Transient Unterdetermination and the Miracle Argument

1
Transient Unterdeterminationand 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

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 lt 8, 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) ltlt 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

• T0Tk? 0.5 k 1.5, Tk F(x) k
• T0ATTk? 0.999 k 1.001, Tk F(x) k
• T0RFTk? 0.5 k lt 0.999 ? 1.001 lt k 1.5,
• Tk F(x) k
• µ(T0AT) 0.002 µ(T0RF) 0.998
• Indeed, µ(T0AT) 0.002 ltlt 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 gt 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) ltlt 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
  latters conclusion about T states a posterior
  probability based on additional information
• technically Hempels 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 t0 with 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) ltlt 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) ltlt 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) ltlt 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 Putnams words The positive argument for
  realism is that it is the only philosophy that
  doesnt 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