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For those who enjoyed the “Memory” session on Monday
Multiplying 10

Digit Numbers Using Flickr: The Power
of Recognition Memory
by Andrew Drucker (my PhD student)
http://people.csail.mit.edu/andyd/rec_method.pdf
9883603368
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FREE WILL
Scott Aaronson
Associate Professor
Without Tenure
(!), MIT
The Looniest Talk I’ve Ever Given In My Life
A SCIENTIFICALLY

SUPPORTABLE
NOTION OF
IN ONLY
6
CONTROVERSIAL STEPS
Introduction
I’ll present a perspective about free will, quantum
mechanics, and time that I’ve never seen before
Compatibilist? Determinist?
Automaton? No problem! You can
listen to the talk too
I’ll place a much higher premium on being original and
interesting than on being right
Thanks
This talk will assume what David Deutsch calls
the
“momentous dichotomy”
:
Example application:
Quantum computing
Either a given technology is possible, or else there’s
some principled reason why it’s not possible.
Conventional wisdom:
“Free will is a hopelessly muddled
concept. If something isn’t deterministic, then logically, it
must be
random
—
but a radioactive nucleus obviously
doesn’t have free will!”
But the leap from “indeterminism” to
“
randomness” here is
total nonsense! In computer science, we deal all the time
with processes that are neither deterministic
nor
random…
Nondeterministic
Finite Automaton
x := x + 5;
// Determinism
x := random(1…10);
// Randomness
x := input();
// “Free will”
Hopelessly

Muddled?
Free will
Determinism
We can easily
imagine
“external inputs” to the giant
video game we all live in: the problem is just where such
inputs could
fit
into the actual laws of physics!
Quantum Mechanics and the Brain:
A Bullshit

Strewn Interdisciplinary Field
Two obvious difficulties:
(1)
The brain isn’t exactly the most hospitable place for
large

scale quantum coherence
(nor is there any clear reason
for such coherence to have evolved)
(2)
Even
if
QM were relevant to brain function, how would
that “help”? Again, randomness
free will
The Deterministic Path of This Talk
1.
A proposed “empirical” notion of free will (based on
algorithmic information theory)
2.
A falsifiable hypothesis about brain function
(Little or no exotic physics needed)
3.
The No

Cloning Theorem
4.
Recent applications of the No

Cloning Theorem
(Quantum money and copy

protected quantum software)
5.
“
Knightian
uncertainty” about the initial quantum
state of the universe
6.
A radical speculation about time
(Independent motivations from quantum gravity?)
How can we define free will in a way that’s
amenable to scientific investigation?
I propose to consider the question, “Can
machines think?” … The original question,
“Can machines think?” I believe to be too
meaningless to deserve discussion.
A. M. Turing, “Computing Machinery and
Intelligence,”
Mind
, 1950
So Turing immediately replaced it with a
different
question:
“Are there imaginable digital computers
which would do well in the
imitation game
?”
1.
For inspiration, I turned to
computer science’s Prophet
In this talk, I’ll propose a similar “replacement”
for the problem of free will
People mean many different things by “free will”:

Legal or moral responsibility

The feeling of being in control

“Metaphysical freedom”
But arguably, one
necessary condition
for “free will” is
(partial)
unpredictability
—
not by a hypothetical Laplace
demon, but by actual or conceivable technologies
(DNA testing, brain scanning…)
The Envelope Argument:
If, after you said
anything, you could open a sealed envelope and
read what you just said, that would come pretty
close to an
“empirical refutation of free will”
!
Obviously, many of your actions
are
predictable, and the
fact that they’re predictable doesn’t make them “unfree”!
Discussion
In general, the better someone knows you, the better they
can predict you … but even people who’ve been married
for decades can occasionally surprise each other!
(Otherwise, they would’ve effectively “melded” into a single person)
If someone could predict
ALL
your actions, it seems to me
that you’d be “unmasked as an automaton,” much more
effectively than any philosophical argument could unmask you
But how do we formalize the notion of “predicting
your actions”?
After all, if your actions were perfectly random,
then
in the sense relevant for us
, they’d also be
perfectly predictable!
I’ll solve that problem using a
“Prediction Game”
It’s the year 3000.
You enter the brain

scanning machine.
The Prediction Game: Setup Phase
The machine
records all the
neural data it can,
without killing
you
The machine outputs
a self

contained
“model” of you
(running on a classical
computer, a quantum
computer, or whatever)
Hardest part of this
whole setup to
formalize!
Q #34: Which physicist would you
least
want to be stranded at sea with: Paul
Davies, Sean Carroll, or Max Tegmark?
The Prediction Game: Testing Phase
“Max Tegmark”
0
0.2
0.4
0.6
0.8
Paul
Sean
Max
Q #35: Multiverse: for or against?
FEEDBACK
LOOP
The Prediction Game: Scoring Phase
The Questions: Q
1
,…,Q
n
Your Answers: A
1
,…,A
n
Predictor’s Guessed Distributions: D
1
,…,D
n
where C = some small constant (like 0.01),
B = the number of bits in the shortest computer
program that outputs A
i
given Q
1
,…,Q
i
and D
1
,…,D
i
as
input, for all i
{1,…,n}
We’ll say the predictor “succeeds” if:
Justification
Beautiful Result from Theory of Algorithmic Randomness
(paraphrase):
Assume you can’t compute anything that’s
Turing

uncomputatable. Then the inequality from the last
slide can be satisfied with non

negligible probability, in the
limit n
,
if and only if
you’re indeed choosing your
answers randomly according to the predictor’s claimed
distributions D
1
,…,D
n
.
Note:
B is itself an uncomputable quantity! Can
falsify
a
claimed Predictor by computing upper bounds on B, but
never prove absolutely that a Predictor works.
(But the same issue arises for separate reasons, and even arises in QM itself!)
If you don’t like the uncomputable element, can replace B
by the number of bits in the shortest
efficient
program
Crucial Point
In retrospect
, looking back on your entire
sequence of answers A
1
,…,A
n
, the predictor could
always decompose the sequence into (1) a part
that has a small Turing

machine description and
(2) a part that’s “algorithmically random.”
But when it’s forced to guess your answers
one by
one
, it might see a third, “fundamentally
unpredictable” component.
So,
can
the Prediction Game be won?
An “aspirational question” that could play a similar
role for neuroscience as the Turing Test plays for AI!
Argument for “yes”:
All information relevant for cognition
seems macroscopic and classical. Even if quantum effects
are present, they should get “washed out as noise”
But this is by no means obvious! Consider the following…
Falsifiable Hypothesis (H):
The behavior of (say) a
mammalian brain, on a ~10s timescale, can be (and
often is) sensitive to molecular

level events
2.
If
you believe Hypothesis H, then there would appear to be a
fundamental obstacle to winning the Prediction Game…
“Penrose Lite”:
No speculations here about the brain
as quantum computer, noncomputable QG effects in
microtubules, objective state

vector reduction, etc …
just the standard No

Cloning Theorem!
3.
The No

Cloning Theorem
There’s no general procedure to copy an unknown
quantum state, even approximately
Simple 1

Qubit Model Situation
VANILLA
CHOCOLATE
BOXERS
BRIEFS
But can the No

Cloning Theorem actually be used to get
quantum states that are both unclonable and
“functional”
?
Recent work in quantum computing theory illustrates that
the answer is yes…
Putting Teeth on the No

Cloning Theorem
Quantum Money
(Wiesner 1969, A. 2009,
Farhi et al. 2010, A.

Christiano 2011…):
Quantum state 
that a bank can
prepare, people can verify as legitimate,
but counterfeiters can’t copy
Quantum Copy

Protected Software
(A. 2009)
: Quantum
state 
f
that a software company can prepare, a customer
can use to compute some function f, but a pirate can’t use to
create more states that also let f be computed
4.
While these proposals raise separate issues (e.g.,
computational complexity), they’re analogous to what we
want in one important respect: if you don’t know how the
state 
or 
f
was prepared, then you can copy it, but
only with
exponentially

small success probability
(just like if you were trying to guess the outputs by chance!)
Suppose the Prediction Game can’t be won, even by a being
with unlimited computational power who knows the
dynamical laws of physics (but is constrained by QM).
Then such a being’s knowledge
must
involve Knightian
uncertainty either about the initial state of the universe (say,
at the big bang),
or
about “indexical” questions (e.g., “our”
location within the universe or the Everett multiverse)
For otherwise, the being could win the Prediction Game!
Knightian Uncertainty
5.
In economics,
Knightian uncertainty
means
uncertainty that one can’t even accurately quantify
using probabilities. There are formal tools to
manipulate such uncertainty
(e.g., Dempster

Shafer theory)
Poetically, we could think of this
Knightian
uncertainty about initial
conditions as “a place for free will
(or something like it) to hide in a
law

governed world”!
“Look, suppose I believed the Prediction Game was
unwinnable. Even so, why would that have
anything
to do
with free will? Even if I don’t know the initial state 
0
,
there still
is
such a state, and combined with the dynamical
laws, it still probabilistically determines the future!”
A Radical Speculation About Time
6.
If the Prediction Game was unwinnable, then it
would seem just as logically coherent to speak about
our decisions determining the initial state, as about
the initial state determining our decisions!
“Backwards

in

time causation”
, but crucially,
not
of a
sort that can lead to grandfather paradoxes
0
簰
†††
簰
簱
簰
簰
簰
† †
簰
INITIAL HYPERSURFACE (AT THE BIG BANG?)
MACROSCOPIC
AMPLIFICATION

㵼=
†
䉯戠慳B猠䅬A捥c
潮摡te

㵼=
†
Alice says yes
MACROSCOPIC
AMPLIFICATION


簱
簫
There’s a “dual description” of the whole spacetime history
that lives on an initial hypersurface only, and that has no
explicit time parameter
—
just a partially

ordered set of
“decisions” about what the quantum state on the initial
hypersurface ought to be.
A decision about particle
A
’s initial state
gets made
“before” a decision about particle
B
’s initial state, if and
only if, in the spacetime history,
A
’s amplification to
macroscopic scale occurs in the causal past of
B
’s
amplification to macroscopic scale
Are there independent reasons, arising from
quantum gravity, to find such a picture attractive?
(Now comes the speculative part of the talk!)
“The Black Hole Free Will Problem”:
You jump into a black
hole. While falling toward the singularity, you decide to wave.
According to
black hole
complementarity
, there’s a “dual
description” living on the event horizon. But how does the event
horizon “know” your decision? Could a
superintelligent
predictor,
by collecting the Hawking radiation, reconstruct your decision
without having ever seen
either
“your” past or “your” future?
The account of free will I’m suggesting can not
only
accommodate
a dual description living one
dimension lower; in some sense, it
demands
such
a description
Two Principles That I Held Inviolate
1.
Evolution from initial to later states is
completely
determined by the Hamiltonian:
there’s no room for free will to “hide” there
2.
Classical memories and records, once written,
can’t be “magically altered” by tinkering with
the universe’s initial state
Without
quantum mechanics
(or some other source
of unclonability)
, my account would have required
abandoning at least one of the principles above!
Conclusions
On the other hand, the idea that the Prediction Game
can
be won
also
strikes me as science fiction!
(For then how could you ever know you were “you,” rather than
one of countless simulations being run by various Predictors?)
I admit: the idea that the Prediction Game can’t be won
(because of, e.g., quantum mechanics and Knightian uncertainty
about the initial state)
strikes me as science fiction
By Deutsch’s “Momentous Dichotomy,”
one of these two
science

fiction scenarios has to be right!
Crucially, which scenario is right is not just a metaphysical
conundrum, but something that physics, CS, neurobiology,
and other fields can very plausibly make
progress
on
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