1. 1 Cosmic Confusions Not Supporting versus Supporting Not- John D. Norton Department of History and Philosophy of Science Center for Philosophy of Science University of Pittsburgh www.pitt.edu/~jdnorton CARL FRIEDRICH VON WEIZSÄCKER LECTURES UNIVERSITY OF HAMBURG June 2010

2. 2 This Talk B ayesian probabilistic analysis conflates neutrality of evidential support with disfavoring evidential support. Wrong formal tool for many problems in cosmology where neutral support is common. F ragments of inductive logics that tolerate neutral support displayed. Non-probabilistic state of completely neutral support. A rtifacts are introduced by the use of the wrong inductive logic. “ Inductive disjunctive fallacy.” Doomsday argument.

3. 3 Completely Neutral Evidential Support

4. 4 Unconnected Parallel Universes: Completely Neutral Support Same laws, but constants undetermined. h = ? c = ? G = ? … h = ? c = ? G = ? … h = ? c = ? G = ? … Background evidence is neutral on whether h lies in some tiny interval or outside it. 0 123 4 5 h

5. 5 Parallel Universes Born in a Singularity: Disfavoring Evidence Stochastic law assigns probabilities to values of constants. P(h 1 ) = 0.01 … P(h 2 ) = 0.01 … P(h 3 ) = 0.01 … Background evidence strongly disfavors h lying in some tiny interval; and strongly favors h outside it. 0 123 4 5 h very improbable very probable

6. 6 How to Represent Completely Neutral Evidential Support

7. 7 Probabilities from 1 to 0 span support to disfavor P(H|B) + P(not-H|B) = 1 No neutral probability value available for neutral support. P(H|B) P(not-H|B) Large. Strong favoring. Small. Strong disfavoring. P(H|B) P(not-H|B) Large. Strong favoring. Small. Strong disfavoring.

8. 8 Logic of all evidence Underlying Conjecture of Bayesianism… Logic of physical chances …Fails

9. 9 Completely Neutral Support [A|B] = support A accrues from B “indifference” “ignorance” [ |B] = I any contingent proposition Argued in some detail in John D. Norton, "Ignorance and Indifference." Philosophy of Science, 75 (2008), pp. 45-68. "Disbelief as the Dual of Belief." International Studies in the Philosophy of Science, 21(2007), pp. 231-252. 0 123 4 5 h IIIII I I I

10. 10 [ h in [0,1] OR h in [1,2] | B] = [ h in [0,1] | B] = [ h in [1,2] | B] The principle of indifference does not lead to paradoxes. Paradoxes come from the assumption that evidential support must always be probabilistic. I. Invariance under Redescription using the Principle of Indifference Justification… Equal support for h in equal h-intervals. 0 123 4 5 h IIIII 0 123 4 5 h’ II I II rescale h to h’ = f(h) Equal support for h’ in equal h’-intervals.

11. 11 II. Invariance under Negation Justification… Equal (neutral) support for h in [0,1] and outside [0,1]. 0 123 4 5 h I I [ h in [0,1] OR h in [1,2] | B] = [ h in [0,1] | B] Equal (neutral) support for h in [0,2] and outside [0,2]. 0 123 4 5 h I I

12. 12 Neutrality and Probabilistic Independence

13. 13 Probabilistic independence vs. Neutrality of (total) support For a partition of all outcomes A 1, A 2, … P(A i |E&B) = P(A i |B) all i For incremental measures of support * inc (A i, E, B) = 0 * e.g. d(A i, E, B) = P(A i |E&B) - P(A i |B) s(A i, E, B) = P(A i |E&B) - P(A i |not-E&B) r(A i, E, B) = log[ P(A i |E&B)/P(A i |B) ] etc. Tertiary function Presupposes background probability measure. [A i |B] = I all contingent A i Binary function Presupposes NO background probability measure.

14. 14 Objective vs Subjective

15. 15 Neutrality and Disfavor Ignorance and Disbelief or Objective Bayesianism degrees of support Subjective Bayesianism degrees of belief Initial “informationless” priors? Impossible. No probability measure captures complete neutrality. Pick any. They merely encode arbitrary opinion that will be wash out by evidence. Many conditional probability represents opinion + the import of evidence. Only one conditional probability correctly represents the import of evidence. In each evidential situation, Bruno de Finetti mad dog

16. 16 Pure Opinion Masquerading as Knowledge 1. Subjective Bayesian sets arbitrary priors on k 1, k 2, k 3, … Pure opinion. 2. Learn richest evidence = k 135 or k 136 3. Apply Bayes’ theorem P(k 136 |E&B) P(k 135 |E&B) P(k 136 |B) P(k 135 |B) 0.00005 0.00095 = = P(k 135 |E&B) = 0.95 P(k 136 |E&B) = 0.05 Endpoint of conditionalization dominated by pure opinion.

17. 17 Inductive Disjunctive Fallacy

18. 18 Completely neutral support conflated with Strongly disfavoring support a 1 a 1 or a 2 a 1 or a 2 or a 3 … a 1 or a 2 or … or a 99 Neutral support III…IIII…I Disjunction of very many neutrally supported outcomes is NOT a strongly supported outcome. prob = 0.01 prob = 0.02 prob = 0.03 … prob = 0.99 Disfavoring

19. 19 van Inwagen, “ Why is There Anything At All?” Proc. Arist. Soc., Supp., 70 (1996). pp.. 95-120. One way not to be. Infinitely many ways to be. … Probability zero. “As improbable as anything can be.” Probability one. As probable as anything can be.

20. 20 Our Large Civilization Ken Olum, “Conflict between Anthropic Reasoning and Observations,” Analysis, 64 (2004). pp. 1-8. Fewer ways we can be in small civilizations. Vastly more ways we can be in large civilizations. … “Anthropic reasoning predicts we are typical…” “… [it] predicts with great confidence that we belong to a large civilization.”

21. 21 Our Infinite Space Informal test of commitment to anthropic reasoning. Fewer ways we can be observers in a finite space. Infinitely more ways we can be observers in an infinite space. … Hence our space is infinitely more likely to be geometrically infinite.

22. 22 Inductive Logics that Tolerate Neutrality of Support

23. 23 If T 1 entails E. T 2 entails E. P(T 1 |B) = P(T 2 |B) then P(T 1 |E&B) = P(T 2 |E&B) Discard Additivity, Keep Bayesian Dynamics Bayesian conditionalization. Conditionalizing from Complete Neutrality of Support If T 1 entails E. T 2 entails E. [T 1 |B] = [T 2 |B] = I then [T 1 |E&B] = [T 2 |E&B] Postulate same rule in a new, non-additive inductive logic. equal priors equal posteriors

24. 24 Pure Opinion Masquerading as Knowledge Solved “Priors” are completely neutral support over all values of k i. [k 1 |B] = [k 2 |B] = [k 3 |B] =… = [k 135 |B] = [k 136 |B] = … = I No normalization imposed. [k 1 |B] = [k 1 or k 2 |B] = [k 1 or k 2 or k 3 |B] =… = I Bayesian result of support for k 135 over k 136 is an artifact of the inability of a probability measure to represent neutrality of support. Apply rule of conditionalization on completely neutral support. E = k 135 or k 136 [k 135 |B] = [k 136 |B] = I [k 135 |E&B] = [k 136 |E&B] Nothing in evidence discriminates between k 135 or k 136.

25. 25 The Doomsday Argument

26. 26 Doomsday Argument (Bayesian analysis) time = 0 time of doom T we learn time t has passed What support does t give to different times of doom T? Bayes’ theorem p(T|t&B) ~ p(t|T&B). p(T|B) p(T|t&B) ~ 1/T Support for early doom For later: which is the right “clock” in which to sample uniformly? Physical time T? Number of people alive T’?… p(t|T&B) = 1/T Compute likelihood by assuming t is sampled uniformly from available times 0 to T. Variation in likelihoods arise entirely from normalization. Entire result depends on this normalization. Entire result is an artifact of the use of the wrong inductive logic.

27. 27 Doomsday Argument (Barest non-probabilistic reanalysis.) time = 0 time of doom T we learn time t has passed What support does t give to different times of doom T? Take evidence E is just that T>t. T 1 >t entails E. T 2 >t entails E. E = T>t [T 1 |B] = [T 2 |B] = I Apply rule of conditionalization on completely neutral support. [T 1 |E&B] = [T 2 |E&B] The evidence fails to discriminate between T 1 and T 2.

28. 28 Doomsday Argument (Bayesian analysis again) time = 0 time of doom T we learn time t has passed What support does t give to different times of doom T? Consider only the posterior p(T|t&B) Require invariance of posterior under changes of units used to measure times T, t. Invariance under T’=AT, t’=At Days, weeks, years? Problem as posed presumes no time scale, no preferred unit of time. Disaster! This density cannot be normalized. Infinite probability mass assigned to T>T *, no matter how large. Evidence supports latest possible time of doom. Unique solution is the “Jeffreys’ prior.” p(T|t&B) = C(t)/T

29. 29 A Richer Non-Probabilistic Analysis time = 0 time of doom T we learn time t has passed What support does t give to different times of doom T? Consider the non-probabilistic degree of support for T in the interval [T 1,T 2 |t&B] Presume that there is a “right” clock-time in which to do the analysis, but we don’t know which it is. So we may privilege no clock, which means we require invariance under change of clock: T’ = f(T), t’ = f(t), for strictly monotonic f. [T 1,T 2 |t&B] = [T 3,T 4 |t&B] = I for all T 1,T 2, T 3,T 4

30. 30 Inductive inference the right way

31. 31 A Warrant for a Probabilistic Logic Ensemble Randomizer + No universal logic of induction Material theory of induction: Inductive inferences are not warranted by universal schema, but by locally prevailing facts. The contingent facts prevailing in a domain dictate which inductive logic is applicable. An ensemble alone is not enough. Mere evidential neutrality over the ensemble members does not induce an additive measure. Some further element of the evidence must introduce a complementary favoring-disfavoring.

32. 32 Probabilities from Multiverses? Gibbons, Hawking, Stewart (1987): Hamiltonian formulation of general relativity. Additive measure over different cosmologies induced by canonical measure. Gibbons, G. W.; Hawkings, S. W. and Stewart, J. M. (1987) “A Natural Measure on the Set of All Universes,” Nuclear Physics, B281, pp. 736-51. Just like the microcanonical distribution of ordinary statistical mechanics? No: there is no ergodic like behavior and hence no analog of the randomizer. Ensemble without randomizer “Giving the models equal weight corresponds to adopting Laplace’s ‘principle of indifference’, which claims that in the absence of any further information, all outcomes are equally likely.” Gibbons, Hawking, Stewart, p. 736

33. 33 Read "Deductively Definable Logics of Induction." Journal of Philosophical Logic. Forthcoming. “What Logics of Induction are There?” Tutorial in Goodies pages on my website. “Deductively definable logics of induction” Inductive strength [A|B] for propositions A, B drawn from a Boolean algebra is defined fully by deductive structure of the Boolean algebra. Large class of non-probabilistic logics. No-go theorem. All need inductive supplement. No universal formal logic of induction. Formal results: Independence is generic. Limit theorem. Scale free logics of induction.

34. 34 Winding Up

35. 35 This Talk B ayesian probabilistic analysis conflates neutrality of evidential support with disfavoring evidential support. Wrong formal tool for many problems in cosmology where neutral support is common. F ragments of inductive logics that tolerate neutral support displayed. Non-probabilistic state of completely neutral support. A rtifacts are introduced by the use of the wrong inductive logic. “ Inductive disjunctive fallacy.” Doomsday argument.

36. 36 Read all about it…

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39. 39 C ommercials

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41. 41 Finis

42. 42 Appendices

43. 43 The Self-Sampling Assumption Penzias and Wilson measure 3 o K cosmic background radiation. Level I multiverses. Many clones of Penzias and Wilson measure 3 o K cosmic background radiation in other parts of space. Which is our Penzias and Wilson? Self-Sampling Assumption: “One should reason if as one were a random sample from the set of of all observers in one’s reference class.” (Bostrom, 2007, p. 433) Evidence on which is our PW is neutral. No warrant for a probability measure. The self-sampling assumption imposes probabilities where they do not belong by mere supposition.

44. 44 Why have the Self-Sampling Assumption? P( | ) measure 3 o K back- ground is 100 o K = q <<1 P( | ) someone somewhere measures 3 o K back- ground is 100 o K is (near) one. “(L)” A physical chance computed in a physical theory. Very many trials carried out in the multiverse. Introduce self-sampling to reduce this probability by allowing that our PW is probably not the “someone somwhere.” P( | ) Our PW measure 3 o K back- ground is 100 o K = P( | ) i-th PW measure 3 o K back- ground is 100 o K P( ) i-th PW is our PW  i = q q 1/n If n = infinity, the computation fails. “1/n = 1/infinity = 0” The failure is an artifact of the probabilistic representation and its difficulties with infinitely many cases. Recover the same result without sampling or calculation just by applying (L) directly to case of “our PW.”

45. 45 A Tempting Fallacy in Modern Cosmology Prior theory is neutral on the values of some fundamental cosmic parameters: Non-inflationary cosmology provides no reason to expect a very flat space. Fundamental theories give no reason to expect h, c, G, … to be the values that support life. No reason to expect observed values. The values are improbable and therefore in need of explanation. We cannot demand that everything be explained on pain of an infinite explanatory regress. How do we decide what is in urgent need of explanation and what is not? We decide post hoc. Only after we have the new explanatory theory do we decide the cosmic parameter in urgent need of explanation.