Proof verification within set theory

The proof-checker \uc6tnaNova, aka Ref, processes proof scenarios to establish whether or not they are formally correct. A scenario, typically written by a working mathematician or computer scientist, consists of definitions, theorem statements and proofs of the theorems. There is a construct enabling one to package definitions and theorems into reusable proofware components. The deductive system underlying Ref mainly first-order, but with an important second-order feature: the packaging construct just mentioned is a variant of the Zermelo-Fraenkel set theory, ZFC, with axioms of regularity and global choice. This is apparent from the very syntax of the language, borrowing from the set-theoretic tradition many constructs, e.g. abstraction terms. Much of Ref\u2019s naturalness, comprehensiveness, and readability, stems from this foundation; much of its effectiveness, from the fifteen or so built-in mechanisms, tailored on ZFC, which constitute its inferential armory. Rather peculiar aspects of Ref, in comparison to other proof-assistants (Mizar to mention one), are that Ref relies only marginally on predicate calculus and that types play no significant role, in it, as a foundation. This talk illustrates the state-of-the-art of proof-verification technology based on set theory, by reporting on \u2018proof-pearl\u2019 scenarios currently under development and by examining some small-scale, yet significant, examples of use of Ref. The choice of examples will reflect today\u2019s tendency to bring Ref\u2019s scenarios closer to algorithm-correctness verification, mainly referred to graphs. The infinity axiom rarely plays a role in applications to algorithms; nevertheless the availability of all resources of ZFC is important in general: for example, relatively unsophisticated argumentations enter into the proof that the Davis-Putnam-Logemann-Loveland satisfiability test is correct, but in order to prove the compactness of propositional logic or Stone\u2019s representation theorem for Boolean algebras one can fruitfully resort to Zorn\u2019s lemma

Bisimilarity, Hypersets, and Stable Partitioning: a Survey

Since Hopcroft proposed his celebrated $n \log n$ algorithm for minimizing states in a finite automaton, the race for efficient partition refinement methods has inspired much research in algorithmics. In parallel, the notion of bisimulation has gained ground in theoretical investigations not less than in applications, till it even pervaded the axioms of a variant Zermelo-Fraenkel set theory. As is well-known, the coarsest stable partitioning problem and the determination of bisimilarity (i.e., the largest partition stable relative to finitely many dyadic relations) are two faces of the same coin. While there is a tendency to refer these topics to varying frameworks, we will contend that the set-theoretic view not only offers a clear conceptual background (provided stability is referred to a non-well-founded membership), but is leading to new insights on the algorithmic complexity issues

Rethinking Leonardo for the Anthropocene

The aesthetic grace of Leonardo da Vinci’s depictions of nature—the backgrounds of his master paintings and the meticulous drawings in the preserved codices—has a significance that goes beyond mere visual pleasure. The Renaissance’s naturalism was enormously important for the development of a practice-oriented scientific culture rooted in empirical observation. A broad range of research on the practical foundations of science has highlighted this insight, research that spans from Marxist sociology to newer scholarship on practical knowledge in the history of art and science. This essay discusses Leonardo from the perspective of the current debate on the Anthropocene

The Struggle for Objectivity: Gramsci’s Historical-Political Vistas on Science against the Background of Lenin’s Epistemology

This contribution interprets the intertwined issues of science, epistemology, society, and politics in Gramsci’s Prison Notebooks as a culturalist approach to science that does not renounce objectivity. Gramsci particularly criticized the scientist positions taken by the Bolshevik leader Nikolai Bukharin in Historical Materialism (1921) and the conference communication he delivered at the International Congress of History of Science and Technology in London in 1931. Gramsci did not avoid, at least implicitly, engaging with the theses of Lenin’s Materialism and Empiriocriticism (1909). Gramsci’s reception of these Russian positions was twofold: on the one hand, he agreed with the centrality of praxis (and politics) for a correct assessment of the meaning of epistemological positions; on the other hand, he disagreed with the reduction of the problem of epistemology to the dichotomy of materialism and idealism at the expense of any consideration of the ideological dimension of science

A Diophantine representation of Wolstenholme's pseudoprimality

As a by-product of the negative solution of Hilbert\u2019s 10th problem, various prime-generating polynomials were found. The best known upper bound for the number of variables in such a polynomial, to wit 10, was found by Yuri V. Matiyasevich in 1977. We show that this bound could be lowered to 8 if the converse of Wolstenholme\u2019s theorem (1862) holds, as conjectured by James P. Jones. This potential improvement is achieved through a Diophantine representation of the set of all integers p >= 5 that satisfy the congruence C(2 p,p) 61 2 mod p^3. Our specification, in its turn, relies upon a terse polynomial representation of exponentiation due to Matiyasevich and Julia Robinson (1975), as further manipulated by Maxim Vsemirnov (1997). We briefly address the issue of also determining a lower bound for the number of variables in a prime-representing polynomial, and discuss the autonomous significance of our result about Wostenholme\u2019s pseudoprimality, independently of Jones\u2019s conjecture

Heavenly Animation as the Foundation for Fracastoro’s Homocentrism: Aristotelian-Platonic Eclecticism beyond the School of Padua

This essay deals with the ensouled cosmology propounded by the physician and philosopher Girolamo Fracastoro. His Homocentrica sive de stellis (1538), which propounded an astronomy of concentric spheres, was received and discussed by scholars who belonged to the cultural environment of the Padua School. Paduan Aristotelians generally explained heavenly motions in physical terms as the effect of heavenly souls and intelligences. Since the time of the polemics over the immortality of the human soul, which had famously opposed Pomponazzi to Nifo, all psychological discussions—including those about heavenly spheres’ souls—raised heated controversies. In the wake of these controversies, Fracastoro discussed the foundations of his homocentric planetary theory in a dialogue on the immortality of the soul entitled Fracastorius, sive de anima (1555). This work also included a cosmogonic myth which was, however, not published in early-modern editions of the dialogues in order to avoid theological censorship. Fracastoro had already discussed problems of celestial physics and the physical problems linked with mathematical modeling in relation to physical causation in an exchange with Gasparo Contarini which took place in 1531. In this exchange Contarini expressed his doubts over Fracastoro’s lack of consideration of the Aristotelian viewpoints on heavenly souls and intelligences. Fracastoro offered a full account of cosmic animation in his later dialogue ‘on the soul’ by taking a different path than his Paduan teachers and philosophical interlocutors. He picked up the Platonic idea of the world soul, which animates the whole, and freely connected it with Aristotelian views about the ensouled cosmos of concentric spheres. Thus, his cosmology resulted from an eclectic composition of Platonic, Aristotelian and Averroistic elements. He aimed to create a renewed mathematical astronomy that would explain planetary motions as the result of the movements of concentric spheres. Fracastoro grounded this renewed astronomy on an understanding of the cosmos as a living whole. Such an animated homocentric cosmos represented, at the same time, both a development based on Aristotelian premises and a step beyond this legacy