Gianluigi Oliveri obtained a Laurea (BA) in philosophy from the University of Bari (Italy). After that, he received a DPhil in philosophy from the University of Oxford (GB), and a PhD in cognitive science from the University of Messina (Italy).His research interests range from the philosophy of mathematics, artificial intelligence, and the philosophy of science, to metaphysics and the history of philosophy.He has held teaching and research posts at the universities of Reading, Leeds, Keele, Oxford and Palermo, and has been Visiting Scholar at the Sydney Centre for the Foundations of Science of the University of Sydney, Australia. He is, currently, Associate Professor of Logic and the Philosophy of Science at the university of Palermo (Italy), Corresponding Member of the Accademia Nazionale di Scienze, Lettere ed Arti di Palermo (National Academy of Sciences, Letters and Arts of Palermo), and Chairman of the Centro Interdipartimentale per leTecnologie della Conoscenza (Interdepartmental Centre for the Technologies of Knowledge) of the university of Palermo.Claudio Ternullo (PhD Liverpool, 2012), is currently Beatriu de Pinós (Marie-Skłodowska Curie Actions COFUND) Post-Doctoral Fellow at the University of Barcelona. Previously, he has held positions as post-doc at the Kurt Gödel Research Center for Mathematical Logic at the University of Vienna and at the University of Tartu. His research interests lie in logic and the philosophy of mathematics, in particular in the philosophy of set theory. His work focuses on the set-theoretic multiverse, new axioms (and their justification) and mathematical platonism (in particular, Gödel’s Platonism).  He has also done work on issues in ancient and medieval philosophy. Stefano Boscolo (PhD Palermo, 2016), is an IT Solution Architect at Volkswagen Group. His current areas of expertise are advanced analytics, cloud computing, and machine learning. After receiving his PhD in Logic and Philosophy of Science, he worked at Ca' Foscari University of Venice on truth pluralism. Throughout his academic career, he worked on the philosophy of mathematics, in particular on the debate between platonism and anti-platonism. His current research interests range from natural-language processing to deep neural networks.

• 1. Introduction.- Part 1. Mathematical Objects.- 2. Aristotle's Problem (Zanetti, L).- 3.Hofweber's Nominalist Naturalism (Snyder, E., Samuels, R., Shapiro, S).- 4. Exploring Mathematical Objects from Custom-Tailored Mathematical Universes (Blechschmidt, I.).- 5. Rescuing Implicit Definition from Abstractionism (Waxman, D.).- Part 2. Structures and Structuralisms.- 6. Structural Relativity and Informal Rigour (Barton, N.).- 7. Ontological Dependence and Grounding for a Weak Mathematical Structuralism (Bianchi, S.).- 8. The Structuralist Mathematical Style: Bourbaki as a Case Study (Marquis, J.-P.).- 9. Grothendieck Toposes as Unifying Bridges: a Mathematical Morphogenesis (Caramello, O.).- Part 3. Logics and Proofs.- 10. Game of Grounds (Catta, D., Piccolomini D'Aragona, A.).- 11. Predicativity and Constructive Mathematics (Crosilla, L.).- 12. Truth and the Philosophy of Mathematics (Cantini, A.).- 13. On Lakatos's Decomposition of the Notion of Proof (Moriconi, E.).- 14. A CategoricalReading of the Numerical Existence Property in Constructive Foundations (Maschio, S.).