The Way Not Taken or How to Do Things with an Infinite Regress

1. The Way Not TakenorHow to Do Things with anInfinite Regress

2. Two Methodological Paradigms Certainty “perfect support”

3. Problem of induction . . . Two Methodological Paradigms Certainty “perfect support”

4. Two Methodological Paradigms Confirmation Partial support Certainty “perfect support”

5. Two Methodological Paradigms Confirmation Partial support Eureka! Certainty “perfect support” Halting with the right answer

6. Two Methodological Paradigms Confirmation Learning Partial support Convergent procedure Eureka! Certainty Entailment by evidence Halting with the right answer

7. Two Methodological Paradigms Confirmation Learning Partial support Convergent procedure Certainty Entailment by evidence Halting with the right answer

8. Two Methodological Paradigms • Relational state • “Internal” • Relation “screens off” process • Computation is extraneous • Dynamical stability • “External” • Process paramount • Computational perspective Confirmation Learning Partial support Convergent procedure Certainty Entailment by evidence Halting with the right answer

9. The Ultimate Question Methods justify conclusions. What justifies methods?

10. The Ultimate Question What justifies methods? Confirmation What justifies the confirmation relation?

11. Options Coherentism Methods justify themselves . . . . . . . . . Foundationalism A priori justification Regress Always respond with a new method

12. Alternative Approach What Justifies Science? Confirmation Learning What justifies the confirmation relation? How can we learn whether we are learning?

13. Learning Theory Primer Putnam Weinstein Gold Freiwald Barzdin Blum Case Smith Sharma Daley Osherson Etc. epistemically relevant worlds K correctness H hypothesis M 1 0 1 0 ? 1 ? 1 0 ? ? ? data stream conjecture stream method

14. Convergence finite With certainty: M ? ? 0 ? 1 0 1 0 halt! finite forever In the limit: M ? ? 0 ? 1 0 1 0 1 1 1 1…

15. Reliability = Guaranteed convergence to the right answer 1-sided 2-sided worlds conjectures data H K M

16. Some Reliability Concepts Two-sided One-sided

17. Example: UniformitariansmMichael Ruse, The Darwinian Revolution Uniformitarianism (steady-state) Complexity Stonesfield mammals Catastrophism (progressive): Complexity creation

18. Example: UniformitariansmMichael Ruse, The Darwinian Revolution • Uniformitarianism is refutable in the limit: • Side with uniformitarianism each time the current progressive schedule is violated. • Eventually slide back to progressionism after the revised schedule stands up for a while.

19. Historicism Explained • Articulations refutable only in a paradigm. • Articulations crisply refutable with a paradigm. • No time at which a paradigm must be rejected. • Method inputs needn’t be cognized. • Propositions forced by inputs relative to a paradigm may be paradigm-relative.

20. Underdetermination = Degrees of Unsolvability Verifiable in the limit Refutable in the limit Decidable in the limit Verifiable with certainty Refutable with certainty Decidable with certainty

21. Underdetermination = Degrees of Unsolvability • Catastrophism • The coin is unfair • Computability • Uniformitarianism • The coin is fair • Uncomputability Verifiable in the limit Refutable in the limit Decidable in the limit Verifiable with certainty Refutable with certainty • Phenomenal laws • Observable existence Decidable with certainty • It will rain tomorrow

22. Underdetermination = Complexity AE EA • The coin is fair • Computability • Catastrophism • The coin is unfair • Uncomputability • Uniformitarianism Verifiable in the limit Refutable in the limit Decidable in the limit E A Verifiable with certainty Refutable with certainty • Universal laws • Existence claims Decidable with certainty clopen/recursive • It will rain tomorrow

23. Refinement: Retractions You are a fool not to invest in technology Retractions NASDAQ 0 1 1 0 ? 1 1 ? ? ? ? ? Expansions

24. AE EA Retractions asComplexity Refinement Verifiable in the limit Refutable in the limit Decidable in the limit E A A E v v . . . 2 retractions starting with 0 2 retractions starting with 1 • Exactly n… Boolean combinations of universals and existentials 1 retraction starting with ? E A = verifiability with certainty 1 retraction starting with 0 1 retraction starting with 1 = refutability with certainty 0 retractions starting with ? = decidability with certainty

25. Standard Objection • Very nice, but every method is reliable only under background presuppositions. • How do we know that the background assumptions are true? • Every assumption should be subject to empirical review.

26. Plausible Fallacy • If you could learn whether your pressupositions were true, you could chain this ability together with your original method to learn without them. • Therefore, if you need the presuppositions, you can’t reliably assess them. • Therefore, learning theory is an inadmissible version of foundationalism.

27. A Learnability Analysisof Methodological Regress • Presupposition of M = the set of worlds in which M succeeds. worlds conjectures data H success presupposition M

28. P1 P2 P3 H M1 M2 M3 M4 Methodological Regress Same data to all

29. No Free Lunch Principle • The instrumental value of a regress is no greater than the best single-method performance that could be recovered from it without looking at the data. … Regress achievement Single-method achievement Scale of underdetermination

30. P1 P2 H M1 M2 M3 M Empirical Conversion • An empirical conversion is a method that produces conjectures solely on the basis of the conjectures of the given methods.

31. Methodological Equivalence • Reduction: B<A iff There is an empirical conversion of an arbitrary group of methods collectively achieving A into a group of methods collectively achieving B. • Methodological equivalence = inter-reducibility.

32. P1 H M1 M2 Simple Illustration • P1 is the presupposition under which M1 refutes H with certainty. • M2 refutes P1 with certainty.

33. Worthless Regress • M1 alternates mindlessly between acceptance and rejection. • M2 always rejects a priori.

34. Pretense • Mpretends to refute H with certainty iff M never retracts a rejection. • Duhem: no hypothesis is refutable in isolation. • Popper: to avoid coddling a false hypothesis forever, establish rejection conditions in advance even though the hypothesis is not really refutable.

35. P1 H M1 M2 Modified Example • Same as before • But now M1 pretends to refute H with certainty.

36. M Reduction H Starts not rejecting 2 retractions in worst case

37. M Reliability H

38. Converse Reduction • M decides H with at most 3 retractions starting with acceptance. • Choose: • P1 = “M retracts at most once” • M1 accepts until M uses one retraction and rejects thereafter. • M2 accepts until M retracts twice and rejects thereafter. • Both methods pretend to refute.

39. Reliability

40. P1 H H M1 M2 M1 Regress Tamed method regress 2 retractions starting with 1 Pretends to refute with certainty Refutes with certainty Complexity classification

41. P0 P1 M M1 M2 Mk+1 Finite Regresses Tamed P0 Pretends : n1 retractions starting with c1 Sum all the retractions. Start with 1 if an even number of the regress methods start with 0. Pretends : n2 retractions starting with c2 P2 Pn . . . n2 retractions starting with c2 H

42. P0 P1 M M1 M2 Mk+1 Infinite Popperian Regresses UI P0 UI Ever weaker presuppositions P2 UI Refutes with certainty over UiPi Pn . . . Each pretends to refute with certainty UI Pn+1 . . .

43. Pi-1 P0 Mi M1 Example Regress of deciders: “2 more = forever” Halt after 2 and project final obs. Halt after 2i and project final obs. K P2 P3 P1 . . . Grue3 Grue4 Grue5 Grue0 Grue1 Grue2 P0 Green

44. M Example Equivalent single refuting method P0 K Grue3 Grue4 Grue5 Grue0 Grue1 Grue2 P0 H Green

45. P0 P1 M M1 M2 Mk+1 Other Infinite Regresses UI P0 UI Ever weaker presuppositions P2 Each pretends to Verify with certainty Use bounded retractions Decide in the limit Refute in the limit UI Refutes in the limit over UiPi Pn . . . UI Pn+1 . . .

46. Example: UniformitariansmMichael Ruse, The Darwinian Revolution Uniformitarianism (steady-state) Complexity Stonesfield mammals Catastrophism (progressive): Complexity creation

47. Pi-1 P0 M1 Mi Example: UniformitariansmMichael Ruse, The Darwinian Revolution Regress of 2-retractors equivalent to a single limiting refuter: Pi = any number of surprises except i. H

48. EAE AEA The Powerof NestedInfiniteEmpiricalRegresses Gradual refutability Gradual verifiability Similar results AE EA Verifiable in the limit Refutable in the limit Decidable in the limit A E Verifiable with certainty Refutable with certainty Decidable with certainty

49. Naturalism Logicized • Unlimited Fallibilism: every presupposition is held up to the tribunal of experience. • No free lunch: capturesobjective power of empirical regresses. • Progress-oriented: convergence is the goal. • Feasibility: reductions are computable, so analysis applies to computable regresses. • Historicism: dovetails with a logical viewpoint on paradigms and articulations.