A1 - A21 (opravy z Ea01-32y.doc potvrzeny dle stavu 9.4.1997, dodan opraveny Preface)


Ivan M. Havel

Translated from Czech by Josef P. Skala

Translator's note: The translation of Ivan M. Havel's reflections encompassing a variety of inter-disciplinary areas posed a number of challenges. The principal one was to somehow retain the original's special flavor of language witout unduly compromising comprehension. It was felt appropriate to respect the author's unique phraseology even at the cost of the English text appearing somewhat unusual at places.

Author's (later) note: The English version was thoroughly revised in collaboration of the author (the meaning) and Nancy Bishop (English language).

The following chapters were not included in the Czech version submitted for publication: A12, A13, B1, B3, B18. A1 was shifted between A3 and A4. On the other hand, the Czech version includes additional chapters A33 ("The Stone of the Year"), A34 ("The Surgeon and The Magician") whose translation is not yet finished.

Prague, April 1997

To my wife

Preface(to the Czech edition)

Unaware of what I was doing and suspecting nothing about what was expected of me, several years ago I agreed to write a monthly editorial column for the journal "Vesmír" (The Universe). And now, as it became a habit, I made the decision to collect these columns, modify them, rewrite some of them, delete others, supply notes, and publish them as a book. You have it in your hands.

I did not bother too much with the ordering of the articles. I divided the texts into two parts according to the themes. The first (A) is more concerned with the scientific reflection on the world, the second (B) with my reflection on science. I admit I often hesitated about what to place where. Within each of the two parts I have chosen the chronological order (with a few exceptions) in which the editorials appeared in the journal.

Since my essays were written in synchronny with the journal, they often reacted some topic from that particular issue. However, for this book I tried to eliminate those topicalities or particularities that could mislead the reader, usually by filling in some of the relevant material. A careful reader, who will also read the endnotes, will of course notice the relationships to "Vesmír" - references to literature give preference to articles in the journal. Besides the references, I used the endnotes also for supplementary and more technical informatin for a more meticulous reader.

I would like to apologize in advance to all who might have the feeling that their opinions and philosophical attitudes are belittled, ironicized, or even attacked. There should not be too many people offended - I am born in Libra and according to Daniela Fischerová my life motto is "Audiatur et altera pars", let the other side be listened to.

It is the custom to make acknowledgements. I decided not to do so, because all of those, whose advice, ideas, motivations, and influence may be acknowledged, are sufficiently realized through their own work, and to mention them here would not add anything more. The educated reader will guess who they are anyway. Perhaps there is one exception: without Ivan Boháček this book would not exist - and not only this book, even the Universe would cease to exist.

Prague, end of 1996

Ivan M. Havel


Part A. On Our World and Sciences

xn + yn = zn31
1, 2, 3, 4, 5, 6, 7,...

Part B. On Science and Scientists


Part A

On Our World and Sciences


In the South-western United States live Zuni Indians who are known for their beautiful animal-like amulets and fetishes. Much less known, however, is their understanding of the world. They believe that an all-encompassing, great, fully conscious and interrelated life system encompasses everything - the sun, moon, stars, sky, earth and oceans, all elements and natural phenomena, all inanimate objects as well as all plants, animals and humans. The starting point of this system is man, who is the most finished and, at the same time, the lowest form of life. The lowest because he is the most dependent, the most mortal and the least mysterious. Animals occupy a higher level, higher still are the natural phenomena and at the top are the mythical beings since they are least dependent, immortal, and totally mysterious.

I was surprised by the uncommon depths of logic expressed in the natural philosophy of these Indians. Lightning is beyond our understanding because it does not behave as a human being. I understand a tree slightly more: it takes care of itself and its offspring. A fox behaves much more like a man; it hunts, experiences fear, it feels hunger --nevertheless, I don't understand it as well as I understand myself. I understand humans the best because I am one. Understanding of the world by "civilized" science is exactly the opposite: because scientific comprehension involves objective description, generalization, classification and projection, a man is more mysterious than a fox, a fox is more mysterious than a tree, a tree is more mysterious than lightning -- and the supernatural is mysterious not at all, because it simply does not exist.

When comparing the world according to Zunis with the scientific conception of the world, it becomes clear that the key difference is in the way they approach consciousness, or the conscious experience. Until recently (Western) science preferred to consider consciousness only as a by-product, an over-shoot of brain processes, or even as a subjective delusion which is of no concern to sciences. In contrast, Zunis, if they want to be in harmony with their natural surrounding, have to strive for at least rudimentary understanding of its beings -- foxes, trees, and lightning. How should they do so otherthan to endow them with consciousness and soul? For Zunis, understanding might mean to identify with them, to enter into their consciousness.

Iit is remarkable to note that scientific disciplines which used to avoid problems related to human consciousness, now all seem to circle around this subject along an ever diminishing spiral. It is here, that a physicist meets a philosopher, a philosopher visits with a biologist and a biologist talks to a psychiatrist.

Let us consider the last example. Psychiatry is, in fact, the only applied science which has always concerned itself with consciousness; even when it was considered unfashionable to do so. It could, however, seldom solve specific illnesses of the soul. Only recently pronounced and surprising interrelationships surfaced between some psychoses and some already defined (and in some cases perhaps curable) pathological brain processes. There is hope yet that we may enter the new millennium in full psychic and intellectual health!

Scientists concern themselves not only with psychic and intellectual health, but also with understanding. The understanding of the world and its mysterious inner relationships. They should not, however, overlook the mysterious interrelationship between the brain and mind, or if you prefer, between body and soul.

Admittedly, this is not an easy area of scientific enquiry. Chemistry can "explain the existence of living organisms but cannot understand theirbeing"; similarly, the behavior of neurons may some day explain the function of consciousness, but it will hardly contribute to our understanding of its internal nature. Science has simply chosen another path, which is its advantage as well as drawback. Unlike Zunis, the more science will be concerned with a man and its inward nature, the more fragile its conclusions will be. What I really experience, that is alien to science. Andyet: this is the least mysterious thing, because to us humans it is indeed the closest and the most intimate one!



Different techniques are used by different scientific disciplines in order to diminish the variability introduced into a body of knowledge by different observers using different tools and occupying different points of view. As may be expected, mathematics is the discipline least affected in this respect. The landscapes which mathematics opens to us may in some respect resemble the structures, relationships and events of the real world, but at the same time, its landscapes are separated from our world to an extent which effectively precludes any concerns about their susceptibility to an infection by the ever-so-human variety of viewpoints.

Why is it called an "infection"?, you may argue. After all the principal support for pure mathematics comes from an emphasis on its usefulness. In other words, on its proximity to humans and their intentions.

I have to confess, however, that in spite of my ancient degree in engineering, I have never been elated by the practical, and perhaps industrial usefulness of mathematics. Infinitely more enticing for me is its invitation to a voyage into mysterious and uncharted lands, which open up only to those who are willing to submit themselves to the rigorous trials of abstract thinking.

Discoveries of foreign lands always result in some benefits which may or may not impact industry; perhaps they only enhance our experience. Even experiencing exotic tastes may be of benefit: it teaches us how to better appreciate domestic tastes. An example of a new experience offered by theoretical mathematics is the encounter with infinity.

Of all the scientific disciplines it is traditional mathematics which has outdistanced all others in its strife for objectivity and liberation from mankind's blindness and limitations. Mathematics typically uses the process of abstraction not only to narrow its vision to only characteristics shared by many objects, but also to widen its horizon to encompass even that which shall never be seen by human eyes. The notion of geometrical space does not take into consideration our inability to see objects too small and at the same time as objects too big. Likewise, the notion of a real numerical axis does not take into consideration our inability to calculate with numbers which are too small or too large.

A mathematician, frequently working with objects infinitely small or infinitely large, distances himself (with surprising benefits) from our natural experience. Is that, however, really necessary? Is it not possible, even in the realm of mathematics, to subjectivize the concept of the world as it is actually seen through the human eye?

The work of Petr Vopìnka and his Prague group, the so called Alternative Set Theory is an example of such a direction in mathematics. It is called "alternative" because it differs significantly from the classical set theory in its approach to the distinction between simple and complex collections of objects. The starting criterion of complexity of a set, used by classical theory, is its size the number of its elements and this number's natural generalization which enables us to talk about finite and infinite sets; a finite set can never contain an infinite part (logical monstrosities, i.e. a collection of all sets that are not their own elements, are not considered to be sets). The alternative theory, in contrast, considers manageability, surveyability, and "describability" to be the criteria of complexity: only collections which can be described in a given context (by cataloguing their elements, for example) can be considered sets. Collections that somehow, usually indirectly, formalize our often inaccurate images and idealizations (a "heap", for instance) are not considered to be sets. At the same time, it is recognized that a manageable set may contain unmanageable parts, as it often happens in the natural world. In addition, all these categorizations are also subjected to relativization.

The Alternative Set Theory even promotes as principles many situations that have traditionally been pushed aside by the conceptual system of classical mathematics; such situations were considered to be no more than human limitations, from which it is obligatory to distance ourselves. One such human limitation is the fact, for instance, that we can never simultaneously see all elements in very large sets of real objects (grains of sand, stars in the sky). Whether we try to look from up close or from afar, we manage to see less or more, never all. The range of our vision is restricted. We encounter the phenomenon of the horizon, which I consider a basic intuition necessary to comprehend the Alternative Set Theory (it distinguishes it from the fashionable concept of fuzzy sets, a concept which reminds me of a club with different classes of membership).

The horizon is not the absolute border of the contemplated world. It is the mobile and ever-escaping edge of our vision. It is possible to move it, sometimes even to widen it, but never to eliminate it. It is everywhere - we experience it wherever there is more or less of something, wherever something is closer or farther, smaller or larger, stronger or weaker, richer or poorer. We may contemplate things beyond the horizon, but we have to anchor our thoughts in our knowledge of what lies in front of it. On a higher level, it also applies to mathematics itself whose notion of infinity could be anchored in its natural materialization - the insurveyable sets.

Because of its philosophy, rather than because of some direct applications, the alternative theory of sets is of considerable relevance to all sciences. It allows us to return from a voyage into the exotic spheres of infinity, enriched by a better comprehension of that which can be seen by the human eye.

(1991, 1995)


                                                   Jakub Deml

Perhaps it is our destiny that we find ourselves in the midst of an ecological crisis which makes it possible for all of us to realize that we live in a land that can be healthy or sick, alive or dead, beautiful or ugly.

On the pages of newspapers, in magazines, on radio and TV, almost daily there appear ecological warnings, discussions, suggestions and analyzes. Our government, until recently the owner of everything, has gradually surrendered its ownership. Let us hope that it will in time become a caretaker of its citizens, their freedoms and their properties as well as the freedoms and possessions of their descendants. The state should also become the guardian of entities which do not belong to anybody, yet are of great concern to all: the earth, water, air, nature, countryside.

There exist wild uninhabited countrysides, and those where people are only few and far between. Those countrysides take care of themselves, and they do it well. Our countryside is, on the other hand, an inhabited one; humans with their cultivated fields, roads and residences constitute a significant part of it. On such a countryside it is possible not only to "listen to the greatness of nature, but also to see the results of labors of past generations" A 5], I read in a treatise by an ecologist on the inhabited countrysides (or on " a regional ecosystem with super-imposed socio-economical and cultural elements", as it is referred to by experts). The article is sad, because our countryside is also sad. It fell victim to the carelessness, insensitivity and arrogance of the rulers and their officials. It gradually lost its beauty - and even its names.

I don't know exactly why it is, but I feel most offended by the cruelty of people towards trees, woods, forests. If we had an ocean, I would surely be also offended by cruelty towards whales. A dwarf attacking a defenseless giant looks somehow particularly repulsive to me. Forests can not defend themselves; nevertheless the offending man will not go unpunished. He knows that, but pays no attention; his grandchildren are of no concern to him.

A countryside, whether inhabited or not, always has something in common with a living organism. I don't hesitate to say that it actually is a living organism. Land, or any of its relatively independent components (ecosystems), for instance a floodplain , is a complicated dynamic system encompassing an entire hierarchy of events - from the fast ones (the river-flow, wind, dances of insects) to the slower (seasons of the year, plant growth, population changes) to the slowest (erosion, sedimentation).

There are events which are cyclical, avalanche-like, chaotic, stabilizing and evolutionary. Events macroscopical as well as microscopical.

Everything is interconnected by a dense and fragile web of ties and interrelations. The more complex a behavior of a component, for example that of a biological species, the more difficult it becomes to study it in isolation from the behavior of all the other components.

There exists among the biological species of the ecosystem a variety of relationships; partnership and competition, symbiosis and parasitism, cooperation and abuse. Different species of plants and animals feed, fertilize, give shelter, provide a hiding-place or refuge to each other. There are even instances when one species provides care, cleans or protects the other. Different species mimic each other's coloration, shape and behavior for one reason or another. They help each other to fertilize and to reproduce. Often such behavior involves not only a couple, but a whole chain of many species.

We can even consider as partnerships some relationships between the life forms and the inanimate components of the ecosystem: water, wind, even fire. There are bushes of the Banksia genus growing in the Australian outback where bush-fires occur from time to time. The fires are necessary for the plant's reproduction: its seeds are imprisoned in hard wooden capsules where they wait, sometimes for many years, to be liberated by the heat of a fire. Seeding thus occurs at the most advantageous time when the freshly burned Out-Back does not harbor many competitors. Is it just cunning or a contract with the devil?

A countryside with its inhabitants has a tremendous regenerative potential. There may appear, however, someone stronger (or someone just considering himself to be stronger) who would carelessly tear apart the delicate web of interrelations. This someone stronger could well be man. Man, who forgets that even he is no more than a part of the countryside, and that by harming it he harms himself.

Perhaps it is our destiny to experience the ecological crisis (like the Banksia bush experiences a bush-fire) in order to liberate a seed of interconnection from the capsule of indifference.




There are two kinds of truths. Those that we all agree on and those in which someone unshakingly believes. I even dare to say, that the more someone is certain about his truth the less it conforms with even his own definition of truth as such.

What exactly does it mean "to be certain about one's own truth"? Here I refer to the certainty from within, which is immediate, un-problematic and non-hesitant. It could be anything, from the obsessive delusions of a paranoiac to the trust in one's own experience, which is a life's privilege and necessity for all of us. Of course, we all have our rock-solid beliefs and deeply imbedded fixed ideas (no mind can exist without them) and the reader probably expects me now to begin musings on the divide between tolerance and arrogance. I won't; let us try something different. I would like to ask the reader to kindly chose one of his or her own fixed truths and use it to test himself or herself regarding to what it really means to be certain about something. Quite seriously: grab that truth by your consciousness and, entirely disregarding what it is about, concentrate fully only on the way you experience it. Can one actually do this? Is it possible to analyze one's own certainty without spoiling it? In any case, what certainly is valid, is the notion that my certainty about my own truth is entirely different from my perception of the certainties of others.

Let us illustrate this by a simpler example: While I know very well (or, more precisely, I remember) how it feels when I have a headache, I cannot imagine a headache felt by somebody else using any means other than remembering again my own aching head. There is, however, another difference: when I am actually experiencing a headache (fortunately not the case right now), the ache felt in my head is straight-forward and unproblematic, and I don't really see any need to objectively test the possibility that I may be mistaken about it. Could I be mistaken? Even if all the neurologists on Earth would insist that I do not have a headache, the matter would not change - I know that I have a headache, and my consciousness does not fool me because my consciousness is me. Something entirely different is, however, your headache, my dear reader (my apologies - it is only an example). Should you complain about your headache to me, I may chose to either believe you or not. It would even make sense to consult neurologists for their "objective" advice about your head (after all they may know what's happening in there if it really hurts).

How is it then? In addition to the realities of the outside world which lend themselves to objective scientific enquiry, there exist worlds of our inner experiences, aches, truths and certainties about those truths - worlds which could only be perceived by our own consciousness. We all know that methods of measuring, observing and imaging by the human brain are being improved upon with each passing day; shouldn't we then consider the physical/chemical/biological processes known to occur in the brain to be equal with the events occurring in the consciousness? That is an opinion adopted by some. Never, insist others, neither a sign of equality nor a sign of inequality could be used - nothing which happens in the mind exists in reality, all of it is just scientifically uninteresting "optical" dellusions accompanying the brain processes. But it does exist, others suggest, and that what is really doubtful are the brain processes themselves, because, let us face it, isn't even scientific enquiry into those processes itself really just a fabrication? Both really do exist, propose others still, but they are two separate kingdoms which communicate with each other via some mystical means. There even exist, in fact, many other opinions.

In this context it is interesting to note the ever-increasing number of publications in the field of cognitive sciences (comprising psychology, artificial intelligence, neurosciences, linguistics, philosophy of the mind etc.) which in one way or another address the problem of consciousness. Even modern physics is beginning to pay attention to it. Is then the objective enquiry into the inner universe of the mind possible? Could there really be a sound scientific enquiry into our consciousness?

D.C. Dennett, a well-known American philosopher working in the field of cognitive science, suggests that such an enquiry is indeed possible. In his book, with the boastful title "Consciousness Explained" , he proposes the following methodological approach: to analyze the testimony of others about the experiences in their mind. Granted - we would never be able to verify their testimonies. It would not make sense to even attempt any such verification inasmuch as it does not make any sense to consider verification of what has been described in novels, fairytales and myths. In spite of that, proposes Dennett, it is possible to theorize objectively about consciousness as well as it is possible to objectively interpret and analyze the content of novels, fairytales and myths.

Well then, let us allow ourselves to be curious about the contributions of this new science.




Le temps de s'engager dans la simplicité royale des sentiers étroits

Le monde est dioque par définition : que la lumiære soit : et corpusculaire et ondulatoire

Surtout ne jamais opter pour l'un ni l'autre osciller entre deux demeurer en dehors

Vera Linhartova [A 8a]

Our century will soon submit the final accounts of its accumulated wisdom and I believe that the greatest historical contributions will undoubtedly include physics; quantum mechanics having a star billing.

To describe quantum mechanics in an easily comprehensible way is certainly not easy; after all, one can almost define it as a science on the inaccessible. "It is possible to say without any hesitation that noone understands quantum mechanics" said R. Feynman, the author of a best known textbook on physics. It is true: quantum physics certainly does not derive its fame from explaining to us, ordinary people, the secrets of how the world was made. Its success lies elsewhere - in its ability to rely on the language and skills of mathematics in areas where the language of natural experience and with it, our power of imagination fail us. Mathematics in its applied form is sometimes mistakenly considered to be only a supplementary tool which brings exactitude, correlations and predictions to matters already well understood intuitively. Quantum physics can serve as an example that the opposite may also be true.

Consequently, modern physics frequently talks about interpretation - i.e. the process by which meaning may be attributed to mathematical formulas. Many books were published and many hours spent in discussions on the interpretation of quantum mechanics; it is neither my calling nor my intent to contemplate the subject here. There are occasions, however, when novel ideas appear, for example, such things as "quantum erasure" - and I have never been able to resist reading without playing around with at least some new ideas. My physicist friends will forgive me for the following, I trust.

First, what is it about? The most inconceivable notion of quantum mechanics, namely its proposal that micro-particles (photons, electrons, atoms) sometimes behave as particles and sometimes as waves, is usually demonstrated by an experiment with two slits separating the source of the particles from the detection screen. Under normal circumstances, each well-behaved particle reaching the screen and leaving its print there should pass through just one of the slits. The location of its trace should therefore be independent of the presence of the unused second slit. That's not, however, how micro-particles obedient to the laws of quantum mechanics behave: they prefer certain places on the screen while avoiding others. Exactly like a propagating wave on a water surface they would form a separate wave-front behind each slit. These two waves then exhibit an interference pattern - at some points reinforcing, at others cancelling each other.

The interference disappears and everything returns to a classical order as soon as one of the slits is closed. Moreover, and that is where the mystery of quantum mechanics begins, instead of closing one of the slits it suffices to determine which slit a particle had used on its way. No interference occurs and such a particle suddenly behaves as a classical particle would; simply just because we have assumed it to be a classical particle.

The explanation used to be that an uncontrolled influence of the detector, coincidental with determining the trjectory of the particle, was responsible. Recently, a new way of determining the particle's transit through a slit has been developed, which the particles do not seem to mind at all. Nonetheless - the results are identical: just the information on the particles' path is enough to make the particle behave as a particle rather than a wave. Furthermore, it works even if the information is hidden somewhere for possible future reference! But wait - should we succeed in subsequently erasing that information (completely and irreversibly) the interference re-appears. Everything happens in accordance with the famous principle of complementarity ((and the paradox, probably already suspected by the reader, is only fictitious: it is impossible to guarantee beforehand a complete erasure of the information on a specific transit. Even the erasure obeys the quantum laws).

We still talk about the wave-particle duality, as it has been since the birth of quantum physics. I have always known what waves are; anyone who has splashed in water knows about waves. I have also always known what particles are: I used to play with marbles and to jingle pellets.

Are those experiences of any use, however, when contemplating unimaginable small scales which cannot be approached by any tools other than mathematical tricks and equations? Does the fact that laws of mathematics remain valid always and everywhere - above or bellow, near or far, in detail or in whole - imply that also my images and allegories can be fearlessly moved anywhere I choose?

Apparently not; at least not if the destination of my choice is the microcosm: the strange chameleon-like behavior of the local inhabitants prevents it. It is almost as if these little creatures paid close attention to where and when I was looking in order to change their faces accordingly. Well, if I really try, I can manage to digest that: my face in a mirror also changes depending on whether, and how, I look at it. As we learn today, however, the matter is even more complicated. Inhabitants of the microcosm seem to have conspired to behave according not only to the fact of whether I was looking or not, but also according to my intention to look, or even just my reluctance to pre-empt the possibility of a future glance!

Is there any other way of explaining the fact that a quantum mechanics system can distinguish whether unused information is still hidden somewhere or whether it had been honestly and definitely erased from existence, and to behave accordingly? Well, here we have probably met with a key expression which is essential for our understanding of the strange behavior of the microcosm: the concept of information.

All natural conceptions of information require the terms "on", "about", "for whom". The every-day information - about weather, stock exchange, speed of aircraft - relates, with its "about what", to our natural world, i.e. the macro-world. By its "for whom" it relates to the inhabitants of the said macro-world, e.g. to us humans. It is entirely different in the case of information pertaining to the micro-world, e.g. on the presence or absence of a photon. In the case that we perceive such information as a message being intended for us, it becomes a link between the two entirely separate worlds. A link between the world of the photons, where it was written, and our world, where it is read. Who knows what adventures such a message had to experience on its way from there to here; what traps it had to avoid.

As I said, quantum physics could be considered a scientific enquiry into the inaccessible. Its success depends on its ability to transpose events occurring in the inaccessible world of extremely small things into the accessible world of big things, of measuring instruments, of ourselves. That is a skill bordering on magic, and thus we shouldn't be surprised by the stubborn reluctance of quantum physics to share its tricks with us.

(1992, 1995)



"Death is our eternal companion," don Juan said with a most serious air. "It is always to our left, at an arm's length."


If science is not omniscient, it obviously must encounter some problems on its frontiers. Here I don't refer to problems with supernatural and mysterious things, but rather to situations where no general consensus has been reached in regard to either a feasible method of enquiry, or in the way science could be called upon to help in making important decisions. The former, i.e. the search for an appropriate scientific approach to be used when a true explanation of a certain reality is sought, is the realm of epistemology, which deals with the Truth.

Let us contemplate, another question: whether, and how should science influence human behavior. That is the realm of ethics, which deals with the Common Good. And it is in this realm that we are overwhelmed by problems. The most prevalent and the most emotionally charged are problems related, directly or indirectly, to something which is on one hand ever-present and feared, while sometimes liberating, and on the other hand incomprehensible, namely death.

A source of our problems is the lack of a "value" which could be assigned to death and which would make possible comparisons with other things, such as personal suffering (controversy on euthanasia) or the security of others (controversy on capital punishment). We cannot agree on whether death prior to birth is truly death (controversy on abortion) and what degree of interference with the human mind constitutes death (controversy on brain surgery). We don't know how to relate the death of a human being to that of an animal. Science is at a loss in these matters; it either lacks any arguments or it has so many that any solution could be validated by them.

Euthanasia is the termination of life (a murder, if you prefer) of somebody suffering from a painful and incurable disease. It is usually assumed that such an end of life is brought about gently by a qualified physician, and with consent or upon a direct request of the patient. No wonder that a variety of opinions on euthanasia have existed ever since antiquity. In today's Holland, for example, while euthanasia remains in principle a criminal offense (punishable by prison terms of up to 12 years), the Dutch courts yielded to precedents and established a precisely-defined set of conditions under which a physician may be exempted from criminal prosecution.

Euthanasia pertains usually to the passing away of somebody whose life is already behind him. Abortion, on the other hand, is the death of somebody (or something) with his (its) life still awaiting. It is therefore quite typical, that those involved in the abortion controversy usually call science to their side. The problem here is not whether to kill, but whether it may be considered killing yet; some expect the science to express an opinion on the "yet". What will science say? It is difficult for me to imagine that one or another scientific definition of life could decide on the question of what is sinful and what is not. Furthermore, I cannot imagine any scientific definition to solve any problem related to an area where Good and Bad compete for our favour.

It is customary to assume that among living organisms only humans are capable of knowing about and expressing opinions on life and death. Why then are we horrified by the infanticide carried out by members of same animal species or even by parental animals themselves? The theory of natural selection offers a variety of believable explanations in specific instances. For example, a new leader of a monkey harem gets rid of the offspring of his predecessor in order to make it possible for the females to devote their undivided attention to only his, i.e. the superior descendants. Such explanations quench our scientific curiosity and reassure us about the benefits of such behavior for the species; nevertheless a slight hesitancy seems to linger on in our mind (mind you, probably not our scientific mind).

The theme of aggression as such is certainly a most interesting subject which lies within the frontier region of our discussions on both nature and human society. True, even this theme lends itself to some "reasonable" explanations. Take the theory, for example, that hatred among individuals of the same species results in their dispersion over a large area and therefore makes them less vulnerable to a natural disaster. I am not sure whether or not that's indeed the case. I would probably consider such a scattering of individuals only one of the side-effects of hatred; after all, simple reclusiveness would result in the same benefit. Something different is perhaps at play here, something like a dynamic equilibrium between hatred and love, between evil and good. Fairy tales certainly teach us this.

What does science have to say about it all? The reputed ethical neutrality of science is actually considered to be evil by some. I would suggest the term "dangerous" rather than evil. A fire is not evil, it is dangerous. "Good" or "evil" refers to somebody who exploits the fire; not the one who fuels it.

In its description of death science would most likely characterize it as the termination of a certain process. If a process is faulty, it may be advantageous to stop it. There is, however, an intermediate step between such a general statement and the specific termination of a specific process: we, who must make the decision. Neither science nor law will ever alleviate our responsibility because the decision is ours and ours only.

Each of us is blessed by the awareness that death, our own death, is our eternal companion . This knowledge is the only privilege we have. "It is not the outside circumstances but ourselves who could make death to be such, as it ought to be: a willingly accepted end of life" wrote D. Bonhoeffer. To take a man's life means to rob him of his death. On that, science is silent.

(1992, 1995)



Democracy is a political system which uncomfortably hovers at the **border between chaos and order; let us add that the politics of chaos is anarchy and the politics of order is totality. In the former, everybody has a say about everything and does whatever he wants, in the latter everybody obediently keeps quiet and does things he doesn't want. At the border separating the two systems is an interesting area of confluence of opinions; [**cf. manuscr.] it is those opinions which should constitute the essence of democratic election campaigns instead of the usual electioneering practise of buying loyalty by promises of offices.

Conflicting opinions are not only natural but necessary: if everybody held the same opinion we would freeze up in an ideological totality. At the same time, it is necessary that all the discord is overridden by a certain higher agreement, or consensus, within which all individual opinions flow together.

The systems exhibiting behavior which move along the border separating chaos from order posses many interesting characteristics. As an example, we can consider some small local anomaly (e.g. a mutation, idea, intent, slogan, joke, virus) which has the potential to spread; particularly if it is timely and lucky. At the same time, we have to recognize that an anomaly could hardly spread throughout the whole system, particularly if it is untimely or unlucky. Another crucial characteristic: some systems with behaviour at the edge of chaos seem to become evolutionary attractors: evolutionary pressures continue to keep such systems at the edge.

As demonstrated by a well-known American biologist/physicist/mathematician S. A. Kauffman on networks of simple logical elements, the above notion can also be applied to artificial mathematical models.

It would not surprise me in the least if the human brain itself balances at the edge of chaos. The brain also generates opinions, ideas, thoughts, perceptions, images and decisions, all of which are known to exhibit at least some sharp conflicts and disagreements. In a sane mind, however, there exists above these conflicts some higher consolation or consensus in which everything flows into harmony. A harmony perceived as the flow of consciousness. Such a consensus develops and disperses spontaneously throughout the brain without any central election committees, referees or witnesses, as pointed out by D.C. Dennett mentioned above. True, everybody thinks that he or she (via their own consciousness) is the only necessary referee or witness; that's everybody's private business (as Dennett would probably say), his or her own virtual reality.

Let us take another example. There exist conflicts even among us, scientists. Like in the brain or in politics, it could not be otherwise. Let us put the activity occurring in any scientific discipline under a microscope. We will see a wild dance and whirl in which individual scientists, teams, schools of thought, streams and paradigms collide, jostle, infiltrate, fight and pull, catch up and overtake each other. All of that happens just below the surface of a seemingly unified and smooth flow of knowledge; it is that surface flow which eventually becomes the subject of our education in schools, the content of popular literature, and which future historians will label an evolutionary period of a scientific discipline. Paying attention only to this surface flow, and ignoring the underlying conflicts, means dealing with collective consciousness of a specific scientific community.

Where is such a collective consciousness located, or better still, where is the grand canyon through which it flows? It is probably a multitude of different locations; it includes the minds of all of us when we think about, read about, or listen to a particular scientific discipline. During such times each of us becomes a miniature particle of the great current of collective consciousness. The degree of our contribution certainly varies depending on the individual and also on the specific time. Any creative scientist in any discipline serves as such a particle. For such a scientist, however, I would like to propose a double-role: his first role is simply being one of the many particles of the common consensus. His second role is to serve as a source of new discoveries, ideas, and opinions, with which he actively partakes in the conflicts and jostling buzz under the quiet surface of consensus.

These two roles do not have to be necessarily in harmony with each other. They do, however, influence each other. The second role participates in the formation of the first, the first role determines the behavior exhibited within the second one. It is like that also in politics. And in any other collective activity, provided that it is of a higher order than a simple sum of individual activities.

Some years ago I studied the newly developing field of neual networks (another example of a system exhibiting a collective participation of multiple elements). In the library, by chance I had opened a historical book on ritual dances. Its motto, a citation of A.T. Macrobius from the fifth century, caught my eye:

The voices of individuals are hidden in a chorus while the voices of all are heard, and thus a harmony emerges...(1992)



Why are horses afraid of humans in spite of being so much bigger? One of the many obtrusive questions with which even I, as a child, used to bother those around me. Somebody gave me a remarkable answer: horses have big eyes and therefore they see us as much bigger than we really are. I believed it for a while and then it started to annoy me. Where is the difference in size if everything else that horses see is also magnified? And what about, for example, the ants? Do they see the world smaller or bigger than we do? Most likely, you may say, they see what we would see while looking through a magnifying glass: an identical world shifted into another scale - the world of "Ferda - the ant".

Let us try to forget geometry and physics for a while and ask ourselves what is man's most natural space. The space of a wanderer, let us say. A countryside stretches in front of him in its width and length. There is even the third dimension of height; that one is, however, of no relevance for the wanderer's freedom of movement. The wanderer's world is thus essentially two-dimensional. Similar to that of an ant. Proximity is more important for a wanderer than great distances, if only because a nearby dog is of a greater concern than a distant bear. The proximity, taken as a measure, shapes and deforms the wanderer's space: it diminishes with distance and disappears completely behind the horizon. In addition, the proximity moves with the wanderer while he walks through a countryside: it is therefore not a measure of the countryside but of the wanderer's position within it.

The wanderer's mind gradually develops another picture of a countryside - a larger one, because it reflects his recollection of all other places he has walked through. The new countryside is no longer dependent on the wanderer's actual location. His memory has created it. Certain parts of his memory are clear, other parts are hazy. There are places overflowing with events and also places he has never visited. All of these images together create the wanderer's cognitive map of the countryside; in some sense it partially creates even his own self.

Let us imagine that the cognitive maps of many people are totalled up and combined into a single map in which any single deformation would be cancelled out. The result will be an "objective" map, which could be drawn, printed and sold to tourists.

With a bit of imagination we may repeat the above scenario for the world of an ant. A map for ant-tourists will emerge. It will differ from the human one principally in its scale. No big deal, we all know that maps may be drawn with different scales; large for ants, smaller for tourists, smaller yet for airlines. Furthermore, there is obviously an infinite number of possible scales in both directions, and there are scales that differ from each other only to an infinitely small degree. Consequently, there is nothing to prevent us from imagining all the scales to be on a common axis, similar to the real number axis. Let us call it a scale axis. And now we should let our imagination go completely wild and stack all the possible maps (two-dimensional) of a region, superimposed one on top of each other according to decreasing scales. The result will be a three-dimensional super-map with two space axes and one scale axis. (Try, if you could, to imagine three space axes and one scale axis). If we imagine moving "up" along the scale axis, everything will shrink or concentrate, diminish and eventually disappear; in the "down" direction everything will increase in size and become more diluted.

Our home is somewhere around the middle of the scale axis (around the 1:1 scale); here is our natural place, the realm of human size. We can, however, see a certain distance in both directions; something like a "proximity of the scales" and "horizon of the scales" applies even here. The world of horses and that of ants are relatively close to us. In order to see any further, we have to use either telescopes or microscopes. Even with these tools we could, however, never reach everywhere. We can only learn about those more distant places from the tales of theoretical scientists, who have the courage to talk to us about things nobody has seen and, worse yet, nobody will ever see.

I said "talk to us". That implies language; and there lies a hitch. Our language has developed as a language of human dimensions, i.e. a vocabulary reflecting upon a relatively small range of the scale axis. Our intuition also derives from that same area. Scientific theories which invite us on mental voyages into worlds unimaginably small or immeasurably large must be extremely cautious about their use of language. They must not fail to correctly identify situations when it is appropriate to use a natural language, situations when a metaphorical transposition takes us beyond our horizon of empirical experience, and situations when a new language must be invented.

We enjoy our voyages - sometimes travelling in the direction of the microworld of quantum physics, sometimes going in the direction of macroworld of astronomical dimensions of the entire Universe. In both directions we discover shapes and events which are new, mysterious and mystically beautiful. Our imagination cannot expand, however, to encompass everything; we always carry with us the burden of our limited horizon of reference. Looking at trees we do not to see the forest, and looking at cells we fail to see the tree.

There is an old Czech fairytale about three fellows. One is extremely tall, one extremely wide and one extremely sharp-sighted. If only we possessed the last fellow's story-book sharp-sightedness, we would certainly be able to simultaneously see molecules, trees and forests. We would even see shapes and events most mysterious: those which stretch along the entire scale axis and, at the same time, form together a single organic entity. But that's a subject for another time.




Let us imagine a large group of individuals which undergoes changes with time; something like a club or association, or to be up-to-date, like a political party. From time to time some members leave and some join. Let us presume that each member exhibits certaintraits, some of which are shared among many individuals and some of which are rare. Let us also assume that the traits of those who are allowed to join relate in some specific way to the frequency of identical traits among the individuals who are already members. At certain times entirely new traits may appear - produced either by an error or by a fusion of already existing traits - at other times some other traits may disappear completely.

Readers are probably confused at this point as to the intent behind the above less than precise and abstractscenario. Well, here it is: the Darwinian theory of evolution may be applied to any group which fits such a scenario.

Let us consider individual members of the same biological species to be the elements of our set. Some of their traits improve their chances to survive and consequently to have a larger number of descendants; other traits exert no such effect or even have a detrimental effect on survival. Traits are inherited and therefore the future number of individuals possessing a certain trait depends directly on the present number of such individuals and, in addition, on their ability to survive and to produce offspring. The latter dependency takes care of the expression of precisely those traits which increase survival rate. It is called the selection pressure.

In this form (the reader should excuse its simplicity) Darwinian theory is exclusively mechanistic: survival is reduced to the simple and irrevocable logic of natural selection. I confess that, being a living organism myself, I do not derive any particular pleasure from such a simplification. Are all my properties and abilities (including my current uneasy feeling) only the logical products of blind mechanisms of collective evolution? It seems that biologists themselves have some uneasiness in this respect and that biology therefore tries to gently shake itself free from the clutches of neo-Darwinism.

It is impossible to uproot the inherent logic of natural selection (at least when considering a group large enough on a time scale long enough for its evolution to take place). Of course, that does not necessarily mean that natural selection is the only exclusive driving force behind evolution. Theoretically, we can expand on it or we can combine it with any other concept: from the concept of directed mutagenesis to the notion of the survival instinct of Gaia, our living planet.

There is yet another point. Using the abstract scheme mentioned above, it is possible to replace the group of individuals and the set of traits by anything: each individual element may represent a group of living individuals, a community or a colony; it could be a whole ecosystem as well as a single species and even higher taxons. Traits may correspond to phenotypes or genotypes, either static or dynamic (as, for example, some patterns of behavior); they may even represent propensities towards certain traits and their variability. The Dawkinsian "inversion" represents an extreme in this context (albeit a logical one): traits may become "individuals" and their frequencies may become "traits". The mistake of orthodox (neo)Darwinism lies perhaps in its narrow application of the evolutionary scheme within the framework of only one from a possible multitude of entangled and interacting conceptual systems.

If our imagination does not fail us, it becomes possible to apply the theory of natural selection also to any inanimate system. There always seems to exist in every system a self-reassuring and self-preserving "ability" which is bound to improve under selection pressure. It could be the ability of an industrial product to be useful, the ability of the manufacturer to produce such a product, and the ability of the economical system to support such manufacturers. It may be the ability of a novel to become a best-seller, the ability of a cook-book to teach us how to prepare a delicious meal, the ability of a joke to make people laugh. It may also be the ability of an idea, a discovery or a hypothesis to succeed in the intellectual market-place, and to become one of the accepted "truths" of the educated class.

It would be extremely interesting to know more about whether and how the mechanisms of natural selection apply to ideas. For instance, the prevailing fashion of determining the value of research work by the number of citations it generates, even with all its associated controversies, seems to indicate that the survival of an idea is dependent on the number of other ideas which have been fertilized or influenced (either positively or negatively) by it. There even exist places where living scientific ideas meet, are introduced to each other, multiply, compete, defeat and cannibalize each other, all in clear view. Those places are scientific meetings. I'd like to propose these meetings as a starting point for anybody who would like to study the dynamics of evolution of scientific ideas.




We have already reflected on the idea of scale dimensions of the world: a coordinate axis where different points represent a measure of zooming in and out rather than a position in space. Such an axis serves as our guide when shifting our focus of interest. One direction leads us towards the micro-world, the other towards the macro-world.

The classical geometry (of Eucleidean space) is uniform not only in respect to the location but also in respect to the scale: its objects behave uniformly regardless of their size. Because of that, we can use our geometrical intuition in the limiting definitions of mathematical analysis (for example when defining continuity and derivative of a function). We can also invent shapes occupying an infinite number of scales without needing to change their structure or form - the ever-popular fractals.

Newtonian mechanics takes for granted similar uniformity in the real world: large objects must move according to the same laws as small objects. We have learned already, however, that the world is in fact much more complicated than that. In the micro-world our language and intuition fail us; our efforts to peek into the microcosmos from the vista point of our real world are heavily paid for by experiencing some very paradoxical situations. The world of cosmic scales is no different.

And how about time? I trust that the reader has already figured out that if it is possible to talk about the spatial scale dimension, it must also be possible to talk about a temporal scale dimension. We can envisage, for example, a hierarchic continuity of time aarrows, each on a different scale: one in microseconds, next ones in minutes, hours, years, centuries - and all the other imaginable scales below and above.

Similar to space, there is a range of time scales which are familiar to us: it stretches somewhere between fractions of seconds (the time necessary to recognize a friend) and decades (time after which I would still recognize him). Our consciousness is at work here, as well as the sum of our experiences. Man's space always has its "here", and his time always has its "now". Because of our background in geometry we have, under normal circumstances, a tendency to envisage both terms as points: one is located in space and the other in time. In the realm of small scales the "here" and "now" become blurry: where is my "here" in the realm of millimeters and my "now" in the realm if milliseconds?

Science is objective and consequently it does not pose such subjective questions. There are, however, other questions: how short must something last to be perceived as anevent rather than a thing? How long must it last and how little must it change to be considered a thing rather than an event? More generally, what span of the spatial and temporal scales can be objects of our examination? On what scales do processes occur related to these objects? Is it at all possible to separate time and its scales from space and its scales? Does not every existence occur in a space and does not every space exist at a time?

Any human enquiry tends to separate and classify its objects of study. We don't yet appreciate what an important role scales play in this process. Let us take the steam engine. It "exists" on a small spatial scale range (decimeters and meters), because it is in that range that it has been created and it is there that it fulfills its useful function. The fact that it fulfills its function only because of certain properties of molecules, does not change our every-day perception of a steam engine. Molecules are simply regarded as occurring on a different level of description than steam engines.

A level of description - that's an expression which deserves more attention. It is sometimes related to the scale dimension used (the example of a steam engine) and we take pride in being able to explain phenomena at higher levels by reducing them to phenomena occurring at lower levels (thermodynamics being a valuable example). At times we even refer to a whole hierarchy of levels of description. For example, in the case of living organisms.

Here we approach the key question - is it not its existence on a long continuous span of scales of space and time that is an essential characteristic of a living organism and what principally distinguishes it from a steam engine? Is this not also the major obstacle experienced in our efforts to explain the behavior of living organisms by simply reducing their description from a higher to a lower level? The function, consciousness and existence of a living organism should not be related to a preferred level of description (or scale). They occur at the level of molecules as much as they do at the levels of cells, organs, individuals, societies, and whole ecosystems. For a true understanding of living organisms it will be necessary to uncover influences, connections and interrelationships, oftenmutual, among the different levels - both close to and distant from one another.

Perhaps also the reverse is true: every natural structure (in contrast to an artificial one) that must, in order to exist, consist of similar influences, connections and interrelationships occurring among its different levels is, in fact, a sort of a living organism.

Often what is referred to is a grandiose organism endowed by a surprisingly perfect behavioral pattern of self-preservation, self-stabilization and self-improvement. It is Gaia - our mother planet. She also inhabits a great span of spatial and temporal scales. It consists of many symbiotic processes, from photosynthesis in cells of microscopic seaweed to the green-house effect, from winter gales and deep-sea salt water currents - all the way to even ice-ages alternating on the scale of hundreds of thousands of years [A 22a].

Regardless of how well such a scale-extended being spanning a multitude of scales and levels can take care of itself, it is still possible to harm it - the smaller and faster the intruder is the more treacherous he becomes. Life has survived on Earth under tolerable conditions for half a billion years - yet a few years would be enough for man to destroy it.




We know quite well that our, i.e. human, communication with the outside world is aided principally by our ability to see and, secondly, by our ability to hear. The remaining three senses (and a multitude of different "sixth" senses as referred to here and elsewhere) appear to be less important, to exhibit a lesser degree of differentiation and to be less frequently called upon. We sometimes even consider the remaining three senses to be atavistic rather than to be genuine characteristics of the most advanced and proudest of all creatures.

Even for the purpose of communication with each other, the homo sapiens has relied on hearing and (later - for reading) on vision. (By the way, does anybody know why people are more ashamed of impaired hearing than of impaired vision?)

Theatre is a metaphorical reflection of our world; in a theatre we quietly sit, watch and listen. Theatre is not, however, a good metaphor of the real world for many other living creatures. For some species, for example, scents and tastes play a much more important role than do lights and sounds. Among such chemicals the pheromones - chemical signals used by individuals of the same species to warn against danger, to announce territorial claims, to instigate aggressiveness or to pacify, to influence sexual behavior and to identify food - play an especially important role.

Communication by chemical means is found throughout nature; it occurs in simple organisms such as sea fungi and brown weeds, in insects, birds, fish, reptiles, mammals and even in man. Males of some sub-species of North American garden snakes mark their females to repel competing males. The bolasso spiders use the pheromone of female moths to attract the naive male moths who subsequently fall victim to the spider's harpoon of a sticky thread.

The reader should not take the following very seriously - my intention is to spread the wings of my phantasy a bit despite my lack of expertise in either chemistry or zoology. Let us imagine that the destiny of humans were slightly different. That the evolution created us half-blind and half-deaf, but retaining from "lower" life forms a perfect sense of smell and the ability to secrete a variety of pheromones easily distinguishable by our perfect olfactory system.

As far as the olfactory system is concerned, our phantasy is not that far-fetched: we still possess all the receptors, only our olfactory brain centers became somewhat lazy. Oliver Sacks, a well-known neurologist, described a patient who gained a phenomenal sense of smell because of some brain pathology. He suddenly became capable of distinguishing an unbelievable wealth of fragrances and stenches. His keen sense of smell made it possible to recognize even different moods in other people. (It remains questionable whether or not we might all subconsciously accomplish the same deed.)

What form would the "world as theatre" metaphor assume for such a being, i.e. for a man whose only highly developed sense would be his ability to smell pheromones? The universal matrix of meanings would surely change: we would "smell" each other (in other words, our olfactory sense would be much more prominent than our visual sense). We would preferentially recognize the smells of the members of our own species; everything else would become an unimportant, grayish and hardly noticeable background. Our loved ones would not appear to us as flesh and skin, but would be perceived as large fields of smells, delicate networks of traces, territories representing not only the present but also the past. We would probably perceive even ourselves as such fields of fragrances rather than bodies reflected by the "here" and "now". Our identity would assume external dimensions. It would become spatial and territorial, and would lose its dimension of time (inner, memory-related). The fields of smell of different people would fuse and overlap; the degree of overlap might even reveal the nature of people's relationships.

What is the purpose of my phantasy? Nothing more than to point out how dependent we have become on our view of the world which relies entirely on the senses bestowed upon us (perhaps only coincidentally) by Nature. The above speculation also teaches us to exercise caution when addressing "objective" facts while relying on our fragmented perception (so very much human) of the world around us.

There are two kinds of objectivity. One is the "absolute" pobjectivity, i.e. the much sought-after but unattainable God-like conception of the world yearned for by post-Enlightment science. There is, however, also an objectivity which emphasizes its distance from any specific, individual, and therefore always somewhat biased subject, while operating within the framework of the universe visible to the human eye. The latter could be called "inter-subjectivity". Scientific disciplines, and scientists working in them, sometimes fail to recognize which of the two types of objectivity they discuss.




We have already distinguished the two concepts of objectivity (the absolute and the intersubjective). Neither of the two concepts could encompass matters which are completely related to a subjective inner experience of an individual. All of us are well aware of such things because we ourselves live through them. We have already discussed the feasibility of scientific enquiry into consciousness, i.e. into that part of our subjective experiences which make it possible for us (I should say "me") to even perceive life as such. Would we encounter similar difficulties when addressing some more specific functions of our mind, such as thinking, reasoning and inferring?

Yes and no. It is true that the inner experience of thinking is the intimate priviledge of the one who thinks. On the other hand, it is possible to share with others the results and even the process (under normal circumstances) of our thinking. Together with others we may even analyze the thought process and to evaluate its correctness. A long time ago Aristotle proposed the fundamental principals of logic - a science on thinking. More precisely, a science on correct thinking. More precisely still, science on logical thinking, where the term "logical" is used in its common meaning (e.g. logical behaviour in the sense that it is understandable).

I have always held logic, both in its scientific and common meaning, in great regard. A feeling for logic often makes it easier for us to comprehend complicated matters, to better communicate with others, and sometimes to purposely mislead them. It could also, however, confuse us. The common use of language is associated principally with the meaning, i.e. with the topic of discussion, with the "why" and "how" a subject is discussed, rather than with the form and correctness of what is said. A girl, who asks her boyfriend in front of a jewelry store display "Isn't that a beautiful ring?" is looking for a glittering gift rather than for a correct answer, such as "I have seen better" or "I am not an expert".

Logic always extracts one aspect of human expression (and hence of the thought process) and purposely limits itself to it. It is the formal aspect (as opposed to the content aspect). Instances when logic concerns itself with content occur only when the content may be objectified and formalized - and therefore extracted - from situational contexts in the real world under specific circumstances and thus from our inner experience.

I am far from blaming logic for this limitation - it can not be avoided while retaining objectivity and trying to achieve a common, communicable and lasting body of knowledge. Students of logic should not, however, entirely forget all that was put aside, assuming that it has no further relevance for correct thinking. A principal question has not been answered yet: is there always a sharp division between the formal aspects and the content aspects? We should at least examine situations where the divide is blurred or evasive.

Attractive tools for such examinations are certain paradoxes and word-games. Note, for example, that the statement

"A good wine is not cheap"

is logically identical to the statement

"A cheap wine is not good",

but each statement actually has a different meaning. Another example (while talking about wine):

"There is at least one man about whom it holds that if he likes wine then everyone likes wine".

It sounds strange, but it is a pure logic.

The most famous is the liar's paradox:

"This sentence in quotation marks is not true"

If the above sentence tells the truth, it lies, if it lies it must be true.

Contradiction is the principal enemy of logic - yet, I dare to say, healthy thinking must afford itself the luxury of contradiction from time to time. For example: I am sure that among all the statements which I believe to be valid there must be some which in fact are not. Even those, however, belong among the ones I still consider truthful.

There is a scientific discipline to which logic is easier to apply than to anything else. It is mathematics, a science completely formal in its principle, and therefore immune to accidental events of the natural world. Logic dealing with mathematical thinking (and itself using mathematical methods) is called mathematical logic or metamathematics.

Among the principal objects of attention of mathematical logic are the concepts of proof, of (mathematical) truth and of formal theory. Proof is a formal procedure which allows deduction of propositions from other propositions; if the original ones were considered valid, we should (even must) consider the deduced propositions to be valid as well. It is possible, for example, to select a few elementary "truths" on natural numbers and to use them for defining a formal number theory as a collection of everything that can be proven by the initial truths.

No such number theory will, however, contain all truths about numbers (i.e. everything which is valid about real numbers). That is the essence of the famous Gödel's incompleteness theorem - incidentally a fine example of a contribution to human knowledge made by mathematical logic.

Mathematics is a formal science and its concept of validity is purely logical and thus open to examination. Other disciplines find themselves in a more delicate situation when examining the validity of theirstatements. They strive to observe, analyze and measure the real "external" world while being armed only by tools invented by man. They strive to interpret, explain and evaluate their observations while using nothing better than a language shaped by human experience.

Obviously, the above should not imply that methods of logic are not applicable to such disciplines. Logic belongs to the culture of every science because it belongs to the culture of thinking.




A student of geometry faces a page of paper covered with lines of various shapes, lines both straight and curved, lines intertwined and crossing each other at different points. His eyes rest on the picture in front of him. His vision, however, penetrates the picture and ventures out of the real world into the world of geometry.

This is the opening of the first chapter of reflections on geometry by Petr Vopenka. Anybody who has ever been at least a bit interested in geometry, even in its high-school variety, must have felt the gentle touch from the mystical world of fairytales, where one finds himself as soon as he steps over the wandering root. It is a world which seems to be eery in its simultaneous emptiness and fullness. In it, some crystal-clear objects, expressions and truths materialize themselves out of the darkness; on the one hand these objects seemingly obey our rules while, on the other hand, they obviously live their own, independent, ever-lasting and untouchable existence.

From all the theoretical sciences, geometry should be the most congenial one. In geometry one penetrates through the wall of reality and escapes into the world of concepts and constructs while still being anchored by one's imagination and natural experience. Educated and elated, a traveller to the world of geometry returns back to the real world only to realize how geometrical the nature of our reality truly is.

Throughout their existence the two reigning queens of sciences, geometry and physics, have always had romantic affairs with each other. In the beginning they perhaps just admired each other, envious of characteristics of each other - concern with dynamism of one as opposed to the stability of the other, concern with material reality of one as opposed to immateriality of the other.

It would be worth a separate reflection to consider when and how geometry has helped a despondent physics as well as when and how physics has exploited even the strangest whims of geometry. Today, many consider the former to be part of the latter, many vouch for the reverse.

How shocking it must have been for scientists to learn at the beginning of the last century from the discovery of non-Eucleidean geometry, that there may be distinct geometrical worlds; worlds peacefully co-existing yet mutually incompatible. Which one of them, if any, best corresponds to the space we inhabit?

Some readers probably consider geometry to be outside their sphere of interest and believe that a study of geometry is no longer "cool". I recommend wholeheartedly that such readers take an excursion into the world of geometry at their earliest convenience. It will certainly not be effortless: it will confront you with the need not only to reconsider some important assertions, but also to re-think the evidence supporting their validity. To be absolutely honest, I must agree that some proofs may sometimes be skipped without affecting significantly our superficial comprehension of a subject. In spite of that, I do strongly recommend that the reader does not only read but also contemplates at least some of the arguments.

It can be said, and it certainly is my own experience, that it is almost impossible to read a mathematical proof without contemplating it. A mathematical text differs from a normal text: the latter may be read without any concentration and the meaning of its words and sentences seems to carry the reader along under its own power. To think about a proof, on the other hand, requires us to slow down, to reach for a pencil and a piece of paper, to draw lines, and to continue reading only when we become certain that what was written does indeed make sense. If a proof is well presented, I find it quite easy, while contemplating any single sentence, to recognize the exact moment at which I become convinced. I can usually almost hear the sigh "I see!". I urge you to try it, my patient reader.

There will be rewards: e.g. a discovery that even the unexpected may be possible. The fact that two non-parallel lines can share a perpendicular line, for example. A straight line may have an infinite number of parallel lines passing through the same point. Non-Eucleidean geometry, in which such phenomena occur, reminds us how easy it is to be derailed in our thought processes by our customary intuition. How easy it is to consider something possible at one time and impossible at another time, without ever trying to support our opinion by a precise method of proof.

Lines on a paper. Magic lines on paper. Lines magically creating seemingly impossible worlds. Worlds of multidimensional spaces, spaces that are crooked, curved, bent, twisted and riddled. Spaces which make it possible to disappear at one point only to simultaneously re-appear at another. Spaces which on an extremely tiny scale resemble foam and others which acquire additional degrees of freedom. Spaces which have swallowed time and whose geometry is shaped by matter and measured by light.

Which one of those spaces provides shelter to our real world, the natural world we live in? I know of no other scientific discipline in which imagination diffuses into reality as much as it does in geometry. I know of no other discipline which would ask with such depth of authority the question "Where are we?"

Let us recall the warning sign above the gate of Platon's Academy:"Let no one enter who does not know geometry!"

(1993, 1995)



Almost anything in nature and society which is complex, and therefore interesting, exhibits a common characteristic: it consists of a multitude of elements or components which either influence, support and complement each other, or which fight, push and stamp each other out. It may be atoms in large molecules, molecules in matter, matter in continents. It may be cells in organs, organs in organisms, organisms in species, species in ecosystems. It may also be citizens innations, nations inregions. It may be elements in logical circuits, circuits in computers, computers in communication networks. Individual data within scientific hypotheses, hypotheses in scientific disciplines, disciplines in the body of knowledge.

In spite of continuing interest by system theoreticians, students of synergy, statisticians and other interdisciplinary specialists, there have yet to be described any universal principles providing us with a unifying view of the above mentioned variety of collective systems. I cannot help but believe that such principles must exist. I am also certain that, when uncovered, we will be astonished by their simplicity. Finally, I don't think that we have to wait much longer.

Statistical physics seem to be a discipline most advanced in that direction. Ludwig Boltzmann and his followers have already created in the last century a conceptual system which makes it possible to talk about characteristics of macro-systems (systems which surround us) in terms of the common or collective properties of their elements (atoms and molecules). The principal trick used is the following: if the immense number of individual elements prevents us from dealing with a complete understanding of the so called micro-state (determined by the state of each individual element), never mind - it suffices to step away, look from a distance and allow the immense number of micro-states to fuse into one macro-state. If done properly, nothing gets lost (nobody will ever see any micro-state anyway) and a better understanding of reality may even be achieved.

Let us try the following simplified example. Consider a system of molecules and let us assume that each micro-state of that system relates to one of all possible distributions of molecules among imaginary compartments of a container (in order to understand any micro-state it would be therefore necessary to know which molecules are in which compartment). In contrast, one macro-state shall be determined by the relative occupancy of the compartments (and therefore all we have to know in this case is the total number of molecules in each compartment). By the spontaneous movement of molecules from one compartment to another the system would randomly drift from one micro-state to another. In other words, given an infinite time span, all the possible micro-states - the different distributions of molecules among compartments - will have statistically an identical chance or probability to occur. As a result, those macro-states corresponding to a larger number of micro-states will become statistically more probable (we say that such a macro-state exhibits a higher level of entropy); the reverse also applies. For instance, the macro-state with all the molecules occupying only one pre-determined compartment is associated in our system with only one micro-state; it is therefore extremely improbable.

From the outside the above system will be discerned by exhibiting a gradual shift away from the improbable macro-states (if indeed the system ever got there in the first place, which is another issue) to the more probable ones, i.e. it could be described as a system of an ever-increasing entropy. The reverse evolution - i.e. a collective drift of the molecules towards one compartment - even though it is logically possible - is immensely improbable (particularly so, if it lasts for an observable time span). The asymmetry of nature in its time dimension may therefore be perceived, at least in this case, in terms of an emergent phenomenon of the macro-world. As a matter of fact, this phenomenon is to a certain degree determined by ourselves since it reflects the way we view multitudes of micro-states as single macro-states.

An interesting example of topics addressed by statistical physics are changes occurring in the overall order of a system - the so called phase transitions. The best known phase transition occurs between solid, liquid and gaseous phases of water. When ice melts a minuscule temperature change causes the macroscopical properties (clearly visible) to undergo profound and abrupt changes. At the same time, the molecules and forces exerted among them, remain identical.

Grimmett in his book on percolation presents another intuitive example of a completely different phase transition. In the spring the leaves of water lilies covering the surface of a pond are small and therefore inadequate for a snail to use when crossing to the other side. As the leaves grow, one day they conglomerate together as to form a continuous path across the pond. (With an identical original density of the leaves and an identical rate of growth, that path is completed each year on the same day.) Even here the individual characteristics (size or density of the leaves) change very little from year to year and from day to day, even though the radical global transition (important for a snail) occurs suddenly in a single great leap.

A phase transition is a typical example of a "trans-disciplinary" phenomenon: in each case it involves the collective behavior of a great number of elements. The nature of the elements may vary from case to case - not only atoms or molecules, but also for example leaves of water lilies, burning trees, rabid foxes and indecisive voters.

The way in which the behavior of individuals projects into the characteristics of a system depends, among other parameters, on the nature of interaction or communication among the individual components. I often happens that each element "sees" only the behavior of a few of its immediate neighbours and therefore only those behaviors influence its own behavior. Then any distant interaction can only occur indirectly, via a sequence of neigbouring elements. An important parameter is usually the (average) number of neighbours surrounding a single element. A larger number makes the system generally more permeable. In that respect, the dimensions of the system are obviously important: in a multi-dimensional network the number of neighbours (under an appropriate definition of a neighbour) is always higher than it is in a two-dimensional network. For instance, if a three-dimensional space were filled with inflammable things, fires would spread much easily then if they were displaced on a surface (as trees are in the forest), even if the probability of fire jumping were the same.

In one of the previous chapters we discussed some interesting phenomena occurring in collective systems at the borderline between chaos and order. Such phenomena occur even in human society. Chaos and order, it can now be said, are in fact two phases of a system comprised of many individuals interacting with each other in some way. Undoubtedly, there are many people and they do interact with each other. It would certainly be a simplification to overlook individual differences among people (differences in behavioral patterns), and to disregard the fact that people communicate with each other using a language based on meaning; in any case, such simplifications would only be a first step. We do know, for example, that there are two rather well-defined kinds of recognition relationships: some individuals know each other, some do not. If we were to prevent any form of communication other than among acquaintances (as was the case in our country under the communists regarding any communication on politically-sensitive topics) then natural clusters of mutually informed individuals emerge.

I shall leave it to the reader to ponder the characteristics of such clusters and, more generally, to decide what common characteristics could be recognized in the behavior of, e.g. bacterial colonies, beehives and human societies.




When the primeval matter had congealed but breath and form had not yet appeared, there were no names and no action.Kojiki(Japan, 8th Century)

Whenever we were lectured in school on the evolution of life in terms of the blind, stale, and in fact very primitive and transparent mechanism of selection of the most advantageous errors, I used to envisage the beautiful cobwebs of a cross-spider. What an elaborate sequence of genetical errors had to happen in his predecessors to teach the cross-spider how to build such fantastic creations! How naively helpful the flies must have been who let themselves be caught in the spider's first clumsy trial cobwebs; it must have been only because of the generous collaboration of these victims that the spider ever had a chance to survive and thus to undergo further genetic errors!

True, the earlier imperfect cobwebs could have served other purposes than to catch flies - even a single thread may become a tool of survival - just ask any mountain climber. Then again, how many such side benefits would have to exist in order for the cross-spider to diligently and profitably invest his energy into the development of a complicated weaving mill carried on his belly.

The old myths of the Far East recognize three principal attributes of life: matter, form and breath. Materialism in its most primitive version recognizes only matter, while the latter two are taken just as side effects or coincidences. A more sophisticated type of materialism acknowledges that the form (of atoms and molecules) provides matter with properties enabling its creative forces. Another, even wiser variety of materialism proposes that the primeval matter already contained (by some hidden way of encoding) all of its future forms.

What then is the third attribute - the breath? As it is a norm in old myths, a large variety of interpretations is possible (even if the importance of breathing in yoga exercises is put aside). The concept of breath is metaphorically linked with ideas of movement, life,action. It also brings with it a concept of mutual belonging: breath silently and tirelessly connects every living creature to the surrounding world.

Breathing in, inspiration, symbolizes an intent, breathing out, expiration, symbolizes relief. Breathing involves (to the same extent) both the consciousness and unconsciousness, both the willingness and unwillingness, and both freedom and necessity. It serves both the body and the spirit. It also fuels the voice - and consequently speech and song. In summary - the breath represents those attributes which could be created neither by matter nor by form.

It would suit me quite well if the triad of matter, form and breath would apply not only to living organisms, but would also represent, somewhat symbolically, the three facets of everything that exists. Breath would symbolize the content, purpose, sense, reason, intent - in one word, a meaning. Meaning, which diffuses throughout everything of form (because form gives matter its meaning), and which is creative (because meaning gives matter its form). It is not a coincidence that such a symbiosis of form and meaning reminds us of the symbiosis of form (syntax) and meaning (semantics) of language.

I began with a cobweb. A cobweb appears to be almost a pure form: who would seriously talk about a matter of something three meters long and weighing less than a milligram! All we see is the form - and what a form it sometimes is! For a spider, however, the cobweb's major attribute is its purpose: a cobweb is principally the spider's hunting weapon. We may even consider a cobweb to be an extension of the spider's body - in the same way as tools and other devices are extensions of the human body.

In spite of all of that, a cobweb represents an exemplary interplay of matter, form and meaning. Its typical form derives from properties of the matter (which provides the fibres with a remarkable strength and elasticity). Its form is responsible for the cobweb's ability to fulfil its purpose or meaning. In reverse - its appropriate form is determined by its meaning or purpose, and the form itself enforces the appropriate characteristics of its matter. Even we, non-spiders, may sometimes guess the cobweb's purpose from its form. We say that the form contains information (in-formation, i.e. the process of forming, shaping into).

The form and meaning make it possible to recognize and differentiate objects. Recognition of objects is in fact linked to naming them. Without the former there would be no latter, that is obvious. The reverse is also true: without words there would be no recognition. Development of terminology always goes hand in hand with creation. It serves a double purpose: on the one hand different names distinguish different objects (sometimes the assignment of a name itself creates the difference), on the other hand a name establishes (or even creates) a lasting identity of unstable objects. (The second case applies principally to generic identities, rather than to any individual identity: for example, our cross-spider taken as a species. That was probably the main reason why Adam bequethed living creatures with different names.)

Thus form and meaning differentiate things from each other. They are, however, attributes not only of things but also of action. The majority of cosmological myths consider the primeval whirl to lack both form and meaning - therefore there were "...no names and no action".

So far I have failed to mention one thing: a perfect cobweb is beautiful.




In the winter people sometimes complain that it really is no winter at all, and at the beginning of summer they sometimes complain that the summer is already gone. If it were not for the weather and its whims the Brits would be even less verbal and the physicists would lose an exemplary illustration of a deterministic chaos. At the same time, if it were not for the weather and its regular seasonal rhythm the Japanese poets would be left with nothing to glorify and humanity would be deprived of the direct and vitally important experience of a cyclical order occurring on a scale of years.

I would like to attempt to imagine how a human being, let us say of the Stone Age, would perceive the world and its order. To be more precise, how would I perceive it, if I were such a natural being, unburdened by any previously acquired knowledge but endowed with an emerging consciousness.

The first things I would recognize, if only because of their vital importance, would be two cycles: a day and a year. I call them cycles because I would probably perceive repetitions more keenly than durations. The shorter of the two cycles - day and night - I would probably correctly attribute to the striking round being which irradiates heat and light and which moves along its heavenly path with remarkable regularity. (It is perhaps because of the sun's travels that we customarily call anything which is repeated at regular intervals a "cycle".)

After some time I would probably also attribute the changing seasons to the same being (let us call it the Sun) - I would have noticed that the position on the horizon, from which the Sun wakes me up each day, moves in accordance with some higher order and it does so in remarkable correlation with the cycles occurring in the weather and the natural environment around me. (I assume that I would have been living in an open countryside - if I lived in a dense forest, I would have had to deduce my conclusions from the length of shadows.)

Note that in the previous two paragraphs I have used the term "attribute" when relating earthly occurrences to the Sun. I did not say "explain" and did not mention anything about causal relationships. My knowledge, as a Stone Age man, had been based on coincidences and correlations only, not on influences. Had I already mastered abstract thinking I would have chosen the cyclical order to be the basic principle of the Universe and a search for congruencies to be a major method of discovery. Efforts to explain coincidences, to uncover effects and hidden causes would be left to scientists of a far-away day; as would be the pleasures derived from such discoveries.

We can continue ad libitum in our make-believe games with Stone-Age wisdom. To celebrate the order of the world, we would erect massive stone structures which would allow the rays of rising Sun to penetrate its walls only on the solstice day. With great admiration we could glance into the night sky and uncover other cyclical occurences, from the phases of the Moon to the mystical travels of planets between different constellations along paths mapped by the Sun. An image of the Universe without repetition would be entirely foreign to us.

There are other astronomical cycles which a Stone Age man could have known nothing about. For example, the travel of the Sun around the center of gravity of the solar system (or, if you prefer, the travel of the center of gravity around the Sun), in which orderly phases alternate with chaotic phases. I am particularly elated by the hypothesis that the two cycles of this complicated movement of the Sun are somehow related to the cycles in the Sun's activity, and thus, indirectly, also to the periodicity of geophysical, meteorological, biological and even social events here, on Earth.

Why am I pleased? Because it throws a different light on some snap judgements about astrology; such as "It is absolutely unthinkable that planets could influence human destiny!" I very much dislike people who insist that something does not exist while "insinuating that their curses are based on strong and straightforward arguments". Well then, changes in the relative positions of major planets directly correlate (as could be mathematically shown) with the movement of the Sun around the solar system's center of gravity. Consequently, there is nothing absurd in an assumption that such a position exerts even a causal (albeit indirect) effect on events occurring on Earth (mercurially volatile state of the human mind not exempted).

I should emphasize that the above mentioned hypothesis, even if irrevocably proven, would never validate or invalidate astrological assertions. All I am saying is that it does, even in its yet unproven form, lighten the weight of the above quoted negative judgement.

Here I cannot help but make two small remarks, one directed to astrologers and one to their adversaries (I belong to neither group and I do not use the term astrology to refer to the sleazy boulevard variety). To the former, do not take it personally when we, scientists attack you from time to time. It is because we do not understand you and anything which we don't understand makes us irritable. And do not try to become a scientific discipline at any cost - that only raises our tempers. You'd be better off considering yourselves poets.

To the adversaries of astrology: You should not be surprised by the fact that astrologers do not seek causal explanations for their assumptions. They have no need for them: they perceive the Universe in its cyclical order and therefore they seek congruencies, not explanations. After all, isn't a mixture of cyclical order with a bit of chaos a common principle behind events of such an immense variety as to include movements of planets, quakes of earth, whims of weather, surges of tides, breathing of animals, outbreaks of wars and changes in my own moods? And please do not rush your condemnation. There are events which could well belong to the realm of superstition and yet they have a simple physical explanation. The fact that the highest tides coincide with a full moon provides a good example.



xn + yn = zn

Mathematicians are always excited by anything which is easy to ask but difficult to answer. Take, for example, the above equation and ask whether it could be satisfied for x, y, z, n, all in the realm of natural numbers (1,2,3,4,...). The question is crystal clear to everybody - failing to understand it would indicate a lack of even the most basic knowledge of addition and multiplication. To understand the question, however, does not mean to know the answer.

In our case, if n = 2, the affirmative answer to the question has been known ever since the days of ancient Greece. The simplest example, 9 + 16 = 25, represents the Pythagorean theorem for a rectangular triangle with sides of 3, 4, and 5 (other examples are 36 + 64 = 100, 25 + 144 = 169).

It was on Wednesday, June 23, 1993, just before lunch, that Andrew Wiles, a forty years old Englishman and a professor of mathematics at Princeton University, ended a series of lectures at a conference in Cambridge by presenting proof of a certain assumption, namely the 40 year old postulate on elliptical curves. It was perhaps because of his shyness, respect, or fear that he did not utter a single word in reference to something that was absolutely obvious both to him and to his keen audience: the fact that from this assumption derives Fermat's "Last Theorem", one of the most famous mathematical puzzles of all times. According to Fermat's theorem our equation has no solution in natural numbers for other exponents than for n = 2.

It was sometime in 1637 when a Toulose lawyer and amateur mathematician Pierre de Fermat scribbled on a page margin in his copy of Diophantus Arithmetic (a page where the Pythagorean theorem is discussed):

On the other hand it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, and generally any power except a square into powers with the same exponent. I have discovered a truly marvelous proof of this, which however the margin is not large enough to contain.He never wrote anything else on the subject.

Such a statement is an ever-lasting source of provocation for any mathematician. The greatest mathematicians have attempted to prove or disprove Fermat's conjecture and because of it they developed entirely new fields of mathematics, awards were offered, thousands of erroneous and false proofs were mailed to scientific institutions - and no answer. The most powerful computers were employed because a chance existed, that by discovering at least one solution for the equation, Fermat's theorem would be disproved. Numbers were fed into computers all the way up to n = 4,000,000 - and still no answer.

Andrew Wiles finished his lecture. Dead silence first spread throughout the lecture hall only to be followed by tumultuous ovations. The news hit the electronicmail. Anybody who happened to sit at his computer at the time learned in a few minutes the solution awaited by humanity for 350 years.

Mathematicians are not, however, easily persuaded. It was estimated that it would take about a year to carefully examine all of the steps of Wiles' proof. And sure enough, reviewers found an error. At first it seemed that the error could be easily corrected, but even by August 1994, during the International Congress of Mathematics, Wiles could not announce his ultimate victory. It took an additional two months for him and R. Taylor to complete the missing part of the proof.

Many a reader will perhaps say at this point, "Well, the question is answered, so what? What good is it?" Similar questions could be asked about any nontrivial results in number theory. What good is it to know, for example, the hidden laws of how prime numbers are distributed in number series? I hope that the reader will not ask whether there is a practical application of such knowledge but is interested in the reason why the natural numbers (and prime numbers in particular) keep human curiosity alive.

In the first place, as pointed out by many before, the natural numbers are the least artificial of all mathematical objects (and therefore the most natural - even though I could not help but feel that natural character somehow disappears in very large numbers). The fact that we can count the concrete things surrounding us is one of the only things we can truely count on. It is possible, however, to count even objects of questionable reality, be it quarks, black holes, or dreams. It is also easy to reach an agreement on the count of every-day things: a simple re-count solves any dispute. By examining natural numbers (particularly the small ones) we examine in fact the natural world. No wonder that questions about properties of numbers are easy to ask and difficult to answer!

It is entirely possible that Fermat's brain possessed a special ability to "guess" correct answers to questions about numbers with no need for the painful process of sifting through evidence. We know of other such brains, e.g. the brain of the East Indian mathematical genius Ramanumjan (1887 - 1920). It is said that "on rising from bed, he would frequently note down results and verify them, though he was not always able to supply a rigorous proof".

Lessrare, but even more fascinating are the brains of human "calculators". These prodigious savants play effortlessly in the jungle of immense numbers where a normal person would never venture without a pocket calculator. Curiously in this case genius meets in some strange way with mental defectedness. The afore-mentioned neurologist O. Sacks described the case of autistic and mentally retarded twins who amused themselves by reciting to each other six-digit prime numbers - eventually they managed even twenty-digit prime numbers. For them it was perhaps no more difficult than it is for us to recognize a friend's face in a glance. A question comes to my mind: is it possible to "see" numbers without counting?

Imagine yourself facing a heap of pebbles. Do not count them. Draw rather upon your inherited preference for orderly things, and start to organize the pebbles into a rectangle (several columns, several rows). Eventually you may discover that some pebbles could not be organized other than into a single row. That would happen whenever there is a prime number of pebbles. Who knows, maybe the Sacks' miracle autistic twins perceived numbers in such a way - like two-dimensional (perhaps even multi-dimensional) geometrical shapes. They could then distinguish prime numbers "in a glance", in the same way the majority of us could see in a glance, and without counting, which of the six numbers on a dice is up.

(1993, 1995)



"That's the beetle, the beetle

that's true the beetle that's not"

Jiøí Kubìna

As a boy I used to be a keen beetle collector. I hunted them in hundreds, pinned them down or glued the small ones onto paper labels. I got hold of Klapálek's Atlas and Javorek's Handbook, determined which family, genus and species they belong to and sorted them accordingly. I gave my hobby up only at the age when one begins to hesitate before killing a helpless creature. I found the process of determining and sorting to be interesting in itself. It was, in fact, the second phase of hunting - first to hunt a beetle, then to hunt for its name. It has never been clear to me, and I am afraid not to biologists either, what exactly a species is. Let us take, for example, the common rosebeetle, Cetonia aurata L. We all know it, it is a beetle, which

with its green color and gold or copper-like lustre pleasingly invites our glance when discovered on the bloom of a rose,

according to the old Obenberger's Natural History of Insects. There is, however, a large variety of rosebeetles - how does the one we look at differ from all the others? The green color and the gold or copper-like lustre are not its principal characteristics. Neither are the leaf-like widenings of its antennae, its digger's legs and its habitation on roses. All of those are characteristics of individuals which marely help in sorting them out into different groups, sub-groups, super-groups (such groups are generally called taxonomical units or taxons).

The most basic and most important taxonomic unit is the species. According to biologists an important distinction of species as taxons is the fact that individuals of the same species are capable of mating and producing viable and fertile offspring, whereas individuals of different species are not (provided that mating is physically possible). Furthermore the sex of the individuals should also be taken into consideration, however for our purposes a simple assumption that there are always two parents will suffice. Two aspects of the above distinction are of interest. First, in contrast to e.g. color, lustre and shape of antennae, it is a characteristic somewhat "more objective" and less dependent on what we see and notice. Secondly, it is no longer a characteristic which can be applied to a single individual but rather to an entire group: in order to be precise in our definition of a species in terms of a group of individuals, we would have to first examine each and every individual! (Fortunately nature shows its kind face to systematic biologists: individuals of the same species look usually more or less similar to each other.)

The above characteristics of a species do not provide a complete definition by themselves, should we prefer to be exact. It is, for example, impossible to guarantee the applicability of the reproductive criterion throughout the entire species - e.g. it is quite possible to imagine that one individual can successfully mate with the second, the second with the third, etc. but the first with the last in long enough a sequence would not mate at all (they may even belong to separate biotypes). On the other side of the coin, even inter-species mating can occasionally succeed. The situation becomes much more blurry in lower organisms, not to mention viruses.

In their striving for exactness the modern biologists turn to molecules which carry genetic information. How elegant it would be to define a species in terms of a sequence of nucleotides in a strand of DNA! Exactness which matches that of chemistry in its definitions of chemical compounds, but even that is of no help! When talking about biological species, of viruses for example, we have to acknowledge the fact that an unavoidable proportion of any proper population consists of mutants, i.e. of individuals with minor or major errors in their genetic code. I say "errors" just because of my inertia - in fact, they are not errors but variations - variations which in fact may, under adverse conditions, even save the entire population or species from extinction.

This seems an appropriate place to introduce the term **"quasi-species" coined by the German biochemist Manfred Eigen. I shall mention only one aspect of this concept. Namely, the way in which the term can be exactly described by statistical geometry which, in addition provides us with a possibility to form the term's intuitive geometrical image. The only pre-requisite is a certain abstraction of the term "space" which differs from the common Eucleidean space. On one hand we shall only consider a "discrete" set of points (rather than a continuum), on the other hand we shall define the distance between these points differently than it is usually done in our three-dimensional world.

Individuals ore genetically characterized by their genoms. Genomes are represented by sequences of nucleotides of four types called bases. These sequences may be relatively very long and therefore the number of all possible variants is astronomical - even if finite (we consider all formally possible alternatives, disregarding their biological feasibility). For example, the number of sequences of length n = 10 is 410, i.e. about one million (to compare: the length of genom of the HIV virus is n = ~10,000) To make our further discussion simpler, let us presume that there are only two types of nucleotides ( giving 2n possible sequences).

Our "space" (called sequential space) will have exactly 2n points; each point would represent one specific sequence, i.e. one possible genom. The "distance" between these points will be defined as the number of places in which the two sequences differ from each other. Each point would therefore have exactly 3n of immediate neighbours, i.e. of points at the distance of 1.

The points of equential space represent all combinatorially possible nucleotide sequences. Only a relatively small part of the space would therefore be occupied by genomes created by mutations of a chosen sequence. Consider a population in which different mutants appear at a frequency determined by their ability to reproduce in a stable state. Points corresponding to these mutants form an irregular "cloud" surrounding the point representing the initial chosen sequence (genom); each point within the cloud is also assigned a certain weight - i.e. the relative frequency of the respective mutant within the whole population. The weight distribution pattern would exhibit characteristics of a probability distribution with higher values typically located near the selected sequence, lower at the cloud periphery. Formally, a quasi-species is nothing else than such a cloud with the related probability distribution. It suffices to talk about the probability distribution only - the shape of the cloud is determined by its non-zero values.

If it were not for the conservation (preservation) effect of the selection pressure, the points representing the mutants would behave like an ideal gas - they would spread out evenly throughout the entire sequential space at an extremely low density.

A change in the external environment or in the parameters of reproduction will be reflected by changes in the probability distribution inclusive of the shape and size of the cloud within the sequential space. A quasi-species can therefore evolve; that raises questions about its identity and individuality. To define a species (of organisms, molecules or anything else) extensionally (i.e. by listing all its members) is as useless as to define a walker by listing all people who walk. We have tried some other types of definitions at the beginning of this article and were equally unsuccessful.

It was mathematical statistics and geometry's power of imagination which finally helped us out of a tight spot. Why could not a species be perceived as such a "cloud" of points exhibiting non-null values of probability within some multi-dimensional space? Each point is one of the possibilities. The cloud is malleable - it behaves like a strange amoeba whose shape, position and density change dynamically in accord with changes in external conditions, variability and tolerance of its own errors. Within a long enough time, the cloud may change into something entirely different, it may shift, may break up into several new clouds, or even disperse and disappear completely.

It is almost like the cloud itself were an organism, "living" in that strange multi-dimensional space. And that is the Cetonia aurata L. It does not sit on a rose. That object on the bloom of a rose, which pleasingly invites our glance, is something which, during its life-time, provides a non-zero value of probability for one specific point somewhere within the cloud.




...a painter's glance sometimes touches the distant mountains and thence they may be depicted in the margins of his paintings; what the mountains really are, however, and what world stretches beyond them could only be told by someone who has climbed them.

J.R.R. Tolkien

The dome of the heavens is a meadow of stars no more. It has become a cosmic space - a space described by geometry and occupied by bodies consisting of solid matter; bodies behaving in accordance with the same laws as those governing falling apples and clock pendulums. Modern science has annulled the divide which used to distinguish the two orders - the heavenly one and the earthly one.

"The World Ages" whose beginnings we witness has made another major stride: man's foot has been set onto the Universe [A 39c]. The moon and the planets are no longer only distant bodies in the sky, they have assumed the attributes of a landscape. It has become possible to walk on them or to observe them in a close-up.

The conquests of the great explorers of the beginning of our century, such as Peary, Amundsen or Scott, were the North and South poles. Those adventurers were driven by their thirst for knowledge, a yearning similar to that of today's scientists and engineers who build space crafts and probes to be sent to the distant planets of the Solar system. Crafts as yet unmanned, but capable of providing us with images perfectly equal to eye-witness accounts.

How did the astronomers of the beginning of our century envisage the planets? I read with amusement the book "Heavenly Wonders" by Camille Flammarion, a French astronomer famous for his popularizations of science. On Venus, for example:

"That star makes its entrance in either the mornings just before daybreak or at evenings just prior to nightfall. [...and it is] the star of all of those who like to daydream at dusk, being it a shepherd returning from his fields or those love-birds who touch each other with their hearts and whose souls meet at night"

Nevertheless we must defend Venus against

"malicious tongues who insist that if Venus appears so beautiful from a distance, it surely must be ugly from up close. I am certain that no reader of mine, especially none of my lovely lady-readers, would share such an opinion: it surely is entirely possible to be beautiful both from a distance and from close-up; do you not agree?"

While the ever-changing Venus symbolizes femininity and a sense of beauty, the reserved Saturn became a symbol of stability and perseverance.

"Saturn was not favored by the old poets who of course could not suspect its grandeur and its wealth. It used to be considered, in the days prior to the discovery of Uranus, to be the farthest extremity of the planetary system and thence to be the coldest and slowest of all planets. It symbolized the God of Time who had been toppled from his throne and relegated into exile. Sorry are those souls who were born under his influence!"

Saturn had suffered even worse than Venus:

"there are those who look upon it through a malicious eye [...as to] a place where the punishment of bad souls takes place while the good souls float happily from world to world."

Let us hope, together with Flammarion, that

"the world of his is less dreadful than it may appear to the prejudiced eyes. It does not lack for wealth and if only we were bestowed with an opportunity to visit it, we might find it even more beautiful than Earth itself and thus would wish to forever reside in that royal and majestic kingdom" [A 39d]

Venus and Saturn, the two exceptional and mystical planets, have always drawn the attention of scientists, poets, dreamers and mystics to a much larger extent than has any other planet. (Venus became admired even by the language philosophers, its two renowned costumes, one for the evening and one for the morning, rendered it to become an eternal illustration of the fact that the denotation of a word is not identical to its sense.)

"If only we were bestowed with an opportunity to visit" Saturn or another planet, how would we feel? It is not surprising that in our research of the Solar System our desire for knowledge (devoid of all subjective perspectives and emotions) paradoxically fuses together with the ever so \human compulsion to rejoice our experience of an unknown and exotic landscape while viewing its strangely wrinkled mountains.

A while ago I visited the well-known Jet Propulsion Laboratory in Pasadena, California. As part of the tour, I was taken into the cosmic flight control room. A very interesting room indeed. It looks exactly as we know its twin in Houston from TV broadcasts: gentlemen in white shirts lounge around sipping a brownish liquid (called coffee in America), watching panels full of screens and screens full of numbers, taking down an occasional note or pushing a key on a keyboard.

These gentlemen are the explorers of our time. Invisible threads of radio signals connect them with several cosmic probes wandering among the planets; some of them already among the stars (the oldest one - Pioneer - has been out there in the Universe for 25 years).

Do these gentlemen facing the screens realize how incredibly powerful they are in spite of their insignificant size? A small pull on the invisible thread could change the behavior of an object several light-hours away! To be in a room where all of the threads converge was a strange experience even for me. In addition, my visit made me realize that we, the coincidental audience (and tax-payers), are taken very seriously here. There is a balcony provided for us with comfortable armchairs, and huge screens on the opposite wall are used to project pictures or video-recordings upon our request. Other rooms in the building serve as offices for the computer experts whose only task is to fish out anything interesting from the immense flood of data continuously arriving on Earth from the cosmic probes. This information flood has been accumulating in the memory discs of computers for a number of years without anybody being able to analyze the data in time. Who knows what future discoveries are already hidden in the discs! A solution to this problem has recently become available. Using the techniques of virtual reality, the data is transformed into animated video-films which create an illusion of a low-altitude flight above a distant planet's landscape. We fly around mountains and enter craters, we fly at different altitudes and velocities according to a pre-selected set of parameters.

Much has already been written on the usefulness of cosmic research; whether, how and for what good is it, whether directly or indirectly, practically or politically.

I still feel that there is something amiss in all of that. I am sure that it is terrific to know (with the greatest degree of precision) the paths, movements, sizes and densities of one or another planet with all its moons, to know the temperature of its surface and its nucleus, the chemical composition and the origins of its formations, and the dynamics of its atmosphere. I cannot help but feel that I would gladly exchange all that information for a single short walk on a planet's surface.

Meanwhile I sit in the auditorium of the Pasadena's laterna magica and while watching the screen on the opposite wall I experience a bird's flight over the bizarre wrinkled surface of Miranda, the smallest moon of Uranus.

And thus I would wish to forever reside in that royal and majestic kingdom.

(1991, 1994, 1996)



There has been a long and relentless debate among scholars and thinkers on the question of whether or not there is a substantial difference between the mind and the body, and if so, whether the mind and the body influence each other, and if so, in what way [A 40]. If it were true that the more competing theories we have the better off we are, we would surely be in great shape.

The situation is, however, quite confusing. The confusion creeped in principally because the words we use to refer to the mind are used in at least four different meanings: in reference to our inner and immediate experience, in reference to the behavior of others, in reference to processes going on in the nerve tissues of our brain, and in reference to artificial systems, e.g. computers. Furthermore, it is possible to view the mind-body problem ontologically (how something is), epistemologically (how we learn about it), or analytically (how we discuss it).

There is a persistent tendency in Western philosophy, established psychology, and brain sciences to reduce, eliminate, bagatelize or in some other way suppress the concepts of mind, soul, spirit and psyche, and to oppose any notion that these concepts could pertain to anything other than to strictly material nature. Science wants to explain things; the most preferred explanation is a reference to causal connections. Physics provides an excellent example; its reduction of everything to its basic physical phenomena exemplifies scientific goals. The primitive variety of reductionism (a reduction of everything to a whirl of atoms) is nevertheless already retreating, if only because modern physics itself has its own problems: the micro-world behaves strangely to say the least, and non-linear dynamics introduces chaos even into the macro-world. Reductionism experiences a crisis even within physics itself.

A well-known physicist, Silvan Schweber, recently pointed out a hierarchical stratification of the physical universe into separate levels represented each by its own idiosyncratic concepts and principles which are not reducible to neighbouring levels [A 41]. Interrelationships between the distinct levels exhibit emergent (i.e. non-causal) nature.

A majority of today's materialistic or physicalistic theories of the mind, whether they happen to be reductionistic or not, are based on an implied (often non-verbalized) notion that thinking occurs at a single and sufficiently complex level, well "above" the basic level of neurophysiological and biochemical processes. What remains to be uncovered by these theories is the nature of the relationship between these levels (whether it is causal, non-causal, emergent, deterministic, statistical, bottom-up or top-down, direct or indirect).

I don't believe that the above concept is correct, or, to be more precise, that it is the only one possible - there certainly are many other possible (and also materialistic) theories. Why should we chose the level of neural processes to be the truly basic one? Why should we place it either "under" or "above" the level of mind within some assumed hierarchy? Both these levels could easily be independently and differently related to yet another, deeper level. There are increasing numbers of scientists who support a hypothesis that such a deeper level could be a level where quantum physics already applies [A 42].

Further still, one may polemise about whether any sharp division of levels used in our concept of body and mind does in fact exist at all. It is my personal opinion that in order to understand life (and even more so when trying to understand the mind) it is unwise to attach ourselves to a particular single level. Doing so may be adequate, or even desirable, when trying to comprehend the physical world (see the above mentioned stratification of the physical universe into separate levels). The very existence and harmony of a variety of interacting processes on many levels - that of moleculs, cells, organs, organisms, societies, ecosystems - is after all the principal characteristic which distinguishes living organisms (and brains) from physical systems (and machines). If it were possible to label such an approach "physicalistic", it would result in a new concept of the mind. A concept described in terms of an emergent phenomenon of a higher order: no longer only a phenomenon restricted to a single level and supported by processes occupying a lower level (worse yet, a phenomenon which could be reduced down to be no more than a combination of those lower-level processes) but a phenomenon emerging from a global (and far from causal only) interaction of the entire hierarchy of different levels.

We have very little natural experience with such a concept; consequently it is difficult to discuss it and even think about it. Yet!




There exist innocent questions such as "What is a house?", or "What is a rosebeetle?", or perhaps "What is science?", "What is a game?", even "What is that?" (the last question requires us to point a finger). It would appear that such questions do not really presume anything by themselves; nothing more seems to be reflected by such questions than the fact of a somewhat elementary ignorance.

It is not entirely so. All of the above questions, and all of the similar ones, could not exist in the absence of the most common, and at the same time, the most treacherous little word: "is". Consequently, simply because they exist at all, they do make one assumption (at least under normal circumstances): they assume that something in fact "is", or that its existence is at leastthinkable. We therefore deal with something, which we can either name (house, rosebeetle, science, game) or can point a finger at. A thing which is worthy of an enquiry.

A question "What is that...?" may elicit a variety of different answers. Most frequently an answer will teach us the composition of the thing we enquire about, what it comprises, what is it good for, how it could be otherwise called, possibly how it can be distinguished from something which it is not. Only the "is" part of the question is usually, and undeservedly, forgotten. Because of that, we in fact never get (and actually never expect) an answer to an essential part of our question: how something is, what is its modus of being.

We do not experience much of a problem when asking questions about things; a house for example. A house has a **matter, form, sometimes even meaning. Other attributes, such as the fact that it has its origins, that it will last a certain time and that it does undergo changes, seem to play an inferior role. Let us chose, however, a different example - say the game. What is a game, such as chess? Is it a thing? You would be surprised how many "things" are around us which are not things at all!

Let us start with an important distinction (which in every-day talk becomes clear from the context): a game - let us imagine chess - as determined by a set of rules on how to play it (and, in addition, perhaps by theories on how to play it well) differs substantially from a particular and unique game (match) of chess, which can be described as a longer or shorter sequence of moves complying with the rules.

Another distinction: an abstract chess game (match) is one thing, and its actualization in a real time and space, and with concrete players under specific circumstances, is yet another thing. Similarly, it is possible to distinguish a game defined as a set of rules (and theories) from the identical game perceived in its historical context as something which appeared at a certain time, had evolved and could continue evolving as a historic or diachronical entity (by changing its theories and even rules). We have thus four components:

game as a set of rules game as historical entity

sequence of moves specific match

When talking generally about a game in terms of its existence, and when enquiring about how it is, what is the nature of its being, we have to pay attention to all of these four parts. The two on the right refer to the actual realization of the thing (in time and space), the two on the left determine formal and timeless characteristics of a game.

Of interest is also the relationship between the upper and lower parts of the above diagram. A completely different time scale applies to a game perceived as a historical process as compared to a game perceived as a specific match. The former is measured in years and centuries, the latter in minutes and hours. There is, nevertheless, a bi-directional interaction between the two levels: specific matches obey the theories and rules prevailing at the respective historical moment and, reciprocally, the rules and theory evolve over longer periods of time as a consequence of (many) specific matches. Because of these inter-relationships we have to recognize our distinction of different components only as a heuristic tool. They are in fact only different facets or manifestations of the sameentity, a "thing" which lacks the simplicity of beings, of things we can actually touch.

I have chosen the game as an example only. An identical situation is encountered in the case of many other entities which always express themselves at two levels - a coarser one which is less precise and corresponds to larger scales of time and space, and a more precise fine level which corresponds to smaller scales. Several examples: natural language (vs. specific utterances), biological species (vs. living individuals), science (vs. scientific discoveries), memory (vs. recollections), influenza epidemic (vs. flu occurrences), a fashion of long skirts (vs. long skirts), artistic style (vs. works of art), law (vs. legal acts), political system (vs. governments). On the lower and finer level we encounter specific, concrete and usually isolated occurrences or expressions of the entity in question, whereas at the higher level the entity itself exhibits a latent, non-material but lasting existence, stored in the properties, knowledge or abilities of a substrate.

It is by virtue of such a continuous nature of being that we may consider the occurrences at the finer level to be expressions of a certain single entity and therefore be able to even assign a name common to them.