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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
Reflections on Skolem's Paradox
"... In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project  i.e., the half which tries to expose an initial tension between Cantor's theorem and the Lö ..."
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In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project  i.e., the half which tries to expose an initial tension between Cantor's theorem and the LöwenheimSkolem theorem. I argue that, even on quite naive understandings of set theory and model theory, there is no such tension. Hence, Skolem's Paradox is not a genuine paradox, and there is very little reason to worry about (or even to investigate) the more extreme consequences that are supposed to follow from this paradox. The heart of my...
Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
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
The evolution of types and logic in the 20th century: A journey through Frege, Russell and . . .
 ILLC ALUMNI EVENT, AMSTERDAM 2004
, 2004
"... ..."
century: A journey through Frege, Russell and
, 2004
"... The evolution of types and logic in the 20th ..."
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).
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.