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This book, Type Theory and Formal Proof: An Introduction, is a gentle, yet profound, introduction to systems of types and their inhabiting lambda-terms. The book appears shortly after Lambda Calculus with Types (Barendregt et al., 2013). Although these books have a partial overlap, they have very different goals. The latter book studies the mathematical properties of some formalisms of types and lambda-terms. The book in your hands is focused on the use of types and lambda-terms for the complete formalisation of mathematics. For this reason it also treats higher order and dependent types. The act of defining new concepts, essential for mathematical reasoning, forms an integral part of the book. Formalising makes it possible that arbitrary mathematical concepts and proofs be represented on a computer and enables a machine verification of the well-formedness of definitions and of the correctness of proofs. The resulting technology elevates the subject of mathematics and its applications to its maximally complete and reliable form.
The endeavour to reach this level of precision was started by Aristotle, by his introduction of the axiomatic method and quest for logical rules. For classical logic Frege completed this quest (and Heyting for the intuitionistic logic of Brouwer). Frege did not get far with his intended formalisation of mathematics: he used an inconsistent foundation. In 1910 Whitehead and Russell introduced types to remedy this. These authors made proofs largely formal, except that substitutions still had to be understood and performed by the reader.
Lambda calculus is a formalism introduced by Church in 1932 that was intended to be used as a foundation for mathematics, including its computational aspects. Supported by his students Kleene and Rosser – who showed that the prototype system was inconsistent – Church distilled a consistent computational part and ventured in 1936 the Thesis that exactly the intuitively computable functions could be captured by it. He also presented a function that could not be captured by the λ-calculus. In that same year Turing introduced another formalism, describing what are now called Turing Machines, and formulated the related Thesis that exactly the mechanically computable functions are able to be captured by these machines. Turing also showed in the same paper that the question of whether a given statement could be proved(from a given setofaxioms) using the rules of any reasonable system of logic is not computable in this mechanical way. Finally Turing showed that the formalism of λ-calculus and Turing Machines define the same class of functions.
Together Church's Thesis, concerning computability by homo sapiens, and Turing's Thesis, concerning computability by mechanical devices, using formalisms that are equally powerful and that have their computational limitations, made a deep impact on the 20th century philosophy of the power and limitations of the human mind. So far, cognitive neuropsychology has not been able to refute the combined Church-Turing Thesis. On the contrary, that discipline also shows the limitation of human capacities.
This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs. In this book, the authors focus on three classes of typing for lambda terms: simple types, recursive types and intersection types. It is in these three formalisms of terms and types that the unexpected mathematical beauty is revealed. The treatment is authoritative and comprehensive, complemented by an exhaustive bibliography, and numerous exercises are provided to deepen the readers' understanding and increase their confidence using types.