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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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Cited by 6 (5 self)
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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...
Functions and Types in Logic, Language and Computation ∗
, 2003
"... The introduction of a general definition of function was key to Frege’s formalisation of logic. Selfapplication of functions was at the heart of Russell’s paradox. Russell introduced type theory in order to control the application of functions and hence to avoid the paradox. Since, different type s ..."
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The introduction of a general definition of function was key to Frege’s formalisation of logic. Selfapplication of functions was at the heart of Russell’s paradox. Russell introduced type theory in order to control the application of functions and hence to avoid the paradox. Since, different type systems have been introduced, each allowing different functional power. Despite the extensive use of types in many applications, there remains many “non believers ” in type theory. In this talk, I will briefly review the evolution of types from the time of Euclid (325 B.C.) to the mid of the 20th century. Then, I will introduce de Bruijn’s formulation of functions and types in Automath, his famous system for automating mathematics. De Bruijn’s formulation is a living example which illustrates that while type theory is useful, there are many other considerations that need to be accommodated when attempting to “computerize ” a system. This talk is of interest for anyone
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
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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HeriotWatt University Edinburgh, Scotland
, 2005
"... The evolution of types and logic in the 20th century ∗ ..."
Um Ceclo de Computeraçao
"... Brasiliá 2010Welcome to the fastest developing and most influential subject: Computer Science • Computer Science is by nature highly applied and needs much precision, foundation and theory. • Computer Science is highly interdisciplinary bringing many subjects together in ways that were not possible ..."
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Brasiliá 2010Welcome to the fastest developing and most influential subject: Computer Science • Computer Science is by nature highly applied and needs much precision, foundation and theory. • Computer Science is highly interdisciplinary bringing many subjects together in ways that were not possible before. • Many recent scientific results (e.g., in chemistry) would not have been possible without computers. • The Kepler Conjecture: no packing of congruent balls in Euclidean space has density greater than the density of the facecentered cubic packing. • Sam Ferguson and Tom Hales proved the Kepler Conjecture in 1998, but it was not published until 2006. • The Flyspeck project aims to give a formal proof of the Kepler Conjecture.
MathLang, a framework for computerising
, 2007
"... Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the content of the original mathematical text is respected) ..."
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Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the content of the original mathematical text is respected). 2. The formalised text can be fully manipulated and searched in ways that respect its mathematical structure and meaning. 3. Steps can be made to do computation (via computer algebra systems) and proof checking (via proof checkers) on the formalised text. 4. This formalisation of text is not much harder for the ordinary mathematician than L ATEX. Full formalization down to a foundation of mathematics is not required, although allowing and supporting this is one goal. (No theorem prover’s language satisfies these goals.) University of West of England, Bristol 1A brief history • There are two influencing questions:
MathLang, a framework for computerising
, 2007
"... Saarbruecken, GermanyWhat is the aim for MathLang? Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the cont ..."
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Saarbruecken, GermanyWhat is the aim for MathLang? Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the content of the original mathematical text is respected). 2. The formalised text can be fully manipulated and searched in ways that respect its mathematical structure and meaning. 3. Steps can be made to do computation (via computer algebra systems) and proof checking (via proof checkers) on the formalised text. 4. This formalisation of text is not much harder for the ordinary mathematician than L ATEX. Full formalization down to a foundation of mathematics is not required, although allowing and supporting this is one goal. (No theorem prover’s language satisfies these goals.) Saarbruecken, Germany 1A brief history • There are two influencing questions: