The Origins of Type Theory ryotaro612 December 31, 2024

Outline 1. Introduction 2. Non-Euclidean Geometry 3. Naive Set Theory 4. Mathematical Logic 1

Objective An introduction to areas related to type theory Fields relevant to this presentation: ∘ Logic ∘ Set theory ∘ Semiotics ∘ Structuralism ∘ Philosophy of language 2

Outline 1. Introduction 2. Non-Euclidean Geometry 3. Naive Set Theory 4. Mathematical Logic 3

Geometry in Ancient Greece Geometry in ancient Greece was developed using deductive reasoning. Images adapted from “Philosophy for Beginners”6 4

Euclid’s Elements Euclid’s Elements is a collection of axioms, postulates, propositions, and proofs. Definition 1 A point is that which has no part. Definition 2 A line is length without breadth. Definition 3 A surface is that which has length and breadth only. Axiom 5 The whole is greater than the part. 5

Euclid’s Postulates The fifth postulate never seemed self-evident. 1. Let it have been postulated to draw a straight-line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To draw a circle with any center and radius. 4. All right-angles are equal to one another. 5. If a straight-line falling across two straight-lines makes internal angles on the same side less than two right-angles, then the two (other) straight-lines, being produced to infinity meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side). 6

The Fifth Postulate: The Parallel Postulate Given a line and a point not on it, there is exactly one line parallel to the original line passing through the point. A parallel line and a point 𝑝 not on the line Figure adapted from 「数学の世界史」 10 7

Investigations of the Parallel Postulate ∘ Numerous attempts were made to derive the parallel postulate using the other postulates. ∘ Alhazen (965-1039) and Khayyam (1048 – 1131) explored these investigations. ∘ The Saccheri-Legendre Theorem states that if the parallel postulate is not assumed, then the sum of the angles of a triangle is less than or equal to 180 degrees. ∘ The negation of the parallel postulate does not result in a contradiction with the other postulates. ∘ Lobachevsky (1792-1856) and János Bolyai (1802-1860) independently proposed a system in which multiple parallels exist, leading to a consistent geometry. 8

Non-Euclidean Geometries In spherical geometry, for example, great circles intersect at exactly two points. Spherical and hyperbolic spaces In spherical geometry, a great circle corresponds to a straight line in Euclidean geometry. Figure adapted from ” はじめての構造主義”9 9

Outline 1. Introduction 2. Non-Euclidean Geometry 3. Naive Set Theory 4. Mathematical Logic 10

Development of Set Theory The discovery of non-Euclidean geometry inspired a rethinking of mathematical foundations. ∘ This discovery led mathematicians to realize that mathematics could be constructed on various axiom systems. ∘ Georg Cantor developed set theory as a foundational framework, providing a unified way to handle different mathematical concepts. 11

A Unified Approach A sketch of how to constrcut mathematical concepts using only the empty set. 1. Begin with the empty set, denoted by ∅ = {𝑥|𝑥 ≠ 𝑥}. 2. Define the successor function 𝑆(𝑛) = 𝑛 ∪ {𝑛}, which essentially adds one to a number by creating a new set. 3. Define ∅ as 0: 0 = ∅. 4. 1 is defined as 𝑆(0) = {∅}. 5. 2 is 𝑆(1) = {∅, {∅}}. 6. Continue this process to generate all natural numbers. 12

Outline 1. Introduction 2. Non-Euclidean Geometry 3. Naive Set Theory 4. Mathematical Logic 13

Logicism Frege developed predicate logic and used it along with set theory to provide a logical basis for arithmetic. ∘ In ”The Foundations of Arithmetic,” Frege aimed to show that arithmetic truths are derived from logical axioms. ∘ In ”On Sense and Reference,” Frege distinguished between sense and reference. ∘ Terms ”Morning Star” and ”Evening Star” have different senses but share the same reference. Figure adapted from Philosophy for Beginners6 14

Leibniz’s pioneering work in symbolic manipulation The symbolic calculation of inferences was explored even in the 17th century. ∘ Calculus derivatives ∘ The binary number system ∘ Characteristica universalis ∘ A universal system of signs aimed at eliminating natural language ambiguity by developing an ”alphabet of human thoughts” Figures adapted from Philosophy for Beginners6 15

Boole Turned Logic into Algebra Boole employed symbols to represent classes of individuals in propositions. ”The Laws of Thought”2 Let us then agree to represent the class of individuals to which a particular name or description is ap- plicable, by a single letter, as x. If the name is “men,” for instance, let x represent “all men,” Figure adapted from ”The Universal Computer: The Road from Leibniz to Turing” 16

Propositional Logic Propositional logic involves using statements and deducing their truth based on logical connectives. 𝜑, 𝜓 (∧ I) 𝜑 ∧ 𝜓 𝜑 ∧ 𝜓 (∨ E) 𝜑 𝜑 ∧ 𝜓 (∨ E) 𝜓 [𝜑] ⋮ 𝜓 (→ I) 𝜑 → 𝜓 𝜑 𝜑 → 𝜓 (→ E) 𝜓 ⊥ (⊥) 𝜑 [¬𝜑] ⋮ ⊥ (RAA) 𝜑 A proposition is a declarative statement that is either true or false, but not both. 17

Conceptual Notation Frege formalized statements using predicates. ∘ Statements such as “𝜑 is ” are represented by predicates 𝑃(𝜑), mapping an argument 𝜑 to truth values, and extending propositional logic. ∘ Introduces the universal quantifier ∀ meaning “for any 𝜙,” and the existential quantifier ∃ meaning “there exists 𝑥.” ∘ For example, the statement “Pets at home are either dogs or cats” is represented as ∀𝑥.𝑃(𝑥) → 𝐷(𝑥) ∨ 𝐶(𝑥). ∘ Complex propositions can be expressed as combinations of simpler ones. 18

Russell’s Paradox The paradox reveals a fundamental problem with naive set theory. 1. Consider the set 𝑅 defined as: 𝑅 = {𝑥 ∣ 𝑥 ∉ 𝑥}, where 𝑅 includes all sets that do not contain themselves. 2. If 𝑅 ∈ 𝑅, then by definition, 𝑅 should not contain itself. 3. If 𝑅 ∉ 𝑅, then according to the definition, 𝑅 must contain itself. 19

Russell’s Paradox in ”Foundations of Arithmetic” Axiom V, 𝜖𝑃 = 𝜖𝑄 ≡ ∀𝑥[𝑃(𝑥) ≡ 𝑄(𝑥)], leads to contradiction. ∘ 𝜖𝑃 denotes the set of elements satisfying predicate 𝑃. ∘ 𝑅 can be expressed as ∃𝑃[𝑥 = 𝜖𝑃 ∧ ¬𝑃(𝑥)] in predicate logic. ∘ Substituting 𝑥 as 𝜖𝑅, we find 𝑅(𝜖𝑅) ≡ ¬𝑅(𝜖𝑅). ∘ In Frege’s work, there is a distinction between objects, predicates, and predicates of predicates. This hierarchy can be seen as an early notion of type theory. 20

References [1] Maria Rosa Antognazza. Leibniz: A Very Short Introduction. Oxford University Press, 2016. [2] George Boole. THE LAWS OF THOUGHT. 1854. URL: https://plato.stanford.edu/entries/settheory-early/. [3] Martin Davis. The Universal Computer: The Road from Leibniz to Turing. CRC Press, 2018. [4] Epistemology of Geometry. 2021. URL: https: //plato.stanford.edu/entries/epistemology-geometry/. [5] Richard Fitzpatrick. EUCLID’S ELEMENTS OF GEOMETRY. 2008. URL: https: //farside.ph.utexas.edu/Books/Euclid/Elements.pdf. 21

References [6] Richard Osborne. Philosophy for beginners. For Beginners, 2007. [7] Type Theory. 2022. URL: https://plato.stanford.edu/entries/type-theory/. [8] 竹内 外史. 新装版 集合とはなにか―はじめて学ぶ人のために. 講談社, 2001. [9] 橋爪 大三郎. はじめての構造主義. 講談社, 1988. [10] 加藤 文元. 数学の世界史. KADOKAWA, 2024. [11] 上垣 渉. はじめて読む 数学の歴史. ベレ出版, 2006. [12] 宮岡 礼子. 曲がった空間の幾何学 現代の科学を支える非ユークリッド幾 何とは. 講談社, 2017. 22

References [13] 寺阪 英孝. 非ユークリッド幾何の世界 新装版. 講談社, 2014. 23