Abstract
This brief historical survey is written from a logical point of view. It is a rational reconstruction of the genesis of some interrelations between formal logic and mathematics. We examine how mathematical logic was conceived: as the abstract mathematics of logic or as the logic of mathematical practice.
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Notes
- 1.
Yet, some “purification” of formal logic did take place. The various interpretations of Boole–Schröder algebra showed that the logic of concepts and one-place propositional functions could be founded on the logic of propositions. Since this algebra eventually remained in the background of the logical researches and was not being frequently connected with the later logic of propositions and propositional functions, it was only recently that this important result was used to prove, in a simple way, the decidability of monadic logic in a student textbook (cf. [22] ch. 24.).
- 2.
The construction of real numbers by Dedekind’s cuts (i.e., the arithmetization of the continuum) should have shown the logical definability of real numbers by rational numbers. Logical definition of rational numbers by natural numbers is also to Dedekind’s merit, as well as the reduction of the natural number concept to the logical concept of chain (by abstracting the nature of its elements). The possibility of the logically founded process of abstraction by the concept of the similarity of sets is due to Cantor. It is interesting that Dedekind’s concept of chain (in [5]) anticipates Peano’s axiomatization of natural numbers from [16]. However, each of these two works represents just one of the two above-stated aspects of the critical movement in mathematics. Thus, Peano postulates the principle of the mathematical induction by formally expressing it, while Dedekind has informally proved it as a consequence of purely logical characteristics of chains. Peano’s axiomatization is not original. It can be found informally and in broader lines in Grassmann’s [11] and, of course, in Dedekind [5], as quoted by Peano himself. He is one of the few mathematicians who noticed and publically promoted the importance of Grassmann’s work, particularly in geometry.
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Šikić, Z. (2020). Mathematical Logic: Mathematics of Logic or Logic of Mathematics. In: Skansi, S. (eds) Guide to Deep Learning Basics. Springer, Cham. https://doi.org/10.1007/978-3-030-37591-1_1
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