Cosmic Confusions
Not Supporting versus
Supporting NotJohn D. Norton
Department of History and
Philosophy of Science
Center for Philosophy of Science
University of Pittsburgh
www.pitt.edu/~jdnorton
CARL FRIEDRICH VON WEIZSÄCKER LECTURES
UNIVERSITY OF HAMBURG
June 2010
1
This Talk
Bayesian probabilistic analysis
conflates neutrality of evidential support
with disfavoring evidential support.
Fragments of inductive logics that
tolerate neutral support displayed.
Artifacts are introduced by the use of
the wrong inductive logic.
Wrong formal tool for
many problems in
cosmology where neutral
support is common.
Non-probabilistic state of
completely neutral support.
“Inductive disjunctive fallacy.”
Doomsday argument.
2
Completely
Neutral
Evidential
Support
3
Unconnected Parallel Universes: Completely Neutral Support
Same laws, but
constants
undetermined.
h=?c=?
G=?…
h=?c=?
G=?…
h=?c=?
G=?…
h0
1
2
3
4
Background
evidence is
neutral on
whether h lies
in some tiny
interval
or
outside it.
5
4
Parallel Universes Born in a Singularity: Disfavoring Evidence
Stochastic law assigns
probabilities to values
of constants.
P(h1) = 0.01
…
P(h2) = 0.01
…
P(h3) = 0.01
…
very
probable
h0
1
very
improbable
2
3
Background
evidence strongly
disfavors h lying
in some tiny
interval; and
strongly favors h
outside it.
very
probable
4
5
5
How to
Represent
Completely
Neutral Evidential
Support
6
Probabilities from 1 to 0 span support to disfavor
P(H|B) + P(not-H|B) = 1
Large.
Strong
favoring.
Small.
Strong
disfavoring.
Small.
Strong
disfavoring.
Large.
Strong
favoring.
No neutral probability value available for neutral support.
7
Underlying Conjecture of Bayesianism…
Logic of
physical chances
Logic of
all evidence
…Fails
8
Completely Neutral Support
[
h0
|B] = I
any contingent
proposition
I
I
1
I
I
“indifference”
“ignorance”
I
2
I
3
I
[A|B] = support
A accrues from B
4
5
I
Argued in some detail in
John D. Norton, "Ignorance and Indifference." Philosophy of Science, 75 (2008), pp. 45-68.
"Disbelief as the Dual of Belief." International Studies in the Philosophy of Science, 21(2007), pp. 231-252.
9
Justification…
I. Invariance under Redescription
using the Principle of Indifference
Equal support
for h in equal
h-intervals.
h0
I
I
1
I
2
I
3
I
5
4
rescale h
to h’ = f(h)
Equal support
for h’ in equal
h’-intervals.
h’0
I
I
I
1
2
I
3
I
4
5
[ h in [0,1] OR h in [1,2] | B] = [ h in [0,1] | B] = [ h in [1,2] | B]
The principle of indifference does not lead to paradoxes.
Paradoxes come from the assumption that evidential support must always be probabilistic.
10
Justification…
II. Invariance under Negation
Equal (neutral)
support for h in
[0,1] and
outside [0,1].
Equal (neutral)
support for h in
[0,2] and
outside [0,2].
h0
h0
I
I
1
2
4
5
I
I
1
3
2
3
4
5
[ h in [0,1] OR h in [1,2] | B] = [ h in [0,1] | B]
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Neutrality and
Probabilistic
Independence
12
Probabilistic
independence
vs.
Neutrality of
(total) support
For a partition of all outcomes
A1, A2, …
P(Ai|E&B) = P(Ai|B)
all i
For incremental measures of support*
inc (Ai, E, B) = 0
Tertiary function
Presupposes background
probability measure.
[Ai|B] = I
all contingent Ai
Binary function
Presupposes NO background
probability measure.
* e.g. d(Ai, E, B) = P(Ai|E&B) - P(Ai|B)
s(Ai, E, B) = P(Ai|E&B) - P(Ai|not-E&B)
r(Ai, E, B) = log[ P(Ai|E&B)/P(Ai|B) ]
etc.
13
Objective
vs
Subjective
14
mad dog
Neutrality and
Disfavor
or
Ignorance
and Disbelief
Bruno de Finetti
Subjective Bayesianism
degrees of belief
Objective Bayesianism
degrees of support
Only one conditional
In each evidential
situation,
represents opinion + the import
of evidence.
probability correctly
represents the import of
evidence.
Impossible.
No probability measure
captures complete neutrality.
Many conditional probability
Initial
“informationless”
priors?
Pick any.
They merely encode
arbitrary opinion that will be
wash out by evidence.
15
Pure Opinion Masquerading as Knowledge
1. Subjective
Bayesian sets
arbitrary priors on
k1, k2, k3, …
Pure opinion.
2. Learn richest evidence
= k135 or k136
3. Apply Bayes’ theorem
P(k135|E&B)
=
P(k136|E&B)
0.00095
P(k135|B)
=
0.00005
P(k136|B)
P(k135|E&B) = 0.95
P(k136|E&B) = 0.05
Endpoint of conditionalization
dominated by pure opinion.
16
Inductive
Disjunctive
Fallacy
17
Completely
neutral support
conflated
with
Strongly
disfavoring
support
Disfavoring
Neutral support
I
I
I
…
I
a1
a1 or a2
a1 or a2 or a3
…
a1 or a2 or … or a99
Disjunction of very many
neutrally supported outcomes
prob = 0.01
prob = 0.02
prob = 0.03
…
prob = 0.99
a strongly supported
is NOT
outcome.
18
van Inwagen, “Why is There Anything At All?”
Proc. Arist. Soc., Supp., 70 (1996). pp.. 95-120.
One way
not to be.
Probability zero.
“As improbable as
anything can be.”
Infinitely many ways to be.
…
Probability one.
As probable as anything can be.
19
Our Large Civilization
Ken Olum, “Conflict between Anthropic Reasoning and Observations,”
Analysis, 64 (2004). pp. 1-8.
Fewer ways
we can be in small
civilizations.
“Anthropic
reasoning predicts
we are typical…”
Vastly more ways
we can be in large civilizations.
…
“… [it] predicts with great confidence that we
belong to a large civilization.”
20
Our Infinite Space
Informal test of commitment to anthropic reasoning.
Fewer ways
we can be
observers in a
finite space.
Infinitely more ways
we can be observers in an infinite space.
…
Hence our space is infinitely more
likely to be geometrically infinite.
21
Inductive Logics that Tolerate
Neutrality of
Support
22
Discard Additivity, Keep Bayesian Dynamics
Bayesian
conditionalization.
If
T1 entails E. T2 entails E.
P(T1|B) = P(T2|B)
then
P(T1|E&B) = P(T2|E&B)
equal
priors
equal
posteriors
Postulate same rule in a new,
non-additive inductive logic.
Conditionalizing from
Complete Neutrality of
Support
If
T1 entails E. T2 entails E.
[T1|B] = [T2|B] = I
then
[T1|E&B] = [T2|E&B]
23
Pure Opinion Masquerading as Knowledge Solved
“Priors” are completely
neutral support over all
values of ki.
No normalization
imposed.
Apply rule of
conditionalization on
completely neutral
support.
[k1|B] = [k2|B] = [k3|B] =… = [k135|B] = [k136|B] = … = I
[k1|B] = [k1 or k2|B] = [k1 or k2 or k3|B] =… = I
E = k135 or k136
[k135|B] = [k136|B] = I
[k135|E&B] = [k136|E&B]
Nothing in evidence discriminates
between k135 or k136.
Bayesian result of support for k135 over k136 is an artifact of the
inability of a probability measure to represent neutrality of support.
24
The
Doomsday
Argument
25
Doomsday Argument (Bayesian analysis)
Bayes’ theorem
time = 0
we learn
time t has
passed
time of doom
T
What support does
t give to different
times of doom T?
p(T|t&B) ~ p(t|T&B) . p(T|B)
Compute likelihood by
assuming t is sampled
uniformly from available
times 0 to T.
p(t|T&B) = 1/T
For later: which is the right “clock” in which to
sample uniformly? Physical time T? Number of
people alive T’?…
p(T|t&B) ~ 1/T
Support for early doom
Variation in likelihoods
arise entirely from
normalization.
Entire result depends on
this normalization.
Entire result is an artifact
of the use of the wrong
inductive logic.
26
Doomsday Argument (Barest non-probabilistic reanalysis.)
Take evidence E is just that T>t.
T1>t entails E. T2>t entails E.
time = 0
E = T>t
we learn
time t has
passed
[T1|B] = [T2|B] = I
[T1|E&B] = [T2|E&B]
Apply rule of
conditionalization on
completely neutral
support.
The evidence fails to
discriminate between T1
and T2.
time of doom
T
What support does
t give to different
times of doom T?
27
Doomsday Argument (Bayesian analysis again)
Consider only the posterior
p(T|t&B)
time = 0
Require invariance of posterior under changes
of units used to measure times T, t.
Invariance under T’=AT, t’=At
Days, weeks, years? Problem as posed presumes no
time scale, no preferred unit of time.
we learn
time t has
passed
time of doom
T
What support does
t give to different
times of doom T?
Unique solution is the “Jeffreys’ prior.”
p(T|t&B) = C(t)/T
Disaster! This density
cannot be normalized.
Infinite probability mass
assigned to T>T*, no matter
how large.
Evidence supports latest
possible time of doom.
28
A Richer Non-Probabilistic Analysis
Consider the non-probabilistic degree of support
for T in the interval
time = 0
we learn
time t has
passed
time of doom
T
What support does
[T1,T2|t&B]
Presume that there is a “right” clock-time
in which to do the analysis, but we don’t
know which it is. So we may privilege no
clock, which means we require invariance
under change of clock:
T’ = f(T), t’ = f(t),
for strictly monotonic f.
[T1,T2|t&B] = [T3,T4|t&B] = I
for all T1,T2, T3,T4
t give to different
times of doom T?
29
Inductive
inference
the right way
30
No universal logic of induction
Material
theory of
induction:
Inductive inferences are not warranted by universal
schema, but by locally prevailing facts.
The contingent facts prevailing in a domain dictate which
inductive logic is applicable.
A Warrant for a Probabilistic Logic
Ensemble
Randomizer
+
An ensemble alone is
not enough.
Mere evidential neutrality over the ensemble
members does not induce an additive measure.
Some further element of the evidence must introduce
a complementary favoring-disfavoring.
31
Probabilities from Multiverses?
Gibbons, Hawking, Stewart (1987):
Hamiltonian formulation of general relativity.
Additive measure over different cosmologies induced by
canonical measure.
Gibbons, G. W.; Hawkings, S. W. and Stewart, J. M. (1987) “A Natural
Measure on the Set of All Universes,” Nuclear Physics, B281, pp. 736-51.
Just like the microcanonical distribution
of ordinary statistical mechanics?
No: there is no ergodic like behavior and
hence no analog of the randomizer.
“Giving the models equal weight corresponds to adopting Laplace’s
‘principle of indifference’, which claims that in the absence of any
further information, all outcomes are equally likely.”
Ensemble without randomizer
Gibbons,
Hawking,
Stewart,
p. 736
32
No universal formal logic of induction.
Inductive strength
“Deductively
definable logics
of induction”
[A|B]
for propositions A, B
drawn from a Boolean
algebra
Large class of non-probabilistic
logics.
No-go theorem.
deductive
structure
is defined
fully by of the Boolean
algebra.
Formal results:
Independence is generic.
Limit theorem.
Scale free logics of induction.
All need inductive supplement.
Read
"Deductively Definable Logics of Induction." Journal of Philosophical Logic. Forthcoming.
“What Logics of Induction are There?” Tutorial in Goodies pages on my website.
33
Winding Up
34
This Talk
Bayesian probabilistic analysis
conflates neutrality of evidential support
with disfavoring evidential support.
Fragments of inductive logics that
tolerate neutral support displayed.
Artifacts are introduced by the use of
the wrong inductive logic.
Wrong formal tool for
many problems in
cosmology where neutral
support is common.
Non-probabilistic state of
completely neutral support.
“Inductive disjunctive fallacy.”
Doomsday argument.
35
Read all about it…
36
37
38
Commercials
39
40
Finis
41
Appendices
42
The Self-Sampling Assumption
Penzias and Wilson
measure 3oK cosmic
background radiation.
Which is
our Penzias
and Wilson?
Level I multiverses. Many clones of Penzias and Wilson measure 3 oK
cosmic background radiation in other parts of space.
Self-Sampling Assumption: “One should reason if as one were
a random sample from the set of of all observers in one’s
reference class.” (Bostrom, 2007, p. 433)
Evidence on which is our PW
is neutral. No warrant for a
probability measure.
The self-sampling assumption
imposes probabilities where they do
not belong by mere supposition.
43
Why have the Self-Sampling Assumption?
P(
measure
3oK
P(
|
someone
somewhere
measures
3oK
background is
100oK
|
)
background is
100oK
= q <<1
“(L)” A physical chance computed
in a physical theory.
)
is (near) one.
Very many trials carried out
in the multiverse.
Introduce self-sampling to reduce this probability by allowing
that our PW is probably not the “someone somwhere.”
P(
Our PW
measure
3oK
|
background is
100oK
)
=

i
P(
i-th PW
measure
3oK
|
background is
100oK
) P(
q
Recover the same result without
sampling or calculation just by applying (L)
directly to case of “our PW.”
i-th PW
is our PW
)
=q
1/n
If n = infinity, the computation fails.
“1/n = 1/infinity = 0”
The failure is an artifact of the probabilistic representation
and its difficulties with infinitely many cases.
44
A Tempting Fallacy in Modern Cosmology
Prior theory is neutral on the
No reason to expect observed values.
values of some fundamental cosmic
parameters:
• Non-inflationary cosmology
provides no reason to expect a very
flat space.
The values are improbable and
therefore in need of explanation.
• Fundamental theories give no
reason to expect h, c, G, … to be the
values that support life.
We cannot demand that
everything be explained on
pain of an infinite
explanatory regress.
How do we decide
We decide post hoc. Only
what is in urgent need
of explanation and
what is not?
after we have the new
explanatory theory do we
decide the cosmic parameter in
urgent need of explanation.
45