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A theory of recursive and corecursive definitions has been developed in higher-order logic (HOL) and mechanized using Isabelle. Least fixedpoints express inductive data types such as strict lists; greatest fixedpoints express coinductive data types, such as lazy lists. Wellfounded recursion expresses recursive functions over inductive data types; corecursion expresses functions that yield elements of coinductive data types. The theory rests on a traditional formalization of infinite trees. The theory is intended for use in specification and verification. It supports reasoning about a wide range of computable functions, but it does not formalize their operational semantics and can express noncomputable functions also. The theory is illustrated using finite and infinite lists. Corecursion expresses functions over infinite lists; coinduction reasons about such functions. Key words. Isabelle, higher-order logic, coinduction, corecursion Copyright c fl 1996 by Lawrence C. Paulson Content...

The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.

Abstract A categorical ontology of space and time is presented for emergent biosystems, super-complex dynamics, evolution and human consciousness. Relational structures of organisms and the human mind are naturally represented in nonabelian categories and higher dimensional algebra. The ascent of man and other organisms through adaptation, evolution and social co-evolution is viewed in categorical terms as variable biogroupoid representations of evolving species. The unifying theme of local-to-global approaches to organismic development, evolution and human consciousness leads to novel patterns of relations that emerge in superand ultra- complex systems in terms of colimits of biogroupoids, and more generally, as compositions of local procedures to be defined in terms of locally Lie groupoids. Solutions to such local-to-global problems in highly complex systems with ‘broken symmetry ’ may be found with the help of generalized van Kampen theorems in algebraic topology such as the Higher Homotopy van Kampen theorem (HHvKT). Primordial organism structures are predicted from the simplest metabolic-repair systems extended to self-replication through autocatalytic reactions. The intrinsic dynamic ‘asymmetry ’ of genetic networks in organismic development and evolution is investigated in terms of categories of many-valued, Łukasiewicz–Moisil logic algebras and then compared with those obtained for (non-commutative) quantum

Frege is highly valued as a logician by Russell and Wittgenstein, the latter nonetheless concludes in his Tractatus that one of Frege’s central notions, the judgment stroke, is “logically quite meaningless”. In order to see why Wittgenstein thinks so, we will investigate the ‘indirect interpretation thesis’, which says that Wittgenstein’s interpretation of Frege was strongly influenced by the reading Russell gives of the Begriffsschrift in Principia Mathematica and Principles of Mathematics. This is done by analyzing the different conceptions of logic, focusing on the representations of judgment and assertion in Frege, Russell and the early Wittgenstein. Stong similarities can be found between the interpretations of Russell and Wittgenstein, this makes the indirect interpretation thesis plausible, although Russell’s influence cannot be the only reason why Wittgenstein rejected the judgment stroke as a logical symbol.