Old Book: Chapter 4
Much Ado About Nothing
In order to justify my idea of assumption switching, I've tried to make the case that all explantions involve a foreground and a background, every assertion needs a context. That is, there is no logical argument without a set of postulates in the Euclidean sense. In contrast to this, the world itself is presumably indifferent to foregrounds and backgrounds; the world simply is. Maps have an orientation; the territory doesn't. This fact is the heart of the MT problem. The choice of what constitutes the background as opposed to the foreground ultimately leads to the limitations of the explanation so constructed. Here I want to try to take the idea of the background as far as it can go to see if that will shed light on the issues. It's a fun ride for me.
Well, what is the ultimate background when one is trying to explain everything? What is left when everything is stripped away? Nothing! Nothingness is the ultimate background, the final context. Boy howdy, however, nothingness is hard to talk about and is rife with paradoxes and natural puns.
In fact, I assert that the concept of nothingness can't quite make sense. In the history of philosophy there is a very deep question (it can actually be interpreted at many levels of depth), "Why is there something rather than nothing.'' My reply to the question: "What the hell is this nothing to which you refer?" What is utter absence? How does one go about stripping away all the somethings until there is nothing to strip away? We feel that there has to be a nothing that's the opposite of something, but to me it's utterly unimaginable.
We can imagine various quasi-nothings, substitutes-for-nothing rather than the (un)real (un)thing itself, and each of them turns out to be a kind of thing, a something. All our apparent examples of nothingness are just canceled out somethings. A vacuum, empty space, etc.
Consider silence in a radio broadcast. It isn't the total lack of a signal; it's a signal which designates an emptiness. The true lack of a signal manifests as random static. The chaos of static is overridden in order to communicate a rest. Thus, apparent emptiness can convey order. I think this example may be onto something. Maybe the best opposite of somethingness is random chaos rather than nothingness. My favorite contradiction of "things only change when caused to do so" is "everything is trying to happen at once but fails to because of cancellation." Static is the sound of everything trying to happen at once.
The silent spaces in a radio broadcast have meaning and provide information as much as the sound before and after. They aren't really in the background but have equal ontological status with the sound parts. A written message is in the something of black ink and the nothing of white paper, in the letters and the spaces between them.
Profoundly blind people don't live in darkness, don't see blackness. They live without sight, black or otherwise. Our minds automatically substitute trumped up relative nothings to stand for the deepest nothings. We picture the Void as empty space rather than as a non-thing without the attributes of space nor anything else.
Many philosophies from various forms of mysticism to existentialism have held nothingness in a central place. We will see what a rich and elusive idea this nothingness is.
Zero, in the meantime, is our quantitative correlative of nothingness. Since math epitomizes certainty and precision, we'd expect something as simple as zero to be an unambiguous and definite object, but... zero has issues. Let's start there.
He doesn't have any!
Math Problem: A man has two apples. He then eats one of them and gives the other to a friend. Now how many apples does the man have?
Solution #1: He has zero apples.
Solution #2: He doesn't have any!
Is there a difference between these two solutions? They certainly feel different. One sounds like math and the other like just plain common sense. Solutions like #1 have always bugged me. Our schooling has helped make zero seem a real and natural thing, but something deep inside me says that it is not.
People have been counting since long before the dawn of recorded history and so the counting numbers 1, 2, 3 and so on are very old. The name given to the counting numbers by mathematicians, namely the natural numbers, indicates their special status and presumed priority. "God made natural numbers; all else is the work of man." Zero, on the other hand, is a relatively recent invention. There is no acknowledgement of it at all in Greek mathematics. Wow! When Arab mathematicians and accountants first started using zero in the middle ages — literally to keep the books balanced -- the symbol 0 had no meaning as a quantity. It was just used to distinguish, for instance, 308 from 38. It was, as we called it in elementary school, a place holder, meant to indicate that we should skip the tens place — three hundreds, no tens and eight ones.
The Babylonians, who invented place value numeration, where the value of a digit depends on its position, used a space to serve the same purpose as the Arab 0. The lack of an actual symbol suggests it may never have occurred to them that they could treat nothing the way they treated something. A lack must have seemed to them, as it seems to me, to be in a different category than a number. Spaces ultimately lost out to zeros not for abstract mathematical reasons but because spaces led to ambiguities when they were placed at the end of the number (380) or when there was more than one space (3008). Thus zero arose as a lexicographic convenience, a mark rather than a quantity or even a concept.
In the 17th century, zero acquired a second equally important role. René Descartes may have been the first to explicitly use the idea that locations on a line could be labeled with numbers. This simple labeling idea led directly to the invention of the most important bit of technology of the last thousand years, analytic geometry, which exploits the insight that algebra and geometry are equivalent. Spatial relationships can be represented as equations and vice versa. You may never have thought about that when you learned about graphing in middle school. This realization paved the way for Newton and the rest of quantitative science. Geometry-algebra equivalence may be the first triumph of Descartes' mechanistic thought, adding confidence to the notion that the MT relationship is simple and one-to-one: Space and spatial imagery, the seeming antithesis of linear verbal expression, had turned out to be expressible in "words." Perhaps, therefore, everything is expressible in words.
Since lines go on indefinitely in both directions, locations to the left as well as the right of 1 need labels. Zero and the negative numbers had been around long enough in other contexts that they were the obvious choices to be used here.
Zero therefore has important and clear meanings as a place holder and as a location on a number line, but can it be interpreted as a quantity? Can one have zero apples? The difficulty boils down to zero's strange status as a sign which stands for absence when all other numerals stand for presence. Zero makes "that which is not a thing" a thing. The nothingness that can be named is not the true nothingness.
The strategy mathematicians have used to avoid this paradoxical situation about naming nothingness is to say that zero is not really a quantification of nothing at all but rather refers to a simple object called the empty (or null) set. That is, numbers are ultimately only tokens which refer to abstract sets (or collections), and zero is the token for the special set which has no members or elements. The bag is empty, but at least there is still a bag. Zero is not an absolute nothing. It does have appended to it the units of that set being examined — apples, for example. But what can we possibly mean by a set with no elements when "set" seems to be defined as something that has elements, as nothing but its elements? The bag was never really there; it was a contrivance.
When conferring a value of zero on some measurement we are committing what Whitehead called the fallacy of misplaced concreteness. We are reifying a mental construct rather than a "thing" in any reasonable sense of the word. When we have no apples, the quality of "appleness" is missing and there is nothing to measure. One cannot sensibly say "I counted the apples and there were zero of them." We can get as far as preparing to count, but we never quite commit the act of counting. Zero, interpreted as "nothing with units", has a quality of readiness or potential without content, as if something were about to appear.
I've written a lot about the unbridgeable separation (and the extraordinary connection) between words and what they signify. It may seem a small thing to adopt an MT model that ignores the distinction between a name and the supposed thing it names, but there seems to be a far greater gap between nothingness and a name for nothingness. Names are labels for things and not for non-things. The substitution of "zero things" for "not anything" is a misleading reduction (or inflation) of the situation as much as the substitution of "seeing blackness" for "not seeing anything" misrepresents the experience of blind people.
Since nothingness is not a thing, if we are to reason with it or around it, some sort of "approximate" substitutes must be invented. Nothingness has no numeric properties (or any properties at all), so we have conjured up zero as a numeric substitute for nothingness, just as we created the image of total darkness to stand in place of the nothingness of sightlessness.
Zero is the exception to many mathematical rules. We all know, for example, that you can't divide by zero. That is, there is no answer to the questions "How many slices will a pie yield if each slice is of zero size?" or alternatively "How big will each portion be if we divide a pie into zero portions?" From simple arithmetic through calculus, rule after rule makes an exception for zero.
From the middle of the 19th century to the present, one of the strongest trends in mathematics has involved stretching the idea of a number system to its most abstract forms. In doing so, mathematicians have looked for the most general properties of numbers systems and the operations such as addition and multiplication that we use to combine the elements (numbers) of the system. One such property is closure. A system is said to be closed under an operation if the result of performing the operation on two elements of the system results in another element of the system. The counting numbers, for example, are closed under addition since the sum of any two counting numbers is also a counting number. On the other hand, that system is not closed under subtraction because differences like 3 minus 5 do not yield counting numbers. If we expand our system to include negative whole numbers, then the system is closed for subtraction.
By further expanding the system to include rationals, irrationals and complex numbers, we can arrive at systems that are closed for multiplication, exponentiation and other more exotic operations. Because of the exception we must make for zero, however, no simple system which includes zero will be closed under division. That is, the result of dividing a number by zero is never in the system. In order to create a system closed under division, we have to tack on a specially contrived extra infinity element. Why infinity? Well, look at what happens to 1/ X as we substitute smaller and smaller values for X:
1/1 = 1
1/.1 = 10
1/.01 = 100
1/.001 = 1000
The closer the divisor gets to 0 the larger the dividend gets. It seems almost reasonable then to say when the divisor gets to 0, the dividend will have gotten infinitely large. Thus we can think of zero and infinity as reciprocals. This formulation does lead to a little problem however. If 1/0 = ¥ and 3/0 = ¥, then ¥/ 0 = 1 and ¥/0 = 3. Also, if we approach zero from the negative side, the dividend heads toward - ¥.
Zero as a Measurement
So far we have looked at the problems associated with considering zero as a counting number. Similar difficulties arise when we consider it as a real measurement. Suppose, for example that we look at the controversial proposition that there exist massless particles. Assigning a value of zero to the mass of the particle means that the concept of mass does not apply to the particle. It will neither exert nor be subject to gravitational effects, it won't transfer momentum in a collision, etc. In other words, to say that a particle has zero mass is a little like saying that the spirit of democracy takes up zero milliliters; volume has nothing to do with the conceptual world, and mass has nothing to do with a particle that doesn't have any.
Similarly, when we say that the electric field potential in a portion of space is zero, what that really means is that the portion of space has no electrical character. You can't take a measurement and get zero for an answer; you can only fail to get a measurement. Perhaps a counterexample springs to mind -- a temperature of 0° Celsius, but 0° is not a measure of quantity at all but rather uses zero in its number line sense — a marking on an arbitrary scale. It is only by convention that we label the freezing point of water as 0°. A measurement of 0 degrees Kelvin might count as a sort of quantity (related to the amount of heat in a sample), except that absolute zero has never been achieved and can probably never be achieved. What seems like a hard stop at the end of the temperature scale turns out to be fuzzy and open-ended like the set of real numbers between 0 and 1 that does not include the endpoints. It is instructive to imagine that such an open-ended set is really like an infinitely deep well. No matter how close to the end you think you've come, there's always plenty of room left to travel downward.
Physicists managed to produce a mathematical theory that pushes our knowledge of the state of the universe back to a small fraction of a second after the supposed Big Bang, but there are theoretical and logical obstacles to reaching all the way back precisely to Zero Hour. One current response to the powerfully anti-intuitive notion of a hard beginning to time is the idea that, like the temperature scale, time fuzzes out toward the presumed endpoint. The theory of Hawking and Hartle says that, working backwards toward a tiny fraction of a second after the imaginary Big Bang, time becomes indistinguishable from space in just such a way that the hard endpoint, the beginning of time, smears out completely. There may be no hard zeroes at the ends of our "absolute" scales.
Zero also is used to describe the size of a spatial point or a point in time, and this usage is also fraught with difficulties. First, there are no events of zero duration and no camera capable of capturing a single instant and there is nothing that exists in space but doesn't take up any.
A line segment of length one inch ostensibly consists of nothing but points. If each of these points have zero width, how can a collection of them, no matter how numerous, fill up that angry inch?
This is not to say that most uses of zero as an accounting convention or as a measure are not clear and unambiguous. A batting average of zero means no hits in some number of at bats. Zero apples means no apples. I'm not trying to say that everything we know about zero is wrong. It is my intention only to peel back your sense that zero is a simple and easy concept. It is sophisticated relative to one, two, and three. It is especially sophisticated in that it is a map without a corresponding territory, even in principle.
Even mathematics, this bastion of Platonic perfection in which science has laid its faith as a kind of ultimate reality, itself is no more than a map of something which ultimately is not perfectly mappable. Math can't get very far without a zero concept, but we can't act as if nothingness is something without losing certainty and definiteness. Also, if nothingness is considered an ultimate and absolute background (one that cannot ever be a foreground), as a mere stage on which all phenomena act out their parts, then you automatically lose half the story. My modified nothingness substitute makes no pretense of being absolute nothingness and that may allow for a Foreground-Background Switch.
Ghosts of departed quantities
The ancient Greeks had no zero concept. Greek mathematicians, for the most part, reasoned geometrically and geometry yields no perfect counterpart to zero. In Euclid's geometry, the closest thing to a zero is the point. Points are purposely left undefined in the modern treatment of geometry, but they are to be conceived of as locations without size or dimension and as the constituent parts of lines, planes and space. We might be tempted to say that points have zero length (as well as zero area and volume), and, in fact, that is what many of us have been taught in school, but we are about to look at a paradox that arises from such an equivalence that will shed light on the difficulties we have in treating zero as a quantity.
If points have no width (or widths of zero units), then no matter how many of them we lay side by side, even an uncountable infinity, we will never get a line segment of finite length. And yet we know that lines are somehow "made of" points. On the other hand, if a point has a width greater than zero, then there is more than one location represented, and we therefore have more than a single point. To put the paradox plainly, points seem to fill space but to take up none.
The closer we look at this situation, the more confusing it becomes. Zero width implies we can't make lines while nonzero width implies the nonuniqueness of points. It is interesting to note that one of the famous paradoxes of the Greek philosopher Zeno described this problem with points hundreds of years before Euclid wrote and compiled his Elements. Euclid simply side-stepped the issue. I read somewhere that awareness of such paradox steered Greek mathematics away from arithmetic and number toward geometry.
Before we go on to think about solutions for the point-width problem, I'd like to bring up an additional absurdity built into the concept of a point. To ask what space would be like without the attribute of extension is like asking what the world would be like without the property of time. What would happen if there were no time? Nothing I presume. Clearly, time and the world are inseparable. They arise mutually. Likewise a bit of space without extension is difficult to imagine. We tend to think it is okay to separate out qualities which arise mutually, and, as I've said, creating these artificial foreground-background orientations that impose linear order on a nonlinear world is what intellectualization is all about, but in the long run such separations are bound to lead to difficulties. This absurdity is for me very much like that implicit in the idea of zero. What would number be like if we removed the attribute of quantity? Whatever it is, it wouldn't be much like a number.
A Small Suggestion
Mathematicians have devised many ways to patch up the problem about the widths of points. The one I will present is far from the received version, but it has a certain intuitive appeal. It's my alternative quantitative version of zero. We postulate the existence of a new kind of number, called an infinitesimal or an indivisible, which bridges the gap between the hard stop of zero and tiny finite numbers (like .0000001). We can think of infinitesimals (rather than zero) as reciprocals of infinity. These infinitesimals are not all identical in size and thus we avoid the problem with zero as a reciprocal of infinity that was mentioned earlier. I mentioned a moment ago pie slices of size zero. Imagine instead that each slice has an infinitesimal size. In that case we would get an infinite number of slices.
We will say that points have infinitesimal width. Any sum of a finite number of these infinitesimals will still be infinitesimal, but an (appropriately large) infinite collection will break through into finitehood. That is, we would get a finite sum such as 3.
It may help you understand the way infinitesimals sum to finites to look at the analogous "level-breaking" we see with finite numbers. A sum of finitely many finites will always give a finite result, but an appropriate infinite collection of them, for example, produces an infinite sum. That is, finites too can break through to the "next" level. Thus, infinitesimals are to finites as finites are to infinities. There is a continuum of levels
. . .2nd order infinitesimals, infinitesimals, finites, infinities, . .
extending in both directions.
We will see shortly what these numbers have to do with our zero problem, but notice now the similarity between the zero/not zero ambiguity of infinitesimals and the quantity/no quantity ambiguity of zero.
The history of the infinitesimals and the debate over their existence is fascinating and sordid, having stirred up more passion and acrimony than you would expect. One vestige of the debate is the often lampooned scholastic question about the number of angels that can dance on the head of a pin. I think the question is not whether 10 or 100 fit but whether a finite or infinite number fit. That is, are there infinitesimal embodiments? Assuming that angels can manifest themselves only in physically possible ways, can there possibly be infinitely many in a finite space?
These tiny numbers have been reinvented several times by scientists and mathematicians in order to solve problems but have been consistently discredited by critics who see them as abhorrent, strange and unnecessary. There is no question that infinitesimals lead to correct answers when used properly, but mathematicians have often gone to great lengths to avoid them. Their strangeness and consequent controversial character account for some of that aversion, but there is also the fact that reasoning with them can easily get you into a muddle of inconsistencies.
It is well known that the logic and arithmetic of infinity is very different from ordinary logic. For example, even though the set of all natural numbers {1,2,3,...} contains the set of even numbers {2,4,6,...} the two sets have the same size. That is, they can be put into one to one correspondence—each number in the first set matching up with its double in the second set. So we would expect that the logic of infinitesimals will also require special attention.
As an example of reasoning by the free use of infinitesimals, I will give a demonstration of the area formula for a circle which is attributed to Nicholas of Cusa, a 16th century scholar. Divide a circle of radius R into infinitely many pie slices. It is impossible to picture one such slice precisely, but we will represent it like this:
Since a circle is like an infinite sided polygon (!), we can imagine that the arc of each of the infinitely many slices is actually a straight line segment. Thus the slices are triangles with an infinitesimal base (whose length I will designate with the symbol ·). Since the triangle is infinitely skinny, its altitude is the same as its side which is R. Thus the area of one slice is ½R· (or half the height times the base). Now let's add up all the slices of the pie to get the total area
A = ½R· + ½R· + ½R· + ½R· + ½R· + ...
Factoring out the ½R from each term, we get
A = ½R (· + · + · + · + ...).
The sum of the infinitesimals in the parentheses is the sum of the arcs which make up the circle and thus equals the circumference of the circle. By the definition of p, the circumference is 2pR, so
A = ½R (2pR) = ½ 2 p R·R = pR2 QED
You can probably imagine the sort of reaction this sloppy looking reasoning would provoke from rigor-minded mathematicians. Clearly there are many possible objections. What does it mean to divide a pie into an infinite number of slices? Are we sure we can consider these things as triangles rather than sectors? How can the altitude of the isosceles triangle be equal to the side? Can you factor from infinitely many terms? On the other hand, it can't be a coincidence that we got the right answer. Something essentially right must be going on here.
Infinitesimals seem to have first been used in Greece but much of that work is lost. There is some evidence that Democritus used infinitesimals to derive several volume formulas. None of his written works have survived, but references to his books in the writings of others make it appear more than likely that infinitesimals are prototypes for his better known invention, atoms, which he believed are the constituents of all things. His concept was not much like our current model where atoms are strictly finite.
Infinitesimals have tremendous practical significance. They were the central concept in the development of calculus, though they are rarely mentioned in today's calculus courses. Calculus "reduces" curves to many straight line segments of infinitesimal length (differential calculus) and reduces areas and volumes to infinitely many infinitesimally thick slices (integral calculus). We saw both of these aspects in the above demonstration. Calculus and its offshoot, differential equations, are the most widely used bits of mathematics in the sciences, so it is a little scandalous that it rests on such dubious entities.
The debate over whether infinitesimals really exist as mathematical objects parallels the equivalent and better known quarrels about the existence of actual, as opposed to potential, infinities. All mathematicians agree that, for example, one can talk about continuing the sum 1/2 + 1/4 + 1/8 +... indefinitely, but many balk at the idea that we can speak of an actual completed sum (which equals 2) that includes all of the infinitely many terms. We are asserting that such actual sums exist when we say that .9999999999999... equals 1, since µ § is just a shorthand for the completed infinite sum µ § An infinitesimal thinker, by the way, would say that .999... and 1 differ by an infinitesimal.
Belief in infinitesimals has always been associated with mystical thought and thus has been consistently disparaged by the scientific orthodoxy. It is believed (weasel words) that Zeno's paradoxes were formulated partly to prove that infinitesimals could not exist.
In an extraordinary treatise called "On the Method", rediscovered early in the 20th century, and written by Archimedes, the greatest of all Greek mathematicians, we see how he used infinitesimals to arrive at some of his most famous discoveries. He used them to find the volume of a sphere, for example. Archimedes, however, was an early leading proponent of rigor in mathematical proof, and he therefore carefully expunged any trace of these suspect quantities from his completed demonstrations, generally replacing their use with his famous and aptly named method of exhaustion, a far less intuitive approach.
Newton too was suspicious of the validity of infinitesimals, despite their prominent place in his discoveries. His worries about the controversy their use would cause may have persuaded him to delay for many years publishing the reasoning behind the ideas of the Principia. In the Principia itself all demonstrations have been made rigorous in the fashion of Archimedes. When Newton's calculus of "fluents" and "fluxions" was finally published (because Leibniz was about to get all the credit), his fears of controversy were realized. In an infamous critique of the new science, Bishop Berkeley assailed infinitesimals as the "ghosts of departed quantities." and claimed that belief in their existence required as much faith as the most dubious point of theology.
Leibniz, who independently developed many of Newton's mathematical results, had no such qualms about infinitesimals. He took up mathematics as an adult and in just a few short years was producing results of the greatest importance. Perhaps his rapid arrival on the scene or his background in philosophical thinking allowed him to ignore the traditions and taboos of mathematics. He was clearly captivated by the mysterious "zero/not zero" quality of infinitesimals which, I believe, inspired some of his deepest philosophical thought. His Monadology, which owes much to Democritean atomism, conceives of nature as an abstract and infinite collection of point-like consciousnesses in mystical communication and interaction. These atoms formed wholes or beings in the same ineffable fashion that points form lines. We "consist in" these monads, but we are not "made of" them, just as lines consist in points but cannot be made from them. One of the models of reality I will ultimately propose as a complement to our usual view shares a great deal with this vision.
For the most part, the conservative elements in mathematics seem to have prevailed in the case of infinitesimals. In the 19th century, embarrassed by the dirty little secret that they thought infinitesimals to be, several mathematicians put calculus on a firm foundation by replacing infinitesimals with the perfectly rigorous and elegant but unintuitive idea of limits. That is, they replaced actual infinitesimals with variables that take on values which approach, but never reach, zero, just as the sequence 1/2, 1/4, 1/8, ... approaches but never reaches zero. Many claimed, perhaps rightly, that limits were what Newton had in mind all along.
Infinitesimals can't be counted out yet, however. In the 1960's, mathematician Abraham Robinson showed that infinitesimals could be made rigorous. Using wonderfully inventive structures derived from modern logic, he showed that limits and infinitesimals are equivalent formulations of one phenomenon. Unlike in the past, mathematicians nowadays, rarely even address the question of actual existence for the tools of mathematics. They feel free to bestow mathematical existence on objects so long as they pass the test of consistency, if they lead to no contradictions. Robinson's concept of infinitesimals leads to no contradictions and, because fresh approaches can create fresh insights even in fully mined veins, infinitesimals have increasing become an active area of mathematical research.
Sweet nothings
It may not be clear what is meant by the question of the existence of infinitesimals, but we can well ask whether infinitesimals might ultimately be found to have a more conventional kind of reality. Like so many other ideas that started out as mathematical abstractions, could infinitesimals have a place in the physical world? As far as I know, no theoretical physicist has seriously considered the idea of actual infinitesimals, but here are a few vague possibilities.
First of all, infinitesimal physical quantities could achieve finite embodiment through a kind of summing or amplification provided by a feedback process, not unlike the one shown in the last chapter to illustrate the idea of mathematical attraction. The difference is that this amplification would involve infinitely many iterations or terms with each step being performed in infinitesimal time. One can imagine a quantum-like spontaneous event produced from the amplification of an infinitesimal "seed" without ever violating the law of conservation of energy, much as the "zero energy" information in a book or periodical can be amplified in the brain to bring about huge changes in the world. A relative nothing produces or causes a something. A quantum itself (as represented by Planck's Constant) may be the finite "arrival point" of such an amplification. Remember in our demonstration of the area formula for a circle that the sum · + · + · + · + ... arrived at 2pR without ever passing through 1 or .1 or .01. Here's a little BASIC computer program, for those of a mathematical turn of mind, that outlines one possible path of amplification.
FOR N = 1000 TO 100000 STEP 1000
X = 0: Q = 1
FOR I = 1 TO N
LET X = 1/N * Q + (N-1)/N * X
NEXT I
PRINT X
NEXT N
(revise, expand, explain)
We are told that virtual particles possess only a quasi-reality. They bubble in and out of existence in the quantum froth of space, like the "cloud of not doing" from the beginning of chapter 2, apparently at the exact moments they are needed to complete interactions. To me they have the feel of infinitesimals, simultaneously existing and not existing, both zeros and not zeros. They once again show the lack of a hard stop in the physical world. Stuff percolates out the relative nothingness of space, so there must be a soft edge to nothingness that I'm saying can be crudely pictured in terms of infinitesimals. The quantities that are now associated with virtual particles are not infinitesimal, but I believe an infinitesimal substratum which is again amplified by feedback may give rise to these strange objects. It is significant that infinitesimals, like virtual particles, can be had for free.
As of today (1995), astrophysicists feel that they have only accounted for 10% of the universe's apparent mass. The search for the "missing mass" is a major area of study. Infinitely many particles of infinitesimal mass could make up some of the other 90%. The assumption that there are finitely many particles in the universe is deep-seated but not motivated by any important a priori considerations except the limits on our ability to imagine the infinite. Perhaps it would be useful to lay aside our assumptions about the finite nature of reality in general. Infinities crop up as singularities in fields and black holes. Why not find infinitesimals out there as well?
All areas of science have their "dirty little secrets" like infinitesimals in math. Most theories contain large gaps that tend to be glossed over. Evolutionary biology has its missing links, and physics has the so-called renormalization process, in which practitioners remove pesky infinities, arising from the considerations of singularities, from the equations of quantum mechanics and miraculously arrive at correct answers. The alternative mathematical techniques offered by an infinitesimal approach may provide a key to legitimizing the renormalization process.
Many physicists have argued that time and space are not continuous but quantized like energy. Infinitesimals may offer a kind of reconciliation between the two hitherto irreconcilable domains of the continuous and the discrete, between smoothness and graininess. They can be a substrate for either realm. Continuity can be seen as infinitesimal change in infinitesimal time, and, as I said a while back, quanta can be arrival points of feedback with infinitely many recursions or iterations in finite time.
Infinitesimals have been trotted out occasionally, although not as often as Heisenberg's Uncertainty Principle and Gödel's Incompleteness Theorem, by those of a mystical frame of mind to serve as loopholes through which consciousness can slip into our presumably computer-like brains. God has long been associated with the infinite, and there is a certain appeal in the idea that the reciprocal of infinity is a bit of the Godhead enfolded in us, the macrocosm in the microcosm. We could imagine, for example, that the representations of the world in our mind's eye have an infinitesimal nature. Memories could be modeled as actual infinitesimal residues of experience rather than symbolic representations, which would provide essentially limitless storage space in a finite space. This model could explain the consistent failure of scientific models of memory. Rather than seeing the brain as a computation and storage devise, this point of view sees it as a modulator/demodulator between finite and infinitesimal realms, an amplifier of some sort. On the face of it, the hardware of the brain seems as suited to being a radio or modem as it is to being a computer.
It is of course more than a little troubling to scientists to posit the existence of things which are in principle undetectable, but, on the other hand, lack of detection has not hurt the popularity of gravitons or quarks or superstrings.
Infinitesimals for Zero
How are infinitesimals related to the Zero Paradox where a nothing is treated as a something? I want to suggest that infinitesimals can act as surrogates for zero in cases of counting and, especially, of measurement. These relative nothings which can sum to something clearly avoid the paradox. They are somewhere between absolute (and thus nonsensical) nothingness and something. To begin with, I will try to show that some problems presented by assigning an absolute "nothing" value to zero can be ameliorated by inserting an infinitesimal value in its place. Rather than being an intermediary value between the quantity zero and finite numbers, it replaces altogether the quantity zero. This approach is just as problematic as the usual zero approach but, in my opinion, no more so. It too provides the insights that come with Assumption Switching.
With such a replacement the division rule, as we have seen, would now have no exceptions. A number divided by an infinitesimal is an infinity and a number divided by an infinity is an infinitesimal.
The most obvious and compelling argument for the substitution of infinitesimals for zeros comes from probability theory (and/or measure theory). A simple definition of the probability of an event is the number of equally likely ways that the event can happen divided by the total number of possibilities:
Thus, for example, the probability of randomly selecting an ace from a standard deck would be 1/13 since there are four aces to be had from among the 52 cards.
We might suppose from this intuitive definition that a probability of zero could only be associated with utterly impossible events, but this is not the case. Suppose we drop a coin on the floor, pick it up and then drop it again. What is the probability that it lands in exactly the same place? Technical definitions involving measure theory yield an answer of zero which is highly anti-intuitive because we sense that it could happen. In fact since every possible position yields a probability of zero, we have that peculiar situation we had with the point-width paradox. How can even an uncountable sum of zeros be more than zero? Looking to our simple definition, however, we get one favorable outcome out of infinitely many possible outcomes or µ § for the probability of repeating the position of the coin. As we have seen, this ratio can be interpreted as an infinitesimal. Thus, despite it being infinitely improbable for the identical position to recur, our intuitive sense that such a thing is possible is satisfied by a infinitesimal rather than zero value. It would seem to be more appropriate to reserve a probability of zero for the truly impossible.
0 vs. 1
Binary arithmetic, in which only the digits 0 and 1 are used as opposed to the ten digits used in the decimal system to represent any quantity at all, is given a great deal of attention these days. This is mainly due to their ubiquitous use in computer science. That use is based on the strong analogy between Boolean logic and the "ons" and "offs" of simple electrical circuits or logic gates. But another aspect of the interest in binaries comes from the simplicity and elegant "ultimateness" of the binary system. It takes the idea of a number system to its logical extreme. What could be more stripped down than the somethingness vs. nothingness of 1 vs. 0?
Leibnitz was among the first mathematicians to take an interest in the binary system. He saw something very special and meaningful in the binaries. Laplace, the 18th century French mathematician, comments:
Leibnitz saw in his binary arithmetic the image of Creation... He imagined that Unity represented God, and Zero the void; that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in his system of numeration...I mention this merely to show how the prejudices of childhood may cloud the vision even of the greatest men!
Rudy Rucker, in his book Mind Tools, points out that it is probably no coincidence that the symbols 0 and 1 have evolved to look a little like an egg and a sperm. The zero egg, far from being a nothing, is dormant but symbolically full of potential. Once it is "fertilized" by the finite, it produces all of the multiplicity of forms. This image of zero as the cosmic egg goes better with soft infinitesimals than a hard zero. [Zero as absolute nothingness is sexist!]
This "relative nothingness", unencumbered as it is by positive or negative qualities, is free to become anything. Like space, it can be seen as a reined in 'everything' (the reciprocal of infinity?) from which 'something' sometimes leaks out. As children we were very aware of the potential for stuff to arise from nothingness. Four year olds, for example, appreciate that the darkness is not just the lack of light, a mere nothing. My son used to worry that the darkness outside his window would come into his room and get him. I remember from my own childhood how monsters were born of the shadows on the wall. Likewise the nothingness of sleep gives rise to the astounding multiplicity of dreams. Nature abhors a vacuum in the sense that as one aspect of the world is suppressed others are amplified. Of absolute nothingness we cannot speak, but it may be that this relative sort of nothing which we experience in the world is better represented by infinitesimals than by zero. Infinitesimals conjure that sense of potential; they give that feeling of readiness for becoming. Remember that any sum of zeros is still zero, but infinitesimals can become finite, can become manifest.
It must seem bizarre to be told, in effect, that there is no such thing as zero. I do not expect to convince you that that is true. Operating from the assumption that many truths can peacefully coexist through Assumption Switching, I have been trying to develop, as I did in the previous chapter, a complementary description that will encourage new insights. The standard zero concept fits better with the Fully Automatic Model of reality which sees space as an absolutely dead, passive non-factor. Infinitesimals seem more appropriate to the complementary holistic approach. Zero goes with clear edges and hard stops. Infinitesimals work better with fuzzy edges and soft landings.
Perfect Cancellation
When we are in the mode of considering the world holistically as a product of the interaction of processes, zeros occur only as various influences cancel each other out perfectly. Two waves will interfere with each other and when they are perfectly out of phase they cancel each other's effects.
Noise-cancelling headphones exploit this fact to protect the hearing of people who must work in extremely loud areas like certain factories or airport runways. The phones detect the soundwaves and produce a counter wave to cancel it in the region around the opening of the ear (and less well elsewhere). A canceled wave is a strange thing, a relative zero. A certain logic would indicate that there is no longer any wave. The vibration has been damped out. On the contrary, outside the area of cancellation the wave "reconstitutes." We have all seen this in the expanding rings produced by raindrops on the surface of a pond. There are places where part of a circular wave seems to disappear only to reappear a moment later. The wave did not cease to exist but was only masked as it continued to propagate. A canceled wave once again has this property of unexpressed potential, like an infinitesimal.
An image of cancellation that has been very meaningful to me in this regard comes from elementary physics and calculus. I will develop this image in some detail because I'll be referring to it later and because I find it truly fascinating. Suppose all of the mass of a "planet" is miraculously concentrated in an infinitesimally thin layer on the surface of the sphere. That is to say, the planet is hollow. Newton's gravitation laws say that we can calculate the total gravitational strength of this planet by summing the tiny force vectors of the infinitely many infinitesimal "patches" of which it is made, point by point if you will. As a result of this summing, we can show that an object outside the planet will be attracted to the planet exactly as if it were solid and, equivalently, as if the mass were all concentrated at the central point of the sphere. People on the surface of the planet would not sense any gravitational difference between this and a solid planet of the same mass. By symmetry we are forced to conclude also that an object placed in the exact center of the hollow planet will not be moved by its gravitational force. All of the force vectors acting on it from the many patches cancel each other out.
Unexpectedly, however, the object would experience the same weightlessness at any point in the interior of the planet, rather than be accelerated toward the nearest point on the surface as intuition suggests. By a strange mathematical quirk, the sum of force vectors at any point in the interior cancel perfectly. This is Newton's Shell Theorem. The patches pulling an interior object toward the near point do so strongly but are fewer in number than the ones pulling the other way, trading-off just so that the forces balance. The forces are not gone, mind you, but no acceleration is experienced. In an imaginary solar system, the interior object would move under the influence of all massive bodies in the system except the body it is within. Until it struck the interior surface, the object would move as if the planet were not there. Here again is a strange kind of nothing, full of potential but not realized unless the object is outside of this special region or until, say, someone digs a hole in the planet and thus throws the forces out of balance. This image has a lot in common with our anti-GLOI theory of darkness vacuums from the last chapter.
A similar thing happens with a light source when we are considering light as a classical wave phenomenon (i.e. without quantum effects). As a flash of light, for example from a camera's flashbulb, bursts out in all directions, what happens is mathematically equivalent to the following description. Each of the infinitely many individual points or patches on the crest of the spherical wavefront acts as if it is a mini-flashbulb, sending out light of diminished intensity in all directions. The light heading back toward the source into the interior of the sphere tends to perfectly cancel with light coming from other points along the surface. Thus we can say that, in this model, the apparent outward movement of light from the source is, in a way, an illusion resulting from this cancellation. Without this cancellation, light from a single source would appear to come from many directions.(diagram)
If we look carefully at the standard mathematical computation that proves these odd gravitational and electro-magnetic facts (it is clear that this could probably never be experimentally verified) there is actually a second order infinitesimal error in the approximation of the area of each patch which can be safely ignored in the standard approach to calculus but not when we give real existence to infinitesimals. It implies that there might be an infinitesimal error in the final sum. I suppose then that the object could suffer an infinitesimal acceleration (whatever that means). In any event, infinitesimals once again better capture the status of this strange nothing than does the standard zero. My working assumption is that there is no such thing as perfect cancellation. All events leave at least an infinitesimal trace. "Ghosts of departed quantities," indeed.
The phenomenon of perfect cancellation may seem like something rather exotic and rare, but, on the contrary, it is very commonplace in quantum computations and elsewhere. One fascinating example, due to Feynman, has to do with quantum waves that, again from the point of view of their mathematical description, seem to spontaneously travel backward in time and affect events in the past. The cool thing is that these backward moving waves stimulate wavelike reactions in these past objects that move forward in time and perfectly cancel the backward moving waves, so that there is no net effect of the present on the past (or of the future on the present). If there is some kind of infinitesimal residue from these events as my approach suggests, it opens up the intriguing possibility of sending messages to the past, an idea exploited in Gregory Benford's brilliant Sci Fi novel Timescape, in which scientists desperately attempt to save the world from an ecological catastrophe by sending a warning to scientists in the past.
In the next chapter I will eventually use the image of canceled waves to produce an unusual model of the world.