The Miracle Argument and The Miracle Argument and Transient UnterdeterminationTransient Unterdetermination

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. Transient underdetermination1. Transient underdetermination

2. The miracle argument2. The miracle argument

3. The miracle argument in the light of 3. The miracle argument in the light of transient underdeterminationtransient underdetermination

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

5. Conclusion5. Conclusion

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Transient Underdetermination (TU)Transient Underdetermination (TU)

Idea: A given (finite) set of data does not Idea: A given (finite) set of data does not unambiguously determine a single theoryunambiguously determine a single theory

Notation:Notation:

Let DLet D11 be a finite set of data be a finite set of data

Let TLet T11 be the set of theories such that be the set of theories such that

TT11 := {T, T is relevant for and consistent with D := {T, T is relevant for and consistent with D11} }

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

TU holds iffTU holds iff

T T (T (T T T11) ) T T (T (T T T11 (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 TUWe 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: NotationTU: Notation

Partition of TPartition of T11 into the two subsets: (approximately) true into the two subsets: (approximately) true

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

TT11ATAT := {T := {T T T11, T is true or approximately true}, T is true or approximately true}

TT11RFRF := {T := {T T T1 1 is radically false}is radically false}

Of course, TOf course, T11ATAT T T11

RFRF = T = T11

Assume the idealization TAssume the idealization T11ATAT T T11

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

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

TU holds iff TTU holds iff T11RFRF ≠ 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

This is supported by the intuitive idea of TU:This is supported by the intuitive idea of TU:

In TIn T11, there are many more approximately true , there are many more approximately true

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

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

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

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

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

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

Possible measure:Possible measure:

μ(μ({T{Tkk T Tkk: F(x) = k with a : F(x) = k with a ≤ k ≤ b,}) := b - a≤ k ≤ b,}) := 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 (and last) attempt (and last) attempt

Let Let μ be a measure on the set of theories Tμ be a measure on the set of theories T11

Definition of TU: Definition of TU:

TU holds iff μ(TTU holds iff μ(T11ATAT) << μ(T) << μ(T11

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

Idea: If a theory correctly predicts something it has not been devised for, Idea: If a theory correctly predicts something it has not been devised for, then this fact cannot be purely accidental: the theory must be at least then this fact cannot be purely accidental: the theory must be at least approximately trueapproximately true

Define: “use-novel predictive success of a theory”: the successful Define: “use-novel predictive success of a theory”: the successful prediction of data by a theory which were not used in its constructionprediction of data by a theory which were not used in its construction

Example: quantitative prediction of light bending in GRTExample: quantitative prediction of light bending in GRT

““Scientific realism”: well-established physical theories are approx. trueScientific realism”: well-established physical theories are approx. true

Miracle argument for scientific realism:Miracle argument for scientific realism:

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

2.2. Use-novel predictive success existsUse-novel predictive success exists

3.3. Therefore, it is reasonable to accept scientific realismTherefore, it is reasonable to accept scientific realism

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)

Recall our notation:Recall our notation:

Let DLet D11 be a finite set of data be a finite set of data

Let TLet T11 be the set of theories such that be the set of theories such that

TT11 := {T, T is relevant for and consistent with D := {T, T is relevant for and consistent with D11} }

TT11ATAT := {T := {T T T11, T is true or approximately true}, T is true or approximately true}

TT11RFRF := {T := {T T T1 1 , T is radically false}, T is radically false}

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

The miracle argument is especially impressive if one The miracle argument is especially impressive if one assumes transient underdetermination:assumes transient underdetermination:

μ(Tμ(T11ATAT) << μ(T) << μ(T11

RFRF))

This means: for any T This means: for any T T T11, it is very probable that , it is very probable that

T T T T11RF RF

This means: due to TU, any theory fitting some data is This means: due to TU, any theory fitting some data is probably radically falseprobably radically false

In other words: TU supports anti-realismIn other words: TU supports anti-realism

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The miracle argument (4)The miracle argument (4)

Here comes the miracle argument:Here comes the miracle argument:

Let N be some use-novel data (relative to DLet N be some use-novel data (relative to D11) )

Let there be a theory T* Let there be a theory T* T T11 capable of capable of

predicting the novel data Npredicting the novel data N

Does T* belong to TDoes T* belong to T11ATAT or to T or to T11

RFRF??

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The miracle argument (5)The miracle argument (5)

If T* If T* TT11ATAT, 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* If T* TT11RFRF, its novel predictive success would be , its novel predictive success would be

surprising because T* lacks all resources for 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* Therefore, it is very probable that T* TT11ATAT

In other words: use-novel predictive success supports In other words: use-novel predictive success supports scientific realism (for the respective theories)scientific realism (for the respective theories)

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TU & MATU & MA

But TU strikes back: But TU strikes back: Apply TU again Apply TU again to the situation to the situation with the new data set Dwith the new data set D22 := D := D11 N N

We haveWe have

DD22 := D := D11 N is a finite set of data N is a finite set of data

TT22 := {T, T is relevant for and consistent with D := {T, T is relevant for and consistent with D22} }

Clearly, T* Clearly, T* T T22

Define in the same way as earlierDefine in the same way as earlier

TT22ATAT := {T := {T T T22, T is true or approximately true}, T is true or approximately true}

TT22RFRF := {T := {T T T2 2 , T is radically false}, T is radically false}

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

TU states: μ(TTU states: μ(T22ATAT) << μ(T) << μ(T22

RFRF))

As T* As T* T T22, we get, we get

T* is very probably radically false, T* is very probably radically false, contrary to what the miracle contrary to what the miracle argument claimsargument claims

STOPSTOP, shouts the scientific realist, you cheated!, shouts the scientific realist, you cheated!

T* is not just an ordinary member of TT* is not just an ordinary member of T2 2 to which μ(Tμ(T22ATAT) << ) <<

μ(Tμ(T22RFRF) applies) applies

T* is special in that it was able to predict N on the basis of DD1 1

whereas the ordinary member of T22 was fitted to DD22 := D := D11 N N

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TU & MA (3)TU & MA (3)

However, this difference of T* from the other members of THowever, this difference of T* from the other members of T22

is purely is purely pragmatic pragmatic and not intrinsicand not intrinsic

Every member of TEvery member of T22, once you happen to hit upon it by , once you happen to hit upon it by

looking for a theory that fits data Dlooking for a theory that fits data D11, can be used to predict , can be used to predict

N!N!

Thus, T* is an ordinary member of TThus, T* is an ordinary member of T22

Because of TU: μ(TBecause of TU: μ(T22ATAT) << μ(T) << μ(T22

RFRF), we really get:), we really get:

T* is very probably radically false, T* is very probably radically false, contrary to what the contrary to what the miracle argument claimsmiracle argument claims

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

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

For any radically false theory fitting some data, it is very For any radically false theory fitting some data, it is very improbable (or even impossible) to make a use-novel improbable (or even impossible) to make a use-novel predictionprediction

is false, given TUis false, given TU

Reason: Given TU, there are far more Reason: Given TU, there are far more radically false radically false T*’s T*’s TT1 1 making a use-novel prediction N than approximately making a use-novel prediction N than approximately

true T*’s true T*’s T T11 making the use-novel prediction N making the use-novel prediction N

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

Question: How come that the Miracle Argument appears to Question: How come that the Miracle Argument appears to be so plausible?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 radically false theory fitting some data, it is very For any radically false theory fitting some data, it is very improbable (or even impossible) to make a use-novel predictionimprobable (or even impossible) to make a use-novel prediction

In Putnam’s words: “The positive argument for realism is that it is 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 the only philosophy that doesn’t make the success of science a miracle”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 of answer, i.e., an answer that is not specific of T, to the question why T is successful regarding use-novel T, to the question why T is successful regarding use-novel predictionspredictions

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 problematic1.1. Why a theory is successful regarding use-novel Why a theory is successful regarding use-novel

predictions may have very different reasons: sheer predictions may have very different reasons: sheer luck, mutual cancellation of erroneous luck, mutual cancellation of erroneous assumptions, the novel predictions only appear to assumptions, the novel predictions only appear to be novel, similarity to more successful theories be novel, similarity to more successful theories (not yet known), approximate truth, etc.(not yet known), 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 highly In general, the miracle argument is a highly problematic argumentproblematic argument

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

3.3. The realist and the antirealist have different The realist and the antirealist have different “intuitions” about this question: For a use-novel “intuitions” about this question: For a use-novel successful theory, what is more likely: that it is successful theory, what is more likely: that it is (approximately) true or that its errors cancel each (approximately) true or that its errors cancel each other out in a most advantageous way?other out in a most advantageous way?

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