Michael Bernstein on Propositions as Types

I’ll (Michael) be talking about Philip Wadler’s paper "Propositions as Types," which starts out with the following sentence:

"Powerful insights arise from linking two fields of study previously thought separate."

And just keeps on going from there. In less than 9 full pages, Wadler assembles an exuberant, hilarious take on the deep, meaningful connections between mathematics, philosophy, and computer science.

I’ll also be showing off some work from Daniel P. Friedman and Carl Eastlund’s recent book "The Little Prover," which implements a small theorem prover in Scheme, to demonstrate some of the connections Wadler discusses in his paper.

Transcript

1. Philip Wadler “Propositions As Types” Michael R. Bernstein Papers We Love NYC / 2015.8.17 / New York, New York
2. Dedicated to: James Lennon Engels Golick
7. 1. This paper’s contributions 2. My favorite parts 3. Discussion
8. “Propositions As Types”
9. Contributions
11. “Powerful insights arise from linking two fields of study previously thought separate.”
12. The history of Computer Science is very short
13. The history of logic and philosophy are its pre-history
14. There is a mechanical connection between these fields
15. It can pretty much be explained with a dotted line
17. Propositions as Types is a notion with many names and many origins
18. Propositions as Types is a notion with depth
19. Propositions as Types is a notion with breadth
20. Propositions as Types is a notion with mystery
21. “Propositions as Types” is exciting, and is excellent at teaching you about Propositions as Types
27. My Favorite Parts
28. Logic and Computation
29. Aristotle Ockham Leibniz
30. Boole De Morgan Frege Peirce Peano
31. Hilbert Gödel Church Kleene Turing
32. “Like busses: you wait two thousand years for a definition of ‘effectively calculable,’ and then three come along at once.”
33. Gentzen Curry Howard
34. “There may, indeed, be other applications of the system than its use as a logic.” - Alonzo Church, on the Lambda Calculus
35. W. A. Howard. “The formulae-as-types notion of construction.”
36. Propositions as Types
37. Conjunction A ∧ B corresponds to cartesian product A x B, Disjunction A ∨ B corresponds to a disjoint sum A + B Implication A ⊃ B corresponds to function space A → B. A proof of the proposition A ⊃ B consists of a procedure that given a proof of A yields a proof of B.
38. Conjunction A ∧ B corresponds to cartesian product A x B, Disjunction A ∨ B corresponds to a disjoint sum A + B Implication A ⊃ B corresponds to function space A → B. A proof of the proposition A ⊃ B consists of a procedure that given a proof of A yields a proof of B.
39. Conjunction A ∧ B corresponds to cartesian product A x B, Disjunction A ∨ B corresponds to a disjoint sum A + B Implication A ⊃ B corresponds to function space A → B. A proof of the proposition A ⊃ B consists of a procedure that given a proof of A yields a proof of B.
40. Conjunction A ∧ B corresponds to cartesian product A x B, Disjunction A ∨ B corresponds to a disjoint sum A + B Implication A ⊃ B corresponds to function space A → B. A proof of the proposition A ⊃ B consists of a procedure that given a proof of A yields a proof of B.
41. Proofs are Programs
42. Normalization of proofs is evaluation of programs
43. Logics & Languages
44. Intuitionistic Logic & Simply Typed Lambda Calculus
45. Second Order Logic & Second Order Lambda Calculus
46. Modal Logic & Monads
47. Linear Logic & Session Types
48. Mechanics
57. *exhale*
58. An Aside / Another Book
59. Friedman and Eastlund “The Little Prover”
60. Axioms
61. (dethm if-true (x y) (equal (if 't x y) x)) (dethm if-false (x y) (equal (if 'nil x y) y)) (dethm if-same (x y) (equal (if x y y) y))
62. Rewriting
63. * Replace a function application with its body * Replace expressions according to axioms * Prove that a function is total * Leverage homoiconic language for maximum theorem proving fun
64. Proof Language
65. (defun defun.first-of () (J-Bob/define (defun.pair) '(((defun first-of (x) (car x)) nil))))
66. (defun dethm.first-of-pair () (J-Bob/define (defun.second-of) '(((dethm first-of-pair (a b) (equal (first-of (pair a b)) a)) nil ((1 1) (pair a b)) ((1) (first-of (cons a (cons b '())))) ((1) (car/cons a (cons b '()))) (() (equal-same a))))))
68. Universality/Outer Space
70. “Propositions as Types informs our view of the universality of certain programming languages.”
71. There is a thriving, proven connection between work in logic and work in computation
72. Discussion
73. Works Cited http://bit.ly/1DTWE8Q
74. *Thank You* w michaelrbernste.in t @mrb_bk