Relations of Ideas are Matters of Fact: A Unified Theory of Theoretical Unification
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Transcript of Relations of Ideas are Matters of Fact: A Unified Theory of Theoretical Unification
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Relations of Ideas Relations of Ideas are are
Matters of Fact: Matters of Fact: A Unified Theory of A Unified Theory of
Theoretical UnificationTheoretical Unification
Kevin T. KellyDepartment of Philosophy
Carnegie Mellon [email protected]
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Les relations d'idées Les relations d'idées sont sont
des questions de fait:des questions de fait:Une théorie unifiée d'unification Une théorie unifiée d'unification
théoriquethéorique
Kevin T. KellyDepartment of Philosophy
Carnegie Mellon [email protected]
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I. Relations of Ideas I. Relations of Ideas and and
Matters of FactMatters of Fact
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Table of OppositesTable of Opposites
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
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ProgressProgress
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
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ProgressProgress
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
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ProgressProgress
Philosophy of MathPhilosophy of Math Philosophy of Philosophy of ScienceScience
CertainCertain UncertainUncertain
A PrioriA Priori A PosterioriA Posteriori
Proofs and Proofs and algorithmsalgorithms ConfirmationConfirmation
Truth-findingTruth-finding ““Rationality”Rationality”
ComputabilityComputability ProbabilityProbability
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Old Wine in New Bottles?Old Wine in New Bottles?
Philosophy of MathPhilosophy of Math Philosophy of Philosophy of ScienceScience
CertainCertain UncertainUncertain
A PrioriA Priori A PosterioriA Posteriori
Proofs and Proofs and algorithmsalgorithms ConfirmationConfirmation
Truth-findingTruth-finding ““Rationality”Rationality”
ComputabilityComputability ProbabilityProbability
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Philosophy of Science 101Philosophy of Science 101
FormalEmpirical
Verifiable
UnverifiableThe sun The sun will will nevernever stop stop rising.rising.
The The given given number number is even.is even.
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Wrong Twice!Wrong Twice!
FormalEmpirical
Verifiable
UnverifiableThe sun The sun will will nevernever stop stop rising.rising.
The The given given number number is even.is even.
The The currentcurrentemerald emerald isisgreengreen..The given The given
computaticomputation will on will never never halt.halt.
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Relations of IdeasRelations of Ideas Matters of factMatters of fact
Aristotelian ExcuseAristotelian Excuse
Rational Animal
Rational ?
Reason Observation
UnboundedexperienceCompleted analysis
??Always black?
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Relations of IdeasRelations of Ideas Matters of factMatters of fact
Leibnizian DoubtsLeibnizian Doubts
Rational ?
Reason Observation
Unboundedexperience
Never white?Always black?
Unboundedanalysis
????
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Synthetic A PrioriSynthetic A Priori A PosterioriA Posteriori
Bounded Kantian SynthesisBounded Kantian Synthesis
Pure intuition Observation
Unboundedexperience
3+2 = 5 ?
3 4 5
Never white?Always black?
boundedsynthesis
23
??
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Synthetic A PrioriSynthetic A Priori A PosterioriA Posteriori
Unbounded Kantian SynthesisUnbounded Kantian Synthesis
Pure intuition Observation
Unboundedexperience
Never cross? Never white?Always black?
Unbounded Synthesis?
????
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Synthetic A PrioriSynthetic A Priori A PosterioriA Posteriori
Kantian AntinomiesKantian Antinomies
Beyond all pure intuition Beyond all possible observation
Physicalcontinuum
Infinite?Dense? Never white?
Infinite?Dense?
Geometricalcontinuum
????
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Formal ProblemsFormal Problems Empirical Empirical ProblemsProblems
GGöödel and Turingdel and Turing
Never halts?
Computer Observation
Unboundedreality
Unboundedcomputation
Input
n
Never white?Always black??? ??
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Wrong First CutWrong First Cut
FormalEmpirical
Verifiable
UnverifiableThe sun The sun will will nevernever stop stop rising.rising.
The The given given number number is even.is even.
The The currentcurrentemerald emerald isisgreen.green.The given The given
computaticomputation will on will never never halt.halt.
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Right First CutRight First Cut
Formal Empirical
Verifiable
UnverifiableThe sun The sun will will nevernever stop stop rising.rising.
The The given given number number is even.is even.
The The currentcurrentemerald emerald isisgreen.green.The given The given
computaticomputation will on will never never halt.halt.
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Unified EpistemologyUnified Epistemology
For both For both formalformal and and empiricalempirical questions:questions:
Justification =Justification = truth-finding truth-finding performanceperformance
Find the truth in theFind the truth in the best best possible sense,possible sense, as as efficiently as possibleefficiently as possible. .
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II.II. VerificationVerification
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Verification as Verification as
Halting Halting with the Truthwith the Truth
Conclusion
Yes!
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Maybe youwill change yourmind.
Verification as Verification as Halting Halting with the Truthwith the Truth
Conclusion
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Verification as Verification as
Halting Halting with the Truthwith the Truth
Conclusion
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Bang!
Verification as Verification as Halting Halting with the Truthwith the Truth
Conclusion
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Verification as Verification as
Halting Halting with the Truthwith the Truth
Conclusion
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VerifiabilityVerifiability
Conclusion true
Yes!
Conclusion false
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Classical Problem of InductionClassical Problem of Induction
Always green?
I won’t present blue until he says yes!
Empirical demon
Empirical scientist
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Always green?
Classical Problem of InductionClassical Problem of Induction
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Always green?
Classical Problem of InductionClassical Problem of Induction
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Always green?
Classical Problem of InductionClassical Problem of Induction
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Always green?
You lose if you never halt with “yes”!
Classical Problem of InductionClassical Problem of Induction
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Always green?
Classical Problem of InductionClassical Problem of Induction
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Yes!
Always green?
Classical Problem of InductionClassical Problem of Induction
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Always green?
Classical Problem of InductionClassical Problem of Induction
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Always green?
Classical Problem of InductionClassical Problem of Induction
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Always green?
You loseanyway.
Classical Problem of InductionClassical Problem of Induction
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““Problem” of UncomputabilityProblem” of Uncomputability
Never halts?
Formal demon Formal scientist
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Maybe I am in an infinite loop.
Or maybe I will halt later…
Similar to the Problem of Induction?Similar to the Problem of Induction?
?
Never halts?
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Apparently Not SimilarApparently Not Similar
I see your program!
Never halts? 666
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Apparently Not SimilarApparently Not Similar
Now I can cleverly calculate a priori what you will do!
Never halts? 666
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Never shoots?
Apparently Not SimilarApparently Not Similar
Never halts? 21456666
I’m beaten!
666
chugchugchug
I’m beaten!
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Never shoots?
But Maybe Similar After All…But Maybe Similar After All…
666Never halts?
666
Aha! His program isalso showing!
Aha! His program isalso showing!
2145623455
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Never shoots?
But Maybe Similar After All…But Maybe Similar After All…
666Never halts?
I won’t halt until he halts with “yes” in mya priori simulation ofhim looking at me!
666
I won’t halt until he haltswith “yes” in mya priori simulation ofhim looking at me!
2145623455
666
I won’t halt until he haltswith “yes” in mya priori simulation ofhim looking at me!
2145623455
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Never shoots?
But Maybe Similar After All…But Maybe Similar After All…
666Never halts?
I won’t halt until he halts with “yes” in mya priori simulation ofhim looking at me!
666
I won’t halt until he haltswith “yes” in mya priori simulation ofhim looking at me!
2145623455
666
I won’t halt until he haltswith “yes” in mya priori simulation ofhim looking at me!
2145623455
Self-reference by Kleene fixed point theorem
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Never shoots?
But Maybe Similar After All…But Maybe Similar After All…
666Never halts? 2145623455
I didn’t halt…
2145623455 666
666
I didn’t halt…
I didn’t halt…
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Never shoots?
But Maybe Similar After All…But Maybe Similar After All…
666Never halts? 2145623455
I still didn’t halt…
2145623455 666
666
I still didn’t halt…
I still didn’t halt…
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Never shoots? 666Never halts? 2145623455
2145623455
I give up! You’ll never halt!
I give up! You’ll never halt!
666
666
But Maybe Similar After All…But Maybe Similar After All…
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Never shoots? 666Never halts?
Ban
g!
23455
23455
TheEnd
TheEnd
But Maybe Similar After All…But Maybe Similar After All…
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Never shoots? 666Never halts?
23455
23455
But Maybe Similar After All…But Maybe Similar After All…
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Never shoots? 666Never halts?
666
23455
Sucker!
23455
But Maybe Similar After All…But Maybe Similar After All…
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666
23455
666
Ban
g!
TheEnd
But Maybe Similar After All…But Maybe Similar After All…
Never shoots?Never halts?
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234556
66
But Maybe Similar After All…But Maybe Similar After All…
Never shoots?Never halts?
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234556
66
R.I.P.TM23455
Fooleda priori
But Maybe Similar After All…But Maybe Similar After All…
Never shoots?Never halts?
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Rice-Shapiro TheoremRice-Shapiro Theorem
Q
Whatever a computable cognitive scientist could verify about an arbitrary computer’s input-output behavior by formally analyzing its program…
cog-sci
666
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Rice-Shapiro TheoremRice-Shapiro Theorem…could have been verified by a behaviorist computer who empirically studies only the computer’s input-output behavior.
Qcog-sci
666
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AnalogyAnalogy
Universal Formal Universal Formal ProblemProblem
Universal Empirical Universal Empirical ProblemProblem
UnverifiableUnverifiable UnverifiableUnverifiable
Computable Computable demon can fool demon can fool
computablecomputable agent agent
Empirical demon Empirical demon can can
fool fool idealideal agent agent
Answer runs Answer runs beyond beyond formalformal
experienceexperience
Answer runs beyond Answer runs beyond empiricalempirical
experienceexperience
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Apparent DisanalogyApparent Disanalogy
Universal Formal Universal Formal ProblemProblem
Universal Universal Empirical ProblemEmpirical Problem
Input Input givengiven Input Input never fully never fully givengiven
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Response: Digested vs. GivenResponse: Digested vs. Given
Universal Formal Universal Formal ProblemProblem
Universal Universal Empirical ProblemEmpirical Problem
Input Input givengiven Input Input never fully never fully givengiven
Input never fully Input never fully digesteddigested
Input never fully Input never fully digesteddigested
. . .Infinitely slow-melting candy Infinitely long noodle
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Another Apparent DisanalogyAnother Apparent Disanalogy
Formal ScienceFormal Science Empirical ScienceEmpirical Science
IncompletenessIncompleteness ErrorError
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Focus on ConsistencyFocus on Consistency
Formal ScienceFormal Science Empirical ScienceEmpirical Science
IncompletenessIncompleteness ErrorError
Add axiomsAdd axioms Conjecture a theoryConjecture a theory
Who knows if they Who knows if they are consistent with are consistent with
low-level low-level combinatorics?combinatorics?
Who knows if it is Who knows if it is consistent with all consistent with all future empirical future empirical
data?data?
Assume they are Assume they are and keep checkingand keep checking
Assume it is and Assume it is and keep checkingkeep checking
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ThesisThesisThe problem of induction and the problem of uncomputability are similar.
So the problems should be understood similarly.
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•Philosophy of Science•“Problem” of induction demands a “solution”.•“Confirmation” and “Bayesian rationality”•No clear connection with truth.
•Philosophy of Math•Uncomputability is a fact, not a “problem”.•Logic and algorithms•Insistence on connection with truth.
A House DividedA House Divided
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III.III. Beyond Beyond VerifiabilityVerifiability
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Drop the Halting RequirementDrop the Halting Requirement
•Ought implies can.•If you can’t halt with the truth, find the truth in the best feasible sense.
Q
(W. James, H. Putnam, E.M. Gold, R. Jeroslow, P. Hájek)
Gun control!Second amendment!
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Focus on Focus on QuestionsQuestions
B
Answers partition the set of possible inputs
A C D
Infinite inputFinite input
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Convergence Without HaltingConvergence Without Halting
In the limit
With halting 1
0
Gradual
1
0
At most finitely many mind-changes
1
0
Ever-closer to full confidence
No possible opinion change after halting
Halts
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SuccessSuccess
Domain over which convergence to the truth is required
BA C D
Otherwise, non-convergence to falsehood is required.
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Success Over Binary PartitionsSuccess Over Binary Partitions
BA BA
BA
Verify Ainfalliblyin the limitgradually
Refute A
infalliblyin the limitgradually
Decide A
infalliblyin the limitgradually
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Catch-all
Success Over Countable PartitionsSuccess Over Countable Partitions
= 0 = 1 = 2 . . .
Compute partial
0 1 2 . . .
Theory selection
Dom(
Formalcase
Empiricalcase
infalliblyin the limitgradually
infalliblyin the limitgradually
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RetractionsRetractions
1
0
•A retraction occurs when an answer is “taken back”.•Retractions are what halting is supposed to prevent.•When you can’t prevent all retractions, minimize them.•Halting condition, when feasible, is special case.
2 retractions
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Retraction EfficiencyRetraction Efficiency
Retraction-efficiency = success in the limit under the least feasible retraction bound in each answer.
BA
5 6
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Retraction EfficiencyRetraction Efficiency
BA BA
BA
Halting-verifiable =1-verifiable
1 0 0 1
0 0
Halting-refutable =1-refutable
Halting-decidable =0-decidable
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Kantian Antinomies Kantian Antinomies RevisitedRevisited
Matter
Finitelydivisible
Infinitelydivisible
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Upper Complexity BoundUpper Complexity Bound
I say infinitely divisible when you let me cut.
Finitelydivisible
Infinitelydivisible
Lim-ref Lim-ver
. . .
. . .
Cut allowed by demon
Answer produced by scientist
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Lower Complexity BoundLower Complexity Bound
Finitelydivisible
Infinitelydivisible
Lim-ref-Lim-ver
Lim-ver-Lim-ref
I let you cut when you say finitely divisible.
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Purely Formal AnaloguePurely Formal Analogue
666 2145623455
Finitedomain
Infinitedomain
Lim-ref-Lim-ver
Lim-ver-Lim-ref
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Purely Formal AnaloguePurely Formal Analogue
666 2145623455
I halt on another input each time he says finitely divisible in my a priori simulation of him looking at my program.
Finitedomain
Infinitedomain
Lim-ref-Lim-ver
Lim-ver-Lim-ref
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Mixed Example: PenroseMixed Example: Penrose
UncomputableComputable
Undergraduate subjectCognitive scientist
.
???
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Mixed Example: PenroseMixed Example: Penrose
Cognitive scientist
You can verify human computabilityin the limit…
NCNCCCCCCC…
Undergraduate subject
UncomputableComputable
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Mixed Example: PenroseMixed Example: Penrose
Cognitive scientist
…but if humans are computers, then you are a computer, in which case you can’t verify in the limit that humans are computers. CNCNCCNCCN…
Undergraduate subject
UncomputableComputable
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Formal ComplexityFormal Complexity
. . .
ArithmeticalHierarchy
2145623455
n
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Formal ComplexityFormal Complexity
. . .
ArithmeticalHierarchy
2145623455
n
grad vergrad ref
lim ver lim ref
halt refhalt ver
lim decgrad dec
halt dec
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Formal ComplexityFormal Complexity
. . .
ArithmeticalHierarchy
n
3-ref3-ver
2-ver 2-ref
halt ref
halt-ver1-ver
1-dec
grad dec
0-dec
grad vergrad ref
Putnam’sn-trialpredicates
lim reflim ver
2-dec
halt-dec
halt-ref 1-ref
2145623455
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Empirical ComplexityEmpirical Complexity
. . .
Borelhierarchy
3-ref3-ver
2-ver 2-ref
halt ref
halt-ver1-ver
1-dec
grad dec
0-dec
grad vergrad ref
Differencehierarchy
lim reflim ver
2-dec
halt-dec
halt-ref 1-ref
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Joint ComplexityJoint Complexity
. . .
EffectiveBorel Hierarchy
3-ref3-ver
2-ver 2-ref
halt ref
halt-ver1-ver
1-dec
grad dec
0-dec
grad vergrad ref
Effectivedifferencehierarchy
lim reflim ver
2-dec
halt-dec
halt-ref 1-ref
2145623455
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Wrong First Cut in Wrong First Cut in PhilosophyPhilosophy
??Empirical Joint Formal
2145623455. . . . . . . . .
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Empirical Joint Formal2145623455
. . . . . . . . .
Right First Cut in PhilosophyRight First Cut in Philosophy
2145623455
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How Complexity Inverts SkepticismHow Complexity Inverts Skepticism•Justification is best possible truth-finding performance.•To establish best possible performance, one must show that better senses of success are infeasible.•Therefore, skeptical arguments justify induction!
. . .
Best possibleperformance
Impossibleperformance
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IV. Bounded IV. Bounded RationalityRationality
Bayesian
2145623455
To be empirically ignorant is human. To be logically ignorant is irrational. ?
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Philosophy of Science 101Philosophy of Science 101•“Science works by deducing theoretical predictions and comparing them against the empirical data”.
TheoremProver
Empirical data n No
t m
t
“Scientific Method”
Q
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What if Predictions are Non-What if Predictions are Non-Computable?Computable?
Empirical data n No
Qt
“Scientific Method”
Poof!
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Computable Refutation with Computable Refutation with Halting Can Still be Possible!Halting Can Still be Possible!
Empirical data n No
Qt
“Scientific Method”
?
?
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Non-computable PredictionsNon-computable Predictions
Everything but h
h
(with Oliver Schulte)
There exists predictive hypothesis {h} such that:
•h is infinitely non-computable;•But a computer can refute hypothesis {h} in the halting sense!
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Inductive Complexity : Deductive Inductive Complexity : Deductive Complexity Complexity
:: :: Implicit Definability : Explicit DefinabilityImplicit Definability : Explicit Definability
Learning complexity of {h}= joint complexity of {h}.
{h}
Deductive complexity of {h}= formal complexity of h.
. . .
. . .
hTypical case
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Singleton Non-Basis Singleton Non-Basis TheoremTheorem
{h} is 1 h is non-arithmetical
There exists h such that:
h is the mu-defined Skolem function for the Delta-2 implicit definition of arithmetical truth (Hilbert and Bernays).
{h}
. . .
. . .
h
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Cor. 1: A Foolish Cor. 1: A Foolish ConsistencyConsistency
•Consistent computers can’t verify or refute {h} even gradually!•So computable truth-seekers should be inconsistent.
Q
BayesianComputabledemon
InconsistencyComputablescientist
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Cor. 2: No FirewallCor. 2: No Firewall•No computer architecture that insulates theorem proving from future empirical data can gradually verify or refute {h}.
TheoremProver
Empirical data n No
Q
t m
t
“Scientific Method”
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Empirical data
Cor. 2: No FirewallCor. 2: No Firewall
TheoremProver
n
m
No
Q
“Improved Scientific Method”
•But a computer that has a time lag before noticing inconsistency can refute {h} with halting.
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V. Unification UnifiedV. Unification Unified
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Which Explanation is Right?Which Explanation is Right?
???
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Ockham Says:Ockham Says:
Choose theSimplest!
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But Maybe…But Maybe…
Gotcha!
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PuzzlePuzzle
A reliable indicator must be A reliable indicator must be sensitivesensitive to what it indicates. to what it indicates.
simple
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A reliable indicator must be A reliable indicator must be sensitivesensitive to what it indicates. to what it indicates.
complex
PuzzlePuzzle
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PuzzlePuzzle
But Ockham’s razor But Ockham’s razor alwaysalways points points at simplicity.at simplicity.
simple
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PuzzlePuzzle
But Ockham’s razor But Ockham’s razor alwaysalways points points at simplicity.at simplicity.
complex
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PuzzlePuzzle
How can a broken compass help How can a broken compass help you find something unless you you find something unless you already know where it is?already know where it is?
complex
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Bayesian ExplanationBayesian Explanation
Ignorance (over simple vs. complex)Ignorance (over simple vs. complex) + Ignorance (over parameter settings)+ Ignorance (over parameter settings) = Knowledge (that simple data can’t be = Knowledge (that simple data can’t be
produced by the complex theory).produced by the complex theory).
Simple Complex
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= The Old Paradox of = The Old Paradox of IndifferenceIndifference
Bayesian
Ignorance (over blue, non-blue)Ignorance (over blue, non-blue) + Ignorance (over ways of being non-+ Ignorance (over ways of being non-
blue)blue) = Knowledge (that the truth is blue as = Knowledge (that the truth is blue as
opposed to any other color)opposed to any other color)Not-BlueBlue
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= The Old Paradox of = The Old Paradox of IndifferenceIndifference
Bayesian
Ignorance (over blue, non-blue)Ignorance (over blue, non-blue) + Ignorance (over ways of being non-+ Ignorance (over ways of being non-
blue)blue) = Knowledge (that the truth is blue as = Knowledge (that the truth is blue as
opposed to any other color)opposed to any other color)Not-BlueBlue
And how does he explain formal simplicity?
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Better Idea: Retraction Better Idea: Retraction EfficiencyEfficiency
Truth
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Truth
Better Idea: Retraction Better Idea: Retraction EfficiencyEfficiency
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Curve FittingCurve Fitting
Data = arbitrarily precise intervals Data = arbitrarily precise intervals around around YY at specified values of at specified values of X.X.
Question: assuming that the truth is Question: assuming that the truth is polynomial, what is the degree?polynomial, what is the degree?
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Curve FittingCurve Fitting
Demon shows flat line until Demon shows flat line until convergent method concludes flat.convergent method concludes flat.
Zero degree curve
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Curve FittingCurve Fitting
Demon shows flat line until Demon shows flat line until convergent method concludes flat.convergent method concludes flat.
Zero degree curve
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Curve FittingCurve Fitting
Then switches to tilted line until Then switches to tilted line until convergent method concludes linear.convergent method concludes linear.
First degree curve
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Curve FittingCurve Fitting
Then switches to parabola until Then switches to parabola until convergent method concludes convergent method concludes quadratic …quadratic …
Second degree curve
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SimplicitySimplicity
ConstantLinear
Quadratic
Cubic
The current simplicity of answer H = the number of distinct answers the demon can force an arbitrary convergent method to produce before producing H.
Game-theoretic, topological invariant.
01
2
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In Step with the DemonIn Step with the Demon
ConstantLinear
Quadratic
Cubic
There yet?Maybe.
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In Step with the DemonIn Step with the Demon
ConstantLinear
Quadratic
Cubic
There yet?
Maybe.
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In Step with the DemonIn Step with the Demon
ConstantLinear
Quadratic
Cubic
There yet?Maybe.
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In Step with the DemonIn Step with the Demon
ConstantLinear
Quadratic
Cubic
There yet?Maybe.
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Ahead of the DemonAhead of the Demon
ConstantLinear
Quadratic
Cubic
There yet?Maybe.
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Ahead of the DemonAhead of the Demon
ConstantLinear
Quadratic
Cubic
I know you’re coming!
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Ahead of the DemonAhead of the Demon
ConstantLinear
Quadratic
Cubic
Maybe.
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Ahead of the DemonAhead of the Demon
ConstantLinear
Quadratic
Cubic
!!!
Hmm, it’s quite nice here…
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Ahead of the DemonAhead of the Demon
ConstantLinear
Quadratic
Cubic
You’re back!Learned your lesson?
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Ockham Violator’s PathOckham Violator’s Path
ConstantLinear
Quadratic
Cubic
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Ockham PathOckham Path
ConstantLinear
Quadratic
Cubic
So Ockham is uniquelyretraction efficient!
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Same Argument Works ForSame Argument Works For•Fewer causes•Unified causes•More Conserved quantities•Hidden particles•Fewer free parameters
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FormalFormal Curve Fitting Curve Fitting
MM is given the index of a computable is given the index of a computable polynomial function on the reals.polynomial function on the reals.
MM must converge to that function’s must converge to that function’s polynomial degree.polynomial degree.
Finite stage of effectivedecimal expansion
M
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““Computational Computational Experience”Experience”
MM alsoalso has to refute each polynomial has to refute each polynomial degree with a special halting state degree with a special halting state for that degree. for that degree.
Finite stage of effectivedecimal expansion
M
Definitely not quadratic!
Halting state 2
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Formal Curve-Fitting DemonFormal Curve-Fitting Demon
Polynomial degree?
For j = k to n-1:act like a polynomial of degree j until:
M refutes degree k-1 and M concludes that I am a polynomial of degree j;
set j = j+1;act like a polynomial of degree n.
21456M
D(M, k, n)
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Formal Ockham’s RazorFormal Ockham’s Razor
•M should not conclude that D(M, k, n) has degree > k when M’s experience has refuted at most degree k.
•If M is Ockham, M retracts at most n - j times on D(M, k, n) on experience refuting degree < k.
•Else, M retracts at least n – j + 1 times. Q
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Formal Causal InferenceFormal Causal Inference
•Given an index of a computable joint probability distribution whose conditional independence relations are represented by a causal network, compute the equivalence class of causal networks representing the distribution.
•N-choose-2 retractions required.•Empirical complexity = # of edges in graph•Ockham’s razor = assume no more causes than necessary!
X
YZ WCausal
network?
W
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Formal Causal InferenceFormal Causal Inference
•Given an index of a computable joint probability distribution whose conditional independence relations are represented by a causal network, compute the equivalence class of causal networks representing the distribution.
•N-choose-2 retractions required.•Empirical complexity = # of edges in graph•Ockham’s razor = assume no more causes than necessary!
X
YZ W
W
Causal network?
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Unified Account of Unified Account of UnificationUnification
•When halting is dropped as a condition for success, a penchant for elegance is truth-conducive (retraction-efficient) in both formal and empirical inquiry.
Q
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DiscussionDiscussion
Q
1. Uncomputability is similar to the problem of induction.
2. In both cases, halting with the truth is infeasible.3. Without the halting requirement, skeptical arguments
justify inductive reasoning by establishing efficiency.4. Truth-finding efficiency explains Ockham’s razor
both in empirical and formal reasoning.5. Insistence on separation of formal and empirical
reasoning restricts the power of computable science.
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THE ENDTHE END