Simplicity, Truth, and Ockham’s Razors

1. Simplicity, Truth, and Ockham’s Razors Kevin T. Kelly Hanti Lin Department of Philosophy Carnegie Mellon University www.formalepistemology.org Currently supported by a 3 year grant from the Templeton foundation.

2. Underdetermination constant linear quadratic Which theory is true?

3. Ockham’s Razor constant linear quadratic Choose the simplest theory compatible with experience!

4. Skepticism Gotcha! constant linear quadratic

5. Two Linked Questions • What is simplicity? • How does Ockham’s razor help us find true theories*?

6. *Theories vs. Models Models predict actual observations. Theories guide counterfactual predictions. The line is a very accurate predictor. accurate predictor Energy

7. *Theories vs. Models Models predict actual observations. Theories guide counterfactual predictions. The line is a very accurate predictor. Turn on the new accelerator. accurate predictor Energy

8. *Theories vs. Models Models predict actual observations. Theories guide counterfactual predictions. Oh. accurate predictor Energy

9. Methodology Gap • Model selection:use theories to obtain accurate predictions that may become very inaccurate if you act on them. • Bayesianism:assign high credence to laws and theories but ask only whether you thinkthey would be accurate. Is it even possible to have a non-circular, performance-basedexplanation of Ockham’s razor in theory choice? Welcome to the real problem of induction. Give up.

10. 1. Simplicity

11. What is Simplicity? • Brevity? • Parsimony • Computational compressibility? • Unity? • Explanatory power (likelihood)? • Testability? • Low dimensionality? • Fewer causes or entities? • Contextual mess?

12. Proposed Answer An axiomatic theory of simplicity relative to a theory choice problem. • “Data-mining = question-mining” • unique simplicity order in many problems.

13. Possible Worlds W

14. Information States • W is the vacuous information state; • each true conjunction of information states is entailed by a true information state. I

15. Information Topology • Let V be the closure of I under arbitrary disjunction. • (W, V) is a topological space with basisI. V

16. Question • QuestionQ is a partition into answersH, H’, ... • Hw = the unique answer true in world w. Q

17. Problem P = (W, I, Q).

18. Interior Worlds • Isolated from complementary answer. • No problem of induction.

19. Boundary Worlds • No answer is ever verified. • Problem of induction.

20. Nested Boundaries • Origin is in boundary of axis. • Axis is in boundary of plane.

21. Skeptical Arrows S(w, X)  w cl(X \ Hw). X w

22. Benign Arrows B(w, X)  • w cl(X); • w cl(X\ Hw); • wX. X w

23. Arrows A(w, X)  S(w, X) B(w, X) .

24. Arrows Between Possibilities • S(D, D’)  (w  D)S(w, D’) ; • B(D, D’)  (w  D) B(w, D’) ; • A(D, D’)  (w  D)A(w, D’) .

25. Arrows Between Possibilities • S(D, D’)  (w  D) S(w, D’) ; • B(D, D’)  (w  D) B(w, D’) ; • A(D, D’)  (w  D) A(w, D’) .

26. Coarse-Graining the Problem • Let F partition W. • Elements of F are coarse-grained possibilities. • We want to represent all the problems of induction in P with arrows between possibilities. • Then we call F a factorization of P.

27. Two Definitions • F|E = the restriction of F to E. • Min(F, E) = the set of all elements of F|E that have no arrows coming in.

28. Factorization Axioms • Axiom 1 All worlds in D have the same, unique arrow status toward D’. • Axiom 2 For each possibility D, there is information E such that DE is in Min(F, E). • Axiom 3 No D in Min(F, E) has an arrow to the disjunction of all possibilities in Min(F, E). • Axiom 4If there is an arrow from a possibility D to a second possibility D’ conjoined with answer H, then D’ H.

29. Formalized Ax 1.w, w’ D [ S(w, D’)S(w’, D’)   B(w’, D’) ]  [ B(w, D’)B(w’, D’)   S(w’, D’) ]. Ax 2.(E) D E Min(F, E). Ax 3. D Min(F, E)   A(D, UMin(F, E)). Ax 4. H  Q  A(D, D’  H)  D’ H.

30. Mathematical Note The axioms are a non-constructive way to invert the Cantor-Bendixson construction to apply to problems with all-boundary answers.

31. Strict Partial Order Prop: Arrows are a strict partial order over possibilities. That is the simplicity order relative to P and F. Note: benign arrows can be necessary for transitivity!

32. Preservation Theorem Theorem: Let E be an information state. F factorizes PF|E factorizes P|E.

33. Example: Law vs. Catchall

34. Law vs. Catchall • Question: Will the binary experiment always yield 0? • Information: Past outcomes 1 1 1 1 1 . . . 0 0 0 0

35. Unique Coarsest Factorization • Question: Will the binary experiment always yield 0? • Information: Past outcomes . . .

36. Example: Point Hypothesis

37. Point Hypothesis • Question: Is the true value of parameter q = 0? • Information: intervals. ( )

38. Unique Coarsest Factorization

39. Epistemic Equivalence Different surface topologies. Isomorphiccoarsest factorizations. Isomorphicepistemology! . . .

40. Example: Half-Line

41. Half Line • Question: Is the true value of parameter q ≤ 0?

42. Riding on Induction ( )

43. Transition to Deduction ( )

44. Ambiguous Representation Bottom cell violates arrow homogeneity. Skeptical arrow

45. Unique Coarsest Factorization Refine into arrow-homogeneous possibilities. Skeptical arrow Benign arrow

46. Example: Free Parameters

47. Parameter Space Question: Which parameters are 0? Information: = open balls. (0, y) (x, y) (0, 0) (x, 0)

48. Unique Coarsest Factorization (x, y) (x, 0) (0, y) (0, 0)

49. Ternary Case (x,y,z) (x,y,0) (x,0,z) (0,y,z) (0,0,z) (0,y,0) (x,0,0) (0,0,0)

50. Dimensionality Question: number of free parameters. (x,y,z) (x,y,0) (x,0,z) (0,y,z) (0,0,z) (0,y,0) (x,0,0) (0,0,0)