BENOIT MANDELBROT
Mathematician, Yale University; Author, The Fractal Geometry of Nature

Wandering through the frontiers of the sciences, and the arts, I have always trusted the eye while leaving aside the issues that elude it. It can mislead—of course—therefore I check endlessly and never rush to print.

Meanwhile, for over fifty years, I have watched as some disciplines exhaust the "top down" problems they know how to tackle. So they wander around seeking totally new patterns in a dark and deep mess, where an unlit lamp is of little help.

But the eye can continually be trained and, long ago, I have vowed to follow it, therefore work "from the bottom up." Like the Antaeus of Greek myth, I gather strength and persist by often touching the earth.

A few of the truths the eye told me have been disproven. Let it be. Others have been confirmed by enormous and fruitful effort, and then blossomed, one being the four thirds conjecture in Brownian motion. Many others remain, one being the MLC conjecture about the Mandelbrot set, in which I believe for no other reason than trust in the eye.

STANISLAS DEHAENE
Cognitive Neuropsychology Researcher, Institut National de la Santé, Paris; Author, The Number Sense

I believe (but cannot prove) that we vastly underestimate the differences that set the human brain apart from the brains of other primates.

Certainly, no one can deny that there are important similarities in the overall layout of the human brain and, say, the macaque monkey brain. Our primary sensory and motor cortices are organized in similar ways. Even in higher brain areas, homologies can be found. In the parietal lobe, using brain-imaging methods, my lab has observed plausible human counterparts to several areas of the macaque brain, involved in eye movement, hand gestures, and even number processing.

Yet I fear that those early successes in drawing human-monkey homologies tend to mask other massive differences. If we compare the primary visual areas of macaques and humans, there is already a two-fold difference in surface area, but in parietal and frontal areas, a twenty-to fifty-fold increase is found. Even such a massive distortion may not suffice to "align" the macaque and human brain. Many of us suspect that, in regions such as the prefrontal and inferior parietal cortices, the changes are so dramatic that they may amount to the addition of new brain areas.

At a more microscopic level, it is already known that there is a new type of neuron which is found in the anterior cingulate region of humans and great apes, but not in other primates. These "spindle cells" send connections throughout the cortex, and thus contribute to a massive increase in long-distance connectivity in the human brain. Indeed, the change in relative white matter volume is perhaps what is most dramatic about the human brain.

I believe that these surface and connectivity changes, although they are in many cases quantitative, have brought about a qualitative revolution in brain function:

Breaking the brain's modularity.

Jean-Pierre Changeux and I have proposed that the increased connectivity of the human brain gives access to a new mode of brain function, characterized by a very flexible communication between distant brain areas. We may possess roughly the same list of specialized cerebral processors as our primate ancestors. However, I speculate that what might be unique about the human brain is its capacity to access the information inside each processor, and make it available to almost any other processor through long-distance connections. I believe that we humans have a much more developed conscious workspace—a set of brain areas that can fluidly exchange signals, thus allowing us to internally manipulate information and to perform new mental syntheses. Using the workspace's long-distance connections, we can mobilize, in a top-down manner, essentially any brain area and bring it into consciousness.

Spontaneous activity and the autonomy of consciousness.

Once the internal connectivity of a system exceeds a threshold, it begins to be dominated by self-sustained, reverberating states of activity. I believe that the human workspace system has passed this threshold, and has gained a considerable autonomy relative to the outside world. The human brain is much less at the mercy of signals from the outside world. Its activity never ceases to reverberate from area to area, thus generating a highly structured spontaneous flow of thoughts that we project on the outside world.

Of course, spontaneous brain activity is present in all species, but if I am correct we will discover that it is both more evident and more structured in the human brain, at least in higher cortical areas where "workspace" neurons with long-distance axons are denser. Furthermore, if human brain activity can be detached from outside stimulation, we will need to find new paradigms to study it, because bombarding the human brain with stimuli, as we do in most brain-imaging experiments, will not suffice. There is already some evidence for this statement: by directly comparing fMRI activations evoked by the same visual stimuli in humans and macaques, Guy Orban and his colleagues in Leuven have found that prefrontal cortex activity is five times larger in macaques than in humans. In their own words, "there may be more volitional control over visual processing in humans than in monkeys".

The profound influence of culture on the human brain.

The human species is also unique in its ability to expand its functionality by inventing new cultural tools. Writing, arithmetic, science, are all very recent inventions—our brains did not have time to evolve for them, but I speculate that they were made possible because we can mobilize our old areas in novel ways. When we learn to read, we "recycle" a specific region of our visual system, which has become known as the "visual word form area", for the purpose of recognizing strings of letters and connecting them to language areas. When we learn Arabic numerals, likewise, we build a circuit to quickly convert those shapes into quantities, a fast connection from bilateral visual areas to the parietal quantity area. Even an invention as elementary as finger counting changes dramatically our cognitive abilities: Amazonian people that have not invented counting are unable to make exact calculations as simple as 6-2.

Crucially, this "cultural recycling" implies that whenever we look at a human brain, the functional architecture that we see results from a complex mixture of biological and cultural constraints. Education is likely to greatly increase the gap between the human brain and that of our primate cousins. Virtually all human brain imaging experiments today are performed on highly literate volunteers—and therefore, presumably, highly transformed brains. To better understand the differences between the human brain and the monkey brain, we will need to invent new methods, both to decipher the organization of the baby brain prior to education, and to study of how it changes with education

TOR NØRRETRANDERS
Science Writer; Consultant; Lecturer, Copenhagen; Author, The User Illusion

I believe in belief—or rather: I have faith in having faith. Yet, I am an atheist (or a "bright" as some would have it). How can that be?

It is important to have faith, but not necessarily in God. Faith is important far outside the realm of religion: having faith in other people, in oneself, in the world, in the existence of truth, justice and beauty. There is a continuum of faith, from the basic everyday trust in others to the grand devotion to divine entities.

Recent discoveries in behavioural sciences, such as experimental economics and game theory, shows that it is a common human attitude towards the world to have faith. It is vital in human interactions; and it is no coincidence that the importance of anchoring behaviour in riskful trust is stressed in worlds as far apart as Søren Kierkegaard's existentialist christianity and modern theories of bargaining behaviour in economic interactions. Both stress the importance of the inner, subjective conviction as the basis for actions, the feeling of an inner glow.

One could say that modern behavioral science is re-discovering the importance of faith that has been known to religions for a long time. And I would argue that this re-discovery shows us that the activity of having faith can be decoupled from the belief in divine entities.

So here is what I have faith in: We have a hand backing us, not as a divine foresight or control, but in the very simple and concrete sense that we are all survivors. We are all the result of a very long line of survivors who survived long enough to have offspring. Amoeba, rodents and mammals. We can therefore have confidence that we are experts in survival. We have a wisdom inside, inherited from millions of generations of animals and humans, a knowledge of how to go about life. That does not in any way imply foresight or planning ahead on our behalf. It only implies that we have a reason to trust out ability to deal with whatever challenges we meet. We have inherited such an ability.

Therefore, we can trust each other, ourselves and life itself. We have no guarantee or promises for eternal life, not at all. The enigma of death is still there, ineradicable.

But we a reason to have confidence in ourselves. The basic fact that we are still here—despite snakes, stupidity and nuclear weapons—gives us reason to have confidence in ourselves and each other, to trust others and to trust life. To have faith.

Because we are here, we have reason for having faith in having faith.

STEVE GIDDINGS
Theoretical Physicist, University of California, Santa Barbara

I believe that black holes do not destroy information, as Hawking argued long ago, and the reason is that strong gravitational effects undermine the statement that degrees of freedom inside and outside the black hole are independent.

On the first point, I am far from alone; many string theorists and others now believe that black holes don't destroy information, and thus don't violate quantum mechanics. Hawking himself recently announced that he believes this, and has conceded a famous bet, but has not yet published the work giving a sharp statement where his original logic went wrong.

The second point I believe, but cannot yet prove to the point of convincing many of my colleagues. While many believe that Hawking was wrong, there is a lot of dissent over where exactly his calculation fails, and none of the arguments previously presented have sharply identified this point of failure. If black holes emit information instead of destroying it, this probably comes from a breakdown of locality. Lowe, Polchinski, Susskind, Thorlacius, and Uglum have argued that the mechanism for locality violation involves formation of long strings. Horowitz and Maldacena have argued that the singularity at the center of a black hole must be a unique state, in effect squeezing information out in a ghostly way. And others have made other suggestions.

But I believe, and my former student Lippert and I have published arguments, that the breakdown of locality that invalidates Hawking's work involves strong gravitational physics that makes it inconsistent to think of separate and independent degrees of freedom inside and outside the black hole. The assumption that these degrees of freedom are separate is fundamental to Hawking's argument. Our argument for where it fails has a satisfying generality that mirrors the generality of Hawking's original work—neither depends on the specifics of what kind of matter exists in the theory.

We base our argument on a principle we call the locality bound. This is a criterion for when physical degrees of freedom can be independent (in technical language, described by vanishing of commutators of corresponding operators). Roughly, a degree of freedom corresponding to a particle at position x with momentum p and another at y with momentum q will be independent only if the separation x-y is large enough that they are outside of a black hole that would form from their mutual energy. I believe this is the beginning of a general criterion (which will ultimately more precisely formulated) for when locality breaks down in physics. This could be the beginning of a deeper understanding of holography. And, it should be relevant to black hole physics because of the large relative energies of the Hawking radiation and degrees of freedom falling into a black hole. But this is not fully proven. Yet.

HOWARD RHEINGOLD
Communications Expert; Author, Smart Mobs

I believe that we humans, who know so much about cosmology and immunology, lack a framework for thinking about why and how humans cooperate. I believe that part of the reason for this is an old story we tell ourselves about the world:  Businesses and nations succeed by competing well. Biology is a war, where only the fit survive. Politics is about winning. Markets grow solely from self-interest. Rooted in the zeitgeist of Adam Smith's and Charles Darwin's eras, the scientific, social, economic, political stories of the 19th and 20th centuries overwhelmingly emphasized the role of competition as a driver of evolution, progress, commerce, society.

I believe that the outlines of a new narrative are becoming visible—a story in which cooperative arrangements, interdependencies, and collective action play a more prominent role and the essential (but not all-powerful) story of competition and survival of the fittest shrinks just a bit.

Although new knowledge in biology about the evolution of altruistic behavior and the role of symbiotic relationships, new understandings of economic behavior derived from experiments in game theory, neuroeconomic research, sociological investigations of institutions for collective action, computation-enabled technologies such as grid computing, mesh networks, and online markets all provide important clues, I don't believe anyone is likely to formulate an algorithm or recipe for human cooperation. I suspect that the complex interdependencies of human thought, behavior and culture entails an equivalent to the limits Heisenberg found to physics and Gödel established for mathematics.

I believe that more knowledge than what we have now, together with a conceptual framework that is neither reductionistic nor theological, could lead to better-designed economic and political policies and institutions. Institutional and conceptual barriers to mounting such an effort are as formidable as the methodological barriers. I am reminded of Doug Engelbart's problem in the 1950s. He couldn't convince computer engineers, librarians, public policy analysts that computing machinery could be used to augment human thinking, as well as performing scientific calculation and business data processing. Nobody and no institution had ever thought about computing machinery that way, and older ways of thinking about what machines could be designed to do were inadequate. Engelbart had to create "A Framework for Augmenting Human Intellect" before the various hardware, software, and human interface designers could create the first personal computers and networks.

By necessity, useful new understandings of how humans cooperate and fail to cooperate is an interdisciplinary task. I don't believe that the obvious importance of such an effort guarantees that it will be successfully accomplished. All our institutions for gathering and validating knowledge—universities, corporate research laboratories, and foundations—reward and support specialization.

LEO CHALUPA
Ophthalmologist and Neurobiologist, University of California, Davis

Here are three of my unproven beliefs:

(i) The human brain is the most complex entity in the known universe;

(ii) With this marvelous product of evolution we will be successful in eventually discovering all that there is to discover about the physical world, provided of course, that some catastrophic event doesn't terminate our species; and

(iii) Science provides the best means to attain this ultimate goal.

When the scientific endeavor is considered in relation to the obvious limitations of the human brain, the knowledge we have gained in all fields to date is astonishing. Consider the well-documented variability in the functional properties of neurons. When recordings are made from a single cell—for instance in the visual cortex to a flashing spot of light—one can't help but be amazed by the trial-to-trial variations in the resulting responses.

On one trial this simple stimulus might elicit a high frequency burst of discharges, while on the next trial there could be just a hint of a response. The same thing is apparent when EEG recordings are made from the human brain. Brain waves change in frequency and amplitude in seemingly random fashion even when the subject is lying in a prone position without any variations in behavior or the environment.
And such variability is also evident when one does brain imaging; the pretty pictures seen in publications are averages of many trials that have been "massaged" by various computer programs.

So how does the brain do it? How can it function as effectively as it does given the "noise" inherent in the system? I don't have a good answer, and neither does anyone else, in spite of the papers that have been published on this problem. But in line with the second of the three beliefs I have listed above, I am certain that someday this question will be answered in a definitive manner.

CARLO ROVELLI
Physicist; Institut Universitaire de France & University of the Mediterraneum; Author, Quantum Gravity

I am convinced, but cannot prove, that time does not exist. I mean that I am convinced that there is a consistent way of thinking about nature, that makes no use of the notions of space and time at the fundamental level. And that this way of thinking will turn out to be the useful and convincing one.

I think that the notions of space and time will turn out to be useful only within some approximation. They are similar to a notion like "the surface of the water" which looses meaning when we describe the dynamics of the individual atoms forming water and air: if we look at very small scale, there isn't really any actual surface down there. I am convinced space and time are like the surface of the water: convenient macroscopic approximations, flimsy but illusory and insufficient screens that our mind uses to organize reality.

In particular, I am convinced that time is an artifact of the approximation in which we disregard the large majority of the degrees of freedom of reality. Thus "time" is just the reflection of our ignorance.

I am also convinced, but cannot prove, that there are no objects, but only relations. By this I mean that I am convinced that there is a consistent way of thinking about nature, that refers only to interactions between systems and not to states or changes of individual systems. I am convinced that this way of thinking nature will end up to be the useful and natural one in physics.

Beliefs that one cannot prove are often wrong, as proven by the fact that this Edge list contains contradictory beliefs. But they are essential in science and often healthy. Here is a good example from 25 centuries ago: Socrates, in Plato's Phaedon says:

"... seems to me very hard to prove, and I think I wouldn't be able to prove it ... but I am convinced ... that the Earth is spherical."

Finally, I am also convinced, but cannot prove, that we humans have an instinct to collaborate, and that we have rational reasons for collaborating. I am convinced that ultimately this rationality and this instinct of collaboration will prevail over the shortsighted egoistic and aggressive instinct that produces exploitation and war. Rationality and instinct of collaboration have already given us large regions and long periods of peace and prosperity. Ultimately, they will lead us to a planet without countries, without wars, without patriotism, without religions, without poverty, where we will be able to share the world. Actually, maybe I am not sure I truly believe that I believe this; but I do want to believe that I believe this.

JOHN McCARTHY
Computer Scientist; Artificial Intelligence Pioneer, Stanford University

I think, as did Gödel, that the continuum hypothesis is false. No-one will ever prove it false from the presently accepted axioms of set theory. Chris Freiling's proposed new (1986) axioms prove it false, but they are not regarded as intuitive.

I think human-level artificial intelligence will be achieved.

JAMES O'DONNELL
Classicist; Cultural Historian; Provost, Georgetown University; Author, Avatars of the Word

What do I believe is true even though I cannot prove it? This question has a double edge and needs two answers.

First, and most simply: "everything". On a strict Popperian reading, all the things I "know" are only propositions that I have not yet falsified. They are best estimates, hypotheses that, so far, make sense of all the data that I possess. I cannot prove that my parents were married on a certain day in a certain year, but I claim to "know" that date quite confidently. Sure, there are documents, but in fact in their case there are different documents that present two different dates, and I recall the story my mother told to explain that and I believe it, but I cannot "prove" that I am right. I also know Newton's Laws and indeed believe them, but I also now know their limitations and imprecisions and suspect that more surprises may lurk in the future.

But that's a generic answer and not much in the forward-looking and optimistic spirit that characterizes Edge. So let me propose this challenge to practitioners of my own historical craft. I believe that there are in principle better descriptions and explanations for the development and sequence of human affairs than human historians are capable of providing. We draw our data mainly from witnesses who share our scale of being, our mortality, and for that matter our viewpoint. And so we explain history in terms of human choices and the behavior of organized social units. The rise of Christianity or the Norman Conquest seem to us to be events we can explain and we explain them in human-scale terms. But it cannot be excluded or disproved that events can be better explained on a much larger time scale or a much smaller scale of behavior. An outright materialist could argue that all my acts, from the day of my birth, have been a determined result of genetics and environment. It was fashionable a generation ago to argue a Freudian grounding for Luther's revolt, but in principle it could as easily be true and, if we could know it, more persuasive to demonstrate that his acts were determined a the molecular and submolecular level.

The problem with such a notion is, of course, that we are very far from being able to outline such a theory, much less make it persuasive, much less make it something that another human being could comprehend. Understanding even one other person's life at such microscopic detail would take much more than one lifetime.

So what is to be done? Of course historians will constantly struggle to improve their techniques and tools. The advance of dendrochronology (dating wood by the tree rings, and consequently dating buildings and other artifacts far more accurately than ever before) can stand as one example of the way in which technological advance can tell us things we never knew before. But we will also continue to write and to read stories in the old style, because stories are the way human beings most naturally make sense of their world. An awareness of the powerful possibility of whole other orders of possible description and explanation, however, should at least teach us some humility and give us some thoughtful pause when we are tempted to insist too strongly on one version of history—the one we happen to be persuaded is true. Even a Popperian can see that this kind of intuition can have beneficial effect.

PAMELA McCORDUCK
Writer; Author, Machines Who Think

Although I can't prove it, I believe that thanks to new kinds of social modeling, that take into account individual motives as well as group goals, we will soon grasp in a deep way how collective human behavior works, whether it's action by small groups or by nations. Any predictive power this understanding has will be useful, especially with regard to unexpected outcomes and even unintended consequences. But it will not be infallible, because the complexity of such behavior makes exact prediction impossible.

MARTIN REES
Cosmologist, Cambridge University; UK Astronomer Royal; Author, Our Final Hour

I believe that intelligent life may presently be unique to our Earth, but that, even so, it has the potential to spread through the galaxy and beyond—indeed, the emergence of complexity could still be near its beginning. If SETI searches fail, that would not render life a cosmic sideshow Indeed, it would be a boost to our cosmic self-esteem: terrestrial life, and its fate, would become a matter of cosmic significance. Even if intelligence is now unique to Earth, there's enough time lying ahead for it to spread through the entire Galaxy, evolving into a teeming complexity far beyond what we can even conceive.

There's an unthinking tendency to imagine that humans will be around in 6 billion years, watching the Sun flare up and die. But the forms of life and intelligence that have by then emerged would surely be as different from us as we are from a bacterium. That conclusion would follow even if future evolution proceeded at the rate at which new species have emerged over the 3 or 4 billion years of the geological past. But post-human evolution (whether of organic species or of artefacts) will proceed far faster than the changes that led to emergence, because it will be intelligently directed rather than being—like pre-human evolution—the gradual outcome of Darwinian natural selection. Changes will drastically accelerate in the present century—through intentional genetic modifications, targeted drugs, perhaps even silicon implants in to the brain. Humanity may not persist as a single species for more than a few centuries—especially if communities have by then become established away from the earth.

But a few centuries is still just a millionth of the Sun's future lifetime—and the entire universe probably has a longer future still. The remote future is squarely in the realm of science fiction. Advanced intelligences billions of years hence might even create new universes. Perhaps they'll be able to choose what physical laws prevail in their creations. Perhaps these beings could achieve the computational capability to simulate a universe as complex as the one we perceive ourselves to be in.

My belief may remain unprovable for billions of years. It could be falsified sooner—for instance, we (or our immediate post-human descendents) may develop theories that reveal inherent limits to complexity. But it's a substitute for religious belief, and I hope it's true.