The Way Not Taken or How to Do Things with an Infinite Regress

The Way Not TakenThe Way Not Takenoror

How to Do Things with anHow to Do Things with an Infinite RegressInfinite Regress

Two Methodological Paradigms Two Methodological Paradigms

“perfect support”

Certainty


Certainty

Problem of induction

. . .



Certainty

ConfirmationConfirmation

Partial support



Halting with the right answer

Certainty


Partial support


Eureka!


Entailment by evidence Halting with the right answer

Certainty

ConfirmationConfirmation LearningLearning

Partial support Convergent procedure

Eureka!



Certainty





Certainty



•Relational state

•“Internal”

•Relation “screens off” process

•Computation is extraneous

•Dynamical stability

•“External”

•Process paramount

•Computational perspective

The Ultimate QuestionThe Ultimate Question

Methods justify conclusions.

What justifies methods?


What justifies the confirmation relation?

The Ultimate QuestionThe Ultimate Question

What justifies methods?

OptionsOptions. . .. . .. . .

FoundationalismA priori justification

CoherentismMethods justify themselves

RegressAlways respond with a new method

Alternative Approach Alternative Approach


What justifies the confirmation relation?

How can we learn whether we are learning?

What Justifies Science?

Learning Theory PrimerLearning Theory Primer

MM

epistemically relevant worlds

method

hypothesis

data stream1 0 1 0 ? 1 ? 1 0 ? ? ?1 0 1 0 ? 1 ? 1 0 ? ? ?

correctness

conjecture stream

H

K

PutnamWeinsteinGoldFreiwaldBarzdinBlumCaseSmithSharmaDaleyOshersonEtc.

ConvergenceConvergence

MMIn the limit: ? ? 0 ? 1 0 1 0 1 1 1 1…

finite forever

MMWith certainty: ? ? 0 ? 1 0 1 0 halt!

finite

= Guaranteed convergence to the right answer

MMKH

Verification Refutation Decision

Converge to 1Don’t

converge to 0Converge to 1

Don’t

converge to 1Converge to 0 Converge to 0

ReliabilityReliability

1-sided 2-sided

wor

lds

data

conj

ectu

res

Some Some Reliability ConceptsReliability Concepts

Decision Verification Refutation

Certain Halt with correct answer

Halt with “yes” iff true

Halt with “no” iff false

Limiting Converge to correct answer

Converge to “yes” iff true

Converge to “no” iff false

One-sidedTwo-sided

Example: UniformitariansmExample: UniformitariansmMichael Ruse, Michael Ruse, The Darwinian RevolutionThe Darwinian Revolution

Uniformitarianism (steady-state)

Catastrophism (progressive):

Com

plex

ity

Com

plex

ity

Stonesfield mammals

creation

Uniformitarianism is refutable in the limit: – Side with uniformitarianism each time the

current progressive schedule is violated.– Eventually slide back to progressionism after

the revised schedule stands up for a while.


Historicism ExplainedHistoricism Explained

Articulations refutable only in a paradigm.Articulations crisply refutable with a

paradigm.No time at which a paradigm must be

rejected. Method inputs needn’t be cognized.Propositions forced by inputs relative to a

paradigm may be paradigm-relative.

UnderdeterminationUnderdetermination = = Degrees of UnsolvabilityDegrees of Unsolvability

Verifiable inthe limit


Refutable in the limit


Decidablewith certainty


Decidable inthe limit


Verifiablewith certainty


Refutablewith certainty














•Catastrophism•The coin is unfair•Computability

•Uniformitarianism•The coin is fair

•Uncomputability

•It will rain tomorrow

•Phenomenal laws•Observable existence

UnderdeterminationUnderdetermination = = Degrees of UnsolvabilityDegrees of Unsolvability

UnderdeterminationUnderdetermination = = ComplexityComplexity













AE EA

AE

•The coin is fair•Computability•Catastrophism

•The coin is unfair•Uncomputability

•Uniformitarianism

•Universal laws•Existence claims

•It will rain tomorrowclopen/recursive

Refinement: RetractionsRefinement: Retractions

You are a fool not to invest in technology

0 1 1 0 ? 1 1 ? ? ? ? ?0 1 1 0 ? 1 1 ? ? ? ? ?NASDAQ

Retractions

Expansions

2 retractionsstarting with 0




0 retractionsstarting with ?

0 retractionsstarting with ?

1 retractionstarting with ?

1 retractionstarting with ?

1 retractionstarting with 0




A E E A

AE

•Exactly n…

v

v







AE EA

= refutability with certainty

= verifiabilitywith certainty

= decidabilitywith certainty

. . .

Boolean combinations

of universals and existentials

Retractions Retractions asas

Complexity Complexity RefinementRefinement

Standard ObjectionStandard Objection

Very nice, but every method is reliable only under background presuppositions.

How do we know that the background assumptions are true?

Every assumption should be subject to empirical review.

Plausible FallacyPlausible Fallacy

If you could learn whether your pressupositions were true, you could chain this ability together with your original method to learn without them.

Therefore, if you need the presuppositions, you can’t reliably assess them.

Therefore, learning theory is an inadmissible version of foundationalism.

A Learnability AnalysisA Learnability Analysisof Methodological Regressof Methodological Regress

Presupposition of M =the set of worlds in which M succeeds.

MMH

successpresupposition

wor

lds

data

conj

ectu

res

Methodological RegressMethodological Regress

Same data to all

M1M1

H

M2M2

P1

M3M3

P2

M4M4

P3

No Free Lunch PrincipleNo Free Lunch Principle

The instrumental value of a regress is no greater than the best single-method performance that could be recovered from it without looking at the data.

…

Regress achievement

Scale of underdetermination

Single-method achievement

Empirical ConversionEmpirical Conversion An empirical conversion is a method that

produces conjectures solely on the basis of the conjectures of the given methods.

M1M1

H

M2M2

P1

M3M3

P2

MM

Methodological EquivalenceMethodological Equivalence

Reduction: B < A iffThere is an empirical conversion of an arbitrary group

of methods collectively achieving A into a group of methods collectively achieving B.

Methodological equivalence = inter-reducibility.

Simple IllustrationSimple Illustration

P1 is the presupposition under which M1 refutes H with certainty.

M2 refutes P1 with certainty.

M1M1

H

M2M2

P1

Worthless RegressWorthless Regress

M1 alternates mindlessly between acceptance and rejection.

M2 always rejects a priori.

PretensePretense

M pretends to refute H with certainty iff M never retracts a rejection.

•Duhem: no hypothesis is refutable in isolation.•Popper: to avoid coddling a false hypothesis forever, establish rejection conditions in advance even though the hypothesis is not really refutable.

Modified ExampleModified Example

Same as before But now M1 pretends to refute H with certainty.

M1M1

H

M2M2

P1

ReductionReduction

MReject when just one rejects

Accept otherwise

H

M1 ? ? ? 1 0 0 0 0 0

M2 ? 1 1 1 1 1 1 0 0

M 1 1 1 1 0 0 0 1 1

2 retractions in worst caseStarts not rejecting


H -H

P1

M1 never rejects

M2 never rejects

M1 rejects

M2 never rejects

-P1

M1 rejects

M2 rejects

M1 never rejects

M2 rejects

MReject when just one rejects

Accept otherwise

H

Converse ReductionConverse Reduction

M decides H with at most 3 retractions starting with acceptance.

Choose:– P1 = “M retracts at most once”

– M1 accepts until M uses one retraction and rejects thereafter.

– M2 accepts until M retracts twice and rejects thereafter.

Both methods pretend to refute.


Retractions

used by M0 1 2

H true false true

M1 never rejects rejects rejects

M2 never rejects never rejects rejects

P1 true true false

Regress TamedRegress Tamed

M1M1

H

M2M2

P1

Refutes with certainty

M1M1

H


Complexityclassification

Pretends torefute with

certainty

regress method

Finite Regresses TamedFinite Regresses Tamed

M1M1

P0

M2M2

P1

Mk+1Mk+1

P2

. . .Pn

M

P0

Pretends : n1 retractionsstarting with c1

Pretends : n2 retractionsstarting with c2

n2 retractions

starting with c2

Sum all the retractions.Start with 1 if an even number of the regressmethods start with 0.

H

Infinite Popperian RegressesInfinite Popperian Regresses

M1M1

P0

M2M2

P1

Mk+1Mk+1

P2

. . .

. . .Pn

Pn + 1

M

P0

UI

UI

UI

Refutes with certaintyover UiPi

Each pretends to refute

with certainty

Ever weaker presuppositions

UI

ExampleExample

Gru

e 1

Gru

e 2

Gru

e 3

Gru

e 4

Gru

e 5

Gru

e 0

P0

P1P2 P3

. . .

Green

M1M1

P0

Halt after 2and projectfinal obs.

MiMi

Pi-1

Halt after 2iand projectfinal obs.

Regress of deciders:“2 more = forever”

K

ExampleExample

M

1 if only green emeralds have been seen.

0 otherwise

Gru

e 1

Gru

e 2

Gru

e 3

Gru

e 4

Gru

e 5

Gru

e 0 K

P0

P0

Green

Equivalent single refuting method

H

Other Infinite RegressesOther Infinite Regresses

M1M1

P0

M2M2

P1

Mk+1Mk+1

P2

. . .

. . .

Pn

Pn+1

M

P0

UI

UI

UIRefutes

in the limitover UiPi

Each pretends to Verify with certaintyUse bounded retractionsDecide in the limitRefute in the limit

Ever weaker presuppositions

UI


Uniformitarianism (steady-state)

Catastrophism (progressive):

Com

plex

ity

Com

plex

ity

Stonesfield mammals

creation


M1M1

P0

MiMi

Pi-1

Regress of 2-retractors equivalent to a single limiting refuter:

Halt with acceptance when surprised by early complexity.

Keep rejecting until then.

Accept before the ith surprise is encountered.

Reject when the ith surprise is encountered.

Accept and halt when the i+1th surprise is encountered.

Pi = any number of surprises except i.

H





Decidable withcertainty

Decidable withcertainty



Verifiable withcertainty

Verifiable withcertainty

Refutable withcertainty

Refutable withcertainty

AE

Gradualrefutability

Gradualrefutability

Gradual verifiability

Gradual verifiability

AE EA

The PowerThe Powerof of

NestedNestedInfiniteInfinite

EmpiricalEmpiricalRegressesRegresses

AEAEAE

Similar results

Naturalism LogicizedNaturalism Logicized

Unlimited Fallibilism: every presupposition is held up to the tribunal of experience.

No free lunch: captures objective power of empirical regresses.

Progress-oriented: convergence is the goal. Feasibility: reductions are computable, so analysis

applies to computable regresses. Historicism: dovetails with a logical viewpoint on

paradigms and articulations.

Documents

The Way Not Taken or How to Do Things with an Infinite Regress