On My Mind X

3. Some Ideas

a. Balance and Layers

Imagine a world of rubberized sheets layered upon each other. On any one sheet there are attractions and repulsions, tensions in various directions, so that at each point there is a balance of forces if there is some sort of equilibrium, or if there are changing forces there is ongoing stretching and compression trying to keep up with the changes. Since the speed of sound is quite finite, it is quite likely that the ongoing stretching and compression may lag behind the changes. Moreover adjacent layers are connected to each other, so that adjustments in one layer will affect the layers above and below it.

Now the layers might be notional, for example on one layer are the masses and energies of matter, and on the other is the rulers that measure the distance between points, both in space and time. Such is the account of gravity provided by general relativity, a measure of the mass energy is on one such layer, and a measure of the space-time is on the other. General relativity says that those two layers are intimately connected, and through that vertical connection horizontal connections on each layer are also present. Put differently, mass-energy affects space-time, and that affects the attractions that masses feel for each other as in Newton’s law of gravitation.  In such a world, what is fundamental is neither mass nor space, but events indexed by their mass-energy and their space-time.

We know that masses gravitate toward each other, and so it happens that the components of a star gravitate toward each other. And as they become closer, the star becomes more dense and the energy of the particles becomes greater, and the temperature of the star increases. Similarly, we might think of people gravitating toward each other to make up a city, and the interactions among them become deeper and more involving. 

When density goes up, when flows become more dense, there is not only more interaction, but those interactions will lead to differentiation, with new forms emerging. In stars, nuclear reactions do the work; in cities, human invention and divisions of labor make for difference and hierarchy. Again, since the speed of sound is quite finite, there is a dynamics of catchup—leads and lags, so that balance is never achieved although always sought—that makes for shocks and waves.

b. Differentiation, Hierarchy, Emergence

Whenever there is differentiation, it would seem that hierarchy appears. Hence, kinship systems are not only linkages, but also hierarchical, with each person having a unique role and label, related to other persons’ role and label, and those labels indicate where they stand in an hierarchy. In such a hierarchy, there is both symmetry and orderliness. Symmetry is a statement that some aspects of unique labels do not matter (say siblings, although birth order and gender say that is not the case). Order is a statement that some aspects of unique labels matter, so that siblings owe fealty to their parents.

In the world of elementary particles, there too is kinship structure (a family of particles, that family part of a larger family), and symmetry and order, where some forces are the same for all particles in a certain family, and order where forces are different between families. 

In the world of mathematics, there are hierarchies of numbers, say beginning with the rational numbers, then adding in square roots of prime numbers or i = √-1, thereby creating such an hierarchy. 

Curiously, numbers that are prime in the rationals, such as 3, 5, 7, may not be prime in the next level of the hierarchy, so that 2=(1+i)(1-i), and numbers may have more than one factorization, so that 6= 2×3 = (1+√5)(1-√5). As curiously, particles that are fundamental at one level in the hierarchy, may prove to be composite at a higher level: so that atoms are shown to be composed of electrons and protons and neutrons.

The ongoing problem is to identify the right objects, whether it be the right prime numbers in a different number system, or the right objects in a particular energy regime and the right components of those objects. If we cool down a liquid, there emerges a crystalline solid, something that is quite unexpected were we to know only about the properties of that liquid. If we add numbers such as √5 to our number system, some primes no longer are prime numbers, again not at all expected. Ideally, we would like to know, from just studying the liquid or the rational numbers, what would happen when we lower the temperature or add that √7, respectively—that is, how to understand what emerges in terms of the background or basic system.  Put differently, given what we know microscopically, can we say what will happen at the next macroscopic level—the theme of reductionism, so prevalent in physics. It turns out, that in physics, we cannot say what will happen until we see it, and then we might explain its appearance. (In mathematics, sometimes we can say what will happen.)

Moreover, what is remarkable is that the primes or the particles form a coherent system. For the primes, there are rules that tell us what will happen to a prime, p, when we are in a system where another prime, q, rules, and vice versa—so called reciprocity relations. And if we package all the primes into a nice function, that nice function has a surprising orderliness. Similarly, if we are given a set of elementary particles, once we see them as parts of families, we can describe them in a coherent theory (the Standard Model, or for atomic energy levels, the quantum mechanics of the electron in the electric and magnetic fields of the nucleus). So what we see in each case, what would appear to be distinct individuals (each prime number, each particle), in fact can be understood as part of system.  And, at any level of a hierarchy, denoted by strength of the interparticle forces or by the kinds of added in numbers, such as that √5,  we can understand what is going on without worrying too much about other forces or other added in numbers.

Put differently, there is differentiation and hierarchy, and at each level of differentiation and hierarchy, there is symmetry and orderliness, at that level. So we are back to kinship systems, where these ideas are instantiated by societies without giving it much thought, much as the numbers and the particles are not expected to give it much thought.

Cities are articulated in status, role, activity, product, and person. That there are so many people in one place allows for specialization, and through exchange people trade the products of their strength for those of others. Adam Smith’s needle factory is here particularly complex and perhaps efficient (given the variety being offered). What is striking is how this diversity becomes hierarchized into those who can accumulate further wealth and goods, and those who get by, and those below the margin. We often attribute such outcomes to individual talents and fortune, or to systematic channeling, or to Providence. Providence surely works in surprising ways, but still we try to get a handle on our world. Economics provides many such mechanisms, including effects of agglomeration and congestion, and of increasing returns to scale when concentration serves efficiency—crucial to the economy of a city. 

So is also provided in a different fashion by the physicists and the mathematicians in how they account for the elements in the Periodic Table and for the elementary particles in the Standard Model, and for number systems beyond the integers (…–3, –2, –1, 0, 1, 2, 3 …) and the rational numbers (integer₁/ineger₂), such as the square root of 2 or the square root of –1, or π. What is remarkable is that it is possible to set up hierarchies of these elements (rows and columns of the Periodic Table) or elementary particles (families and masses) or number systems (integers, rationals, adding in solutions to quadratic equations,... ). While it is surely the case that it is our doing to have created such systems, to the physicists or mathematicians, they appear to have been awaiting our discovery and analysis.

Without making any claim about the sources of hierarchy in cities, although that is a major preoccupation of scholars and politicians, it may be useful to have an articulated model of such hierarchy as provided by the mathematicians and the physicists. For these models are rather more amazing in their hierarchy, the emergence of new levels, and in a sense of how they hold together, than would be imagined by a social scientist or an historian. We want to understand these “periodic tables,” and show how they are organized not only horizontally and vertically, as is the Periodic Table of the Elements, but in other dimensions as well, to create a richer analogy than has been heretofore employed. 

The lessons are:

1. It pays to think of objects and phenomena as encased in a hierarchy of boxes, boxes within boxes.
2. If we choose the right sort of objects, and the right sort of boxes, their organization is nicely hierarchical, again boxes within boxes, Russian-doll-like.
3. When we have such a classification or labeling system, and we enumerate the objects (people, institutions, spaces,…), we may find  that we have something like a smooth distribution--much as coin tosses, when numerous, yields to a bell-shaped curve from probability, even though any single toss is totally unpredictable and is a discrete event.
4. The classification system is not unlike anthropologists’ modes of describing the kindship structure of a society:  who might marry whom and which liaisons are forbidden.
5. In understanding such boxes-within-boxes, one has to create even more commodious boxes to enclose emergent or surprising objects, so that all the individuals have their own unique box sequence and locale. And, correspondingly, one has to make sure each object has its own unique label, a label that accords with the boxes-within-boxes system. If two objects have the same label, that indicates we do not understand the structure of the system—and when we do, those objects will have distinguished labels.
6. The boxes are designated by the orderliness or symmetry of the objects enclosed, that they belong to the same family, for example.
7. If the boxes or categories are of the right sort, one discerns an overarching orderliness, accounting for all the boxes and how they fit into each other. And it accounts for how we might treat each object in each box as distinctive and individuated.
8. And, that smooth, perhaps bell-shaped curve looks the same even if we look at the coin tosses in groups and ask about the distribution of head/tail for each group. We might call this automorphy (same shaped), or perhaps scaling by √N, where N is the number of coin tosses.
9. At least for mathematics, you can predict the structure of the levels of the hierarchy, while in physics you may have to first empirically discover a level and then provide an account of why it is natural and inevitable.
10. BUT, this model does not allow for the world’s being polluted, that is, there is mixture and things are unavoidably out of place. It might indicate the unmixable categories, but that does not mean the mixtures won’t appear, often in ways unimaginable until they occur. That pollution or perversion will be a recurrent tension.

These demands are perhaps much too strict for understanding hierarchy and classification in a city. They do set a standard for what we might mean by such a system. In fact, in cities, as in all of society, there is mixture among the categories and stuff that would seem not to fit into any box. Such pollutions and sports (as in genetics) are in fact what make cities so productive and surprising.

c. Harmonies

In a variety of situations, natural scientists and mathematicians find themselves counting up similar situations. Perhaps they systems all have the same energy but different configurations, or they are the number of solutions to an equation in various number systems, or perhaps they are enumerating the prime numbers. They then package those numbers (for different energies, or different number systems), {a_n}, into a function that might look like L(s)=Σa_n/n^s, what is sometimes called an harmonic series. In thermodynamics, the packaging function is directly related to the chemists’ free energy. More generally, it turns out that L has remarkable properties. For example, it might have shape invariance, much as the bell-shaped curve in probability is virtually independent of the situation. And, if they study the properties of L, say the values of s when L(s)=0, its “zeroes,” they learn a good deal about the system under study. Or, if they expand L into a Fourier series, that is L= Σb_n exp-ins, that is, into its primitive tones or frequencies, the b_n are equal to the a_n! What is even more remarkable is that there may be many ways of doing the counting up. 

Now, rather than counting up the obvious objects, we might be able to discern the “right” independent particles that compose the system, much as we discern the normal modes or resonances of a vibrating system such as that spring mattress.  more

d. The Stages for our Action

All the world’s a stage [A vacuum in fact],

And all the particles and antiparticles merely players;

They have their exits and their entrances;

And one particle in its time plays many parts, … 

[Borrowing from Shakespeare’s As You Like It, Act II, Scene VII]

Cities demand orderliness, if only because there is a density of people, roles, institutions, and activities, all of which are coordinated so that the business of life can go on. That orderliness need not be neat or obvious. All we demand is that our roles imply relationships to others that make sense to all of us. Now, there are many places where we enact our roles, some of them quite intimate, others more public and widespread.  In effect, at each scale of enactment, there needs to be an order that is readily recognized, and that order will be different in a family, an extended family, a community, a work life, a commercial life. So we might think of those orders as emerging as we go from one level or area to one that is more intimate or more public or more particular. Moreover, we know that such orders, or roles, will have people and activities that are “in-between,” or maybe marginal, and we have to be able to recognize such liminality and label it as such. Moreover, there is no guarantee that the levels won’t intersect allowing for mixture.

These roles and orderlinesses could be seen as artificial and arbitrary, or abstractions, much as algebra is used to do geometrical proofs. But if the city is to work, especially when situations are marginal or chaotic, it seems that we must actually believe that these devices of order are authentic and not merely social sleights of hand.  (That is, the algebra is meaningful, not merely moving symbols around.) For then, we can improvise more effectively, and what we do is meaningful and not merely a trick or playing a role, or blindly following a rule.

Much of what we do in cities might be described as exchanges—of money for goods, of time for money, of goods for other goods, of favors for other favors. (Again, what is distinctive here is the intensity of such exchanges.) And much of what we do is a matter of following patterns of flow and path already well worn, our being part of that flow. And some of the time, exchange and flows align us with others so that we are bound to them not only for the moment, but also for more extensive spaces and times.

We discover that such orderliness, meaningfulness, and interactions actually make up our lives, and they help define who we are. They are conceptual notions, but in actual life they are the means by which we enact our being and our relationships with others. In effect, we have a composed world, much as a painter might compose a picture so that it encompasses a world. 

Surely since the advent of photography, but earlier in drawing and painting, the idea that we might represent the city as it really is, in perspective of one sort or another, in levels of detail hardly imaginable in earlier eras, has allowed us to believe that we see the city. But, given the diverse representations, we might better understand what we are doing is composing when we draw, paint, photograph or view an image of the city. Put differently, our eye and visual system is not a mechanical device, even if we can discern its lens, its diaphragm, and sensor, and even if we allow for information processing through our neural system. This sort of observation applies to all of what we view, but for those who plan or study or renew cities it provides a level of opportunity and invention for our imagination and action that is empowering (or destructive). When Charles Marville photographed Paris in about 1858-1877, under the auspices of Baron Haussmann, using large wet-plate negatives that allow for enormous detail—to show the before and after of Hassmann’s éventrement of Paris under Napoleon III—it was surely documentation, but with a purpose and a point of view that was revolutionary and justificatory, a new composition of the city.

What is remarkable is how the city allows for so many different perspectives, so many subjects, so many variations on any one theme—yet it is still that city, allowing  for new perspectives, new styles, all of which echo the past as well.

e. Diego Rivera’s Detroit

A Diego Rivera mural encompasses historical events and activities in different places and times, all now on the two-dimensional surface of a wall. He uses various devices to structure the fresco, so that it can include a range of space and time and social class and events, all in front of the viewer—albeit one must scan the mural, take one’s time and consideration, to explore that mural and the story it is telling. 

The historical sweep is visually presented as a wide panoply of specific people, events, activities, occasions. Individuals are particular, places are actual. There are actors and events, not historical forces. (Even geology is a specific event.?) So Mexico has a wide-ranging history, an older world seen as “new” by the European “old world.”

The murals are didactic, for as your eye travels, as you walk along, you are taught. Space and time are traversed in the subtle divisions of the panels, the almost overlaps, your scanning as you look and go by. Each building’s architecture defines that space of your actual motion, that progression of events and scenes as presented, and nearby or opposing panels might speak to each other.

The historiography is systematic, and stuff is interconnected by the fact of the cheek-by-jowl layout of different places and times. The specificity of events and people would allow Diego Rivera to take sides in political disputes—he cannot avoid doing so.

Work is a recurrent theme, the dignity of the worker, the imperatives of the manager or boss, the echo of slavery everpresent. The worker is specific, not a figure or a type, doing particular work. The hierarchy of the classes, of humans over animals?, is seen as a product if the particular situations and individuals pictured. Various “famous” people appear in cameos, and sometimes in starring roles—portrayed so they are readily recognized.

Crowds of people are bunches of particular people. What is ordinary or everyday is significant within an overarching account of domination, modernization, and violence.

In 1932, with the Great Depression under way, Edsel Ford commissioned Diego Rivera to portray Detroit, through a series of murals in an enclosed courtyard of The Detroit Institute of Arts Museum. Both Ford the capitalist and Diego Rivera the Communist were entranced by modern life, by the machine, by the factory as a machine—the Ford Motor Company River Rouge plant took in raw materials, including ore to make steel and labor to engineer and man and woman the mass production factory developed by Edsel’s  father Henry, and at the other end emerged Ford’s automobiles, to be bought by those same workers—if they could afford an automobile—and others like them.

In this suite of murals, we have analogy on the grandest of scales, presenting the whole world as a series of linked images, to be seen in actual surround much as the world is seen. We have that identity in a manifold presentation of profiles, literally, much as the River Rouge Plant was an identity itself, and was an analogy to the articulated complexity of modern industry and labor. Moreover, the everyday becomes transcendent, for it is presented as whole (of course, it is incomplete, with no home scenes) as a work, as a work of art that draws from a tradition of painting that made visible and accessible what was transcendent, namely, the Bible. That is, Detroit and the Bible are analogized here.

The murals are all-encompassing, from the generativity of women and the soil, to the machinery and work-stations at River Rouge, to the plutocrats who were in charge of and owned the plant and the Institute of Arts. 

Diego Rivera’s images have enormous levels of detail, workers are given actual faces and bodies as are the plutocrats, the actual detailed machinery and processes almost allowing for their reproduction (reminding me of Samuel Slater (~1789) memorizing the plans of textile machinery when he emigrated from England to America—since it was forbidden to leave England with the actual plans on paper). In part, the images are much like Diderot’s for the Encyclopédie (1751-1772) but they are more personal, less mechanical, but have the same level of detail and specificity showing a fascination with the machine and the people who work at the machine, and with the organization of the process of production. Whatever the ideology, Ford’s or Diego Rivera’s, the composition of this world, this River Rouge plant and the environment of generativity and of natural resources needed to make it work, dominates. What is striking is not only any single mural, or any detailed part of the larger murals, is how Diego Rivera composed this world, fit the parts into the spaces available in the walls of the courtyard, made it into a whole. None of what I have been saying is surprising when describing a work of art, no matter how concrete is the reference (the City of Detroit or, historically, the Bible). But we need to keep Diego Rivera in mind as we consider our now dominant mode of representation in photography through cameras rather through painting on wet plaster.

We see in a composition, no matter how mechanical and organizational (for that was Diego Rivera’s doing) the actual context. The appropriate analogy is not the eye but the muralist and for the muralist the appropriate analogy is painting as it had been developed over 600 years and the verisimilitude that photography allowed over the previous almost-century. Yet painting allowed for a composition that presented many scenes and events in one work, in one image, with scenes up front, in the background, and adjacent to each other perhaps separated by a stream. 

Different modes allow for different composition and so different seeing. The technology matters, as does the tradition of its employ and the traditions of allied technologies. As for the city as seen, is it to be a machine, a designed object, a composed work of art, or at least all of these. 

f. Making the World Linear and Summable

Here, I present our current account of elementary particles: the nature of particles, the presence of a scaling symmetry, dealing with very not so linear situations, and, conservation of whatever there is in the world. It is rather more technical than the rest of the text and might be scanned.

1. Our best account of Nature assumes that whatever there is, it may be accounted for in terms of additive or summable parts (such as atoms) and small changes in those parts, where what we are adding-up are those parts’ properties.  We might say that there is a nothing, what is called a vacuum, and particles in that vacuum, much as a theatrical stage is black and it is then lit and the actors appear. Those particles are affected by disturbances (perturbations) due to their interactions with each other and perhaps they are affected by some sort of external force. Interactions may be attractive or repulsive, or maybe they are like two billiard balls bouncing off of each other. 

There is a hierarchy of vacuums, where the details of one vacuum and its particles are articulated in another (smaller-scale) vacuum and its particles. And the transition between vacuums might be seen as a “phase transition” such as that between ice and water. The system becomes more or less energetic as we raise or lower the temperature or, equivalently, the energy per particle. At some point there is so much available energy, so much random disturbance, there will that change in the orderliness, a change in phase, as in a magnet losing its magnetization or ice melting (the change from crystalline ice with its orderliness, to water, which has much less order). Moreover, even if we expect the world to be orderly or exhibit symmetry, it may be more stable for that symmetry to be broken—so that the direction of the ice crystal’s axes is definite even though no direction is otherwise preferred--just what happens when water is supercooled (that is, below 32 degrees Fahrenheit) and then one taps on the container and it immediately crystallizes out into ice

There would seem to be a small number of stages or levels in the hierarchy of vacuums. For each more-orderly vacuum an indication is to be found in the adjacent less-orderly vacuum but that potential interaction is not readily visible there, the potential orderliness in say water not to be seen until it becomes ice. 

2. Particles may be analogized to springs, a collection of which are loosely tied together, and that is what we mean by linearity. Namely, Hooke’s law, Forces = -vx, where v is a measure of the microscopic interaction of much smaller particles that make up a spring (here the net effect is attraction and so there is springy-ness) and x measures how extended is the spring.  In honor of the sound of a tuning fork, those springs are called harmonic oscillators. Moreover, the loose tie of the springs to each other, think of a spring mattress, are also linear in this sense by analogy, now v and x measuring their tie to each other and how much that is strained. The springs may lose some of their springy-ness if they are stretched too far, or those interactions may marginally change the tone or frequency of their ringing, but may not otherwise change them much. 

Sometimes, the interactions among the particles are not so marginal and the springs’ sound is now very different than before. The musical notes need to be relabeled and there is a new scale. 

You expect to account for what you see in nature by combining or adding up those springs or particles (again, adding up their properties), their music different than what you expected from each individual particle since they are interacting with each other, perhaps gently, perhaps less gently leading us to redefine the notes or particles. What you see in Nature are those interacting or dressed springs, not the original bare or naked ones. 

Interactions among particles are mediated by other particles within an environment or vacuum that sets up their original tones (the tones are equivalent to energy or mass). 

3. Curiously, if one adds up the sounds, so that you hear the music of the spheres, so to speak, that is, have all the particles or springs sounding, often that sound exhibits a similarity to a related sound, say one octave up or shifted by a third. 

Formally, combining or counting up particles leads to that tonal similarity. That counting has to be done carefully, so as to count up the right objects, and to avoid double-counting. In effect, the frequencies or tones are expressed in the sound or in that similarity. Even more remarkably, that counting up leads to the shape of the container that contains the springs. It is said that you can hear the shape of a drum. 

There would seem to be a threefold analogy or equivalence, among the counting or packaging function, a function that is shape-similar, and geometry.

4. More generally, rather than counting or adding, we often do better to discern the manifest properties or symmetry of a system, and when that symmetry is not present or broken. Rather than counting up all the notes or resonant tones from hitting a drum, we will do better to measure its area and perimeter. Rather than using an atom’s spectral lines to figure out its basic frequencies, discerning its rotational symmetries will be more efficient. And rather than solving an equation for its roots, you will do better to ask which interchanges of its roots will leave the equation unchanged. That interchange symmetry may be represented by a system of matrices. Even better, there is a smooth function, a cousin of the sine and the cosine, one that exhibits a scaling symmetry or automorphy, around which those matrices may be arrayed. And the harmonies of that smooth function tell you just what the interchanging would tell but now more conveniently. In effect, here, we might explore the deepest features of the prime numbers by studying a cousin of the sines and cosines. 

5. In general, energy is conserved as are other quantities such as electric charge or current, or the leptons’ flavor or the gluons’ color or the quarks’ flavor and color, much as a banker adds up deposits and withdrawals and compares them with what is at hand. And so the system is describable by nice (partial differential) equations—equations that build in the possibility of conservation, of diffusion away from a source (as in heat flow, or the wafting of a scent across a room), and of waves . Their model is James Clerk Maxwell’s equations of 1862 for describing electricity and magnetism, and those equations would appear to describe a fluidic system. Maxwell figured out the equations by imagining a mechanical system of rotating elements with suitable grease and gunk between the gears. Moreover, in the various interactions, close up they appear much as they do for Newton’s gravity or Maxwell’s electricity, namely [{Charge₁ x Charge₂ }/r²].

We expect Nature’s interactions to be local (a particle interacts with a distant one through a particle that goes between them), causal (so that effects come after causes), smooth with isolated bumps, and where something does not appear out of nothing. Moreover, we often find that stronger interactions mirror weaker ones, low temperature interactions mirror high temperature interactions.

During the development of radar in World War II, it turned out to be convenient to abstract away from the exact details in designing a tube for generating radar waves, and consider general properties such as the shape of the tube and locality, causality, …,  mentioned in the last paragraph. Those properties have proved powerful again and again in understanding microscopic physics in black-box terms.

This black-box strategy is more generally employed. Your theory or model is meant to apply to the situation you are dealing with, with exact details deferred to a more microscopic theory. That microscopic theory might explain a property of your situation, and you use that as a given.  You try not to get into the black box, and just work with its features.