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Types in logic and mathematics before 1940
 Bulletin of Symbolic Logic
, 2002
"... Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, thou ..."
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Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λcalculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced
Revisiting the Notion of Function
"... Functions play a central role in type theory, logic and computation. We describe how the notions of functionalisation (the way in which functions can be constructed) and instantiation (the process of applying a function to an argument) have been developed in the last century. We explain how both pro ..."
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Functions play a central role in type theory, logic and computation. We describe how the notions of functionalisation (the way in which functions can be constructed) and instantiation (the process of applying a function to an argument) have been developed in the last century. We explain how both processes were implemented in Frege's Begriffschrift [17], Russell's Ramified Type Theory [42] and the lambdacalculus (originally introduced by Church [12, 13]) showing that the lambdacalculus misses a crucial aspect of functionalisation. We then pay attention to some special forms of function abstraction that do not exist in the lambdacalculus and we show that various logical constructs (e.g., let expressions and definitions and the use of parameters in mathematics), can be seen as forms of the missing part of functionalisation. Our study of the function concept leads...
The evolution of types and logic in the 20th century: A journey through Frege, Russell and . . .
 ILLC ALUMNI EVENT, AMSTERDAM 2004
, 2004
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From the Foundation of Mathematics to the Birth of Computation
, 2011
"... deduction/Logic was taken as a foundation for Mathematics, computation was also taken throughout as an essential tool in mathematics. • Our ancestors used sandy beaches to compute the circomference of a circle, and to work out approximations/values of numbers like π. • The word algorithm dates back ..."
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deduction/Logic was taken as a foundation for Mathematics, computation was also taken throughout as an essential tool in mathematics. • Our ancestors used sandy beaches to compute the circomference of a circle, and to work out approximations/values of numbers like π. • The word algorithm dates back centuries? Algorithms existed in anciant Egypt at the time of Hypatia. The word is named after AlKhawarizmi. • But even more impressively, the following important 20th century (un)computability result was known to Aristotle. • Assume a problem Π, – If you give me an algorithm to solve Π, I can check whether this algorithm really solves Π. – But, if you ask me to find an algorithm to solve Π, I may go on forever trying but without success. HAPOC11: History and Philosophy of Computing 1 • But, this result was already known to Aristotle: • Assume a proposition Φ. – If you give me a proof of Φ, I can check whether this proof really proves Φ. – But, if you ask me to find a proof of Φ, I may go on forever trying but without success. • In fact, programs are proofs: – program = algorithm = computable function = λterm. – By the PAT principle: Proofs are λterms.
Is computerisation a 20th century phenomenon, or is it as old as logic and mathematics?
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De Bruijn's Automath and Pure Type Systems
"... We study the position of the Automath systems within the framework of Pure Type Systems (PTSs). In [2, 22], a rough relationship has been given between Automath and PTSs. That relationship ignores three of the most important features of Automath: definitions, parameters and reduction, because at th ..."
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We study the position of the Automath systems within the framework of Pure Type Systems (PTSs). In [2, 22], a rough relationship has been given between Automath and PTSs. That relationship ignores three of the most important features of Automath: definitions, parameters and reduction, because at the time, formulations of PTSs did not have these features. Since, PTSs have been extended with these features and in view of this, we revisit the correspondence between Automath and PTSs. This paper gives the most accurate description of Automath as a PTS so far.
Logic and Computerisation in mathematics?
, 2009
"... – If you give me an algorithm to solve Π, I can check whether this algorithm really solves Π. – But, if you ask me to find an algorithm to solve Π, I may go on forever trying but without success. • But, this result was already found by Aristotle: Assume a proposition Φ. – If you give me a proof of Φ ..."
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– If you give me an algorithm to solve Π, I can check whether this algorithm really solves Π. – But, if you ask me to find an algorithm to solve Π, I may go on forever trying but without success. • But, this result was already found by Aristotle: Assume a proposition Φ. – If you give me a proof of Φ, I can check whether this proof really proves Φ. – But, if you ask me to find a proof of Φ, I may go on forever trying but without success. • In fact, programs are proofs and much of computer science in the early part of the 20th century was built by mathematicians and logicians. • There were also important inventions in computer science made by physicists (e.g., von Neumann) and others, but we ignore these in this talk. ISR 2009, Brasiliá, Brasil 1An example of a computable function/solvable problem • E.g., 1.5 chicken lay down 1.5 eggs in 1.5 days. • How many eggs does 1 chicken lay in 1 day? • 1.5 chicken lay 1.5 eggs in 1.5 days. • Hence, 1 chicken lay 1 egg in 1.5 days. • Hence, 1 chicken lay 2/3 egg in 1 day. ISR 2009, Brasiliá, Brasil 2Unsolvability of the Barber problem • which man barber in the village shaves all and only those men who do not shave themselves? • If John was the barber then – John shaves Bill ⇐ ⇒ Bill does not shave Bill – John shaves x ⇐ ⇒ x does not shave x – John shaves John ⇐ ⇒ John does not shave John • Contradiction. ISR 2009, Brasiliá, Brasil 3Unsolvability of the Russell set problem
Types and Functions since Principia
"... Abstract. Types were invented by Russell to solve the logical paradoxes that resulted from Frege’s generalisaton of the notion of function. Since, the past 100 years saw new formalisations of the notions of functions and types that extend and put to better use Frege’s and Russell’s inventions. Most ..."
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Abstract. Types were invented by Russell to solve the logical paradoxes that resulted from Frege’s generalisaton of the notion of function. Since, the past 100 years saw new formalisations of the notions of functions and types that extend and put to better use Frege’s and Russell’s inventions. Most such formalisations are extensions of Church’s simply typed λcalculus (Church’s calculus of functions together with Ramsey’s simplification of Russell’s types). Currently, types and functions are the heart of logic and computation and not only are they so closely intertwined, but their evolution demands that they be treated in the same manner. Both are usually constructed, abstracted over and instantiated and the operations for abstraction, construction, instantiation and substitution act alike on both. This paper aims to give a framework where the relationship between functions and types takes into account the similarity of their construction, abstraction, instantiation and substitution. For such a “types as functions ” framework to work, special forms of function abstraction (that do not exist in Church’s λcalculus but exist in Principia and Frege’s work) need to be included. 1 A brief overview of the evolution of types and functions