“History of type theory”的意思、由来-开放百科全书

The type theory was initially created to avoid paradoxes in a variety of formal logics and rewrite systems. Later, type theory referred to a class of formal systems, some of which can serve as alternatives to naive set theory as a foundation for all mathematics.

It has been tied to formal mathematics since Principia Mathematica to today's proof assistants.

1900–1927

Origin of Russell's theory of types

In a letter to Gottlob Frege (1902) Russell announced his discovery of the paradox in Frege's Begriffsschrift.^[1] Frege promptly responded, acknowledging the problem and proposing a solution in a technical discussion of "levels". To quote Frege:

He goes about showing how this might work but seems to pull back from it. As a consequence of what has become known as Russell's paradox both Frege and Russell had to quickly amend works that they had at the printers. In an Appendix B that Russell tacked onto his The Principles of Mathematics (1903) one finds his "tentative" theory of types. The matter plagued Russell for about five years.^[4]

Willard Quine^[5] presents a historical synopsis of the origin of the theory of types and the "ramified" theory of types: after considering abandoning the theory of types (1905), Russell proposed in turn three theories:

Quine observes that Russell's introduction of the notion of "apparent variable" had the following result:

Quine dismisses this notion of "bound variable" as "pointless apart from a certain aspect of the theory of types".^[6]

The 1908 "ramified" theory of types

Quine explains the ramified theory as follows: "It has been so called because the type of a function depends both on the types of its arguments and on the types of the apparent variables contained in it (or in its expression), in case these exceed the types of the arguments".^[6] Stephen Kleene in his 1952 Introduction to Metamathematics describes the ramified theory of types this way:

The axiom of reducibility and the notion of "matrix"

But because the stipulations of the ramified theory would prove (to quote Quine) "onerous", Russell in his 1908 Mathematical logic as based on the theory of types^[7] also would propose his axiom of reducibility. By 1910 Whitehead and Russell in their Principia Mathematica would further augment this axiom with the notion of a matrix — a fully extensional specification of a function. From its matrix a function could be derived by the process of "generalization" and vice versa, i.e. the two processes are reversible — (i) generalization from a matrix to a function (by using apparent variables) and (ii) the reverse process of reduction of type by courses-of-values substitution of arguments for the apparent variable. By this method impredicativity could be avoided.^[8]

Truth tables

Eventually Emil Post (1921) would lay waste to Russell's "cumbersome"^[9] Theory of Types with his "truth functions" and their truth tables. In his "Introduction" to his 1921 Post places the blame on Russell's notion of apparent variable: "Whereas the complete theory [of Whitehead and Russell (1910, 1912, 1913)] requires for the enunciation of its propositions real and apparent variables, which represent both individuals and propositional functions of different kinds, and as a result necessitates the cumbersome theory of types, this subtheory uses only real variables, and these real variables represent but one kind of entity, which the authors have chosen to call elementary propositions".

At about the same time Ludwig Wittgenstein made short work of the theory of types in his 1922 work Tractatus Logico-Philosophicus in which he points out the following in parts 3.331–3.333:

3.332 No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the whole "theory of types").

3.333 A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself...}}

Wittgenstein proposed the truth-table method as well. In his 4.3 through 5.101, Wittgenstein adopts an unbounded Sheffer stroke as his fundamental logical entity and then lists all 16 functions of two variables (5.101).

The notion of matrix-as-truth-table appears as late as the 1940–1950s in the work of Tarski, e.g. his 1946 indexes "Matrix, see: Truth table"^[10]

Russell's doubts

Russell in his 1920 Introduction to Mathematical Philosophy devotes an entire chapter to "The axiom of Infinity and logical types" wherein he states his concerns: "Now the theory of types emphatically does not belong to the finished and certain part of our subject: much of this theory is still inchoate, confused, and obscure. But the need of some doctrine of types is less doubtful than the precise form the doctrine should take; and in connection with the axiom of infinity it is particularly easy to see the necessity of some such doctrine".^[11]

Theory of simple types

In the 1920s, Leon Chwistek^[18] and Frank P. Ramsey^[19] noticed that, if one is willing to give up the vicious circle principle,

the hierarchy of levels of types in the "ramified theory of types" can be collapsed.

The resulting restricted logic is called the theory of simple types^[20] or, perhaps more commonly, simple type theory.^[21] Detailed formulations of simple type theory were published in the late 1920s and early 1930s

by R. Carnap, F. Ramsey, W.V.O. Quine, and A. Tarski. In 1940 Alonzo Church (re)formulated it as simply typed lambda calculus.^[22] and examined by Gödel in 1944. A survey of these developments is found in Collins (2012).^[23]

1940s–present

Gödel 1944

Kurt Gödel in his 1944 Russell's mathematical logic gave the following definition of the "theory of simple types" in a footnote:

He concluded the (1) theory of simple types and (2) axiomatic set theory, "permit the derivation of modern mathematics and at the same time avoid all known paradoxes" (Gödel 1944:126); furthermore, the theory of simple types "is the system of the first Principia [Principia Mathematica] in an appropriate interpretation. . . . [M]any symptoms show only too clearly, however, that the primitive concepts need further elucidation" (Gödel 1944:126).

Curry–Howard correspondence, 1934–1969

The Curry–Howard correspondence is the interpretation of proofs-as-programs and formulae-as-types. The idea starting in 1934 with Haskell Curry and finalized in 1969 with William Alvin Howard. It connected the "computational component" of many type theories to the derivations in logics.

Howard showed that the typed lambda calculus corresponded to intuitionistic natural deduction. (That is, natural deduction without the Law of excluded middle.) The connection between types and logic lead to a lot of subsequent research to find new type theories for existing logics and new logics for existing type theories.

de Bruijn's AUTOMATH, 1967–2003

Nicolaas Govert de Bruijn created the type theory Automath as a mathematical foundation for the Automath system, which could verify the correctness of proofs. The system developed and added features over time as type theory developed.

Martin-Löf's Intuitionistic type theory, 1971–1984

Per Martin-Löf found a type theory that corresponded to predicate logic by introducing dependent types, which became known as intuitionistic type theory or Martin-Löf type theory.

Martin-Löf's theory uses inductive types to represent unbounded data structures, such as natural numbers.

Barendregt's lambda cube, 1991

The lambda cube was not a new type theory but a categorization of existing type theories. The eight corners of the cube included some existing theories with simply typed lambda calculus at the lowest corner and the calculus of constructions at the highest.

References

1. ^Russell's (1902) Letter to Frege appears, with commentary, in van Heijenoort 1967:124–125.
2. ^Frege (1902) Letter to Russell appears, with commentary, in van Heijenoort 1967:126–128.
3. ^van Heijenoort 1967:128
4. ^cf. Quine's commentary before Russell (1908) Mathematical Logic as based on the theory of types in van Heijenoort 1967:150
5. ^cf. commentary by W. V. O. Quine before Russell's (1908) Mathematical logic as based on the theory of types in van Hiejenoort 1967:150–153
6. ^¹Quine's commentary before Russell (1908) Mathematical logic as based on the theory of types in van Heijenoort 1967:151
7. ^Russell (1908) Mathematical Logic as based on the theory of types in van Heijenoort 1967:153–182
8. ^cf. in particular p. 51 in Chapter II The theory of Logical Types and *12 The Hierarchy of Types and the Axiom of Reducibility pp. 162–167. Whitehead and Russell (1910–1913, 1927 2nd edition) Principia Mathematica
9. ^Post (1921) Introduction to a general theory of elementary propositions in van Heijenoort 1967:264–283, in particular p. 265
10. ^Tarski 1946, Introduction to Logic and to the Methodology of Deductive Sciences, Dover republication 1995
11. ^Russell 1920:135
12. ^cf. "Introduction" to 2nd edition, Russell 1927:xiv and Appendix C
13. ^cf. "Introduction" to 2nd edition, Russell 1927:i
14. ^cf. "Introduction" to 2nd edition, Russell 1927:xxix
15. ^The vertical bar " | " is the Sheffer stroke; cf. "Introduction" to 2nd edition, Russell 1927:xxxi
16. ^"The theory of classes is at once simplified in one direction and complicated in another by the assumption that functions only occur through their values and by the abandonment of the axiom of reducibility"; cf. "Introduction" to 2nd edition, Russell 1927:xxxix
17. ^These quotes from "Introduction" to 2nd edition, Russell 1927:xliv–xlv.
18. ^L. Chwistek, Antynomje logikiformalnej, Przeglad Filozoficzny 24 (1921) 164–171
19. ^F. P. Ramsey, The foundations of mathematics, Proceedings of the London Mathematical Society, Series 2 25 (1926) 338–384.
20. ^Gödel 1944, pages 126 and 136–138, footnote 17: "Russell's mathematical logic" appearing in Kurt Gödel: Collected Works: Volume II Publications 1938–1974, Oxford University Press, New York NY, {{ISBN|978-0-19-514721-6}}(v.2.pbk).
21. ^This does not mean the theory is "simple", it means that the theory is restricted: types of different orders are not to be mixed: "Mixed types (such as classes containing individuals and classes as elements) and therefore also transfinite types (such as the class of all classes of finite types) are excluded." Gödel 1944, pages 127, footnote 17: "Russell's mathematical logic" appearing in Kurt Gödel: Collected Works: Volume II Publications 1938–1974, Oxford University Press, New York NY, {{ISBN|978-0-19-514721-6}}(v.2.pbk).
22. ^A. Church, A formulation of the simple theory of types, Journal of Symbolic Logic 5 (1940) 56–68.
23. ^J. Collins, A History of the Theory of Types: Developments after the Second Edition of 'Principia Mathematica'. LAP Lambert Academic Publishing (2012). {{ISBN|978-3-8473-2963-3}}, esp. chs. 4–6.
24. ^Gödel 1944:126 footnote 17: "Russell's mathematical logic" appearing in Kurt Gödel: Collected Works: Volume II Publications 1938–1974, Oxford University Press, New York NY, {{ISBN|978-0-19-514721-6}}(v.2.pbk).