God and the Mathematics of Infinity E-Book

H Chris Ransford

• Herausgeber: ibidem
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch
• Veröffentlichungsjahr: 2017

Seitenzahl: 267

Beschreibung

Drawing on incontrovertible results from the science and mathematics of Infinity, H. Chris Ransford analyzes the traditional concept of Godhood and reaches astonishing conclusions. He addresses humankind's abiding core debate on the meaning of spirituality and God. Using mathematics to explore key questions within this debate, the author is led to counter-intuitive conclusions, including some that had long baffled humanity: For instance, why does evil exist if there is a God? The book fastidiously does not take sides nor proffers opinions, it only follows allowable mathematics wherever it leads. By doing so, it makes a major contribution to an understanding of the nature of reality.

Leseprobe

ibidemPress, Stuttgart

Table of Contents

Introduction

PART 1

Which Nature of Reality?

Theologians

Scientists

Using Mathematics

Emergence

Defining Godhood

Many Infinities

In Search of the Infinite

The Attributes of Godhood

All-knowingness

Omnipresence

Almightiness

The Free Will Theorem

A Conclusion to Part 1

PART 2

He Said, She Said

PART 3

The Separate Realities Within Reality

Category B

An Omega Point?

The Engines of Growth

Historical Perspectives

The Math of Mystical Experiences

The Existential Question

Takeaways

Epilogue

Further Reading

End Notes

Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet überhttp://dnb.d-nb.deabrufbar. Bibliographic information published by the Deutsche NationalbibliothekDie Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet athttp://dnb.d-nb.de.Cover design: Alexey Byckov

ISBN-13: 978-3-8382-7019-7 ©ibidem-Verlag

Stuttgart 2017 Alle Rechte vorbehaltenDas Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung außerhalb der engen Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Verlages unzulässig und strafbar. Dies gilt insbesondere für Vervielfältigungen, Übersetzungen, Mikroverfilmungen und elektronische Speicherformen sowie die Einspeicherung und Verarbeitung in elektronischen Systemen.

Introduction

There is no consensus within society as to the nature of the reality we live in. Most hold that the universe was born from physical processes but don't quite agree exactly how, with some reckoning that a foundational Big Bang brought about by purely physical laws and events gave rise to our universe, and others suggesting alternative scenarios. Some insist that the universe was created by some ineffable Godhead but readily disagree as to when the act of creation took place, with common estimates and/or beliefs ranging from a mere few thousand to 14-odd billion years ago. There is no consensus either on the question of whether ours is the lone universe in existence, or if there are in fact other universes, embedded in a wider multiverse or metaverse.

Put simply, different people hold utterly irreconcilable views of what simple reality is. This lack of consensus also extends to the more abstract areas of human purpose, cause and destiny. Some subscribe to a more spiritual view and see their lives as part of a meaningful grander scheme of things, while others see life as the meaningless outcome of a long series of ultimately random events, circumscribed by laws of physics which got their start when the universe began through some foundational event, itself ultimately explainable by the laws of physics. Because this latter view of ultimately pointless lives playing themselves out in a random and purposeless universe is often perceived as repugnant, and also because many people have experienced at some point or another in their lives what they felt were deeply spiritual experiences, attempts at larger-than-life, transcendental explanations have been routinely sought throughout history. At different times and places in humankind's early history, these attempts at explanation have been retold and collated, and eventually often codified, thus giving rise to the many competing and often mutually incompatible cults and religions which have been with us since the dawn of mankind. Various religions and cults ([1]) still very much thrive in our modern world, and often dominate international events and narratives.

This book is an exploration of why, if there is such a thing as a Godhead, any possible objective approach to understanding It must begin by not bypassing the most factual and objective tool of analysis at our disposal bar none—provided we have first firmly established that tool's suitability and validity within the domains where we will be using it. That tool is pure mathematics.

This book first examines how and why the use of some numbers is a legitimate and indeed indispensable tool to any objective approach of both what Godhead may possibly be, and cannot possibly be, and how the mathematics of numbers can be safely used within its established areas of suitability. At first sight it would be of course natural to doubt that theology and hard science, let alone simple mathematics, would have any relevance to each other. But as I showed in recent a peer-reviewed article entitled 'Immanence vs. Transcendence: A Mathematical View', they do happen to have surprisingly direct relevance: simple math has the capability to incontrovertibly solve in a few strokes conundrums that have vexed Theologians for centuries (as the question in this paper did to wit, if there is a Godhead, is It present everywhere, or does It mostly keep to some hallowed place in space and time, some privileged Eden mostly removed and away from the rest of the universe?) Even whenever science based on simple math demonstrably cannot solve some theological question, as is sometimes the case, then this in itself is of huge interest. As we shall see, a remarkable instance of this latter case is the existential question itself: math by itself cannot possibly prove whether an infinite Godhead exists or not, so that the claims we sometimes hear that science either proves or disproves the existence of a Godhead cannot be supported (or more precisely, to be able to answer the existential question math would have to define the Godhead, in a definition that may not meet consensus.) The narrative briefly examines whether the attribute of some infiniteness should be inherent to Godhood, and then goes on to look at intriguing and often counter-intuitive results that immediately arise from using neutral mathematics.

Part 2 is a brief exploration of what math may say about variously held beliefs and assumptions. By way of illustration, it delves deeper into a few selected dogmas and tenets, said by a number of separate creeds to have been dictated directly by the Godhead. It examines in the light of mathematical analysis whether such are logically tenable or even compatible with an infinite Godhead, and whether, should we look at their logical consequences, some may not unwittingly lead to hidden contradictions and logical impossibilities. It then examines whether godlike Infinity can exist at all in any actual reality, and if it does, what role it can possibly have, and not have, in the mundane affairs of man.

Part 3 starts from an assumption that a Godhead consistent with mathematics exists, and analyses the inescapable consequences of that assumption—some of which may turn out to be quite unexpected, which will shed new light on a few old questions, including the vexed question of how, if an all-powerful Godhead exists, then why does evil still exists (math provides a straightforward and astonishing answer to that question.) It also looks in a new light at the continued prevalence of perceived mystical experiences throughout the ages.

Extensive use of end notes is made, which are used whenever some point calls for further context, buttressing or underpinning, but should its argument be kept in the main text, it would lead to a lengthy, somewhat narrowly specialized discussion and thereby risk losing the thread of the main narrative into a stray off-tangent.

A few passages in the main text are indented. They either consist of short passages quoted from outside sources, or alternatively of some brief background relevant to a point currently under discussion, but which may be however safely skipped without impairing the ability to follow the argument.

PART 1

Which Nature of Reality?

Theologians and, more recently, scientists have traditionally taken on the role of answering the question of what it all means. Their day job is to probe the ultimate nature of reality—to understand what itisthat makes the world tick. These two communities ([2]) approach the issue from vastly different angles and with totally different tools, yet they share a common purpose of understanding and describing reality.

Theologians

The community of theologians goes back thousands of years, and still strongly endures. Many claim to hold special knowledge of Godhood, imparted to them through a variety of ways—meditation, divine revelation past or present, ancient scriptures, and the like. But different theologians routinely offer starkly different and, despite areas of overlap, often mutually contradictory visions of who or what the Godhead may be. Since irreconcilable views of divinity have historically led to severe social disharmony, to crimes and wars, both civil and foreign, and still do so today, some objective means of telling what may possibly be true or at least harbor a measure of truth from what is likely patently wrong is long overdue. Much adding to the confusion, academics with impeccable credentials, from a wide range of reputedly objective disciplines—Richard M. Gale, Michael Martin, Richard Swinburne, Victor Stenger, Peter Russell, and many others, have approached the subject from a variety of supposedly rigorously impartial angles over the years, and yet have still reached opposite conclusions with seemingly metronomic regularity—which further underscores the need for an absolutely objective tool of analysis. Could it be that, much like the faint flapping of butterfly wings may bring about inordinately big effects on distant, virtually unrelated related events—a phenomenon known as the 'butterfly effect'—the slightest subconscious bias may be stealthily determining the eventual outcome of analyses not incontrovertibly fully rooted in pure calculation-driven objectivity?

So whom, and what, are we to believe? And why and how different and incompatible views of Godhood can arise in the first place? Historically, the use of some mathematics and/or logic has been sometimes attempted: so-called ontological arguments were made by some theologians to demonstrate the existence of a Godhead. Such arguments, however, seem flawed ([3]). A far more compelling case for the possible existence of a Godhead has come from a far unlikelier source—a mathematician who was not in the business of seeking answers to queries of a spiritual or theological nature, but who appeared to stumble onto one: Georg Cantor, the pioneer of the formalized study of infinities, demonstrated that the mere existence of infinities ultimately leads to a stark mathematical contradiction, a full-blown breakdown of mathematics and of logic itself. He could only resolve the contradiction—which he called an antinomy—by positing the existence of a super-infinity, something much too vast to ever be approachable through the mathematics of infinity alone, but which required the deployment of much more than mathematics to be even remotely fathomed—an infinity approachable by us, however dimly, only if we use both our left (logical) and right (intuitive) brain. He made an argument that this super-infinity is the Godhead itself. Again, contrarily to the ontologists' approach, Cantor did not set out to find a mathematical definition or proof of Godhead, but, as he saw it, had to invoke the Godhead in order to resolve the intractable contradiction he discovered in the mathematics of infinity.

We will look more in depth at Cantor's arguments below. Generally speaking, everything about a Godhead is about infinities and infinite attributes ([4]), although, surprisingly, a few theologians disagree. The theologian Harold Kushner, for instance, argues in his book 'When Bad Things Happen to Good People' that the Godhead is not infinite. He is led to this puzzling conclusion by his analysis of the question of why a Divinity would see fit to allow 'good people' to undergo 'bad things', from his standpoint as someone who believes in a specific Godhead with precisely defined qualities and attributes, set forth in the narrowly defined framework of a rigidly established dogma. Building on lines of thought first put forward by Gersonides in the fourteenth century and more recently by others, such as Levi Olan, Kushner rather extraordinarily concludes that Godhead is in fact powerless to stop 'bad things' from happening. Under his view, the Godhead is neither infinite nor almighty. We shall keep here with the majority view that any Godhead must be infinite, and that infinity is the very quality that ultimately gives rise to the disruptive, extraordinary phenomenon of divinity. A more in-depth analysis of the question is presented in note ([5]).

The issue of why and how it is legitimate to use simple numbers-based mathematics in this context is a valid question, dealt with under ([6]). The bottom line is that some math is at the very least valid within certain areas and domains relevant to the questions at hand, and we shall restrict ourselves to such domains. By demonstrating incontrovertible facts, math will enable us to tell apart what can possibly be, and what most definitely cannot be. It will enable us to come closer to an understanding of who or what a Godhead could possibly be, if there is such a thing as a Godhead. It will also help in circumscribing the existential question itself—can there possibly be a Godhead to begin with?

A few guidelines on how mathematics can and should be used is appropriate here, so please bear with me as I briefly set them forth here. Broadly speaking, math can only deal with precisely defined words describing sharply delineated concepts. For instance, should we say that Godhead is love, or compassion, these somewhat fuzzy concepts cannot be readily probed or analyzed by numbers or by math. But should we say that Godhead is infinite love, infinite compassion, then the 'infinite' part of the statement is directly amenable to mathematical analysis: indeed, math is the only tool available in the box that objectively deals, or even can deal, with infinities.

Within the framework of a number of possible limitations and provisos, math will thus allow for deploying non-subjective, logical, 'left-brain' approaches, all the more indispensable because the wonted subjective, right-brain approaches seem to unfailingly lead to contradictions. Somewhat unexpectedly, math will also turn out to be helpful in analyzing emotional and right-brain approaches, and it will demonstrate why contradictions inevitably arise when exclusively right-brain approaches are used.

Scientists

Scientists, especially physicists, constitute the second community whose job it is to probe and understand reality, and therefrom to explain it to the rest of us. That there happens to be no more consensus as to what it all means amongst physicists than there is amongst theologians will surprise no one, and only underscores anew the acute need for using the most objective tool of analysis bar none.

A key ongoing debate among physicists today is between the so-called Aristotelian view of reality, and the so-called Platonic view. At its core, the Aristotelian view is a materialistic view ([7]), whereas the Platonic view holds that the ultimate nature of reality is not materialistic, but that abstract concepts, such as, first and foremost, an underlying abstract mathematical structure, play a or indeed the determinant role in weaving the fabric of our reality. In different shadings, this latter view has become more popular of late, in part thanks to published works by the likes of Max Tegmark, Lawrence Krauss, and others. Of course, if a person believes in any deity, that person then necessarily takes the Platonic view of reality, because he or she believes in a Universe which, at the very least, simultaneously accommodates both material reality itself plus some spiritual, immaterial transcendent being. Indeed, as we shall see, not only modern common sense but also straightforward mathematics proves that a Godhead, if It exists, cannot possibly be material.

This modern view may seem self-evident today, but not so long ago the image of Godhead as some avuncular man in the sky was not uncommon. In 1961, the Soviet cosmonaut Yuri Gagarin became the first human to travel in space. The then Soviet leader Nikita Khrushchev was quoted afterwards as saying, in all seriousness, in a speech in support of the USSR's secularist policies, "Gagarin flew into space, but didn't see any God there" (a quote that was later falsely attributed to Gagarin himself.) As recently as 1971, even John Lennon saw fit to write the lyrics: 'Imagine there's no heaven, above us only sky', in apparent reference to the then still surprisingly commonly held view of a material, three-dimensional Godhead resident somewhere in space.

Using Mathematics

Whenever we fail to use a purely mathematical approach, whether from a believing or disbelieving or open standpoint, our views of the Godhead are bound to be almost equally naive. For instance, we may broadly agree on a definition of the Godhead as being infinite and disincarnate. But there are many quite different infinities, so which one is it exactly that we are favoring? We will probably naturally opt for some apex infinity—but, unless we do the math, we won't achieve anywhere near a full understanding or appreciation of the inescapable consequences that must flow from such a definition. Whenever or wherever infinities are involved, an astonishing degree of complexity kicks in and the plots thicken immeasurably, most often well beyond and differently from what we would naturally expect ([8]).

A math-based approach will also prove especially helpful in objectively analyzing received dogmas, i.e. the established foundational doctrines of many organized belief systems. Long-established dogmas are mostly accepted as are today, and reputed to spring from revealed truths. Any request for further justification is often staved off by a demand for a 'leap of faith', or some other demand for unquestioning obedience or acquiescence, often on the basis of the say-so of some (relatively) ancient texts deemed to be unassailable bearers of truth. As we shall however shortly see, dogmas are on the whole fully amenable to mathematical analysis.

As mentioned, math will also prove instrumental in analyzing the consequences of infinity—where the presence of infinity within certain contexts ineluctably leads. We cannot on the one hand accept or assert that a Godhead is infinite, and then blithely attribute traits, thoughts, or properties to that Godhead that would mathematically belie such infinity. We can hardly claim that we believe or for that matter disbelieve in something unless we fully know what it is that we believe or disbelieve in, including the flow-on consequences of such beliefs or disbeliefs. Whenever a contradiction that would belie infinity is uncovered, if we are to retain Its infiniteness there will be no choice but to abandon the corresponding purported trait of the Godhead.

Math will also be used to endow even everyday terms with precise meanings, and we shall endeavor to make such word definitions as broadly consensual as possible. Many concepts and/or realities tend to be loosely defined, and despite the widespread use of the same words, consensual meanings remain elusive, and different people may associate very different meanings to a same term or phrase. Illustrating the point, the age-old question 'Do you believe in God?' is utterly meaningless unless both words 'believe' and 'God' are very precisely defined, yet loose variations of this very question have historically led to all manner of strife, lethal and otherwise. We will therefore adopt here below a first definition for 'Godhead', which may then become further refined as logical analysis proceeds. Likewise, should we for instance say that a Godhead is infinitely intelligent, we must then find some way of defining intelligence appropriately accurately, so that its meaning both meets with consensus and becomes amenable to objective mathematical analysis. Most often, it will be easy to do so: for instance, the definition of 'infinitely intelligent' here would be, simply put, someone with an infinite IQ or IQ equivalent (irrespective of whatever particular IQ measure would be used, see note ([9]). Whenever some quality or attribute seems fuzzy, we will endeavour to find a way of defining it, even if provisionally so, bearing in mind that we could later on be led to a further refining of definition.

Last but not least, numbers will provide a handy way to illustrate certain concepts which would otherwise prove harder to apprehend—first and foremost, infinity itself. For instance, we can quite easily conceptualize how the unbounded series of whole numbers 1,2,3, 4,5,6...... goes on forever and never ends, and as such is infinite. Trying to conceptualize infinity, and infinities, by any other means or within contexts different from mere numbers may sometimes prove less straightforward, so that picturing infinities through the simple expedient of numbers shall provide a helpful shortcut towards visualizing infinity in a variety of other contexts.

What constitutes mathematical proof?

Two separate kinds of proof will be used in this book: first, the purely mathematical variety—proof that requires no further input from any other science to be able to stand on its own—be it physics, chemistry, psychiatry, psychology, or any other scientific discipline. A basic example of such a proof would be the statement that, "if A minus 1 is equal to 0, then necessarily, A is equal to 1." That's it—no further ado, no further discussion nor proof is needed to truthfully and incontrovertibly state that in this above case the value of A is unequivocally equal to 1, period. We'll call such proofs A-type proofs.

Examples of such A-type proofs include, for instance, the above-cited fact that if one believes in a Godhead, then one has no choice but to believe in the existence of some more complex universe or multiverse beyond our simple material universe—i.e., an outside reality that goes beyond the currently known material universe. Should our particular universe be finite (a question which we will revisit below), then the simple statement that a Godhead exists and has some infinite traits proves the existence of something else beyond that universe—because there is simply no mathematical scope or room to accommodate infinity within a finite universe. There are of course ways, as we shall see, whereby our very own universe could harbor infinity, so that a whole new separate universe or a metaverse or multiverse is not needed to accommodate infinity, but at the very least the existence of some infinity within the universe is in some way required if we are to accommodate divinity.

But the plot thickens, because there are many different infinities. Therefore, should we say and accept that a Godhead is infinite, or, say, infinitely good, what does this statement exactly mean? Are we content with some lower-ranking infinity, or shall we insist on a higher ranking, or, if such exists, on a or the apex infinity? We will examine these issues at some length in these pages.

Time is also involved: mathematically, we will see that if a Godhead exists, It by definition is the master of time and space. The only way to do so is to exist out of time, so that both the past and, crucially, the future do not hold sway on a Godhead the way they do on us, and therefore at least some parts of the wider multiverse must be timeless. But how can any place anywhere be timeless? We shall see below that this eventuality is much easier to conceptualize than would first appear.

Beside purely mathematical proofs, there also exists another kind of valid proof. This kind, which we'll call B-type proofs, makes essential use of math but also synergistically draws upon other scientific disciplines to work in full.

Here's an example.

One of your neighbors is a veterinarian doctor who maintains a colony of bonobo apes in his vast back garden. He is authorized to do so by the local authorities, he does it well, and the apes are well cared for.

Now the average IQ of a bonobo is actually roughly measurable—bonobos are, by animal standards, very intelligent. Measured against a human scale, a bonobo's average IQ would be of about 5 or 6 (whereas the average human IQ is by definition 100.) Because the IQ scale is not linear, this does not mean that humans are twenty times cleverer than bonobos (by whatever yardstick of objective intelligence may be applied), but an IQ of 100 is rather something like perhaps ten thousand times or so more intelligent than an IQ of 5. In other words, your neighbor the veterinarian doctor is about ten thousand times or so more intelligent than his wards (the precise figure being of little relevance here, as long as it's clear that the man's IQ, although finite, is very much higher than that of the animals.)

Anyway this neighbor is a bit of an oddball. It turns out that he has been training the bonobos to ... somehow worship him. He has been training the bonobos to behave in a certain way in order to pay homage to him and his glory. When the bonobos neglect to conform to certain rites which the neighbor has devised that he deems reflect his glory, he harshly punishes the animals. He whips them, sometimes kills them. He has taught them to gather in rank and file, to wear certain pieces of cloth when he blows a certain whistle in the evenings, calling them to his worship.

Soon however the behavior of that neighbor came to the attention of the authorities. Word leaked, rumors spread, concerned neighbors espied his weird doings. Soon the authorities intervened; after due process, the good doctor was certified and put away. Once a well-regarded man who enjoyed respect within the community, the good doctor never recovered his reputation. His unexplainable need to be worshipped by lesser life forms was widely seen as totally detracting from his status as a wise and well-respected figure, and also ended his erstwhile renown as a capable veterinarian.

The only bit of mathematics in the above tale is that of the IQ points ([10]). It however lies at the root of how the situation unfolds and of the conclusions drawn: psychiatry, a scientific discipline wholly separate from mathematics, made a consequential numbers-based judgment that the neighbor was mad and put him away. If instead of bonobos, the wards had been human disciples with normal IQ's, the veterinarian would have deemed been a sect leader and not immediately nor necessarily mad. The proof that the man was mad is therefore a category B proof—proof buttressed by the IQ numbers—but not exclusively mathematical.

A key point is that, regardless of the exact IQ distance between the mad neighbor and his wards, loosely estimated at a factor of ten thousand or so, this distance is finite: the neighbor was only something like thousands of times more intelligent than his wards. He was not infinitely more intelligent than they were, yet his need for adoration on the part of lesser life forms clearly marked him as deranged. But by any acceptable definition of a Godhead, the Godhead's IQ is infinite. Mathematically, this means that the distance in intelligence between man and the Godhead is infinite, which in turn also means that this distance is infinitely greater than the finite distance between the mad neighbor and his apes. This throws off the question as to why and how an infinite Godhead would stand in any need of being worshipped by man, or of having us conform to any kinds of rites or strictures as to how we behave or dress or whatever, at least as long as we don't hurt anybody ([11]).

Two categories of considerations could conceivably modify the above conclusion.

The first possible objection is that intelligence is not necessarily the only or even the main criterion. Perhaps something equivalent to an EQ—emotional quotient—should be employed instead, or some other yardstick or combination of yardsticks. But this would not essentially alter the underlying argument: for instance, an infinitely good and compassionate Godhead would also have an infinite EQ. Animals are, like we are, capable of empathy. A Godhead capable of infinite empathy would dwarf both human and animal capacity for empathy, irrespective of any ascertainable distance between the latter two.

Emergence

A second and far more compelling consideration would be that at such very high values of IQ, EQ and any other attributes, the inescapable phenomenon of emergence would occur, and that, in environments where very large metrics or numbers or collections of anything intervene, there is absolutely no way in principle by which we can foretell or second-guess anything, such as behaviours, thoughts or events.

Emergence is the appearance of utterly new and inherently unpredictable properties and phenomena, ineluctably triggered whenever huge numbers, sizes, or any associated metrics start heaping up on top of some previous situation or status, and become involved in some way. Emergence results in novel, utterly unforeseeable phenomena, and often in profound qualitative shifts which cannot in principle be foreseen or anticipated in any way, and which show up—emerge—when the relevant associated metrics, numbers, or characteristics grow to immense proportions, or whenever the envelope of some applicable norm is pushed far beyond its usual moorings.

Quoting briefly from available literature ([12]), emergence is for instance the property that arises when one has one million dollars at one's disposal, as opposed to having one million times a single dollar. When equipped with merely a million times one single dollar without emergence, the most one could buy would be a million times cheap one-dollar (or less) items, such as, say, a million bubble gums. Emergence occurs when the dollars are allowed to pool together into great numbers, and new properties arise: say, a yacht can now be bought rather than a great number of cheap items. Thus, should you be given a million dollars with a stipulation that no emergence is allowed, you'd still be poor—constrained to one-dollar (or less) item purchases. Another example of emergence would be that of a viscous traffic jam, an emergent phenomenon arising from a large number of times the single phenomenon of one car being driven on a given stretch of road. In physics, odd quasiparticles emerge when vast collections of subatomic particles come together within constrained environments in space and time. These emergent quasiparticles typically have strong, measurable effects on the collective properties and behavior of the group of particles.

Emergence can occur both within material contexts and within abstract or mental domains, as long as they build on scales significant enough to trigger the phenomenon. Knowledge is one such area: an early primary school student cannot begin to even conceptualize the contents of, say, an MIT Ph.D.-level course in engineering: the course builds on, and indeed emerges from, too many prior layers to be simply imaginable or in any way predictable from a much deeper layer.

One of the most striking examples of emergence is found in biology, in the odd case of theDictyostelium discoideum,akathe slime mold. When conditions are good, the slime is made up of tens of thousands of independent, single-celledfungusspores. A change of environmental conditions can cause the many spores to congregate and become a single entity—an emergent single, coherent slug-like multi-cellularanimal,which only appears if there are enough spores. In that case, as in the case of the dollars, emergence profoundly shifted the very nature of reality: out of a vast collection of loose independent spores arose one animal. The examples are endless.

Emergent phenomena keep building on rising numbers: the first ones appear at a certain level of magnitude, then further emergence keeps appearing when the resulting operative numbers are in turn vastly further multiplied, and so on. When the relevant numbers however become infinite, then whole new categories and qualities