Transient Unterdetermination and the Miracle Argument

1. Transient UnterdeterminationandtheMiracle Argument Paul Hoyningen-Huene Leibniz Universität Hannover Center forPhilosophyandEthicsof Science (ZEWW)

2. The subject of the talk TU  ¬ MA TU = transient underdetermination MA = miracle argument

3. Outline 1. Notions of underdetermination • Radical underdetermination (RU) • Transient underdetermination (TU) 2. The miracle argument 3. The miracle argument in the light of transient underdetermination 4. Presuppositions of the miracle argument 5. Conclusion

4. Radical underdetermination (RU) “Radical” or “strong” or “Quinean” underdetermination (RU): For any theory T, there are always empirically equivalent theories that are not compatible with T Formally: Let DT be the set of all (possible) data compatible with a given theory T Definition: RU holds iff  T  T [T is compatible with DT  (T  T)] RU seems to kill scientific realism because there is no data on the basis of which we can decide between T and T

5. Transient Underdetermination (TU) “Transient” or “weak” underdetermination (TU) Presuppositions: Let D0 be a finite set of data that is given at time t0 Let T0 be the set of theories such that T0 := {T0(i), i  I, T0(i) is relevant for and consistent with D0} where I is some index set; T0 ≠ Ø

6. Definition of TU: 1st attempt TU holds iff  T (T  T0)  T (T T0(T T))] “(T T)” means that T and T are not compatible Note that there are many possible sources for the incompatibility of theories, including incommensurability! This is too weak as a definition of TU: the existence of two minimally differing theories consistent with the data fulfills the condition It is only a necessary condition for TU We need the possibility of radically false theories that are compatible with the available data

7. TU: Presuppositions Partition of T0 into the two subsets: (approximately) true theories and radically false theories (not even approximately true) T0AT := {T0(i), i V, T0(i) is true or approximately true} T0RF := {T0(i), i W, T0(i) is radically false} where V and W are the respective index sets with V  W = I (which implies T0 = T0AT T0RF) Assume T0AT T0RF = Ø Intuitively, radically false theories operate with radically false basic assumptions in spite of their agreement with the available data (e.g., at some historical time, phlogiston theory or classical mechanics)

8. Definition of TU: 2nd attempt TU holds iff T0RF ≠ Ø For the purposes of my argument, this is still too weak: there must be “quite a few” radically false theories in T0

9. Definition of TU: 3rd attempt Intuitive idea of TU: In T0, there are many more approximately true theories than true theories, and many more radically false theories than approximately true theories (Stanford: unconceived alternatives) In order to formalize this idea, I need the concept of a measure on the space of theories A measure is a generalization of the concept of volume for more general “spaces” The measure says how big a subset of the space is Simplistic example for a theory space and a measure on it: Space of theories: {Tk 0 ≤ k < ∞, Tk: F(x) = k} Possible measure μ({Tk a ≤ k ≤ b, Tk: F(x) = k}) := b - a

10. Simplistic example F(x) b F(x)=k b-a a x

11. Definition of TU: 3rd attempt (2) Let μ be a measure on the set of theories T0 Definition of TU: TU holds iff μ(T0AT) << μ(T0RF) In what follows, I will presuppose transient underdetermination in this form

12. TU: Simplistic example (1)with arbitrary numbers F(x) 1.5 F(x)=k 1.0 true theory F(x)=1 0.5 domain of approximatively true theories x

13. TU: Simplistic example (2)with arbitrary numbers T0:={Tk 0.5 ≤ k ≤ 1.5, Tk: F(x) = k} T0AT:={Tk 0.999 ≤ k ≤ 1.001, Tk: F(x) = k} T0RF:={Tk 0.5 ≤ k < 0.999  1.001 < k ≤ 1.5, Tk: F(x)= k} μ(T0AT) = 0.002; μ(T0RF) = 0.998 Indeed, μ(T0AT) = 0.002 << μ(T0RF) = 0.998

14. The miracle argument (MA) There are several forms of the miracle argument I will discuss the following form: • Scientific realism is the best explanation for novel predictive success of theories; other philosophical positions make it a miracle • Therefore, it is reasonable to accept scientific realism Let us articulate this argument more explicitly

15. The miracle argument (2) Let D0 be a finite set of data that is given at time t0 Let T0 := {T0(i), i I, T0(i) is relevant for and consistent with D0}, I is some index set Let T0AT and T0RF be the partition of T0 into true or approximately true and radically false theories Let N be some novel data (relative to D0) that is discovered at time t1 > t0 Let there be a theory T*  T0 capable of predicting the novel data N already at t0

16. The miracle argument (3) Where does T* belong, to T0AT or to T0RF? If T* belongs to T0AT, its novel predictive success is not surprising because it gets something fundamental about nature (approximately) right If T* belongs to T0RF, its novel predictive success would be surprising because T* lacks all resources for successful novel predictions; it would be a miracle Therefore, it is very probable that T* belongs to T0AT – realism explains the novel predictive success of science

17. Transient underdetermination and the miracle argument First, note the following connection between a measure and the prior probability: What is the prior probability to find an element s of some set S in a subset A of S? It is proportional to the “size” of A, i.e. proportional to μ (A) [technically: p(sAsS) = μ(A)/μ(S)] Common sense: Is the probability of winning the lottery small or large?

18. TU & MA (2) Due to this connection, TU supports antirealism: Argument 1 T0 = T0AT T0RF and T0AT T0RF = Ø TU: μ(T0AT) << μ(T0RF) Therefore for any T  T0, it is very probable that T  T0RF In other words: due to TU, any theory fitting some data is probably radically false, i.e., TU supports anti-realism

19. TU & MA (3) But here comes the miracle argument: Argument 2 T0 = T0AT T0RF and T0AT T0RF = Ø  T*  T0 such that T* makes the novel prediction N For any T  T0RF, it is very improbable (or even impossible) to make prediction N Therefore, it is very probable (or even certain) that T*  T0AT In other words: novel predictive success supports realism

20. TU & MA (4) Note the tension between the conclusions of arguments 1 and 2: Conclusion 1: Therefore for any T  T0, it is very probable that T  T0RF Conclusion 2: Therefore, it is very probable (or even certain) that T*  T0AT Argument 1 is overruled by argument 2 because the latter’s conclusion about T* states a posterior probability based on additional information [technically: Hempel’s requirement of maximal specificity for statistical explanations] In other words: with the help of MA, realism beats antirealism that relies on TU!

21. TU & MA (5) But TU strikes back: Apply TU at t = t1 again, namely to the new situation with the new data set D1 := D0 N At time t1, I will do exactly the same as what I did at time t0with data set D0 and theory set T0: with data set D1 := D0 N and theory set T1

22. TU & MA (6) D1 := D0 N is a finite set of data given at time t1 T1 := {T1(j), j  J, T1(j) is relevant for and consistent with D1} where J is some index set Obviously, T*  T1 Partition of T1: T1AT := {T1(j), j  Y, T1(j) is (approximately) true} T1RF := {T1(j), j  Z, T1(j) is radically false} where Y and Z are index sets with Y  Z = J TU: μ(T1AT) << μ(T1RF)

23. TU & MA (7) On this basis, I can formulate an argument analogous to argument 1: Argument 3  T*  T0 such that T* makes the novel prediction N Therefore, T* is relevant for and consistent with the data D1 = D0 N, i.e., T*  T1 T1 = T1AT T1RF and T1AT T1RF = Ø μ(T1AT) << μ(T1RF) Therefore, for any T  T1, it is very probable that T  T1RF. As T*  T1, it is very probable that T*  T1RF

24. TU & MA (8) The conclusion of argument 2 was: it is very probable (or even certain) that T* T0AT The conclusion of argument 3 is: it is very probable that T* T1RF Note that T1RF T0RF (every radically false theory that is consistent with D1 = D0 N is also consistent with D0) Together with T0AT T0RF = Ø, it follows that T0ATT1RF = Ø Thus, arguments 2 and 3 put T* with high probability into two disjoint sets which is inconsistent

25. TU & MA (9) As both arguments are formally valid, at least one of the premises of at least one argument must be false Let us look at these premises

26. TU & MA (10) Premises of Argument 2 T0 = T0AT T0RF and T0AT T0RF = Ø  T*  T0 such that T* makes the novel prediction N For any T  T0RF, it is very improbable (or even impossible) to make prediction N Premises of Argument 3  T*  T0 such that T* makes the novel prediction N Therefore, T* is relevant for and consistent with the data D1 = D0 N, i.e., T*  T1 T1 = T1AT T1RF and T1AT T1RF = Ø μ(T1AT) << μ(T1RF)

27. TU & MA (11) Thus, the core assumption of the miracle argument: For any T  T0RF, it is very improbable (or even impossible) to make prediction N is inconsistent with transient underdetermination, i.e., is false, given TU In other words: TU kills MA Question: How come that the Miracle Argument appears to be so plausible?

28. Presuppositions of MA Remember the crucial assumption of MA: For any T  T0RF, it is very improbable (or even impossible) to make prediction N In Putnam’s words: “The positive argument for realism is that it is the only philosophy that doesn’t make the success of science a miracle” There are two (hidden) presuppositions in these statements: • There is a uniform answer, i.e., an answer that is not specific of T, to the question why T is predictively successful • There are only two alternative answers of the required kind, namely realism and antirealism

29. Presuppositions of MA (2) Both presuppositions are extremely problematic • Why a theory is predictively successful may have many different reasons: sheer luck, the novel predictions only appear to be novel, similarity to more successful theories (not yet known), approximate truth, etc. • Even among the uniform answers, there are other alternatives, i.e., empirically adequate theories Thus, even without TU, MA is highly problematic

30. Conclusion • In general, the miracle argument is a highly problematic argument • Given transient underdetermination in the form discussed, the miracle argument is definitively invalid