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 λ-calculus 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 λ-calculus (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

This article is a brief review of the type free λ-calculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λ-calculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λ-calculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λ-calculus is introduced and some of its advantages are outlined.

A semilinear space S is a non-empty set of elements called points, provided with a collection of subsets called lines such that any pair of points is contained in at most one line and every line contains at least two points. Two points p and q are said to be collinear if p 6 ˆ q and if the pair f p; qg is contained in a line; the line

A semilinear space S is homogeneous if, whenever the semilinear structures induced on two nite subsets S 1 and S 2 of S are isomorphic, there is at least one automorphism of S mapping S 1 onto S 2 . We give a complete classication of all nite homogeneous semilinear spaces. Our theorem extends a result of Ronse on graphs and a result of Devillers and Doyen on linear spaces. Keywords: semilinear space, polar space, copolar space, partial geometry, automorphism group, homogeneity. Mathematics Subject Classication: 05B25, 51E14, 20B25. 1

We establish the fact that for dimension four there are exactly six diagonal normalized polynomials as has been conjectured for some time. They are determined on the grounds of both theoretical as well as computational results.