KM, which is essentially stronger than ZFC. In many works published in these times and later it was clear that the basic framework is ZFC or some stronger theory, e.g. KM, which was in this case explicitly mentioned.
References
Aristotle: Physics (Circa 350 BC), http://classics.mit.edu/Aristotle/physics.html. English translation by Hardie, R.P. and Gaye, R.K
Cayley, A.: On the theory of groups, as depending on the symbolic equation \(\theta ^n = 1\). In: Philosophical Magazine, vol. 7, pp. 40–47. Taylor & Francis, London (1854)
Chang, C.C., Keisler, H.J.: Model Theory, Studies in Logic and the Foundations of Mathematics Series, vol. 73. North-Holland, Amsterdam (1973)
Czarnecki, M.: Foundations of mathematics without actual infinity. Ph.D. thesis, University of Warsaw (2014)
Epstein, R.L.: Degrees of Unsolvability Structure and Theory. Lecture Notes in Mathematics, 1st edn. Springer, Heidelberg (1979)
Gauss, C.F., Schumacher, H.C., Peters, C.A.F.: Briefwechsel zwischen C.F. Gauss und H.C. Schumacher, vol. 2 in Briefwechsel zwischen C.F. Gauss und H.C. Schumacher (1860)
Gödel, K.: Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatshefte für Mathematik und Physik 37(1), 349–360 (1930). English translation in [9]
Gödel, K.: Über Formal Unentscheidbare Sätze der Principia Mathematica und Verwandter Systeme, I. Monatshefte für Math.u.Physik 38, 173–198 (1931). English translation in [10]
Gödel, K.: The completeness of the axioms of the functional calculus of logic. In: van Heijenoort, J. (ed.) From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, pp. 582–591. Harvard University Press (2002)
Gödel, K.: On formally undecidable propositions of Principia mathematica and related systems i. In: van Heijenoort, J. (ed.) From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, pp. 596–617. Harvard University Press (2002)
Gold, E.M.: Limiting recursion. J. Symbolic Logic 30, 28–48 (1965)
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison-Wesley Longman Publishing Co., Inc., Boston (1994)
Hilbert, D.: On the infinite. In: van Heijenoort, J. (ed.) From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, pp. 367–392. Harvard University Press (2002)
Jockusch, C.G.J., Soare, R.I.: Classes and degrees of theories. Trans. Am. Math. Soc. 173, 33–56 (1972)
Kelley, J.L.: General Topology. Graduate texts in mathematics. Van Nostrand, New York City (1955)
Kleene, S.C.: Introduction to Metamathematics. North-Holland, Amsterdam (1952)
Kronecker, L.: Über den Zahlbegriff. Journal für die reine und angewandte Mathematik 101, 337–355 (1887). English translation in [18]
Kronecker, L.: On the concept of number. In: Ewald, W.B. (ed.) From Kant to Hilbert: A Source Book in the Foundations of Mathematics, vol. 2, pp. 947–955. OUP Oxford (2005)
Mostowski, M.: On representing concepts in finite models. Math. Logic Q. 47, 513–523 (2001)
Mostowski, M.: On representing semantics in finite models. In: Rojszczak, A., Cachro, J., Kurczewski, G. (eds.) Philosophical Dimensions of Logic and Science, pp. 15–28. Kluwer Academic Publishers (2003)
Mostowski, M.: Potential infinity and the church thesis. Fundamenta Informaticae 81(1–3), 241–248 (2007)
Mostowski, M.: Limiting recursion, FM-representability, and hypercomputations. In: Logic and Theory of Algorithms, Fourth Conference on Computability in Europe, CiE 2008, Local Proceedings, 15–20 June 2008
Mostowski, M.: Truth in the limit. Rep. Math. Logic 51, 75–89 (2016)
Mycielski, J.: Analysis without actual infinity. J. Symbolic Logic 46, 625–633 (1981)
Putnam, H.: Trial and error predicates and the solution to a problem of Mostowski. J. Symbolic Logic 30, 49–57 (1965)
Russell, B.A.W., Whitehead, A.N.: Principia mathematica, vol. I.-III. Cambridge University Press, Cambridge (1910–1913)
Shoenfield, J.R.: On degrees of unsolvability. Ann. Math. 69, 644–653 (1959)
Simpson, S.G.: Partial realizations of Hilbert’s program. J. Symb. Log. 53(2), 349–363 (1988)
Stone, M.H.: The theory of representation for boolean algebras. Trans. Am. Math. Soc. 40(1), 37–111 (1936)
Tait, W.W.: Finitism. J. Philos. 78(9), 524–546 (1981)
van der Waerden, B.L.: Moderne Algebra: Unter Benutzung von Vorlesungen von E. Artin und E. Noether, vol. I.-II. Springer, Berlin (1930–1931)
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Mostowski, M., Czarnecki, M. (2017). Concrete Mathematics. Finitistic Approach to Foundations. In: Kennedy, J., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2017. Lecture Notes in Computer Science(), vol 10388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55386-2_19
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