A Dialogue on Mathematics & Physics
yijun
C. S. Calude, University of Auckland, G. J. Chaitin, IBM Research, NY
As a visiting professor in the Department of Computer Science of the
University of Auckland, Greg Chaitin is a frequent visitor in New
Zealand. During his recent visit in July 2006 we had time for a
dialogue about mathematics, physics, and philosophy---C.S.C.
Cristian Calude: I suggest we discuss the question, Is mathematics
independent of physics?
Gregory Chaitin: Okay.
CC: Let's recall David Deutsch's 1982 statement:
The reason why we find it possible to construct, say, electronic
calculators, and indeed why we can perform mental arithmetic, cannot be
found in mathematics or logic. The reason is that the laws of physics
``happen" to permit the existence of physical models for the operations
of arithmetic such as addition, subtraction and multiplication.
Does this apply to mathematics too?
GC: Yeah sure, and if there is real randomness in the world then Monte
Carlo algorithms can work, otherwise we are fooling ourselves.
CC: So, if experimental mathematics is accepted as ``mathematics,'' it
seems that we have to agree that mathematics depends ``to some extent''
on the laws of physics.
GC: You mean math conjectures based on extensive computations, which of
course depend on the laws of physics since computers are physical
devices?
CC: Indeed. The typical example is the four-color theorem, but there
are many other examples. The problem is more complicated when the
verification is not done by a conventional computer, but, say, a
quantum automaton. In the classical scenario the computation is huge,
but in principle it can be verified by an army of mathematicians
working for a long time. In principle, theoretically, it is feasible to
check every small detail of the computation. In the quantum scenario
this possibility is gone.
GC: Unless the human mind is itself a quantum computer with quantum
parallelism. In that case an exponentially long quantum proof could not
be written out, since that would require an exponential amount of
``classical'' paper, but a quantum mind could directly perceive the
proof, as David Deutsch points out in one of his papers.
CC: Doesn't Roger Penrose claim that the mind is actually a quantum
computer?
GC: Yes, he thinks quantum gravity is involved, but there are many
other possible ways to get entanglement.
CC: How can such a parallel quantum proof be communicated and checked
when it exists only in the mind of the mathematician who ``saw'' it?
GC: Well, I guess it's like the design of a quantum computer. You tell
someone the parallel quantum computation to perform to check all the
cases of something, and if they have a quantum mind maybe they can just
do it. So you could publish the quantum algorithm as a proof, which the
readers would do in their heads to verify your claim.
CC: On paper you have only the quantum algorithm; everything else is in
the mind! What about disagreements, how can one settle them ``keeping
in mind'' (no pun!) that quantum algorithms are probabilistic? Aren't
we in danger of loosing an essential feature of mathematics, the
independent checkability of proofs in finite time?
GC: Well, even now you don't publish all the steps in a proof, you
depend on people to do some of it in their heads. And if one of us has
a quantum mind, then probably everyone does, or else that would become
a prerequisite, like a high IQ, for doing mathematics!
CC: Theoretical physics suggests that in certain relativistic
space-times, the so-called Malament-Hogarth space-times, it may be
possible for a computer to receive the answer to a yes/no question from
an infinite computation in a finite time. This may lead to a kind of
``realistic scenario'' for super-Turing computability.
GC: Well, to get a big speed-up you can just take advantage of
relativistic time dilation due either to a very strong gravitational
field near the event horizon of a black hole or due to very high-speed
travel (near the speed of light). You assign a task to a normal
computer, then you slow down your clock so that you can wait for the
result of an extremely lengthy computation. To you, it seems like just
a short wait, to the computer, aeons have passed...
CC: Physicist Seth Lloyd* has found that the ``ultimate laptop,'' a
computer with a mass of one kilogram confined to a volume of one litre,
operating at the fundamental limits of speed and memory capacity
determined by the physics of our universe, can perform 1051 operations
per second on 1031 bits. This device sort of looks like a black hole.
*S. Lloyd, ``Ultimate physical limits to computation,'' Nature (2000)
406, 1047-1054.
GC: And he's just published a book called Programming the Universe. The
basic idea is that the universe is a computation, it's constantly
computing its own time evolution.
CC: What about the Platonic universe of mathematical ideas? Is that
``muddied'' by physics too? To exist mathematics has to be
communicated, eventually in some written form. This depends upon the
physical universe!
GC: Yes, proofs have to be written on paper, which is physical. Proofs
that are too long to be written down may exist in principle, but they
are impossible to read.
CC: Talking about writing things down, logicians have studied logics
with infinitely long formulas, with infinite sets of axioms, and with
infinitely long proofs.
GC: How infinite? ℵ0, ℵ1, ℵ2?
CC: Could it be that such eccentric proofs correspond to something
``real''?
GC: Well, if people had ℵ2 minds, then formulas ℵ0 characters long
would be easy to deal with! There's even a set-theoretical science
fiction novel by Rudy Rucker called White Light in which he tries to
describe what this might feel like. I personally like a world which is
discrete and ℵ0 infinite, but why should Nature care what I think?
In one of his wilder papers, physicist Max Tegmark suggests that any
conceptually possible world, in other words, one that isn't
self-contradictory, actually exists. Instead of conventional Feynman
path integrals summing over all histories, he suggests some kind of
crazy new integral over all possible universes! His reasoning is that
the ensemble of all possible universes is simpler than having to pick
out individual universes!
Leibniz had asked why is there something rather than nothing, because
nothing is simpler than something, but as Tegmark points out, so is
everything. In his approach you don't have to specify the individual
laws for this particular universe, it's just one of many possibilities.
CC: What about constructive mathematics?
GC: Of course the mathematical notion of computability depends upon the
physical universe you are in. We can imagine worlds in which oracles
for the halting problem exist, or worlds in which Hermann Weyl's one
second, half second, quarter second, approach actually enables you to
calculate an infinite number of steps in exactly two seconds. But I
guess computability can handle this, everything relativises, you just
add an appropriate oracle. All the proofs go through as before.
CC: ---Are you talking about a physical Church-Turing Thesis?
GC: Yes I am.---But I think the notion of a universal Turing machine
changes in a more fundamental way if Nature permits us to toss a coin,
if there really are independent random events. (Quantum mechanics
supplies such events, but you can postulate them separately, without
having to buy the entire QM package.) If Nature really lets us toss a
coin, then, with extremely high probability, you can actually compute
algorithmically irreducible strings of bits, but there's no way to do
that in a deterministic world.
CC: Didn't you say that in your 1966 Journal of the ACM paper?
GC: Well yes, but the referee asked me to remove it, so I did. Anyway,
that was a long time ago.
CC: A spin-off company from the University of Geneva, id Quantique,
markets a quantum mechanical random number generator called Quantis.
Quantis is available as an OEM component which can be mounted on a
plastic circuit board or as a PCI card; it can supply a (theoretically,
arbitrarily) long string of quantum random bits sufficiently fast for
cryptographic applications. A universal Turing machine working with
Quantis as an oracle seems different from a normal Turing machine. Are
Monte Carlo simulations powered with quantum random bits more accurate
than those using pseudo-randomness?
GC: Well yes, because you can be unlucky with a pseudo-random number
generator, but never with real random numbers. People have gotten
anomalous results from Monte Carlo simulations because the
pseudo-random numbers they used were actually in sync with what they
were simulating.
Also real randomness enables you, with probability one, to produce an
algorithmically irreducible infinite stream of bits. But any infinite
stream of pseudo-random bits is extremely redundant and highly
compressible, since it's just the output of a finite algorithm.
CC: In a universe in which the halting problem is solvable many
important current open problems will be instantly solved: the Riemann
hypothesis or the Goldbach Conjecture.
GC: Yes, and you could also look through the tree of all possible
proofs in any formal axiomatic theory and see whether something is a
theorem or not, which would be mighty handy.
CC: Talking about the Riemann hypothesis, which is about primes,
there's the surprising connection with physics noticed by Freeman Dyson
that the distribution of the zeros of the Riemann function looks a lot
like the Wigner distribution for energy levels in a nucleus.*
And in an inspiring paper on ``Missed opportunities'' written by Dyson
in 1972, he observes that relativity could have been discovered 40
years before Einstein if mathematicians and physicists in Göttingen
had spoken to each other.
*Andrew Odlyzko and Michael Berry continued this work. And recently Jon
Keating and Nina Snaith, two mathematical physicists, have been able to
prove something new about the moments of the Riemann zeta function this
way.
GC: Well in fact, relativity was discovered before Einstein by
Poincaré---that's why the transformation group for Maxwell's equations
is called the Poincaré group---however Einstein's version was easier
for most people to understand.
But mathematicians shouldn't think they can replace physicists: There's
a beautiful little 1943 book on Experiment and Theory in Physics by Max
Born where he decries the view that mathematics can enable us to
discover how the world works by pure thought, without substantial input
from experiment.
CC: What about set theory? Does this have anything to do with physics?
GC: I think so. I think it's reasonable to demand that set theory has
to apply to our universe. In my opinion it's a fantasy to talk about
infinities or Cantorian cardinals that are larger than what you have in
your physical universe. And what's our universe actually like?
* a finite universe?
* discrete but infinite universe (ℵ0)?
* universe with continuity and real numbers (ℵ1)?
* universe with higher-order cardinals (≥ ℵ2)?
Does it really make sense to postulate higher-order infinities than you
have in your physical universe? Does it make sense to believe in real
numbers if our world is actually discrete? Does it make sense to
believe in the set {0, 1, 2, ...} of all natural numbers if our world
is really finite?
CC: Of course, we may never know if our universe is finite or not. And
we may never know if at the bottom level the physical universe is
discrete or continuous...
GC: Amazingly enough, Cris, there is some evidence that the world may
be discrete, and even, in a way, two-dimensional. There's something
called the holographic principle, and something else called the
Bekenstein bound. These ideas come from trying to understand black
holes using thermodynamics. The tentative conclusion is that any
physical system only contains a finite number of bits of information,
which in fact grows as the surface area of the physical system, not as
the volume of the system as you might expect, whence the term
``holographic.''
CC: That's in Lee Smolin's book Three Roads to Quantum Gravity, right?
GC: Yes. Then there are physical limitations on the human brain. Human
beings and computers feel comfortable with different styles of proofs.
The human push-down stack is short. Short-term memory is small. But a
computer has a big push-down stack, and its short-term memory is large
and extremely accurate. Computers don't mind lots of computation, but
human beings prefer ideas, or visual diagrams. Computer proofs have a
very different style from human proofs. As Turing said, poetry written
by computers would probably be of more interest to other computers than
to humans!
CC: In a deterministic universe there is no such thing as real
randomness. Will that make Monte Carlo simulations fail?
GC: Well, maybe. But one of the interesting ideas in Stephen Wolfram's
A New Kind of Science is that all the randomness in the physical
universe might actually just be pseudo-randomness, and we might not see
much of a difference. I think he has deterministic versions of
Boltzmann gas theory and fluid turbulence that work even though the
models in his book are all deterministic.
CC: What about the axioms of set theory, shouldn't we request arguments
for their validity? An extreme, but not unrealistic view discussed by
physicist Karl Svozil, is that the only ``reasonable'' mathematical
universe is the physical universe we are living in (or where
mathematics is done). Pythagoreans might have subscribed to this
belief.
Should we still work with an axiom---say the axiom of choice---if there
is evidence against it (or there is not enough evidence favouring it)
in this specific universe? In a universe in which the axiom of choice
is not true one cannot prove the existence of Lebesgue non-measurable
sets of reals (Robert Solovay's theorem).
GC: Yes, I argued in favor of that a while back, but now let me play
Devil's advocate. After all, the real world is messy and hard to
understand. Math is a kind of fantasy, an ideal world, but maybe in
order to be able to prove theorems you have to simplify things, you
have to work with a toy model, not with something that's absolutely
right. Remember you can only solve the Schrödinger equation exactly
for the hydrogen atom! For bigger atoms you have to work with numerical
approximations and do lots and lots of calculations...
CC: Maybe in the future mathematicians will work closely with
computers. Maybe in the future there will be hybrid mathematicians,
maybe we will have a man/machine symbiosis. This is already happening
in chess, where Grandmasters use chess programs as sparing partners and
to do research on new openings.
GC: Yeah, I think you're right about the future. The machine's
contribution will be speed, highly accurate memory, and performing
large routine computations without error. The human contribution will
be new ideas, new points of view, intuition.
CC: But most mathematicians are not satisfied with the machine proof of
the four-color conjecture. Remember, for us humans, Proof =
Understanding.
GC: Yes, but in order to be able to amplify human intelligence and
prove more complicated theorems than we can now, we may be forced to
accept incomprehensible or only partially comprehensible proofs. We may
be forced to accept the help of machines for mental as well as physical
tasks.
CC: We seem to have concluded that mathematics depends on physics,
haven't we? But mathematics is the main tool to understand physics.
Don't we have some kind of circularity?
GC: Yeah, that sounds very bad! But if math is actually, as Imre
Lakatos termed it, quasi-empirical, then that's exactly what you'd
expect. And as you know Cris, for years I've been arguing that
information-theoretic incompleteness results inevitably push us in the
direction of a quasi-empirical view of math, one in which math and
physics are different, but maybe not as different as most people think.
As Vladimir Arnold provocatively puts it, math and physics are the
same, except that in math the experiments are a lot cheaper!
CC: In a sense the relationship between mathematics and physics looks
similar to the relationship between meta-mathematics and mathematics.
The incompleteness theorem puts a limit on what we can do in axiomatic
mathematics, but its proof is built using a substantial amount of
mathematics!
GC: What do you mean, Cris?
CC: Because mathematics is incomplete, but incompleteness is proved
within mathematics, meta-mathematics is itself incomplete, so we have a
kind of unending uncertainty in mathematics. This seems to be
replicated in physics as well: Our understanding of physics comes
through mathematics, but mathematics is as certain (or uncertain) as
physics, because it depends on the physical laws of the universe where
mathematics is done, so again we seem to have unending uncertainty.
Furthermore, because physics is uncertain, you can derive a new form of
uncertainty principle for mathematics itself...
GC: Well, I don't believe in absolute truth, in total certainty. Maybe
it exists in the Platonic world of ideas, or in the mind of God---I
guess that's why I became a mathematician---but I don't think it exists
down here on Earth where we are. Ultimately, I think that that's what
incompleteness forces us to do, to accept a spectrum, a continuum, of
possible truth values, not just black and white absolute truth.
In other words, I think incompleteness means that we have to also
accept heuristic proofs, the kinds of proofs that George Pólya liked,
arguments that are rather convincing even if they are not totally
rigorous, the kinds of proofs that physicists like. Jonathan Borwein
and David Bailey talk a lot about the advantages of that kind of
approach in their two-volume work on experimental mathematics.
Sometimes the evidence is pretty convincing even if it's not a
conventional proof. For example, if two real numbers calculated for
thousands of digits look exactly alike...
CC: It's true, Greg, that even now, a century after Gödel's birth,
incompleteness remains controversial. I just discovered two recent
essays by important mathematicians, Paul Cohen and Jack Schwartz.* Have
you seen these essays?
*P. J. Cohen, ``Skolem and pessimism about proof in mathematics,''
Phil. Trans. R. Soc. A (2005) 363, 2407-2418; J. T. Schwartz, ``Do the
integers exist? The unknowability of arithmetic consistency,'' Comm.
Pure & Appl. Math. (2005) LVIII, 1280-1286.
GC: No.
CC: Listen to what Cohen has to say:
``I believe that the vast majority of statements about the integers
are totally and permanently beyond proof in any reasonable system.''
And according to Schwartz,
``truly comprehensive search for an inconsistency in any set of
axioms is impossible.''
GC: Well, my current model of mathematics is that it's a living
organism that develops and evolves, forever. That's a long way from the
traditional Platonic view that mathematical truth is perfect, static
and eternal.
CC: What about Einstein's famous statement that
``Insofar as mathematical theorems refer to reality, they are not
sure, and insofar as they are sure, they do not refer to reality.''
Still valid?
GC: Or, slightly misquoting Pablo Picasso, theories are lies that help
us to see the truth!
CC: Perhaps we should adopt Svozil's attitude of ``suspended
attention'' (a term borrowed from psychoanalysis) about the
relationship between mathematics and physics...
GC: Deep philosophical questions are never resolved, you just get tired
of discussing them. Enough for today!
Tianqinglaaaaa
By BENEDICT CAREY
Scientists studying stroke patients are reporting today that an injury to a specific part of the brain, near the ear, can instantly and permanently break a smoking habit. People with the injury who stopped smoking found that their bodies, as one man put it, "forgot the urge to smoke."
The finding, which appears in the journal Science, is based on a small study. But experts say it is likely to alter the course of addiction research, pointing researchers toward new ideas for treatment.
While no one is suggesting brain injury as a solution for addiction, the finding suggests that therapies might focus on the insula, a prune-size region under the frontal lobes that is thought to register gut feelings and is apparently a critical part of the network that sustains addictive behavior.
Previous research on addicts focused on regions of the cortex involved in thinking and decision making. But while those regions are involved in maintaining habits, the new study suggests that they are not as central as the insula is.
The study did not examine dependence on alcohol, cocaine or other substances. Yet smoking is at least as hard to quit as any other habit, and it probably involves the same brain circuits, experts said. Most smokers who manage to quit do so only after repeated attempts, and the craving for cigarettes usually lasts for years, if not a lifetime.
"This is the first time we've shown anything like this, that damage to a specific brain area could remove the problem of addiction entirely," said Dr. Nora Volkow, director of the National Institute on Drug Abuse, which financed the study, along with the National Institute of Neurological Disorders and Stroke. "It's absolutely mind-boggling."
Others cautioned that scientists still knew little about the widely distributed neural networks involved in sustaining habits.
"One has to be careful not to extrapolate too much based on brain injuries to what's going on in all addictive behavior, in healthy brains," said Dr. Martin Paulus, a psychiatric researcher at the University of California, San Diego, and the San Diego V.A. Medical Center. Still, Dr. Paulus said, the study "opens up a whole new way to think about addiction."
The researchers, from the University of Iowa and the University of Southern California , examined 32 former smokers, all of whom had suffered a brain injury. The men and women were lucid enough to answer a battery of questions about their habits, and to rate how hard it was to quit and the strength of their subsequent urges to smoke.
They all had smoked at least five cigarettes a day for two years or more, and 16 of them said they had quit with ease, losing their cravings entirely.
The researchers performed M.R.I. scans on all of the patients' brains to specify the location and extent of each injury.
They found that the 16 who had quit easily were far more likely to have an injury to their insula than to any other area. The researchers found no association between a diminished urge to smoke and injuries to other regions of the brain, including tissue surrounding the insula.
"There's a whole neural circuit critical to maintaining addiction, but if you knock out this one area, it appears to wipe out the behavior," said Dr. Antoine Bechara, a senior author of the new paper, who is a neuroscientist at the Brain and Creativity Institute at U.S.C. His co-authors were Dr. Hanna Damasio, also of U.S.C., and Nasir Naqvi and David Rudrauf of the University of Iowa.
The patients' desire to eat, by contrast, was intact. This suggests, the authors wrote, that the insula is critical for behaviors whose bodily effects become pleasurable because they are learned, like cigarette smoking.
The insula, for years a wallflower of brain anatomy, has emerged as a region of interest based in part on recent work by Dr. Antonio Damasio, a neurologist and director of the Brain and Creativity Institute. The insula has widely distributed connections, both in the thinking cortex above, and down below in subcortical areas, like the brain stem, that maintain heart rate, blood pressure and body temperature, the body's primal survival systems.
Based on his studies and others', Dr. Damasio argues that the insula, in effect, maps these signals from the body's physical plant, and integrates them so the conscious brain can interpret them as a coherent emotion.
The system works from the bottom up. First, the body senses cues in the outside world, and responds. The heart rate might elevate at the sight of a stranger's angry face, for example; other muscles might relax in response to a pleasant whiff of smoke.
All of this happens instantaneously and unconsciously, Dr. Damasio said — until the insula integrates the information and makes it readable to the conscious regions of the brain.
"In a sense it's not surprising that the insula is an important part of this circuit maintaining addiction, because we realized some years ago that it was going to be a critical platform for emotions," Dr. Damasio said in a telephone interview. "It is on this platform that we first anticipate pain and pleasure, not just smoking but eating chocolate, drinking a glass of wine, all of it."
This explains why cravings are so physical, and so hard to shake, he said: they have taken hold in the visceral reaches of the body well before they are even conscious.
Other researchers have found that the insula is activated in unpleasant circumstances, like a bad smell or the anticipation of a painful shock, or even in shoppers when they see a price that seems too high. Damage to the insula is associated with slight impairment of some social function.
While antismoking treatments based on the new findings are still a long way off, the authors suggest that therapies that replicate some of the physical sensations of the habit, like inhalers, could be useful.
And at least two previous studies suggest that people can reduce the sensation of pain by learning to modulate the activity in an area of their brain.
In experiments, healthy volunteers watched real-time M.R.I. images of a cortical region linked strongly to pain sensation and learned to moderate that neural activity, reducing the pain they felt from a heated instrument pressed to their palms. The same kind of technique could be tried with addicts watching images of their insulas.
"The question is, Can you learn to deactivate the insula?" Dr. Volkow said. "Now, everybody's going to be looking at the insula."