Titlebook: Reflections on the Foundations of Mathematics; Univalent Foundation Stefania Centrone,Deborah Kant,Deniz Sarikaya Book 2019 Springer Nature

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Models of HoTT and the Constructive View of Theoriesnstructive view of theories according to which a theory is essentially characterised by its methods. The constructive view of theories was earlier defended by Ernest Nagel and a number of other philosophers of the past but available logical means did not allow these people to build formal representa

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Set Theory and Structuresnguage focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literatur

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A New Foundational Crisis in Mathematics, Is It Really Happening?overy has generated in the community of mathematicians, philosophers and computer scientists might indicate a new crisis in the foundation of mathematics. By examining the mathematical facts behind HoTT and their relation with the existing foundations, we conclude that the present crisis is not one.

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What Do We Want a Foundation to Do?d every classical mathematical theorem can be proved from the axioms of set theory. This is obviously a remarkable mathematical fact, but it isn’t obvious what makes it ‘foundational’. This paper begins with a taxonomy of the jobs set theory does that might reasonably be regarded as foundational. It

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Formal and Natural Proof: A Phenomenological Approaches. Philosophia Mathematica (III) 15:291–320, 2007) that the formalization of a mathematical proof requires a quality of understanding that subsumes all acts necessary for checking the proof and that, consequently, automatic proof checking cannot lead to an epistemic gain about a proof. We present a

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Varieties of Pluralism and Objectivity in Mathematicsbased on basic principles and ideas (later axioms) to which the rest of mathematics could be reduced. There was supposed to be only one foundational theory and it was to carry the philosophical weight of giving the ultimate ontology and truth of mathematics. Under this conception of ‘foundation’ plu

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Higher Structures in Homotopy Type Theoryse involving an infinite hierarchy of coherence data such as semi-simplicial types, as well as the problem of developing the meta-theory of homotopy type theories in Univalent Foundations. We also discuss some proposed solutions.

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A New Foundational Crisis in Mathematics, Is It Really Happening? We reiterate a pluralist vision of the foundations of mathematics..The article contains a short survey of the mathematical and historical background needed to understand the main tenets of the foundational issues.