1. Transient Unterdetermination and the Miracle Argument Paul Hoyningen-Huene Leibniz Universität Hannover Center for Philosophy and Ethics of Science (ZEWW)

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

3. Outline 1. Notions of underdetermination a)Radical underdetermination (RU) b)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 3

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 D T be the set of all (possible) data compatible with a given theory T Definition: RU holds iff  T  T [T is compatible with D T   (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 4

5. Transient Underdetermination (TU) “Transient” or “weak” underdetermination (TU) Presuppositions: Let D 0 be a finite set of data that is given at time t 0 Let T 0 be the set of theories such that T 0 := {T 0 (i), i  I, T 0 (i) is relevant for and consistent with D 0 } where I is some index set; T 0 ≠ Ø 5

6. Definition of TU: 1 st attempt TU holds iff  T  (T  T 0 )   T (T  T 0   (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 6

7. TU: Presuppositions Partition of T 0 into the two subsets: (approximately) true theories and radically false theories (not even approximately true) T 0 AT := {T 0 (i), i  V, T 0 (i) is true or approximately true} T 0 RF := {T 0 (i), i  W, T 0 (i) is radically false} where V and W are the respective index sets with V  W = I (which implies T 0 = T 0 AT  T 0 RF ) Assume T 0 AT  T 0 RF = 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) 7

8. Definition of TU: 2 nd attempt TU holds iff T 0 RF ≠ Ø For the purposes of my argument, this is still too weak: there must be “quite a few” radically false theories in T 0 8

9. Definition of TU: 3 rd attempt Intuitive idea of TU: In T 0, 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: {T k  0 ≤ k < ∞, T k : F(x) = k} Possible measure μ({T k  a ≤ k ≤ b, T k : F(x) = k}) := b - a 9

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

11. Definition of TU: 3 rd attempt (2) Let μ be a measure on the set of theories T 0 Definition of TU: TU holds iff μ(T 0 AT ) << μ(T 0 RF ) In what follows, I will presuppose transient underdetermination in this form 11

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

13. TU: Simplistic example (2) with arbitrary numbers T 0 :={T k  0.5 ≤ k ≤ 1.5, T k : F(x) = k} T 0 AT :={T k  0.999 ≤ k ≤ 1.001, T k : F(x) = k} T 0 RF :={T k  0.5 ≤ k < 0.999  1.001 < k ≤ 1.5, T k : F(x)= k} μ(T 0 AT ) = 0.002; μ(T 0 RF ) = 0.998 Indeed, μ(T 0 AT ) = 0.002 << μ(T 0 RF ) = 0.998 13

14. The miracle argument (MA) There are several forms of the miracle argument I will discuss the following form: 1. Scientific realism is the best explanation for novel predictive success of theories; other philosophical positions make it a miracle 2. Therefore, it is reasonable to accept scientific realism Let us articulate this argument more explicitly 14

15. The miracle argument (2) Let D 0 be a finite set of data that is given at time t 0 Let T 0 := {T 0 (i), i  I, T 0 (i) is relevant for and consistent with D 0 }, I is some index set Let T 0 AT and T 0 RF be the partition of T 0 into true or approximately true and radically false theories Let N be some novel data (relative to D 0 ) that is discovered at time t 1 > t 0 Let there be a theory T*  T 0 capable of predicting the novel data N already at t 0 15

16. The miracle argument (3) Where does T* belong, to T 0 AT or to T 0 RF ? If T* belongs to T 0 AT, its novel predictive success is not surprising because it gets something fundamental about nature (approximately) right If T* belongs to T 0 RF, 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 T 0 AT – realism explains the novel predictive success of science 16

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? 17

18. TU & MA (2) Due to this connection, TU supports antirealism: Argument 1 T 0 = T 0 AT  T 0 RF and T 0 AT  T 0 RF = Ø TU: μ(T 0 AT ) << μ(T 0 RF ) Therefore for any T  T 0, it is very probable that T  T 0 RF In other words: due to TU, any theory fitting some data is probably radically false, i.e., TU supports anti- realism 18

19. TU & MA (3) But here comes the miracle argument: Argument 2 T 0 = T 0 AT  T 0 RF and T 0 AT  T 0 RF = Ø  T*  T 0 such that T* makes the novel prediction N For any T  T 0 RF, it is very improbable (or even impossible) to make prediction N Therefore, it is very probable (or even certain) that T*  T 0 AT In other words: novel predictive success supports realism 19

20. TU & MA (4) Note the tension between the conclusions of arguments 1 and 2: Conclusion 1: Therefore for any T  T 0, it is very probable that T  T 0 RF Conclusion 2: Therefore, it is very probable (or even certain) that T*  T 0 AT 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! 20

21. TU & MA (5) But TU strikes back: Apply TU at t = t 1 again, namely to the new situation with the new data set D 1 := D 0  N At time t 1, I will do exactly the same as what I did at time t 0 with data set D 0 and theory set T 0 : with data set D 1 := D 0  N and theory set T 1 21

22. TU & MA (6) D 1 := D 0  N is a finite set of data given at time t 1 T 1 := {T 1 (j), j  J, T 1 (j) is relevant for and consistent with D 1 } where J is some index set Obviously, T*  T 1 Partition of T 1 : T 1 AT := {T 1 (j), j  Y, T 1 (j) is (approximately) true} T 1 RF := {T 1 (j), j  Z, T 1 (j) is radically false} where Y and Z are index sets with Y  Z = J TU: μ(T 1 AT ) << μ(T 1 RF ) 22

23. TU & MA (7) On this basis, I can formulate an argument analogous to argument 1: Argument 3  T*  T 0 such that T* makes the novel prediction N Therefore, T* is relevant for and consistent with the data D 1 = D 0  N, i.e., T*  T 1 T 1 = T 1 AT  T 1 RF and T 1 AT  T 1 RF = Ø μ(T 1 AT ) << μ(T 1 RF ) Therefore, for any T  T 1, it is very probable that T  T 1 RF. As T*  T 1, it is very probable that T*  T 1 RF 23

24. TU & MA (8) The conclusion of argument 2 was: it is very probable (or even certain) that T*  T 0 AT it is very probable (or even certain) that T*  T 0 AT The conclusion of argument 3 is: it is very probable that T*  T 1 RF Note that T 1 RF  T 0 RF (every radically false theory that is consistent with D 1 = D 0  N is also consistent with D 0 ) Together with T 0 AT  T 0 RF = Ø, it follows that T 0 AT  T 1 RF = Ø Thus, arguments 2 and 3 put T* with high probability into two disjoint sets which is inconsistent 24

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 25

26. TU & MA (10) Premises of Argument 2 T 0 = T 0 AT  T 0 RF and T 0 AT  T 0 RF = Ø  T*  T 0 such that T* makes the novel prediction N For any T  T 0 RF, it is very improbable (or even impossible) to make prediction N Premises of Argument 3  T*  T 0 such that T* makes the novel prediction N Therefore, T* is relevant for and consistent with the data D 1 = D 0  N, i.e., T*  T 1 T 1 = T 1 AT  T 1 RF and T 1 AT  T 1 RF = Ø μ(T 1 AT ) << μ(T 1 RF ) 26

27. TU & MA (11) Thus, the core assumption of the miracle argument: For any T  T 0 RF, 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? 27

28. Presuppositions of MA Remember the crucial assumption of MA: For any T  T 0 RF, 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: 1. There is a uniform answer, i.e., an answer that is not specific of T, to the question why T is predictively successful 2. There are only two alternative answers of the required kind, namely realism and antirealism 28

29. Presuppositions of MA (2) Both presuppositions are extremely problematic 1. 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. 2. Even among the uniform answers, there are other alternatives, i.e., empirically adequate theories Thus, even without TU, MA is highly problematic 29

30. Conclusion 1. In general, the miracle argument is a highly problematic argument 2. Given transient underdetermination in the form discussed, the miracle argument is definitively invalid 30