Transient UnterdeterminationTransient Unterdeterminationand the Miracle Argumentand the Miracle Argument

Paul Hoyningen-HuenePaul Hoyningen-Huene

Leibniz Universität HannoverLeibniz Universität Hannover

Center for Philosophy and Ethics of Science Center for Philosophy and Ethics of Science (ZEWW)(ZEWW)

The subject of the talkThe subject of the talk

TU TU ¬ MA¬ MA

TU = transient underdeterminationTU = transient underdetermination

MA = miracle argumentMA = miracle argument

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OutlineOutline

1. Notions of underdetermination1. Notions of underdeterminationa)a) Radical underdetermination (RU)Radical underdetermination (RU)

b)b) Transient underdetermination (TU)Transient underdetermination (TU)

2. The miracle argument2. The miracle argument

3. The miracle argument in the light of transient 3. The miracle argument in the light of transient underdeterminationunderdetermination

4. Presuppositions of the miracle argument4. Presuppositions of the miracle argument

5. Conclusion5. Conclusion

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Radical underdetermination (RU)Radical underdetermination (RU)

““Radical” or “strong” or “Quinean” underdetermination (RU):Radical” or “strong” or “Quinean” underdetermination (RU):

For any theory T, there are always empirically equivalent For any theory T, there are always empirically equivalent theories that are not compatible with Ttheories that are not compatible with T

Formally:Formally:

Let DLet DTT be the set of all (possible) data compatible with a given be the set of all (possible) data compatible with a given

theory Ttheory T

Definition: RU holds iffDefinition: RU holds iff

T T T T [T [T is compatible with D is compatible with DTT (T (T T T)])]

RU seems to kill scientific realism because there is no data on RU seems to kill scientific realism because there is no data on the basis of which we can decide between T and Tthe basis of which we can decide between T and T

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Transient Underdetermination (TU)Transient Underdetermination (TU)

““Transient” or “weak” underdetermination (TU)Transient” or “weak” underdetermination (TU)

Presuppositions:Presuppositions:

Let DLet D00 be a finite set of data that is given at time be a finite set of data that is given at time

tt00

Let TLet T00 be the set of theories such that be the set of theories such that

TT0 0 := {T:= {T00(i)(i), i , i I, T I, T00

(i)(i) is relevant for and is relevant for and

consistent with Dconsistent with D00} }

where I is some index set; Twhere I is some index set; T00 ≠ Ø ≠ Ø5

Definition of TU: 1Definition of TU: 1st st attemptattempt

TU holds iffTU holds iff

T T (T (T T T00) ) T T (T (T T T00 (T(T T))] T))]

““(T(T T)” means that T T)” means that T and T are not compatible and T are not compatible

Note that there are many possible sources for the Note that there are many possible sources for the incompatibility of theories, including incommensurability!incompatibility of theories, including incommensurability!

This is too weak as a definition of TU: the existence of two This is too weak as a definition of TU: the existence of two minimally differing theories consistent with the data minimally differing theories consistent with the data fulfills the conditionfulfills the condition

It is only a necessary condition for TUIt is only a necessary condition for TU

We need the possibility of radically false theories that are We need the possibility of radically false theories that are compatible with the available datacompatible with the available data

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TU: PresuppositionsTU: Presuppositions

Partition of TPartition of T00 into the two subsets: (approximately) true theories into the two subsets: (approximately) true theories

and radically false theories (not even approximately true)and radically false theories (not even approximately true)

TT00ATAT := {T := {T00

(i)(i), i , i V, T V, T00(i)(i) is true or approximately true} is true or approximately true}

TT00RFRF := {T := {T00

(i)(i), i , i W, T W, T00(i)(i) is radically false} is radically false}

where V and W are the respective index sets with where V and W are the respective index sets with

V V W = I (which implies T W = I (which implies T00 = T = T00ATAT T T00

RFRF))

Assume TAssume T00ATAT T T00

RFRF = Ø = Ø

Intuitively, radically false theories operate with radically false Intuitively, radically false theories operate with radically false basic assumptions in spite of their agreement with the available basic assumptions in spite of their agreement with the available data (e.g., at some historical time, phlogiston theory or classical data (e.g., at some historical time, phlogiston theory or classical mechanics)mechanics)

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Definition of TU: 2Definition of TU: 2ndnd attempt attempt

TU holds iff TTU holds iff T00RFRF ≠ Ø ≠ Ø

For the purposes of my argument, this is still too For the purposes of my argument, this is still too weak: there must be “quite a few” radically weak: there must be “quite a few” radically false theories in Tfalse theories in T00

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Definition of TU: 3Definition of TU: 3rdrd attempt attempt

Intuitive idea of TU:Intuitive idea of TU:

In TIn T00, there are many more approximately true theories than true , there are many more approximately true theories than true

theories, and many more radically false theories than theories, and many more radically false theories than approximately true theories (Stanford: unconceived alternatives)approximately true theories (Stanford: unconceived alternatives)

In order to formalize this idea, I need the concept of a measure on the In order to formalize this idea, I need the concept of a measure on the space of theoriesspace of theories

A measure is a generalization of the concept of volume for more A measure is a generalization of the concept of volume for more general “spaces”general “spaces”

The measure says how big a subset of the space isThe measure says how big a subset of the space is

Simplistic example for a theory space and a measure on it: Simplistic example for a theory space and a measure on it:

Space of theories: {TSpace of theories: {Tkk 0 0 ≤ k < ∞, ≤ k < ∞, TTkk: F(x) = k}: F(x) = k}

Possible measure μ(Possible measure μ({T{Tkk a a ≤ k ≤ b, ≤ k ≤ b, TTkk: F(x) = k: F(x) = k}) := b - a}) := b - a

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Simplistic exampleSimplistic example

F(x)F(x)

bb

F(x)=kF(x)=k

b-ab-a

aa

xx

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Definition of TU: 3Definition of TU: 3rdrd attempt (2) attempt (2)

Let Let μ be a measure on the set of theories Tμ be a measure on the set of theories T00

Definition of TU: Definition of TU:

TU holds iff μ(TTU holds iff μ(T00ATAT) << μ(T) << μ(T00

RFRF))

In what follows, I will In what follows, I will presupposepresuppose transient transient underdetermination in this formunderdetermination in this form

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TU: Simplistic example (1)TU: Simplistic example (1)with arbitrary numberswith arbitrary numbers

F(x)F(x)

1.51.5

F(x)=kF(x)=k

1.01.0

true theory F(x)=1true theory F(x)=1

0.50.5

domain of approximatively true theoriesdomain of approximatively true theories

xx

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TU: Simplistic example (2)TU: Simplistic example (2)with arbitrary numberswith arbitrary numbers

TT00:={T:={Tkk 0.5 0.5 ≤ k ≤ 1.5, ≤ k ≤ 1.5, TTkk: F(x) = k}: F(x) = k}

TT00ATAT:={T:={Tkk 0.999 0.999 ≤ k ≤ 1.001, ≤ k ≤ 1.001, TTkk: F(x) = k}: F(x) = k}

TT00RFRF:={T:={Tkk 0.5 0.5 ≤ k < 0.999 ≤ k < 0.999 1.001 < k ≤ 1.5, 1.001 < k ≤ 1.5,

TTkk: F(x)= k}: F(x)= k}

μ(Tμ(T00ATAT) = 0.002; μ(T) = 0.002; μ(T00

RFRF) = 0.998) = 0.998

Indeed, Indeed, μ(Tμ(T00ATAT) ) = 0.002 = 0.002 << μ(T<< μ(T00

RFRF) ) = 0.998= 0.998

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The miracle argument (MA)The miracle argument (MA)

There are several forms of the miracle argumentThere are several forms of the miracle argument

I will discuss the following form:I will discuss the following form:

1.1. Scientific realism is the best explanation for Scientific realism is the best explanation for novel predictive successnovel predictive success of theories; other of theories; other philosophical positions make it a miraclephilosophical positions make it a miracle

2.2. Therefore, it is reasonable to accept scientific Therefore, it is reasonable to accept scientific realismrealism

Let us articulate this argument more explicitlyLet us articulate this argument more explicitly

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The miracle argument (2)The miracle argument (2)

Let DLet D00 be a finite set of data that is given at time t be a finite set of data that is given at time t00

Let TLet T0 0 := {T:= {T00(i)(i), i , i I, T I, T00

(i)(i) is relevant for and consistent is relevant for and consistent

with Dwith D00}, I is some index set}, I is some index set

Let TLet T00ATAT and T and T00

RFRF be the partition of T be the partition of T00 into true or into true or

approximately true and radically false theoriesapproximately true and radically false theories

Let N be some novel data (relative to DLet N be some novel data (relative to D00) that is ) that is

discovered at time tdiscovered at time t11 > t > t00

Let there be a theory T* Let there be a theory T* T T00 capable of predicting the capable of predicting the

novel data N already at tnovel data N already at t00

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The miracle argument (3)The miracle argument (3)

Where does T* belong, to TWhere does T* belong, to T00ATAT or to T or to T00

RFRF??

If T* belongs to TIf T* belongs to T00ATAT, its novel predictive success is not , its novel predictive success is not

surprising because it gets something fundamental surprising because it gets something fundamental about nature (approximately) rightabout nature (approximately) right

If T* belongs to TIf T* belongs to T00RFRF, its novel predictive success would , its novel predictive success would

be surprising because T* lacks all resources for be surprising because T* lacks all resources for successful novel predictions; it would be a miraclesuccessful novel predictions; it would be a miracle

Therefore, it is very probable that T* belongs to TTherefore, it is very probable that T* belongs to T00ATAT – –

realism explains the novel predictive success of realism explains the novel predictive success of sciencescience

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Transient underdetermination and Transient underdetermination and the miracle argumentthe miracle argument

First, note the following connection between a First, note the following connection between a measure and the prior probability:measure and the prior probability:

What is the prior probability to find an element s of What is the prior probability to find an element s of some set S in a subset A of S?some set S in a subset A of S?

It is proportional to the “size” of A, i.e. proportional It is proportional to the “size” of A, i.e. proportional to to μ (A)μ (A)

[technically: p(s[technically: p(sAAssS) = μ(A)/μ(S)]S) = μ(A)/μ(S)]

Common sense: Is the probability of winning the Common sense: Is the probability of winning the lottery small or large?lottery small or large?

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TU & MA (2)TU & MA (2)

Due to this connection, TU supports antirealism:Due to this connection, TU supports antirealism:

Argument 1

TT00 = T = T00ATAT T T00

RF RF and Tand T00ATAT T T00

RFRF = Ø = Ø

TU: μ(TTU: μ(T00ATAT) << μ(T) << μ(T00

RFRF))

Therefore for any T Therefore for any T T T00, it is very probable that , it is very probable that

T T T T00RF RF

In other words: due to TU, any theory fitting some data is In other words: due to TU, any theory fitting some data is probably radically false, i.e., TU supports anti-realismprobably radically false, i.e., TU supports anti-realism

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TU & MA (3)TU & MA (3)But here comes the miracle argument:But here comes the miracle argument:

Argument 2Argument 2

TT00 = T = T00ATAT T T00

RFRF and T and T00ATAT T T00

RFRF = Ø = Ø

T* T* T T00 such that T* makes the novel prediction N such that T* makes the novel prediction N

For any T For any T T T00RFRF, it is very improbable (or even impossible) , it is very improbable (or even impossible)

to make prediction Nto make prediction N

Therefore, it is very probable (or even certain) that Therefore, it is very probable (or even certain) that

T* T* T T00ATAT

In other words: novel predictive success supports realismIn other words: novel predictive success supports realism

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TU & MA (4)TU & MA (4)

Note the tension between the conclusions of arguments 1 and 2:Note the tension between the conclusions of arguments 1 and 2:

Conclusion 1: Therefore for any T Conclusion 1: Therefore for any T T T00, it is very probable , it is very probable

that T that T T T00RFRF

Conclusion 2: Therefore, it is very probable (or even certain) Conclusion 2: Therefore, it is very probable (or even certain) that that T* T* T T00

ATAT

Argument 1 is overruled by argument 2 because the latter’s Argument 1 is overruled by argument 2 because the latter’s conclusion about T* states a conclusion about T* states a posteriorposterior probability based on probability based on additional informationadditional information

[technically: Hempel’s requirement of maximal specificity for [technically: Hempel’s requirement of maximal specificity for statistical explanations]statistical explanations]

In other words: with the help of MA, realism beats antirealism that In other words: with the help of MA, realism beats antirealism that relies on TU!relies on TU!

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TU & MA (5)TU & MA (5)

But TU strikes back: But TU strikes back:

Apply TU at t = tApply TU at t = t11 again, namely to the new again, namely to the new

situation with the new data set Dsituation with the new data set D11 := D := D00 N N

At time tAt time t11, I will do exactly the same as what I , I will do exactly the same as what I

did at time tdid at time t00 with data set D with data set D00 and theory set and theory set

TT00: :

with data set Dwith data set D11 := D := D00 N and theory set T N and theory set T11

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TU & MA (6)TU & MA (6)

DD11 := D := D00 N is a finite set of data given at time t N is a finite set of data given at time t11

TT1 1 := {T:= {T11(j)(j), j , j J, T J, T11

(j)(j) is relevant for and consistent with D is relevant for and consistent with D11} }

where J is some index set where J is some index set

Obviously, T* Obviously, T* T T11

Partition of TPartition of T11::

TT11ATAT := {T := {T11

(j)(j), j , j Y, T Y, T11(j)(j) is (approximately) true} is (approximately) true}

TT11RFRF := {T := {T11

(j)(j), j , j Z, T Z, T11(j)(j) is radically false} is radically false}

where Y and Z are index sets with Y where Y and Z are index sets with Y Z = J Z = J

TU: μ(TTU: μ(T11ATAT) << μ(T) << μ(T11

RFRF))

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TU & MA (7)TU & MA (7)

On this basis, I can formulate an argument analogous to On this basis, I can formulate an argument analogous to argument 1:argument 1:

Argument 3Argument 3

T* T* T T00 such that T* makes the novel prediction N such that T* makes the novel prediction N

Therefore, T* is relevant for and consistent with the data DTherefore, T* is relevant for and consistent with the data D11 = =

DD00 N, i.e., T* N, i.e., T* T T11

TT11 = T = T11ATAT T T11

RFRF and T and T11ATAT T T11

RFRF = Ø = Ø

μ(Tμ(T11ATAT) << μ(T) << μ(T11

RFRF))

Therefore, for any T Therefore, for any T T T11, it is very probable that T , it is very probable that T T T11RFRF. .

As T* As T* T T11, it is very probable that T* , it is very probable that T* T T11RFRF

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TU & MA (8)TU & MA (8)

The conclusion of argument 2 was:The conclusion of argument 2 was:

it is very probable (or even certain) that T* it is very probable (or even certain) that T* TT00ATAT

The conclusion of argument 3 is:The conclusion of argument 3 is:

it is very probable that T* it is very probable that T* TT11RFRF

Note that TNote that T11RFRF T T00

RFRF (every radically false theory that is (every radically false theory that is

consistent with Dconsistent with D11 = D = D00 N is also consistent with D N is also consistent with D00))

Together with TTogether with T00ATAT T T00

RFRF = Ø, it follows that = Ø, it follows that

TT00ATAT TT11

RFRF = Ø = Ø

Thus, arguments 2 and 3 put T* with high probability into Thus, arguments 2 and 3 put T* with high probability into two disjoint sets which is inconsistenttwo disjoint sets which is inconsistent

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TU & MA (9)TU & MA (9)

As both arguments are formally valid, at least As both arguments are formally valid, at least one of the premises of at least one argument one of the premises of at least one argument must be falsemust be false

Let us look at these premisesLet us look at these premises

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TU & MA (10)TU & MA (10)

Premises of Premises of Argument 2Argument 2

TT00 = T = T00ATAT T T00

RFRF and T and T00ATAT T T00

RFRF = Ø = Ø

T* T* T T00 such that T* makes the novel prediction N such that T* makes the novel prediction N

For any T For any T T T00RFRF, it is very improbable (or even impossible) , it is very improbable (or even impossible) to to

make prediction Nmake prediction N

Premises of Premises of Argument 3Argument 3

T* T* T T00 such that T* makes the novel prediction N such that T* makes the novel prediction N

Therefore, T* is relevant for and consistent with the data DTherefore, T* is relevant for and consistent with the data D11 = = DD00

N, i.e., T* N, i.e., T* T T11

TT11 = T = T11ATAT T T11

RFRF and T and T11ATAT T T11

RFRF = Ø = Ø

μ(Tμ(T11ATAT) << μ(T) << μ(T11

RFRF))26

TU & MA (11)TU & MA (11)

Thus, the core assumption of the miracle argument:Thus, the core assumption of the miracle argument:

For any T For any T T T00RFRF, it is very improbable (or even , it is very improbable (or even

impossible) to make prediction Nimpossible) to make prediction N

is inconsistent with transient underdetermination, is inconsistent with transient underdetermination, i.e., is false, given TUi.e., is false, given TU

In other words: TU kills MAIn other words: TU kills MA

Question: How come that the Miracle Argument Question: How come that the Miracle Argument appears to be so plausible?appears to be so plausible?

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Presuppositions of MAPresuppositions of MA

Remember the crucial assumption of MA:Remember the crucial assumption of MA:

For any T For any T T T00RFRF, it is very improbable (or even impossible) to , it is very improbable (or even impossible) to

make prediction Nmake prediction N

In Putnam’s words: “The positive argument for realism is that In Putnam’s words: “The positive argument for realism is that it is the only philosophy that doesn’t make the success of it is the only philosophy that doesn’t make the success of science a miracle”science a miracle”

There are two (hidden) presuppositions in these statements:There are two (hidden) presuppositions in these statements:

1.1. There is a There is a uniformuniform answer, i.e., an answer that is not specific answer, i.e., an answer that is not specific of T, to the question why T is predictively successfulof T, to the question why T is predictively successful

2.2. There are There are only two only two alternative answers of the required kind, alternative answers of the required kind, namely realism and antirealismnamely realism and antirealism

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Presuppositions of MA (2)Presuppositions of MA (2)

Both presuppositions are extremely problematicBoth presuppositions are extremely problematic

1.1. Why a theory is predictively successful may have Why a theory is predictively successful may have many different reasons: sheer luck, the novel many different reasons: sheer luck, the novel predictions only appear to be novel, similarity to predictions only appear to be novel, similarity to more successful theories (not yet known), more successful theories (not yet known), approximate truth, etc.approximate truth, etc.

2.2. Even among the uniform answers, there are other Even among the uniform answers, there are other alternatives, i.e., empirically adequate theoriesalternatives, i.e., empirically adequate theories

Thus, even without TU, MA is highly problematicThus, even without TU, MA is highly problematic

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ConclusionConclusion

1.1. In general, the miracle argument is a In general, the miracle argument is a highly problematic argumenthighly problematic argument

2.2. Given transient underdetermination in the Given transient underdetermination in the form discussed, the miracle argument is form discussed, the miracle argument is definitively invaliddefinitively invalid

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