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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
A formalization of the Ramified Type Theory
, 1994
"... In "Principia Mathematica " [17], B. Russell and A.N. Whitehead propose a type system for higher order logic. This system has become known under the name "ramified type theory". It was invented to avoid the paradoxes, which could be conducted from Frege's &am ..."
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In &quot;Principia Mathematica &quot; [17], B. Russell and A.N. Whitehead propose a type system for higher order logic. This system has become known under the name &quot;ramified type theory&quot;. It was invented to avoid the paradoxes, which could be conducted from Frege's &quot;Begriffschrift&quot; [7]. We give a formalization of the ramified type theory as described in the Principia Mathematica, trying to keep it as close as possible to the ideas of the Principia. As an alternative, distancing ourselves from the Principia, we express notions from the ramified type theory in a lambda calculus style, thus clarifying the type system of Russell and Whitehead in a contemporary setting. Both formalizations are inspired by current developments in research on type theory and typed lambda calculus; see e.g. [3]. In these formalizations, and also when defining &quot;truth&quot;, we will need the notion of substitution. As substitution is not formally defined in the Principia, we have to define it ourselves. Finally, the reaction by Hilbert and Ackermann in [10] on the
Fitelson & Zalta Steps Toward a Computational Metaphysics 1
, 2003
"... Car si nous l’avions telle que je la conçois, nous pourrions raisonner en metaphysique et en morale à pue pres comme en Geometrie et en Analyse (Leibniz 1890, 21) If we had it [a characteristica universalis], we should be able to reason in metaphysics and morals in much the same way as in geometry a ..."
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Car si nous l’avions telle que je la conçois, nous pourrions raisonner en metaphysique et en morale à pue pres comme en Geometrie et en Analyse (Leibniz 1890, 21) If we had it [a characteristica universalis], we should be able to reason in metaphysics and morals in much the same way as in geometry and analysis. (Russell 1900, 169) Quo facto, quando orientur controversiae, non magis disputatione opus erit inter duos philosophos, quam inter duos Computistas. Sufficiet enim calamos in manus sumere sedereque ad abacos, et sibi mutuo... dicere: calculemus. (Leibniz 1890, 200) If controversies were to arise, there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, to sit down to their slates, and to say to each other...: Let us calculate. (Russell 1900, 170)