Church’s Thesis and Hume’s Problem: Justification as Performance

Church’s Thesis Church’s Thesis and Hume’s and Hume’s

Problem:Problem:Justification as Justification as PerformancePerformance

Kevin T. KellyDepartment of Philosophy

Carnegie Mellon [email protected]

Further ReadingFurther Reading(With C. Glymour) “Why You'll Never Know Whether Roger Penrose is a Computer”, Behavioral and Brain Sciences, 13: 1990.•The Logic of Reliable Inquiry, Oxford: Oxford University Press, 1996.•(with O. Schulte) “Church's Thesis and Hume's Problem,” in Logic and Scientific Methods, M. L. Dalla Chiara, et al., eds. Dordrecht: Kluwer, 1997.•(with O. Schulte and C. Juhl) “Learning Theory and the Philosophy of Science”, Philosophy of Science 64: 1997.“The Logic of Success”, British Journal for the Philosophy of Science, 51:2000.“Uncomputability: The Problem of Induction Internalized,” Theoretical Computer Science 317: 2004.Computability and Learnibility, Textbook manuscript, 2005.

Fateful First CutFateful First Cut

Relations of IdeasRelations of Ideas Matters of factMatters of fact

AnalyticAnalytic SyntheticSynthetic

CertainCertain UncertainUncertain

A PrioriA Priori A PosterioriA Posteriori

Philosophy of MathPhilosophy of Math Philosophy of Philosophy of ScienceScience

Proofs and Proofs and algorithmsalgorithms ConfirmationConfirmation

Truth-findingTruth-finding ““Rationality”Rationality”

ComputabilityComputability ProbabilityProbability

Painful ProgressPainful Progress


AnalyticAnalytic SyntheticSynthetic







Slight RearrangementSlight Rearrangement







Philosophy of Science 101Philosophy of Science 101

Formal QuestionsFormal Questions Scientific Scientific QuestionsQuestions

UnverifiableUnverifiable

““The sun will neverThe sun will never

stop rising”stop rising”

VerifiableVerifiable

““The given number The given number

is even”is even”

Wrong Twice!Wrong Twice!



““The given The given computation will computation will

never halt”never halt”








““The next emerald The next emerald

is green”is green”


Classical ExcuseClassical Excuse

Rational Animal

Rational ?

Reason Observation

UnboundedexperienceCompleted analysis

??Always black?


But Maybe…But Maybe…

Rational ?

Reason Observation

Unboundedexperience

Never white?Always black?

Unboundedanalysis

????

Synthetic A PrioriSynthetic A Priori A PosterioriA Posteriori

KantKant

Intuition Observation

Unboundedexperience

3+2 = 5 ?

3 4 5

Never white?Always black?

temporal intuition

23

??

Synthetic A PrioriSynthetic A Priori A PosterioriA Posteriori

Oops!Oops!

Intuition Observation

Unboundedexperience

Never cross? Never white?Always black?

spatial intuition

????

Formal ProblemsFormal Problems Empirical Empirical ProblemsProblems

After TuringAfter Turing

Never halts?

Computer Observation

Unboundedreality

Unboundedcomputation

Input

n

Never white?Always black??? ??

Bad First CutBad First Cut














Good First CutGood First Cut














Computational EpistemologyComputational Epistemology

For formal For formal andand empirical empirical questions:questions:

Justification =Justification = truth-finding truth-finding performanceperformance

Find the truth in theFind the truth in the best possible best possible sensesenseAnd then as And then as efficiently as efficiently as possiblepossible. .

Verification as Verification as “Support”“Support”

1.0

.5Deductive verification

Inductive verification

Paradigms EvolveParadigms Evolve

•Objective support by evidence.

Paradigms Evolve…Paradigms Evolve…


•Coherent personal opinion.•“Rational” update.

Paradigms Evolve…Paradigms Evolve…


•Coherent personal opinion.•“Rational” update.

But Leading Metaphors MatterBut Leading Metaphors Matter

•Myopic •Bedrock = “rationality” intuitions•Success = being “rational”•Truth-finding performance deferred or ignored•Problem complexity ignored•Little resemblance to computability

Verification as Verification as HaltingHalting

Conclusion

Data


Conclusion

Data

Yes!


Conclusion

Data


Conclusion

Data


Conclusion

Data

Bang!


Conclusion

Data

Can’t take it back.

VerifiabilityVerifiability

Conclusion true

Yes!

Conclusion false

Laws are Not VerifiableLaws are Not Verifiable

Always green?


Always green?


Always green?


Always green?


Always green?

You must halt with “yes” if it’s true!


Always green?

You must halt with “yes” if it’s true!


Yes!

Always green?


Always green?


Always green?

A Purely Formal ProblemA Purely Formal Problem

Never shoots?

Analogy?Analogy?

Never shoots?

I ain’t a’ shootin’, yeller belly!

Apparently NotApparently Not

Game’s up, J.R.! Yer program’s showin’!

Never shoots? 666


Now shaddup! I’m calculatin’a-preeorey-like what yer gonna do.

Never shoots? 666

Never shoots?


Never shoots? 2145623455666

?!!!

Never shoots?

But Then Again…But Then Again…

666Never shoots?

I got it! Two kin play that opry oary game!

2145623455

Never shoots?


666Never shoots? 2145623455

I won’t shoot ‘til Isee him shootin this here opry oarysim-yooo-laytion!

2145623455

666

Never shoots?



2145623455

666

I ain’t a’ shootin’.

Never shoots?



666

I still ain’t a’ shootin’.

2145623455

Never shoots?



666

You check that there opry oary all you want…

2145623455

Never shoots?



666

An’ all you’ll see is I ain’t a-shootin’

2145623455

Never shoots?



666

2145623455

Aw, shucks, J.R. I reckon you ain’t never gonna shoot.

Aw, shucks, J.R. I reckonYou ain’t never gonna shoot.

Never shoots?


666Never shoots?

666

Ban

g!

23455

23455

Never shoots?


666Never shoots?

666

23455

23455

Never shoots?


666Never shoots?

666

23455

See ya inthe grand ol’ opry oary, sucker!

23455

Never shoots?


666Never shoots?

23455

666

Ban

g!Take that!

Never shoots?


Never shoots?

234556

66

Never shoots?


Never shoots?

234556

66

ROBOSHERIFF

FooledA priori

Rice-Shapiro TheoremRice-Shapiro Theorem

QDocQ45863

Whatever a computable cognitive scientist could verify about an arbitrary computer’s input-output behavior by formally analyzing its program…

Rice-Shapiro TheoremRice-Shapiro Theorem

QDocQ45863

…could also have been verified by a behaviorist computer who empirically studies only the arbitrary computer’s input-output behavior!

Uncomputable InductionUncomputable Induction

Will see a non-halting machine

Yes!

H N

Verifiable by “ideal agent”.

Uncomputable InductionUncomputable Induction

Will see a non-halting machine

?

Q

H N

Not verifiable by computable agent.

SimilaritySimilarity

Halting ProblemHalting Problem Universal LawUniversal Law

UnverifiableUnverifiable UnverifiableUnverifiable

Demon can fool Demon can fool computablecomputable agent agent

Demon can fool Demon can fool

idealideal agent agent

Answer runs Answer runs beyond beyond formalformal

experienceexperience

Answer runs Answer runs beyond beyond empiricalempirical

experienceexperience

A DifferenceA Difference

Halting ProblemHalting Problem Universal LawUniversal Law

Input endsInput ends Input never endsInput never ends

Can handle some Can handle some examples a priori…examples a priori…

““Internal” problem Internal” problem of induction of induction

eventually catches eventually catches up!up!

Problem of Problem of inductioninduction

Give and TakeGive and Take

Formal inputFormal input Empirical inputEmpirical input

Jaw BreakerJaw Breaker NoodleNoodle

Fits in mouth but Fits in mouth but may never meltmay never melt

May never fit in May never fit in mouthmouth

May never swallowMay never swallow May never swallowMay never swallow. . .

Another Difference?Another Difference?

Formal ScienceFormal Science Empirical ScienceEmpirical Science

IncompletenessIncompleteness Problem of Problem of inductioninduction

Not NecessarilyNot Necessarily

Formal ScienceFormal Science Empirical ScienceEmpirical Science

IncompletenessIncompleteness Problem of Problem of inductioninduction

Add more powerful Add more powerful axiomaxiom Conjecture a theoryConjecture a theory

Who knows if new Who knows if new axiom is consistent axiom is consistent

with old?with old?

Who knows if Who knows if theory is consistent theory is consistent

with data?with data?

Assume it is, keep Assume it is, keep checking, and hopechecking, and hope

Assume it is, keep Assume it is, keep checking, and hopechecking, and hope

ThesisThesisThe problem of induction and the problem of uncomputability are essentially similar.

So the two should be understood similarly.

From the ground upward.

•Unverifiability•“Confirmation”•“Support”•“Coherence”•“Rational” update, etc.

•Uncomputability. •Suck it up•Logic vs. engineering•Try to approximate, etc.

Philosophy: A House DividedPhilosophy: A House Divided

?!

Unified Approach: Gun ControlUnified Approach: Gun Control•Find the truth in the best feasible sense.•So drop the halting requirement if infeasible.

Q

Start with Start with ProblemsProblems, Not , Not MethodsMethods

B

Partition Q over set K of inputs

A C D

Infinite inputFinite input

ConvergenceConvergence

Convergencein limit(in limit)

Convergencewith halting

? ? ? ?A

? ? ? ? ? B C C . . .?A C

No more revisions

revisions

SuccessSuccess

Converge to right answer

BA C D

Don’t converge to wrong answer

Special CasesSpecial Cases

BA BA

BA

Verify A Refute A

Decide A

Special CasesSpecial Cases

= 0 = 1 = 2 . . .

Compute partial

0 1 2 . . .

Theory selection

Dom(

Solutions and OptimalitySolutions and Optimality

•Solution: method that succeeds on each input.•Optimal solution: solves in best possible sense•Solvable problem: has solution.•Problem complexity: best sense of solvability

Halting Bounds RevisionsHalting Bounds Revisions

BA

Verifiability convergence with 1 revision ending with “no”.

Refutability convergence with 1 revision ending with “yes”.

Decidability convergence with 0 revisions.

n-Decision =n rev

Generalization to Generalization to nn RevisionsRevisions

n-Refutation =n+1 rev ending with -A

n-Verification =n+1 rev ending with A

BA

0-ref0-ver

0-dec

Lim-refLim-ver

Empirical ComplexityEmpirical Complexity

Lim-dec

. . .

1-ref1-ver

1-dec

. . .

clopen

or and

Un

derd

ete

rmin

ati

on open closed

0-ref0-ver

0-dec

Lim-refLim-ver

Formal Formal ComplexityComplexity

Lim-dec

. . .

1-ref1-ver

1-dec

. . .

decidable

or and

r.e. Co-r.e.Un

com

pu

tab

ilit

y

0-ref0-ver

0-dec

Lim-refLim-ver

Formal + EmpiricalFormal + Empirical ComplexityComplexity

Lim-dec

. . .

1-ref1-ver

1-dec

. . .

decidable

or and

r.e. Co-r.e.Un

com

pu

tab

ilit

y +

Un

derd

ete

rmin

ati

on

. . .

Ship-shape EpistemologyShip-shape Epistemology

Ideal empirical

Effectiveformal

Effective empirical

. . .

. . .. . .

. . .


Topologicalinvariants

Recursiveinvariants

Recursiveinvariants

. . .

. . .. . .

. . .


LowerBound

Lower Bound

Combination

. . .

. . .. . .

. . .

Messy, Odd DistinctionMessy, Odd Distinction

EmpiricalBorel

Formalarithmetical

MixedeffectiveBorel

. . .

. . .. . .

. . .

Neat, Natural DistinctionsNeat, Natural Distinctions

EmpiricalBorel

Formalarithmetical

MixedeffectiveBorel

. . .

. . .. . .

Example: Kantian AntinomyExample: Kantian Antinomy

Matter

Finitelydivisible

Infinitelydivisible

Upper Complexity BoundUpper Complexity Bound

I say inf whenyou let me cut.

Finitelydivisible

Infinitelydivisible

Lim-ref Lim-ver

. . .

. . .

fin fin

inf fin

Lower Complexity BoundLower Complexity Bound

Finitelydivisible

Infinitelydivisible

Lim-ref-Lim-ver

Lim-ver-Lim-ref

I let you cut when you say fin.

. . .

. . .

inf fin

inf inf

Purely Formal AnaloguePurely Formal Analogue

666 2145623455

Finitedomain

Infinitedomain

Lim-ref-Lim-ver

Lim-ver-Lim-ref

Purely Formal AnaloguePurely Formal Analogue

666 2145623455

I halt on another input each time he says fin in my opry oary simulation of him ganderin’ at my program.

Finitedomain

Infinitedomain

Lim-ref-Lim-ver

Lim-ver-Lim-ref

. . .

AnalogiesAnalogies

Ideal empirical

Effectiveformal

Effective empirical

. . .

. . .. . .

Finite divisibility

Finitedomain

Finitedivisibility

Mixed Example: PenroseMixed Example: Penrose

UncomputableComputable

Human subjectCognitive scientist

.

???

Empirical IronyEmpirical Irony



You can verify human computabilityin the limit…

UCUCCCCCCCC…

Empirical IronyEmpirical Irony



But only if you aren’t computable!

UCUCUCCUCCU…

ComputabilityComputability

Lim-refLim-ver

Lim-dec

Ideal

ComputabilityComputability

Lim-refLim-ver

Lim-dec

Computable

We can verify our computability in the limit… only if we are not computable!

Gold/PutnamGold/PutnamEven assuming computability,you can converge to the true computable behavior only if you are not computable!

0 32 243 . . .

Total computable functions

Uncomputable PredictionsUncomputable Predictions

-TT

(with Oliver Schulte)

•There exists T such that •T makes a unique prediction at each stage;•The predictions are very uncomputable…

Uncomputable PredictionsUncomputable Predictions

-TT

(with Oliver Schulte)

•There exists T such that •T makes a unique prediction at each stage;•The predictions are very uncomputable…•But T is refutable by a computable method!

Moral 1: Rational Moral 1: Rational HobgoblinsHobgoblins

So “a foolish consistency” precludes truth-finding!

Ideal rationality

No consistent

computable

convergent method.

Exists

inconsistent

computable

refuting method.

Exists

consistent

computable

non-convergent

method.

Moral 2: Coup de GraceMoral 2: Coup de Grace•The very idea of insulating deduction from empirical data restricts the truth-finding power of effective science.

TheoremProver

Empirical data n Yes

Q

?

Empirical data

Moral 2: Coup de GraceMoral 2: Coup de Grace•The very idea of insulating deduction from empirical data restricts the truth-finding power of effective science.

TheoremProver

n

n

Yes

Q

•Justification = truth-finding performance.

•Performance depends on problem complexity.

•Formal and empirical complexity are similar.

•Computational epistemology is unified

•Standard epistemology is divided.

•Insistence on division weakens truth-finding performance of effective science.

ConclusionsConclusionsQ

Closing ImageClosing Image

Standard epistemology

Computational epistemology

Q

Q




Q

Q

I’m rational

Q




Q

I’m rational

THE ENDTHE END

actual agent

pure structure

actual world How does inquiry proceed?Have we found a correct answer?Will we?Are we guaranteed to?How might we improve?Is guaranteed success achievable?Is guaranteed success achievable in a completely specified, hypothetical problem?Which possible methods achieve solutions?Which maxims preclude solutions?What is necessary for guaranteed success to be achievable?

actual problem

Pure Form of NaturalismPure Form of Naturalism

Every Problem in Its PlaceEvery Problem in Its Place

lim verlim ref

lim dec

cert vercert ref

cert dec

grad vergrad ref

Infinity (Kant’s Antinomy)Quantized conservationUniformitarian geology

Parallel postulateProbability, given existence

Probability existenceConservation laws

FinitenessComputabilityGlobal warmingChaos (M. Harrell)Duhem’s problem

Phenomenal laws Experimental effects

Puzzle-solving adequacy

more“normal”.

more “revolutionary”.

1919thth c. Geology c. Geology

Uniformitarianism (steady-state world)

Catastrophism (progressively cooling world):

Adv

ance

men

tA

dvan

cem

ent

Stonesfield mammals 1814

creation

Stonesfield mammals 1814

1919thth c. Geology c. Geology

Early mammals

new schedule

Early reptiles

new schedule

U

Cschedule

Uniformitarianismschedule new violation.

lim ref lim ver Catastrophism schedule future fossils, the schedule stands

6. The Reliability Regress 6. The Reliability Regress Reliability = success over a broad range of possibilities.But but how can we tell we are in the range of reliability?

actual input stream

“method succeeds”

?

Answer: Answer: Check with Another Check with Another

Method!Method!Same inputs to all

11

H

1 succeeds

22

2 succeeds

33

3 succeeds

Popperian Regress Popperian Regress SolvedSolved

H

Methods never take back a rejection.

Later methods are as reliable as earlier.

Refutes H as reliably as any method in the regress

11

H

1 succeeds

22

2 succeeds

33

3 succeeds

…

Verification Regress Verification Regress SolvedSolved

H

Methods never take back an acceptance.

Later methods are as reliable as earlier.

…

11

H

1 succeeds

22

2 succeeds

33

3 succeeds

…Refutes H in the limit as reliably as any method in the regress.

Observed freq too close to p

“too close” set closer

“too close” set closer

p

= p

Observed freq too close to p

Background: a probability exists.Background: a probability exists. Hypothesis: its value is exactly Hypothesis: its value is exactly pp..

Frequentist ProbabilityFrequentist Probability

lim ref

Refuted with auxiliaries

New auxiliariesNew auxiliaries

H

H

Refuted with auxiliaries

Background: there exist correct Background: there exist correct auxiliaries with respect to which auxiliaries with respect to which isolated isolated HH is refutable. is refutable.

Duhem’s ProblemDuhem’s Problem

lim ver

Ref Ver

……neither [the world’s] finitude nor its infinitude can neither [the world’s] finitude nor its infinitude can be contained in experience, because [the be contained in experience, because [the following are impossible]:following are impossible]:

1.1. experience … of an experience … of an infinite space…infinite space…, ,

2.2. or again, of the or again, of the boundaryboundary of the world by an of the world by an empty space …empty space …

ProlegomenaProlegomena, Section 52c., Section 52c.

Antinomy of Pure Antinomy of Pure ReasonReason

Experience

??

moreInfinite

Finite wait for morewait for more

more

Antinomy of Pure Antinomy of Pure ReasonReason

Infinityposition

Intuition of more space.

lim ref lim ver Boundedness position intuitions, the position is not extended.

Parallel PostulateParallel Postulate

?

new pairnew pair

pp

pp

intersectintersect

lim ref

“unsolvable puzzle”

RearticulateRearticulatead

ad

“unsolvable puzzle”

Puzzle-solving adequacyPuzzle-solving adequacy: some : some articulation resolves by its standards articulation resolves by its standards all puzzles that arise all puzzles that arise withinwithin it. it.

Paradigm ChoiceParadigm Choice

lim ver

Ref Ver

ParadigmsParadigms

ad

ad

A1

A2A3

A1 solves 1st puzzle

A1 never solves 3rd

puzzle

A1 solves 2nd puzzle

grad ref““Internal Puzzle-solving adequacy”Internal Puzzle-solving adequacy”:: articulation such thatarticulation such that puzzle that arises within the paradigmpuzzle that arises within the paradigm Stage by which diligent effort would resolve Stage by which diligent effort would resolve

the puzzle by the paradigm’s standards the puzzle by the paradigm’s standards without rearticulation.without rearticulation.

Kuhnian MoralKuhnian Moral

Gradual refutability of each alternative does not suffice for convergence to an effective paradigm!

gaplim ref lim ver

ConservationConservation

C

C

R1

R2

R3

R1 unbalanced

Grad ver““Q is conserved in each reaction”Q is conserved in each reaction”:: reactionreaction stage at which the stage at which the QQ in = the in = the QQ out and out and Later stage, no more Later stage, no more QQ is observed in or is observed in or

out out

R1 unbalanced

R2 unbalanced

Cmismatch

new programnew programC

mismatch

lim ver

Ref Ver

CognitivismCognitivism

Computability program such that input the program produces the right output.

““Empirical Irony of Cog Empirical Irony of Cog Sci”Sci”

C

C

Prog 1

Prog 2Prog 3

Prog 1 outputs 1st

datum

Prog 2 outputs 2nd

datum Prog 3 never outputs 3rd

datum

Computability program such that input stage by which the right output is produced.

computer

ideal

grad ref

Transcendental DeductionsTranscendental Deductions

•What kind of problem structure is necessary for a given sort of success?

Certain Verifiability and Certain Verifiability and RefutabilityRefutability

Certain verifiability:Certain verifiability: H H is a union of “cones” of serious possibilities.is a union of “cones” of serious possibilities.Certain refutability:Certain refutability: HH is an intersection of cones of serious possibilities. is an intersection of cones of serious possibilities.

“blue will be seen.”

“green forever”

. . .

cone = all possible

extensions

Sextus Empiricus!Sextus Empiricus!

Non-verifiability: Non-verifiability:

Some input stream in Some input stream in HH never enters a fan included in never enters a fan included in HH..

. . .

H

HHH

Limiting VerifiabilityLimiting Verifiability

Limiting verifiability:Limiting verifiability:

HH is a countable is a countable unionunion of of intersectionsintersections of fans. of fans.

“The inputs will eventually stabilize to blue.”

Transcendental SystemTranscendental System

lim verlim ref

lim dec

cert vercert ref

cert dec

grad ref grad ver

Computable Case Similar!Computable Case Similar!

lim verlim ref

lim dec

cert vercert ref

cert dec

grad refgrad ver

Turing-decidable relations

Documents

Church’s Thesis and Hume’s Problem: Justification as Performance