Do You Believe Is True Even Though You Cannot Prove It?"
Psychologist, University of Massachusetts,
Amherst; Author, The Cognitive Brain.
have proposed a law of conscious content which
asserts that for any experience, thought, question,
or solution, there is a corresponding analog
in the biophysical state of the brain. As a
corollary to this principle, I have argued
that conventional attempts to understand consciousness
by simply searching for the neural correlates of
consciousness (NCC) in theoretical and empirical
investigations are too weak to ground a good
understanding of conscious content. Instead,
I have proposed that we go beyond NCC and explore
brain events that have at least some similarity to
our phenomenal experiences, namely, neuronal
analogs of conscious content (NAC). In
support of this approach, I have presented
a theoretical model that goes beyond addressing
the sheer correlation between mental states
and neuronal events in the brain. It explains
how neuronal analogs of phenomenal experience
(NAC) can be generated, and it details how
other essential human cognitive tasks can be
accomplished by the particular structure and
dynamics of putative neuronal mechanisms and
systems in the brain.
large body of experimental findings, clinical findings,
and phenomenal reports can be explained within
a coherent framework by the neuronal structure
and dynamics of my theoretical model. In addition,
the model accurately predicts many classical illusions
and perceptual anomalies. So I believe that the
neuronal mechanisms and systems that I have proposed
provide a true explanation for many important aspects
of human cognition and phenomenal experience. But
I can't prove it. Of course, competing theories
about the brain, cognition, and consciousness can't
be proved either. But I can't prove it. Providing
the evidence is the best we can do—I think.
and Developmental Psychologist; Author, The
believe, though I cannot prove it, that three—not
two—selection processes were involved
in human evolution.
first two are familiar: natural selection, which
selects for fitness, and sexual selection, which
selects for sexiness.
third process selects for beauty, but not sexual
beauty—not adult beauty. The ones doing the
selecting weren't potential mates: they were parents.
Parental selection, I call it.
gave me the idea was a passage from a book titled Nisa:
The Life and Words of a !Kung Woman, by the
anthropologist Marjorie Shostak. Nisa was about
fifty years old when she recounted to Shostak,
in remarkable detail, the story of her life as
a member of a hunter-gatherer group.
of the incidents described by Nisa occurred when
she was a child. She had a brother named Kumsa,
about four years younger than herself. When Kumsa
was around three, and still nursing, their mother
realized she was pregnant again. She explained
to Nisa that she was planning to "kill"—that
is, abandon at birth—the new baby, so that
Kumsa could continue to nurse. But when the baby
was born, Nisa's mother had a change of heart. "I
don't want to kill her," she told Nisa. "This little
girl is too beautiful. See how lovely and fair
her skin is?"
of beauty differ in some respects among human societies;
the !Kung are lighter-skinned than most Africans
and perhaps they pride themselves on this feature.
But Nisa's story provides a insight into two practices
that used to be widespread and that I believe played
an important role in human evolution: the abandonment
of newborns that arrived at inopportune times (this
practice has been documented in many human societies
by anthropologists), and the use of aesthetic criteria
to tip the scales in doubtful cases.
with sexual selection, parental selection could
have produced certain kinds of evolutionary changes
very quickly, even if the heartbreaking decision
of whether to rear or abandon a newborn was made
in only a small percentage of births. The characteristics
that could be affected by parental selection would
have to be apparent even in a newborn baby. Two
such characteristics are skin color and hairiness.
selection can help to explain how the Europeans,
who are descended from Africans, developed white
skin over such a short period of time. In Africa,
a cultural preference for light skin (such as Nisa's
mother expressed) would have been counteracted
by other factors that made light skin impractical.
But in less sunny Europe, light skin may actually
have increased fitness, which means that all three
selection processes might have worked together
to produce the rapid change in skin color.
selection coupled with sexual selection can also
account for our hairlessness. In this case, I very
much doubt that fitness played a role; other mammals
of similar size—leopards, lions, zebras,
gazelle, baboons, chimpanzees, and gorillas—get
along fine with fur in Africa, where the change
to hairlessness presumably took place. I believe
(though I cannot prove it) that the transition
to hairlessness took place quickly, over a short
evolutionary time period, and involved only Homo
sapiens or its immediate precursor.
was a cultural thing. Our ancestors thought of
themselves as "people" and thought of fur-bearing
creatures as "animals," just as we do. A baby born
too hairy would have been distinctly less appealing
to its parents.
I am right that the transition to hairlessness
occurred very late in the sequence of evolutionary
changes that led to us, then this can explain two
of the mysteries of paleoanthropology: the survival
of the Neanderthals in Ice Age Europe, and their
disappearance about 30,000 years ago.
believe, though I cannot prove it, that Neanderthals
were covered with a heavy coat of fur, and that
Homo erectus, their ancestor, was as hairy as the
modern chimpanzee. A naked Neanderthal could never
have made it through the Ice Age. Sure, he had
fire, but a blazing hearth couldn't keep him from
freezing when he was out on a hunt. Nor could a
deerskin slung over his shoulders, and there is
no evidence that Neanderthals could sew. They lived
mostly on game, so they had to go out to hunt often,
no matter how rotten the weather. And the game
didn't hang around conveniently close to the entrance
to their cozy cave.
Neanderthals disappeared when Homo sapiens, who
by then had learned the art of sewing, took over
Europe and Asia. This new species, descended from
a southern branch of Homo erectus, was unique among
primates in being hairless. In their view, anything
with fur on it could be classified as "animal"—or,
to put it more bluntly, game. Neanderthal disappeared
in Europe for the same reason the woolly mammoth
disappeared there: the ancestors of the modern
Europeans ate them. In Africa today, hungry humans
eat the meat of chimpanzees and gorillas.
present, I admit, there is insufficient evidence
either to confirm or disconfirm these suppositions.
However, evidence to support my belief in the furriness
of Neanderthals may someday be found. Everything
we currently know about this species comes from
hard stuff like rocks and bones. But softer things,
such as fur, can be preserved in glaciers, and
the glaciers are melting. Someday a hiker may come
across the well-preserved corpse of a furry Neanderthal.
I can sum my intuition up in five words: we're in for climatic mayhem.
Scientist; Personal Computer Visionary, Senior Fellow,
said "You must learn to distinguish between
what is true and what is real". An apt longer
quote of his is: "As far as the laws of mathematics
refer to reality, they are not certain; and as far
as they are certain, they do not refer to reality".
I.e. it is "true" that the three angles
of a triangle add up to 180 in Euclidean geometry
of the plane, but it is not known how to show that
this could hold in our physical universe (if there
is any mass or energy in our universe then it doesn't
seem to hold, and it is not actually known what our
universe would be like without any mass or energy).
science is a relationship between what we can represent
and are able to think about, and "what's out
there": it's an extension of good map making,
most often using various forms of mathematics as
the mapping languages. When we guess in science
we are guessing about approximations and mappings
to languages, we are not guessing about "the
truth" (and we are not in a good state of
mind for doing science if we think we are guessing "the
truth" or "finding the truth").
This is not at all well understood outside of science,
and there are unfortunately a few people with degrees
in science who don't seem to understand it either.
in math one can guess a theorem that can be proved
true. This is a useful process even if one's batting
average is less than .500. Guessing in science
is done all the time, and the difference between
what is real and what is true is not a big factor
in the guessing stage, but makes all the difference
epistemologically later in the process.
corner of computing is a kind of mathematics (other
corners include design, engineering, etc.). But
there are very few interesting actual proofs in
computing. A good Don Knuth quote is: "Beware
of bugs in the above code; I have only proved it
correct, not tried it."
analogy for why this is so is to the n-body problems
(and other chaotic systems behaviors) in physics.
An explosion of degrees of freedom (3 bodies and
gravity is enough) make a perfectly deterministic
model impossible to solve analytically for a future
state. However, we can compute any future state
by brute force simulation and see what happens.
By analogy, we'd like to prove useful programs
correct, but we either have intractable degrees
of freedom, or as in the Knuth quote, it is very
difficult to know if we've actually gathered all
the cases when we do a "proof".
a guess in computing is often architectural or
a collection of "covering heuristics".
An example of the latter is TCP/IP which has allowed "the
world's largest and most scalable artifact—The
Internet—to be successfully built. An example
of the former is the guess I made in 1966 about
objects—not that one could build everything
from objects—that could be proved mathematically—but
that using objects would be a much better way to
represent most things. This is not very provable,
but like the Internet, now has quite a body of
evidence that suggests this was a good guess.
guess I made long ago—that does not yet
have a body of evidence to support it—is
that what is special about the computer is analogous
to and an advance on what was special about writing
and then printing. It's not about automating past
forms that has the big impact, but as McLuhan pointed
out, when you are able to change the nature of
representation and argumentation, those who learn
these new ways will wind up to be qualtitatively
different and better thinkers, and this will (usually)
help advance our limited conceptions of civilization.
still seems like a good guess to me—but "truth" has
nothing to do with it.
Psychologist & Computer
Scientist; Author, Designing World-Class E-Learning
do not believe that people are capable of rational
thought when it comes to making decisions in their
own lives. People believe that are behaving rationally
and have thought things out, of course, but when
major decisions are made—who to marry, where
to live, what career to pursue, what college to
attend, people's minds simply cannot cope with
the complexity. When they try to rationally analyze
potential options, their unconscious, emotional
thoughts take over and make the choice for them.
an example of what I mean consider a friend of
mine who was told to select a boat as a wedding
present by his father in law. He chose a very peculiar
boat which caused a real rift between him and his
bride. She had expected a luxury cruiser, which
is what his father in law had intended. Instead
he selected a very rough boat that he could fashion
as he chose. As he was an engineer his primary
concern was how it would handle open ocean and
he made sure the engines were special ones that
could be easily gotten at and that the boat rode
very low in the water. When he was finished he
created a very functional but very ugly and uncomfortable
I have ridden with him on his boat many times.
Always he tells me about its wonderful features
that make it a rugged and very useful boat. But,
the other day, as we were about to start a trip,
he started talking about how pretty he thought
his boat was, how he liked the wood, the general
placement of things, and the way the rooms fit
together. I asked him if he was describing a boat
that he had been familiar with as a child and suggested
that maybe this boat was really a copy of some
boat he knew as a kid. He said, after some thought,
that that was exactly the case, there had been
a boat like in his childhood and he had liked it
a great deal.
he was arguing with his father in law, his wife,
and nearly everyone he knew about his boat, defending
his decision with all the logic he could muster,
destroying the very conceptions of boats they had
in mind, the simple truth was his unconscious mind
was ruling the decision making process. It wanted
what it knew and loved, too bad for the conscious
which had to figure how to explain this to everybody
course, psychoanalysts have made a living on trying
to figure out why people make the decisions they
do. The problem with psychoanalysis is that it
purports to be able to cure people. This possibility
I doubt very much. Freud was a doctor so I guess
he got paid to fix things and got carried away.
But his view of the unconscious basis of decision
making was essentially correct. We do not know
how we decide things, and in a sense we don't really
care. Decisions are made for us by our unconscious,
the conscious is in charge of making up reasons
for those decisions which sound rational. We can,
on the other hand, think rationally about the choices
that other people make. We can do this because
we do not know and are not trying to satisfy unconscious
needs and childhood fantasies. As for making good
decisions in our lives, when we do it is mostly
random. We are always operating with too little
information consciously and way too much unconsciously.
University of Pennsylvania; Author, A Matter
Big Bang, that giant explosion of more than 13 billion
years ago, provides the accepted description of our
Universe's beginning. We can trace with exquisite
precision what happened during the expansion and
cooling that followed that cataclysm, but the presence
of neutrinos in that earliest phase continues to
elude direct experimental confirmation.
once in thermal equilibrium, were supposedly freed
from their bonds to other particles about two seconds
after the Big Bang. Since then they should have
been roaming undisturbed through intergalactic
space, some 200 of them in every cubic centimeter
of our Universe, altogether a billion of them for
every single atom. Their presence is noted indirectly
in the Universe's expansion. However, though they
are presumably by far the most numerous type of
material particle in existence, not a single one
of those primordial neutrinos has ever been detected.
It is not for want of trying, but the necessary
experiments are almost unimaginably difficult.
And yet those neutrinos must be there. If they
are not, our whole picture of the early Universe
will have to be totally reconfigured.
Pauli's original 1930 proposal of the neutrino's
existence was so daring he didn't publish it. Enrico
Fermi's brilliant 1934 theory of how neutrinos
are produced in nuclear events was rejected for
publication by Nature magazine as being
too speculative. In the 1950s neutrinos were detected
in nuclear reactors and soon afterwards in particle
accelerators. Starting in the 1960s, an experimental
tour de force revealed their existence in the solar
core. Finally, in1987 a ten second burst of neutrinos
was observed radiating outward from a supernova
collapse that had occurred almost 200,000 years
ago. When they reached the Earth and were observed,
one prominent physicist quipped that extra-solar
neutrino astronomy "had gone in ten seconds
from science fiction to science fact". These
are some of the milestones of 20th century neutrino
the 21st century we eagerly await another one,
the observation of neutrinos produced in the first
seconds after the Big Bang. We have been able to
identify them, infer their presence, but will we
be able to actually see these minute and elusive
particles? They must be everywhere around us, even
though we still cannot prove it.
Institute of Advanced Study
like most human activities, is based on a belief,
namely the assumption that nature is understandable.
we are faced with a puzzling experimental result,
we first try harder to understand it with currently
available theory, using more clever ways to apply
that theory. If that really doesn't work, we try
to improve or perhaps even replace the theory.
We never conclude that a not-yet understood result
is in principle un-understandable.
some philosophers might draw a different conclusion—see
the contribution by Nicholas Humphrey—as
a scientist I strongly believe that Nature is understandable.
And such a belief can neither be proved nor disproved.
undoubtedly, the notion of what counts as "understandable" will
continue to change. What physicists consider to
be understandable now is very different from what
had been regarded as such one hundred years ago.
For example, quantum mechanics tells us that repeating
the same experiment will give different results.
The discovery of quantum mechanics led us to relax
the rigid requirement of a deterministic objective
reality to a statistical agreement with a not fully
determinable reality. Although at first sight such
a restriction might seem to limit our understanding,
we in fact have gained a far deeper understanding
of matter through the use of quantum mechanics
than we could possibly have obtained using only
scientist, IBM's T. J. Watson Research Center; Author, Calculus
we believe that consciousness is the result of patterns
of neurons in the brain, our thoughts, emotions,
and memories could be replicated in moving assemblies
of Tinkertoys. The Tinkertoy minds would have to
be very big to represent the complexity of our minds,
but it nevertheless could be done, in the same way
people have made computers out of 10,000 Tinkertoys.
In principle, our minds could be hypostatized in
patterns of twigs, in the movements of leaves, or
in the flocking of birds. The philosopher and mathematician
Gottfried Leibniz liked to imagine a machine capable
of conscious experiences and perceptions. He said
that even if this machine were as big as a mill and
we could explore inside, we would find "nothing
but pieces which push one against the other and never
anything to account for a perception."
our thoughts and consciousness do not depend on
the actual substances in our brains but rather
on the structures, patterns, and relationships
between parts, then Tinkertoy minds could think.
If you could make a copy of your brain with the
same structure but using different materials, the
copy would think it was you. This seemingly materialistic
approach to mind does not diminish the hope of
an afterlife, of transcendence, of communion with
entities from parallel universes, or even of God.
Even Tinkertoy minds can dream, seek salvation
and bliss—and pray.
Visiting Lecturer, University of the West of England,
Bristol; Author The Meme Machine
is possible to live happily and morally without believing
in free will. As Samuel Johnson said "All theory
is against the freedom of the will; all experience
is for it." With recent developments in neuroscience
and theories of consciousness, theory is even more
against it than it was in his time, more than 200
years ago. So I long ago set about systematically
changing the experience. I now have no feeling of
acting with free will, although the feeling took
many years to ebb away.
what happens? People say I'm lying! They say it's
impossible and so I must be deluding myself to
preserve my theory. And what can I do or say to
challenge them? I have no idea—other than
to suggest that other people try the exercise,
demanding as it is.
When the feeling is gone, decisions just happen with no sense of anyone
making them, but then a new question arises—will the decisions
be morally acceptable? Here I have made a great leap of faith (or the
memes and genes and world have done so). It seems that when people throw
out the illusion of an inner self who acts, as many mystics and Buddhist
practitioners have done, they generally do behave in ways that we think
of as moral or good. So perhaps giving up free will is not as dangerous
as it sounds—but this too I cannot prove.
As for giving up the sense of an inner conscious self altogether—this
is very much harder. I just keep on seeming to exist. But though I cannot
prove it—I think it is true that I don't.
Stanford University; Author, The Millennium
we can answer this question we need to agree what
we mean by proof. (This is one of the reasons why
its good to have mathematicians around. We like to
begin by giving precise definitions of what we are
going to talk about, a pedantic tendency that sometimes
drives our physicist and engineering colleagues crazy.)
For instance, following Descartes, I can prove to
myself that I exist, but I can't prove it to anyone
else. Even to those who know me well there is always
the possibility, however remote, that I am merely
a figment of their imagination. If it's rock solid
certainty you want from a proof, there's almost nothing
beyond our own existence (whatever that means and
whatever we exist as) that we can prove to ourselves,
and nothing at all we can prove to anyone
Mathematical proof is generally regarded as the most certain form of
proof there is, and in the days when Euclid was writing his great geometry
text Elements that was surely true in an ideal sense. But many
of the proofs of geometric theorems Euclid gave were subsequently found
out to be incorrect—David Hilbert corrected many of them in the
late nineteenth century, after centuries of mathematicians had believed
them and passed them on to their students—so even in the case of
a ten line proof in geometry it can be hard to tell right from wrong.
you look at some of the proofs that have been developed
in the last fifty years or so, using incredibly
complicated reasoning that can stretch into hundreds
of pages or more, certainty is even harder to maintain.
Most mathematicians (including me) believe that
Andrew Wiles proved Fermat's Last Theorem in 1994,
but did he really? (I believe it because the experts
in that branch of mathematics tell me they do.)
late 2002, the Russian mathematician Grigori Perelman
posted on the Internet what he claimed was an outline
for a proof of the Poincare Conjecture, a famous,
century old problem of the branch of mathematics
known as topology. After examining the argument
for two years now, mathematicians are still unsure
whether it is right or not. (They think it "probably
consider Thomas Hales, who has been waiting for six
years to hear if the mathematical community
accepts his 1998 proof of astronomer Johannes Keplers
360-year-old conjecture that the most efficient
way to pack equal sized spheres (such as cannonballs
on a ship, which is how the question arose) is
to stack them in the familiar pyramid-like fashion
that greengrocers use to stack oranges on a counter.
After examining Hales' argument (part of which
was carried out by computer) for five years, in
spring of 2003 a panel of world experts declared
that, whereas they had not found any irreparable
error in the proof, they were still not sure it
the idea of proof so shaky—in practice—even
in mathematics, answering this year's Edge question
becomes a tricky business. The best we can do is
come up with something that we believe but cannot
prove to our own satisfaction. Others will accept
or reject what we say depending on how much credence
they give us as a scientist, philosopher, or whatever,
generally basing that decision on our scientific
reputation and record of previous work. At times
it can be hard to avoid the whole thing degenerating
into a slanging match. For instance, I happen to
believe, firmly, that staples of popular-science-books
and breathless TV-specials such as ESP and morphic
resonance are complete nonsense, but I can't prove
they are false. (Nor, despite their repeated claims
to the contrary, have the proponents of those crackpot
theories proved they are true, or even worth serious
study, and if they want the scientific community
to take them seriously then the onus if very much
on them to make a strong case, which they have
so far failed to do.)
you recognize that proof is, in practical terms,
an unachievable ideal, even the old mathematicians
standby of G¤del's Incompleteness Theorem (which
on first blush would allow me to answer the Edge question
with a statement of my belief that arithmetic is
free of internal contradictions) is no longer available.
G¤del's theorem showed that you cannot prove an
axiomatically based theory like arithmetic is free
of contradiction within that theory itself.
But that doesn't mean you can't prove it in some
larger, richer theory. In fact, in the standard
axiomatic set theory, you can prove arithmetic
is free of contradictions. And personally, I buy
that proof. For me, as a living, human mathematician,
the consistency of arithmetic has been proved—to
my complete satisfaction.
to answer the Edge question, you have
to take a common sense approach to proof—in
this case proof being, I suppose, an argument that
would convince the intelligent, professionally
skeptical, trained expert in the appropriate field.
In that spirit, I could give any number of specific
mathematical problems that I believe are true but
cannot prove, starting with the famous Riemann
Hypothesis. But I think I can be of more use by
using my mathematician's perspective to point out
the uncertainties in the idea of proof. Which I
believe (but cannot prove) I have.
With a Slow Student
Student: Hi Prof. I've got a problem.
I decided to do a little probability experiment—you
know, coin flipping—and check some of
the stuff you taught us. But it didn't work.
I'm glad to hear that you're interested. What
did you do?
flipped this coin 1,000 times. You remember,
you taught us that the probability to flip heads
is one half. I figured that meant that if I flip
1,000 times I ought to get 500 heads. But it
didn't work. I got 513. What's wrong?
but you forgot about the margin of error. If
you flip a certain number of times then the margin
of error is about the square root of the number
of flips. For 1,000 flips the margin of error
is about 30. So you were within the margin of
now I get if. Every time I flip 1,000 times I
will always get something between 970 and 1,030
heads. Every single time! Wow, now that's a fact
I can count on.
no! What it means is that you will probably get
between 970 and 1,030.
mean I could get 200 heads? Or 850 heads? Or
even all heads?
the problem is that I didn't make enough flips.
Should I go home and try it 1,000,000 times?
Will it work better?
come on Prof. Tell me something I can trust.
You keep telling me what probably means
by giving me more probablies. Tell me
what probability means without using the word
Well how about this: It means I would be surprised
if the answer were outside the margin of error.
god! You mean all that stuff you taught us about
statistical mechanics and quantum mechanics and
mathematical probability: all it means is that
you'd personally be surprised if it didn't work?
I were to flip a coin a million times I'd be
damn sure I wasn't going to get all heads. I'm
not a betting man but I'd be so sure that I'd
bet my life or my soul. I'd even go the whole
way and bet a year's salary. I'm absolutely certain
the laws of large numbers—probability theory—will
work and protect me. All of science is based
on it. But, I can't prove it and I don't really
know why it works. That may be the reason why
Einstein said, "God doesn't play dice." It probably