Simplicity and Scien.fic Progress Konstan.n Genin, Kevin Kelly - PowerPoint PPT Presentation

Sequen.al Example etc. H 2 = “You will see T exactly twice” is locally closed. H 1 = “You will see T exactly once” is locally closed. H 0 = “You will never see T” is closed. H T H H H H H H H H H H H H T H H H H H H H H H H H

Equa.on Example etc. H 2 = “quadra.c” is locally closed. H 1 = “linear” is locally closed. H 0 = “constant” is closed.

Topology • H is limi9ng open iff H is a countable union of locally closed sets. • H is limi9ng closed iff H c is limi.ng open. • H is limi9ng clopen iff H is both limi.ng open and limi.ng closed.

Proposi.onal Methods • Proposi9onal methods produce proposi.onal conclusions in response to proposi.onal informa.on. H E M

Proposi.onal Methods • M is infallible iff w ∈ M ( E ) , whenever E ∈ I ( w ). • M is monotonic iff M ( F ) ⊆ M ( E ) , whenever F ⊆ E.

Convergence M converges to H in w iff there is E in I ( w ) such that for all F in I ( w | E ) , M ( F ) ⊆ H .

Deduc.ve Methods • A verifica9on method for H is an infallible, monotonic method V such that: 1. w ∈ H c implies V (E) = W for E in I ( w ); 2. w ∈ H implies V converges to H in w .

Deduc.ve Methods • A verifica9on method for H is an infallible, monotonic method V such that: 1. w ∈ H c implies V (E) = W for E in I ( w ); 2. w ∈ H implies V converges to H in w . • A refuta9on method for H is just a verifica.on method for H c . • A decision method for H converges to H or to H c without error.

Deduc.ve Methods • A verifica9on method for H is an infallible, monotonic method V such that: 1. w ∈ H c implies V (E) = W for E in I ( w ); 2. w ∈ H implies V converges to H in w . • A refuta9on method for H is just a verifica.on method for H c . • A decision method for H converges to H or to H c without error. • H is methodologically verifiable [refutable, decidable, etc.] iff H has a method of the corresponding kind.

Induc.ve Methods • A limi9ng verifica9on method for H is a method V such that: w ∈ H iff V converges in w to some true H ’ that entails H .

Induc.ve Methods • A limi9ng verifica9on method for H is a method V such that: w ∈ H iff V converges in w to some true H ’ that entails H . • A limi9ng refuta9on method for H is a limi.ng verifica.on method for H c . • A limi9ng decision method for H is a limi.ng verifica.on method and a limi.ng refuta.on for H.

Topological Complexity limi.ng open limi.ng closed limi.ng clopen closed open clopen

Characteriza.on Theorem limi.ng open limi.ng closed limi.ng limi.ng verifica.on meth. refuta.on meth. induc9on limi.ng clopen limi.ng decision meth. closed open refuta.on meth. verifica.on meth. deduc9on clopen decision meth. Genin and Kelly, 2016

OCKHAM’S TOPOLOGICAL RAZOR

Popper’s Simplicity Order • “ More falsifiable hypotheses are simpler”. A � B , A ✓ cl B. H T H H H H H H H H H H H H T H H H H H H H H H H H H 1 � H 2 � H 3 .

A Big Mistake A � B , A ✓ cl B. 1. Weaker hypotheses are less falsifiable. A ✓ B implies A � B. 2. So suspending judgment violates Ockham’s razor! A � W.

Easy and Natural Fix Lack of falsifiers is bad only if A is false! A � B , A ✓ frntr B H T H H H H H H H H H H H H T H H H H H H H H H H H H 1 � H 2 � H 3 .

A Smaller Issue • Gerrymandered hypotheses can obscure simplicity rela.ons. • E.g., “The true law is linear, or the cat is on the mat” is not simpler than “The true law is quadra.c”.

A Response Simpler theories have simpler ways of being true. A / B , A \ frntr B 6 = ∅ H T H H H H H H H H H H H H T H H H H H H H H H H H H 1 / H 2 / H 3 .

Example: Compe.ng Paradigms Polynomial paradigm Y = P N i =0 a i X i . Trigonometric polynomial paradigm Y = P N i =0 a i sin( iX ) + b i cos( iX ).

Example: Compe.ng Paradigms Polynomial paradigm Y = P N i =0 a i X i . degree Trigonometric polynomial paradigm Y = P N i =0 a i sin( iX ) + b i cos( iX ).

Example: Compe.ng Paradigms Q = which degree and which paradigm is true? I = finitely many inexact measurements . locally locally 3 3 closed closed locally locally 2 2 closed closed locally locally 1 1 closed closed 0 0 closed closed

Example: Compe.ng Paradigms Q = which degree and which paradigm is true? I = finitely many inexact measurements . locally locally 3 3 closed closed locally locally 2 2 closed closed locally locally 1 1 closed closed 0 0 closed closed

Example: Compe.ng Paradigms Q = which degree and which paradigm is true? I = finitely many inexact measurements . locally locally 3 3 closed closed locally locally 2 2 closed closed locally locally 1 1 closed closed 0 0 closed closed

Example: Compe.ng Paradigms Q = which degree and which paradigm is true? I = finitely many inexact measurements . locally locally 3 3 closed closed locally locally 2 2 closed closed locally locally 1 1 closed closed 0 0 closed closed

Ques.ons • A ques.on par..ons W into countably many possible answers (Hamblin 1958) • Relevant responses are disjunc.ons of answers.

Ockham’s Razor Proposi9on (Genin and Kelly, 2016). The following principles are equivalent . 1. Infer a simplest relevant response in light of E . 2. Infer a refutable relevant response compa.ble with E . 3. Infer a relevant response that is not more complex than the true answer .

Empirical Problem P = ( W, I , Q ) .

Empirical Problem Q ( w ) is the answer true in w. w

Solu.ons A solu9on for is a proposi.onal method P = ( W, I , Q ) V such that w ∈ H iff V converges in w to some true H ’ that entails . Q ( w ) A problem is solvable iff it has a solu.on.

Solvability, Characterized. Proposi9on. A problem is solvable iff P = ( W, I , Q ) every answer is limi.ng open. de Brecht and Yamamoto (2009) Baltag, Gierasimczuk, and Smets (2015) Genin and Kelly (2015)

Progressive Solu.ons A solu.on for is progressive iff for all E P = ( W, I , Q ) in I ( w ) and F in I ( w | E ) : if V(E) entails Q ( w ) , then V(F) entails Q ( w ) . That is: the true answer is a fixed point of inquiry.

Progressive Solu.ons Proposi9on. If there exists an enumera.on A 1 , A 2 , … of the answers to Q agreeing with the simplicity order, then Q is progressively solvable.

Epistemic Mandate for Ockham’s Razor Proposi9on (Genin and Kelly, 2016) . Every progressive solu.on obeys Ockham’s razor.

Epistemic Mandate for Ockham’s Razor Proposi9on (Genin and Kelly, 2016) . Every progressive solu.on obeys Ockham’s razor. H c H w H / H c

Epistemic Mandate for Ockham’s Razor Proposi9on (Genin and Kelly, 2016) . Every progressive solu.on obeys Ockham’s razor. H c H w H / H c

Epistemic Mandate for Ockham’s Razor Proposi9on (Genin and Kelly, 2016) . Every progressive solu.on obeys Ockham’s razor. H c H w H / H c

Epistemic Mandate for Ockham’s Razor Proposi9on (Genin and Kelly, 2016) . Every progressive solu.on obeys Ockham’s razor. H c H H w H / H c

Epistemic Mandate for Ockham’s Razor Proposi9on (Genin and Kelly, 2016) . Every progressive solu.on obeys Ockham’s razor. H c H H w H c H / H c

Non-Circular By favoring a complex hypothesis, you lose in a complex world! avoidable H H c unavoidable

Skep.cism That story “… may be okay if the candidate theories are deduc.vely related to observa.ons, but when the rela.onship is probabilis.c, I am skep.cal …” Elliot Sober (2015).

A Worry • Proposi.onal informa.on refutes logically incompa.ble possibili.es. H E

A Worry • Proposi.onal informa.on refutes logically incompa.ble possibili.es. • Typically, sta.s.cal samples are logically compa.ble with every possibility. H H E E

Response Don’t worry! H H E E