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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
Reviewing the classical and the de Bruijn notation for λcalculus and pure type systems
 Logic and Computation
, 2001
"... This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentat ..."
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This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λcalculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λcalculus is introduced and some of its advantages are outlined.
The evolution of types and logic in the 20th century: A journey through Frege, Russell and . . .
 ILLC ALUMNI EVENT, AMSTERDAM 2004
, 2004
"... ..."
Parts of this talk are based on Kamareddine [2001]; Kamareddine et al. [2002]; Kamaredine and Nederpelt [2004], and on joint work with Maarek and Wells in Kamaredine et al. [2004b,a] University of LeipzigA Century of Complexity
, 1900
"... The formalisation and computerization of ..."
MathLang: A language for Mathematics
, 2004
"... Parts of this talk are based on joint work with Nederpelt [4] and Maarek and Wells [5] ..."
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Parts of this talk are based on joint work with Nederpelt [4] and Maarek and Wells [5]
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
Principia Mathematica anniversary symposiumSummary
, 2010
"... • General definition of function 1879 [14] is key to Frege’s formalisation of logic. • Selfapplication of functions was at the heart of Russell’s paradox 1902 [44]. • To avoid paradox Russell controled function application via type theory. • Russell [45] 1903 gives the first type theory: the Ramifi ..."
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• General definition of function 1879 [14] is key to Frege’s formalisation of logic. • Selfapplication of functions was at the heart of Russell’s paradox 1902 [44]. • To avoid paradox Russell controled function application via type theory. • Russell [45] 1903 gives the first type theory: the Ramified Type Theory (rtt). • rtt is used in Russell and Whitehead’s Principia Mathematica [49] 1910–1912. • Simple theory of types (stt): Ramsey [40] 1926, Hilbert and Ackermann [26] 1928. • Church’s simply typed λcalculus λ → [11] 1940 = λcalculus + stt. Principia Mathematica anniversary symposium 1 • The hierarchies of types (and orders) as found in rtt and stt are unsatisfactory. • The notion of function adopted in the λcalculus is unsatisfactory (cf. [29]). • Hence, birth of different systems of functions and types, each with different functional power. • We discuss the evolution of functions and types and their use in logic, language and computation. • We then concentrate on these notions in mathematical vernaculars (as in Automath) and in logic. • Frege’s functions = Principia’s functions = λcalculus functions (1932). • Not all functions need to be fully abstracted as in the λcalculus. For some functions, their values are enough. Principia Mathematica anniversary symposium 2 • Nonfirstclass functions allow us to stay at a lower order (keeping decidability, typability, etc.) without losing the flexibility of the higherorder aspects. • Furthermore, nonfirstclass functions allow placing the type systems of modern theorem provers/programming languages like ML, LF and Automath more accurately in the modern formal hierarchy of types. • Another issue that we touch on is the lessons learned from formalising mathematics in logic (à la Principia) and in proof checkers (à la Automath, or any modern proof checker).