Modern logic emerged from the reform of Aristotle’s syllogisms through algebra and other techniques of symbolic manipulation. It was also shaped by the 19th century
- Modern logic emerged from the reform of Aristotle’s syllogisms through algebra and other techniques of symbolic manipulation. It was also shaped by the 19th century developments of mathematics, specifically the foundations of geometry, the search for rigor in the calculus, and the axiomatization of set theory. By mathematical logic (ML) we mean the branch of logic that has arisen from foundational studies in mathematics.
ML developed two models of mathematical proof (MP): (1) sequence of formulae F1,. . . , Fn, where Fk is either an axiom or is obtained from the previous formulae Fi, Fj by the modus ponens rule, see , (2) twofold composition that includes, on the one hand, a sequence of formulae, and on the other, a sequence of signs explaining the status of each formula in the first sequence in terms of axioms, definitions, and references to other theorems or formulae, see ,. While the first model is rather speculative, it gave rise to a branch of ML called proof theory; the second model seems to emulate mathematical practice.
In this talk, we focus on the historical roots of MP and show their Euclid origins. More precisely, we show how editions and commentaries on The Elements, starting with the Late Renaissance and Early Modern ones, via the Peano Formulario Mathematico program, have paved the way to the second model of MP.
- There are two components of Euclid’s proposition: the text and the lettered diagram. Latin tradition, beside extensive commentaries, has introduced a third part into Euclid’s proposition, namely marginalia, containing references to definitions, axioms and other propositions. Next to marginalia, the tradition of commentaries introduced yet another part into Euclid’s proposition: symbols representing some notions and relations; these symbols were included in the linear structure of the text simply in the place of words.
Peano has introduced a technique of purely symbolic representation of Euclid’s propositions. He has followed the same technique of symbolic representation in the foundations of calculus and geometry. Still, his symbolic propositions, next to a sequence of formulae, have included a system of references. Peano’s technique of symbolic representation of mathematical sentences has been adopted in , and due to the great influence of Prinicipa Mathematica on logic, it has become a standard model of MP.
 Hilbert, D. (1922). Neubegründung der Mathematik. Erste Mitteillung, Abhandlungen aus
dem Mathematischen Seminar der Hamburgischen Universität, 1, 157–177.
 Peano, G. (1956). Opere Scelte, vol. I–III. Roma.
 Russell, B. & Whitehead, A. (1910–1913). Principia Mathematica. Cambridge.
Sekcja LogikiPrzewodnicząca Sekcji: dr hab. Joanna Golińska-Pilarek (UW)
Sekretarz Sekcji: dr Michał Zawidzki (UW)
dr Michał Zawidzki