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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- 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.
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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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