We are all familiar with repulsive forces in the physical world, like the repulsive force between two, like magnetic poles. In May 2020, Quanta Magazine reported on a result in mathematics, from mathematician Vesselin Dimitrov of the University of Toronto, where the proof is understood as a demonstration of what mathematicians call a ‘repulsion’ between numbers. Dimitrov proved a decades-old conjecture known as the Schinzel-Zassenhaus conjecture which concerns the roots of a particular family of polynomials. Kevin Hartnett begins by explaining mathematics’ version of physical repulsive actions:

When mathematicians look at the number line, they see the same type of trend. They look at the tick marks denoting the positive and negative counting numbers and sense a kind of numerical force holding them in that equal spacing. It’s as though, like mountain lions with their wide territories, integers can’t exist any closer together than 1 unit apart.

The spacing of the number line is the most basic example of a phenomenon found throughout the field of number theory. It crops up in the study of prime numbers and in the relationships between solutions to different types of equations. Mathematicians can better understand these important values by

quantifying the force that acts between them.(emphasis added)

Let’s look at the result that is the subject of Hartnett’s article. If it’s been a long time since you’ve thought about polynomials, you may recall from Algebra classes that the roots of a polynomial equation *y=f(x)* are the *x* values that produce 0 for *y*. On the Cartesian plane, they are the *x* values where the graphed curve of the polynomial intersects the *x* axis. These intercepts mark the real roots (the real numbers that produce 0 when plugged into the polynomial), if they exist. Polynomials may also have roots that are complex numbers – numbers represented as the sum of a real number and some multiple of *i * which is the symbol for the square root of *-1, *known as the imaginary unit. Roots plotted on the complex plane will include roots that are complex numbers. Finding geometric relationships among the roots of a polynomial has long been a subject of study in mathematics. The Gauss-Lucas theorem, for example, establishes a geometric relationship, on the complex plane, between the roots of a polynomial and the roots of its derivative (which measures or quantifies the way *y* values change as *x *values vary for that polynomial). Derivatives of polynomials are also polynomials, and the theorem says that the roots of the derivative of the polynomial all lie within the smallest polygon, on the complex plane, that contains all the roots of the polynomial itself.

The result discussed in Quanta has to do with a particular class of polynomials called cyclotomic polynomials. These are polynomials, with integer coefficients, that are irreducible (meaning they cannot be factored), whose roots, on the complex plane, all lie on the unit circle (a circle centered at the origin with a radius of 1). There are an infinite number of such polynomials and there is a formula for producing them. It is striking that all of the roots of all such polynomials lie on this circle. The Quanta article discusses the proof of a conjecture about the relationship between the roots of these cyclotomic polynomials and non-cyclotomic polynomials.

In 1965, Andrzej Schinzel and Hans Zassenhaus predicted that the geometry of the roots of cyclotomic and non-cyclotomic polynomials differs in a very specific way. Take any non-cyclotomic polynomial whose first coefficient is 1 and graph its roots. Some may fall inside the unit circle, others right on it, and still others outside it. Schinzel and Zassenhaus predicted that every non-cyclotomic polynomial must have at least one root that’s outside the unit circle and at least some minimum distance away.

Or, to put the Schinzel-Zassenhaus conjecture in terms of repulsion, it predicted that the smallest roots of a non-cyclotomic polynomial — which might fall within the unit circle — effectively push other roots outside the unit circle, like magnets pushing each other away.

The minimum distance was expected to depend on the degree of the polynomial, specifically it was conjectured to be some constant number divided by the degree of the polynomial (or the power of the leading term). Dimitrov proved that this minimum distance is, in fact, (log2)/4d, where d is the degree of the polynomial. Log2 is a constant, and while the discussion allows for the possibility that the result could, perhaps, be tweaked to be something like (log3)/ 5d, the fact was established that the distance does depend on the quotient of a constant and a multiple of the degree of the polynomial.

One might say that these observations are just observations of the distance between numbers. But it’s more than that. These distances are produced by the numbers themselves, by their interaction in the polynomials. It is not unusual for mathematicians to talk about the ‘behavior’ of mathematical things – the behavior of solutions or, in this case, the behavior of roots. Is it a metaphor, or does this language emerge from an intuition about what a number really is? I suspect the latter is true. Numbers appear to be the names we have given to the elements of things we collect, or the duration of events. But within mathematics they have undergone a significant evolution, forcing us to examine other things, like the notion of a continuum, or the effects of an imaginary unit. Their geometric interpretation opened up whole new worlds of mathematical events. I bring the repulsion principle to your attention to make the point that the nature of mathematical things is just not very clear, and I am convinced mathematics doesn’t belong to us. Don’t misunderstand. I am meaning to be neither romantic nor mystical about these things. I mean to see something more clearly. A correction to our view of mathematics will bring with it a correction to our view of the physical world.

]]>Recently, however, maybe in the last four years, I have become a bit distracted by an alarming abuse of words in our sociopolitical world – what I would otherwise call a frightening disregard for the harm caused by lies. Deceitful narratives are produced by the same thoughtful imagination (our words and concepts) that build the arts and the sciences. But, without any fidelity to the facts to which these thoughts should be tied, their expression acts more like a flood, or a fire, that repeatedly weakens the stability of our cultural achievements and challenges the trustworthiness of all words. It looks to me like the protective walls of our civil systems have been badly damaged and may yet crumble. The helplessness that I have felt in response to this damage has given my interest in the true nature of an idea, or a thought, or a word, new impetus. It’s made my fascination with mathematics all the more striking to me, and led me to believe that there may be a cure for my sense of helplessness.

I think I found my way to mathematics along a zigzagy road that, as a young idealist, I hoped would lead to the truth. And I know that the very mention of ‘truth,’ brings with it a long and profound philosophical history. But as a young person, in my Italian-American family, I was often saddened by the effects of simple deceptions. They were mostly harmless, interpersonal distortions of the truth between my mother and my grandmother, or my mother and my aunts,…. but they fed unhappiness, dissatisfaction, and frustration. College was the first time I created some real distance between myself and my family – not geographically, but intellectually. And, while I may have said any number of things about my academic interests to friends and advisors, the classes I chose, the future I imagined, was probably motivated, by my trying to find a way to see what was ‘true,’ in life, and in people. I devoted a lot of time and energy first to philosophy, then psychology, and when I finally considered physics, I found liberation in mathematics.

There is no value to deception in mathematics. And I might argue that mathematics is probably the most consequential, when speaking pragmatically, of our imaginative efforts. Free of deception and profoundly meaningful, the satisfaction I felt from finding this purely symbolic, yet physically connected intellectual enterprise has never subsided. Mathematics bears witness to the worldly relevance of thought, and the power of deception-free analysis. Conjured up by idealizing experience and reasoning, and then letting it grow with a kind of self-organizing life of its own, mathematics may be the most visible evidence of the fact that the mind’s eye, and the eye’s in our heads, each have their own way of perceiving. This leads me to believe that physical structure and thoughtful structure must hold equal weight in nature.

It may be hard to see how all of this would apply to our current sociopolitical situation, but I have thought about it every time I hear some pundit insist that “words matter.” However, taking a broader look at the situation, my husband (who is an experimental physicist) and I developed an analog. We imagined that the law was like mathematics, and politicians were like physicists. By this we mean that the law is the careful and precise development of ideals, and politicians (or government officials) are charged with finding the ways that these ideals may exist in the world. Figuring this out is what the people and their elected leaders try to do together. Deception contaminates the effort.

For whatever reason, I find this analogy reassuring. Perhaps the body politic can recover, like with the pandemic.

]]>There is no starker illustration of the fact that, in the body politic as in the bodies the virus infects, the host’s response can matter far more to the course of the disease than the direct action of the pathogen itself.”

The Economist, “The year of learning dangerously Covid-19 has shown what modern biomedicine can do.” 23 March 2021

It happens often that Quanta Magazine brings news of profound and novel approaches to any number of new scientific questions. And these reports often make a very positive contribution to the perspective I have been trying to nurture at *Mathematics Rising*. In July, Jordana Cepelewicz wrote an article with the title: *What Is an Individual? Biology Seeks Clues in Information Theory.* Two of the words in this title quickly got my attention – *individual*, and *information*. The word *individual* got my attention because the significance of ‘the individual’ plays an important role in so many things including politics and religion, and *information *because once mathematics was used to define information, it has become a mathematical lens that is enormously useful to so many questions in science, including quantum mechanics and theories of consciousness. Cepelewicz makes the pithy remark that nature has “a sloppy disregard for boundaries,” taking note of the M.O. of viruses, bacteria, insect colonies or “superorganisms,” and the myriad varieties of symbiotic composites that live in our world. “Even humans,” she says, “contain at least as many bacterial cells as ‘self’ cells.”

To emphasize the value of clarifying what we mean by ‘individual,’ she writes:

Ecologists need to recognize individuals when disentangling the complex symbioses and relationships that define a community. Evolutionary biologists, who study natural selection and how it chooses individuals for reproductive success, need to figure out what constitutes the individual being selected.

The same applies in fields of biology dealing with more abstract concepts of the individual — entities that emerge as distinct patterns within larger schemes of behavior or activity. Molecular biologists must pinpoint which genes out of many thousands interact as a discrete network to produce a given trait. Neuroscientists must determine when clusters of neurons in the brain act as one cohesive entity to represent a stimulus.

“In a way, [biology] is a science of individuality,” said Melanie Mitchell, a computer scientist at the Santa Fe Institute.

Biology, many agree, has been under-theorized, but this is no doubt changing. The Stanford Encyclopedia of Philosophy has an entry on Biological Individuals from which one can see the development of conceptual frameworks to address the question.

The Quanta Magazine article is based on the work of David Krakauer, an evolutionary theorist and president of the Santa Fe Institute, and Jessica Flack who studies collective behavior and collective computation (also based at the Santa Fe Institute). They created a group tasked with finding a new working definition of the ‘individual.’

At the core of that working definition was the idea that an individual should not be considered in spatial terms but in temporal ones: as something that persists stably but dynamically through time. “It’s a different way of thinking about individuals,” said Mitchell, who was not involved in the work. “As kind of a verb, instead of a noun.”

How do you create this view? What can we use to see things this way? The lens they chose is information theory.

Krakauer and Flack, in collaboration with colleagues such as Nihat Ay of the Max Planck Institute for Mathematics in the Sciences, realized that they’d need to turn to information theory to formalize their principle of the individual “as kind of a verb.” To them, an individual was an aggregate that “preserved a measure of temporal integrity,” propagating a close-to-maximal amount of information forward in time.

Their formalism begins with propositions:

- Individuality can exist at any level of biological organization (sub cellular to social)
- individuality can be nested (one individual within another)
- and individuality exists on a continuum meaning systems can have quantifiable degrees of individuality.

The last of these might translate the question of whether a virus is alive or not, into the question, how *living* is a virus. In other words, where does it lie on the continuum of individuals.

The abstract of their paper is very clear about what this model hopes to accomplish:

Despite the near universal assumption of individuality in biology, there is little agreement about what individuals are and few rigorous quantitative methods for their identification. Here, we propose that individuals are aggregates that preserve a measure of temporal integrity, i.e., “propagate” information from their past into their futures. We formalize this idea using information theory and graphical models. This mathematical formulation yields three principled and distinct forms of individuality—an organismal, a colonial, and a driven form—each of which varies in the degree of environmental dependence and inherited information. This approach can be thought of as a Gestalt approach to evolution where selection makes figure-ground (agent–environment) distinctions using suitable information-theoretic lenses. A benefit of the approach is that it expands the scope of allowable individuals to include adaptive aggregations in systems that are multi-scale, highly distributed, and do not necessarily have physical boundaries such as cell walls or clonal somatic tissue. Such individuals might be visible to selection but hard to detect by observers without suitable measurement principles. The information theory of individuality allows for the identification of individuals at all levels of organization from molecular to cultural and provides a basis for testing assumptions about the natural scales of a system and argues for the importance of uncertainty reduction through coarse-graining in adaptive systems.

(Coarse-graining is a simplification of the details in a system that is as true to the system as the details themselves – the way that temperature represents the average speed of particles in a system)

There is no doubt that this will be a lucrative diversion from the noun-like way we have identified living things. It is, as I see it, one of many efforts in a broad scheme related to the provocative ideas brought to light by biologists Francisco Varela and Humberto Maturana in the 1980s. Rather than list the properties of living things, Maturana and Varela observed that living things are characterized by the fact that they are *continually self-producing* – not reproducing but self-producing. The cell, for example, what we have long thought of as the fundamental living thing, is a network of processes that are *organized as a unity*, where the interaction of these processes continuously and directly *realizes the unity itself.* The cell * is* what the cell

For me, these discussions always bring to mind the thoughts I had when I read Thomas Mann’s *The Magic Mountain*. I read the novel in the absence of any commentary about it, and I remember feeling an unexpected affection for the story’s protagonist, Hans Castorp. Mann began writing the story in 1912, but he completed it after World War I, in 1924. Unexpectedly restricted to a tuberculosis infirmary high in the Alps, a young Castorp is an innocent and eager explorer of biology and medicine. Encouraged by the cold and by his solitude, he rested and read from the library of physicians, fully self-training in the language and images of biology. I remember thinking that Castorp’s look at science and medicine was guileless, without the prejudices created by the pragmatism of fixing things, or the desire for useful knowledge. And, I thought, that contemporary ideas in medicine and biology, that are built on these early observations, and that we tend to think of as just true, were not the only ideas that could grow from the kinds of insights to which Castorp was privileged. Here are just a couple of his reflections:

This body, then, which hovered before him, this individual and living I, was a monstrous multiplicity of breathing and self-nourishing individuals, which through organic conformation and adaptation to special ends, had parted to such an extent with their essential individuality, their freedom and living immediacy, had so much become anatomic elements that the functions of some had become limited to sensibility…

What then was life? …It was the existence of the actually impossible-to-exist, of a half-sweet, half-painful balancing, or scarcely balancing, in this restricted and feverish process of decay and renewal, upon the point of existence. It was not matter and it was not spirit, but something between the two, a phenomenon conveyed by matter, like the rainbow on the waterfall, or like flame.

Maturana and Varela, Frist, Krakauer and Flack, are just some of the explorers that now confirm my hunch that there’s never just one way to see.

]]>….where she studied how the brain uses incoming signals of the velocity of head movement to control eye position. For example, if we want to keep our gaze fixed on a particular location while our head is moving, the brain must continuously calculate and adjust the amount of tension needed in the muscles surrounding the eyes, to compensate for the movement of the head.”

Later, at the University of California at Santa Barbara, Fiete began working on grid cells, a system of neurons I have written about on more than one occasion. These cells, located in the entorhinal cortex of the brain, actually create a grid-like, neural representation of the space around us that allows us to know where we are. Grid cell firings correspond to points on the ground that are the vertices of an equilateral triangular grid. It happens with surprising regularity.

In a Dec 2014 article in *Neuron*, Nobel laureate Neil Burgess said the following:

Grid cell firing provides a spectacular example of internally generated structure, both individually and in the almost crystalline organization of the firing patterns of different grid cells. A similarly strong organization is seen in the relative tuning of head-direction cells. This strong internal structure is reminiscent of Kantian ideas regarding the necessity of an innate spatial structure with which to understand the spatial organization of the world.

I have written on this topic before:*The mathematical nature of self-locating* and*Grid cells and time cells in rats, continuity, and th*e monkey’s mind

At MIT again, Fiete has continued to explore her PhD thesis topic, specifically, how the brain maintains neural representations of the head’s direction, where it is pointed, at any given time. Now, in *a paper published in Nature*, she explains how she identified a brain circuit in mice that produces a one-dimensional ring of neural activity that acts like a compass. It allows the brain to calculate the direction of the head, with respect to the external world, at any given moment.

MIT’s report on that paper explains her approach:

Trying to show that a data cloud has a simple shape, like a ring, is a bit like watching a school of fish. By tracking one or two sardines, you might not see a pattern. But if you could map all of the sardines, and transform the noisy dataset into points representing the positions of the whole school of sardines over time, and where each fish is relative to its neighbors, a pattern would emerge. This model would reveal a ring shape, a simple shape formed by the activity of hundreds of individual fish.

Fiete, who is also an associate professor in MIT’s Department of Brain and Cognitive Sciences, used a similar approach, called topological modeling, to transform the activity of large populations of noisy neurons into a data cloud in the shape of a ring.

More than one aspect of these studies impresses me. The approach is interesting. Topological modeling, in some sense, is a shape-directed computation rather than a purely numerical one. And the simplicity of the ring impresses me. The brain somehow isolates, from a flood of sensory data, the variables that produce this simple representation of the head’s position. And the value of a circle, something we think of as a purely abstract idealization, is something the body seems to know, constructing it, as it does, outside what we call the mind.

“There are no degree markings in the external world; our current head direction has to be extracted, computed, and estimated by the brain,” explains Ila Fiete, … “The approaches we used allowed us to demonstrate the emergence of a low-dimensional concept, essentially an abstract compass in the brain.”

This abstract compass, according to the researchers, is a one-dimensional ring that represents the current direction of the head relative to the external world.

Their method for characterizing the shape of the data cloud allowed Fiete and colleagues to identify the variable that the circuit was devoted to representing.

My first reaction to this story was that it was beautiful. On the one hand, Fiete seems to be making increasingly creative use of the mathematics for which she has always had an affinity. On the other hand that simple ring, or compass, that tells us which way we are looking, highlights the presence of inherent mathematical tools that just exist in the body, in how brain processes just work. And the ring is stable, even through sleep.

Her lab also studies cognitive flexibility — the brain’s ability to perform so many different types of cognitive tasks.

“How it is that we can repurpose the same circuits and flexibly use them to solve many different problems, and what are the neural codes that are amenable to that kind of reuse?” she says. “We’re also investigating the principles that allow the brain to hook multiple circuits together to solve new problems without a lot of reconfiguration.

It would not surprise me if, when viewed from a careful, neuro-scientific perspective, we would find some resemblance between the way the brain makes new use of already configured circuits and the way mathematicians consistently build novel mathematical structures with tested strategies that have built other structures, like ordered pairs, symmetries, compositions, closure properties, identities, homotopy and equivalence, etc….

]]>They call it the “unreasonable effectiveness of mathematics.” Physicist Eugene Wigner coined the phrase in the 1960s to encapsulate the curious fact that merely by manipulating numbers we can describe and predict all manner of natural phenomena with astonishing clarity…

The article by Michael [...]]]>

They call it the “unreasonable effectiveness of mathematics.” Physicist Eugene Wigner coined the phrase in the 1960s to encapsulate the curious fact that merely by manipulating numbers we can describe and predict all manner of natural phenomena with astonishing clarity…

The article by Michael Brooks has the title *Is the universe conscious? It seems impossible until you do the maths.*

As I expected, the primary focus of the article is **Integrated Information Theory**, a way to understand consciousness proposed by neuroscientist Giulio Tononi in 2004. More than one of my previous posts refers to the development of this proposal, in particular there is *Where does the mind begin*? posted in 2017.

Neuroscientist Giulio Tononi proposed the Integrated Information Theory of Consciousness (IIT) in 2004. IIT holds that consciousness is a fundamental, observer-independent property that can be understood as the consequence of the states of a physical system. It is described by a mathematics that relies on the interactions of a complex of neurons, in a particular state, and is defined by a measure of integrated information. Tononi proposes a way to characterize experience using a geometry that describes informational relationships. In an article co-authored with neuroscientist Christof Koch, an argument is made for opening the door to the reconsideration of a modified panpsychism, where there is only one substance from the smallest entities to human consciousness.

Brooks describes Tononi’s idea with broad strokes, saying that the theory tells us that a system’s consciousness arises from the way information moves between its subsystems.

One way to think of these subsystems is as islands, each with their own population of neurons. The islands are connected by traffic flows of information. For consciousness to appear, Tononi argued, this information flow must be complex enough to make the islands interdependent. Changing the flow of information from one island should affect the state and output of another. In principle, this lets you put a number on the degree of consciousness: you could quantify it by measuring how much an island’s output relies on information flowing from other islands. This gives a sense of how well a system integrates information, a value called “phi.”

If there is no dependence on a traffic flow between the islands, phi is zero and there is no consciousness. But if strangling or cutting off the connection makes a difference to the amount of information it integrates and outputs, then the phi of that group is above zero. The higher the phi, the more consciousness a system will display.

ITT has attracted many proponents but at the same time has come under quite a bit of critical scrutiny. The mathematics of the theory is very complex. The language of the mathematics is borrowed from information theory, and informational relationships are characterized geometrically as shapes. These shapes, Tononi says, *are* our experience.

Brooks points to a number of computational issues that require attention as well as the problem of “explaining why information flow gives rise to an experience such as the smell of coffee.” But proponents of the idea are committed to resolving the technical issues, and continue to have faith in the model.

Rather than abandoning a promising model, he [mathematician Johannes Kleiner] thinks we need to clarify and simplify the mathematics underlying it. That is why he and Tull [mathematician Sean Tull] set about trying to identify the necessary mathematical ingredients of IIT, splitting them into three parts. First is the set of physical systems that encode the information. Next is the various manifestations or “spaces” of conscious experience. Finally, there are basic building blocks that relate these two: the “repertoires” of cause and effect.

They posted a preprint paper in February that describes the work.

“We would be glad to contribute to the further development of IIT, but we also hope to help improve and unite various existing models,” Kleiner says. “Eventually, we may come to propose new ones.”

A philosophical difficulty with the theory, however, raises questions close to my heart. The calculations involved in IIT imply that inanimate things possess some degree of consciousness. This may be considerably difficult for many to accept, but I’m happy to report that not everyone has a problem with the possibility. From the Brooks article:

“Particles or other basic physical entities might have simple forms of consciousness that are fundamental, but complex human and animal consciousness would be constituted by or emergent from this,” says Hedda Hassel Mørch at Inland Norway University of Applied Sciences in Lillehammer.

The idea that electrons could have some form of consciousness might be hard to swallow, but panpsychists argue that it provides the only plausible approach to solving the hard problem. They reason that, rather than trying to account for consciousness in terms of non-conscious elements, we should instead ask how rudimentary forms of consciousness might come together to give rise to the complex experiences we have.

With that in mind, Mørch thinks IIT is at least a good place to start.

How does it happen? How to the cells that grow and specialize in embryonic development produce what we call the mind? I became preoccupied with questions like this when I tried to understand the degree to which lesions in the frontal lobe of my mother’s brain changed her experience (and her response to the reality that her new experience created) I began to wonder, in a new way, about the nature of the relationship between her body and her mind. Where was the person in the body? One of the things I heard over and over was “she isn’t there anymore.” And I continued to think, “of course she is.” I now find that the kinds of questions I wanted to ask are increasingly present among researchers, in both neuroscience and mathematics. I have also found that my own hunch that mathematics can shed some light on the mystery is also supported.

IIT is particularly interesting to me because it isn’t just proposing a mathematical description of consciousness, it’s actually suggesting a fundamental or natural relationship between mathematical things and experience. It doesn’t just quantify aspects of consciousness (given by the amount of integrated information), it is also aimed at specifying the quality of experience with informational relationships that are defined by geometric shapes. The points in this geometry are determined using the probability distributions for the different states that a complex of neurons may be in.

I think Tononi himself gets at the significance of this point of view when he writes this in his 2008 Provisional Manifesto:

We are by now used to considering the universe as a vast empty space that contains enormous conglomerations of mass, charge, and energy—giant bright entities (where brightness reflects energy or mass) from planets to stars to galaxies. In this view (that is, in terms of mass, charge, or energy), each of us constitutes an extremely small, dim portion of what exists—indeed, hardly more than a speck of dust.

However, if consciousness (i.e., integrated information) exists as a fundamental property, an equally valid view of the universe is this: a vast empty space that contains mostly nothing, and occasionally just specks of integrated information —mere dust, indeed—even there where the mass-charge–energy perspective reveals huge conglomerates. On the other hand, one small corner of the known universe contains a remarkable concentration of extremely bright entities (where brightness reflects high levels of integrated information), orders of magnitude brighter than anything around them. Each bright “star” is the main complex of an individual human being (and most likely, of individual animals). I argue that such a view is at least as valid as that of a universe dominated by mass, charge, and energy.

Like many of the articles about how mathematics is shedding new light on things, the *New Scientist* article is *not* highlighting the mathematics itself, nor what it has the potential to show us. As I see it, mathematics has the potential to show us how we are likely *misreading* our reality, by showing us how perspectives are built in the mind, or how perspectives are built with the arrangement of concepts and not just the arrangements of neurons firing. Mathematics is in a unique position to show us the bridge between the physical and the conceptual. But this recent *New Scientist* article still reports good news – specifically that ITT continues to attract the interest of neuroscientists and mathematicians alike.

*Category Theory and the extraordinary value of abstraction**More on category theory and the brain**Quantum Mechanical Words and Mathematical Organisms *(for Scientific American)

But the inspiration for this post is something I heard from mathematician Eugenia Chang. It was in a talk she gave, at the School of the Art Institute of Chicago, on *The Power of Abstraction*. Early in her presentation, Chang uses a turn of phrase that I like very much. Mathematics is useful, she says,

…because of the general light that it sheds on all aspects of our thinking.

Notice she doesn’t say, “on all aspects of *things*,” but rather “on all aspects of our* thinking*.” I believe this is important. There is an old tradition among educators to tell reluctant students that, while learning mathematics seems to have nothing to do with their day-to-day lives, or the issues they hope to explore, it’s value lies in the fact that it teaches us how to think. But what Chang is saying is bigger and more important than that. Shedding light on ‘thinking’ is not the same as teaching us how to think. Shedding light on thinking means that mathematics is telling us something about ourselves.

To clarify the value of abstraction Chang uses illumination again:

It’s just like when you shine a light on something (and that’s what mathematics is always doing – trying to illuminate the situation)…if we shine the light very close up, then we will have a very bright light but only a very small area. But if we raise the light further up, then we get a dimmer light, but we illuminate a broader area, and we get a bit more context on the situation…Abstraction enables us to study more things, maybe in less detail, but with more context.

Category theory, as she discusses it, is about relationships among things, the notion of sameness, universal properties, and the efficacy of visual representation. About sameness Chang makes the observation that nothing is *actually the same* as anything else, and that the old notion of an equation is a lie. I haven’t heard anyone apply the term ‘lie’ to a mathematical thing since my first calculus teacher complained about a popular (thick and heavy) calculus text! But the value of an equation, she explains, is that, while it identifies the way two things are the same, equality also points to the way they are different. 2 + 2 = 4 tells us that, in some way, the left side of the equation is the same as the right side, but it other ways, it is not. Equivalences in category theory are understood as sameness *within a context*.

When first introduced to the notion of equivalence classes in topology, I thought of it as a powerful offspring of equality, not a correction. But, either way, the broad applicability of category theory (even within mathematics itself) is certainly fueling its development. The Stanford Encyclopedia of Philosophy says this about it:

Category theory has come to occupy a central position in contemporary mathematics and theoretical computer science, and is also applied to mathematical physics. Roughly, it is a general mathematical theory of structures and of systems of structures. As category theory is still evolving, its functions are correspondingly developing, expanding and multiplying. At minimum, it is a powerful language, or conceptual framework, allowing us to see the universal components of a family of structures of a given kind, and how structures of different kinds are interrelated. Category theory is both an interesting object of philosophical study, and a potentially powerful formal tool for philosophical investigations of concepts such as space, system, and even truth.

Chang also wrote the account of the concept *category* in Princeton’s *Companion to Mathematics*. There she says the following:

An object exists in and depends upon an ambient category…There is no such thing, for instance, as the natural numbers. However, it can be argued that there is such a thing as the concept of natural numbers. Indeed, the concept of natural numbers can be given unambiguously, via the Dedekind-Peano-Lawvere axioms, but what this concept refers to in specific cases depends on the context in which it is interpreted, e.g., the category of sets or a topos of sheaves over a topological space.

If you look back at the earlier posts to which I referred, you will see how the simplicity of the abstractions can serve situations where traditional mathematical approaches contain some ambiguity. I’ve chosen to return to it all today because Eugenia Chang’s language has encouraged me to see mathematics the way I do, as a reflection of thought itself, among other things. Contrary to expectations, she says:

]]>Mathematics is not definitive. It says, here is a point of view.

Probability is inextricably bound to our experience of uncertainty. When, in the 17th century, Pascal explored the calculation of probabilities, his efforts were aimed at finding ways to predict the results of games of chance. But the use of these strategies was fairly quickly adopted to address questions of law and insurance, as these concerned chance (or random) events (like weather or disease) in the day-to-day lives of individuals. The mathematics of probability provided a way to think about future events, about which we are always uncertain. I read in a Britannica article that in the early 19th century, LaPlace characterized probability theory as “good sense reduced to calculation.”

By the 18th century, Bayes’ theorem was already getting a lot of attention. It was beginning to look like the best calculation of likelihoods also relied on the experience of the individual doing the calculation. Bayes’ Theorem is a formula for calculating conditional probabilities, probabilities that are changed when conditions are altered. One of the conditions that could become altered is what the observer knows. This brings attention back to the subject, which is different from the way we understand the likelihood of heads or tails in a coin toss. Since a coin toss can only yield one of two possible outcomes, we have come to understand that there is a 50/50 chance of either. The more times we toss the coins the closer we get to seeing that 50/50 split in the outcomes. What we expect of the coin toss is entirely dependent on the nature of the coin. But conditional probabilities are not so clear. So how should physicists view our reliance on probabilities in quantum mechanical theory. This is what Carroll addresses.

There are numerous approaches to defining probability, but we can distinguish between two broad classes. The “objective” or “physical” view treats probability as a fundamental feature of a system, the best way we have to characterize physical behavior. An example of an objective approach to probability is frequentism, which defines probability as the frequency with which things happen over many trials.

Alternatively, there are “subjective” or “evidential” views, which treat probability as personal, a reflection of an individual’s credence, or degree of belief, about what is true or what will happen. An example is Bayesian probability, which emphasizes Bayes’ law, a mathematical theorem that tells us how to update our credences as we obtain new information. Bayesians imagine that rational creatures in states of incomplete information walk around with credences for every proposition you can imagine, updating them continually as new data comes in. In contrast with frequentism, in Bayesianism it makes perfect sense to attach probabilities to one-shot events, such as who will win the next election.

In an *aeon* article about Einstein’s rejection of unresolved randomness in any physical theory, Jim Baggott say this:

In essence, Bohr and Heisenberg argued that science had finally caught up with the conceptual problems involved in the description of reality that philosophers had been warning of for centuries. Bohr is quoted as saying: ‘There is no quantum world. There is only an abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.’ This vaguely positivist statement was echoed by Heisenberg: ‘[W]e have to remember that what we observe is not nature in itself but nature exposed to our method of questioning.’ Their broadly antirealist ‘Copenhagen interpretation’ – denying that the wave function represents the real physical state of a quantum system – quickly became the dominant way of thinking about quantum mechanics. More recent variations of such antirealist interpretations suggest that the wave function is simply a way of ‘coding’ our experience, or our subjective beliefs derived from our experience of the physics, allowing us to use what we’ve learned in the past to predict the future.

But this was utterly inconsistent with Einstein’s philosophy. Einstein could not accept an interpretation in which the principal object of the representation – the wavefunction – is not ‘real’.

Today, proponents exist for more than one model of the universe. There are models where probability is “fundamental and objective,” as Carroll says.

There is absolutely nothing about the present that precisely determines the future…What happens next is unknowable, and all we can say is what the long=term frequency of different outcomes will be.

In other theories, nothing is truly random and probability is entirely subjective. If we knew, not just the wave function, but all the hidden variables, we could predict the future exactly. As it stands, however, we can only make probabilistic predictions.

Finally there is the many-worlds resolution to the problem, which is Carroll’s favorite.

Many-worlds quantum mechanics has the simplest formulation of all the alternatives. There is a wave function, and it obeys Schrödinger’s equation, and that’s all. There are no collapses and no additional variables. Instead, we use Schrödinger’s equation to predict what will happen when an observer measures a quantum object in a superposition of multiple possible states. The answer is that the combined system of observer and object evolves into an entangled superposition. In each part of the superposition, the object has a definite measurement outcome and the observer has measured that outcome.

Everett’s brilliant move was simply to say, “And that’s okay” — all we need to do is recognize that each part of the system subsequently evolves separately from all of the others, and therefore qualifies as a separate branch of the wave function, or “world.” The worlds aren’t put in by hand; they were lurking in the quantum formalism all along.

I find it foolish to ignore that probability theory keeps pointing back at us. Christopher Fuchs, a physicist at the University of Massachusetts, is the founder of a school of thought dubbed Qbism (for quantum bayesianism). In an interview published in Quanta Magazine, Fuchs explains that QBism goes against a devotion to objectivity “by saying that quantum mechanics is not about how the world is without us; instead it’s precisely about us in the world. The subject matter of the theory is not the world or us but us-within-the- world, the interface between the two.” And later:

QBism would say, it’s not that the world is built up from stuff on “the outside” as the Greeks would have had it. Nor is it built up from stuff on “the inside” as the idealists, like George Berkeley and Eddington, would have it. Rather, the stuff of the world is in the character of what each of us encounters every living moment — stuff that is neither inside nor outside, but prior to the very notion of a cut between the two at all.

The effectiveness of our thoughts on likelihood is astounding. Cognitive neuroscientists suggest that statistics is part of our intuition. They argue that we learn everything through probabilistic inferences. Optical illusions have been understood as the brain’s decision about the most likely source of a retinal image. Anil Seth, at the University of Sussex, argues that all aspects of the brain’s construction of our world are built with probabilities and inferences. The points in the geometry of Tonini’s Integrated Information Theory of consciousness are defined using probability distributions. Karl Friston’s free energy principal, first aimed at a better understanding of how the brain works, defines the boundaries around systems (like cells, organs, or social organizations) with a statistical partitioning – things that belong to each other are defined by the probability that the state of one thing will effect another. Uncertainty defines Claude Shannon’s information entropy and Max Tegmark’s laws of thermodynamics. It’s also interesting that a thought experiment, proposed by James Clerk Maxwell in 1871 and known as Maxwell’s demon, was designed to examine the question of whether or not the second law of thermodynamics is only statistically certain.

As Carroll sees it, “The study of probability takes us from coin flipping to branching universes.” So what’s me and what’s not me? Mathematics has a way of raising this issue over and over again. Maybe we are beginning to look to it for guidance.

]]>The song describes the ‘transfer of information,’ if you will, that moves through the wind to the lambs, from the lambs to a shepherd, from the shepherd to the king, and finally from the king to the people. It goes like this:

Said the night wind to the little lamb

Do you see what I see

Way up in the sky little lamb

Do you see what I see

A star, a star

Dancing in the night

With a tail as big as a kite

With a tail as big as a kite

Said the little lamb to the shepherd boy

Do you hear what I hear

Ringing through the sky shepherd boy

Do you hear what I hear

A song, a song

High above the trees

With a voice as big as the sea

With a voice as big as the sea

Said the shepherd boy to the mighty king

Do you know what I know

In your palace wall mighty king

Do you know what I know

A child, a child

Shivers in the cold

Let us bring him silver and gold

Let us bring him silver and gold

Said the king to the people everywhere

Listen to what I say

Pray for peace people everywhere

Listen to what I say

The child, the child

Sleeping in the night

He will bring us goodness and light

He will bring us goodness and light

The wind *perceives* and communicates what it sees to the lamb. The lamb *hears* the wind, as a song, a formulation, and somehow communicates what he hears to the boy. The boy then *knows* something, has a fully conscious perception, which he brings to the king (the one responsible for organizing the human world) and from there it is broadcast so that everyone knows.

I was raised Catholic and so I remember the birth of Jesus described to us as the marriage of heaven and earth, which may be said to be the reconciliation of the eternal and the temporal, or the ideal and the instantiated. It’s the last of these that has gotten considerable attention from me, in these past many years, as I have worked to square conceptual reality with physical reality through a refreshed look at mathematics. And so the song got my attention because it suggests a continuum of knowing, from the wind to the King, and a oneness to the world of the physical and the devine. The idea that sensation and cognition are somehow in everything reminds me of the polymath and mathematician Leibniz’s monads for one thing, and cognition as understood by biologists Francisco Varela and Humberto Maturana. The rigor of Leibniz’s work in logic and mathematics, together with what he understood about the physical world, and his faith in reason, he dissociated ‘substance’ from ‘material’ and reasoned that the world was not built from passive material but from fundamental objects he called monads – simple mind-like substances equipped with perception and appetite. But the monad takes up no space, like a mathematical point. I wrote about these things in 2012 and made this remark:

All of this new rumbling about mathematics and reality encourages a hunch that I have had for a long time – that the next revolution in the sciences will come from a newly perceived correspondence between matter and thought, between what we are in the habit of distinguishing as internal and external experience, and it will enlighten us about ourselves as well as the cosmos. New insights will likely remind us of old ideas, and the advantage that modern science has over medieval theology will wane. I expect mathematics will be at the center of it all.

For Varela and Maturana, every organism lives in a medium to which it is structurally coupled and so the organism can be said to already have knowledge of that medium, even if only implicitly. Living systems exist in a space that is both produced and determined by their structure.Varela and Maturana extend the notion of cognition to mean all effective interactions – action or behavior that accomplishes the continual production of the system itself. “All doing is knowing and all knowing is doing,” as they say in *The Tree of Knowledge*. I wrote about some of the implications of this idea last year.

There is certainly mystery in Christmas images, from the return of the life-giving presence of the sun on the solstice, to the generous red-suited giver of gifts who lives where there is no life, to the unexpected marriage of the heaven and earth. The song *Do You Hear What I Hear? *has an interesting history. It was written in 1962 by Noel Regney and Gloria Shayne. Regney wrote the lyrics. He was a French-born musician and composer forced into the German army by Hitler’s troops during World War II. He became a member of the French underground and, while in the required German uniform, he collected information and worked in league with the French resistance. He moved to Manhattan in 1952 and continued his career as a composer. Although he once expressed that he had no interest in writing Christmas songs, amidst the the distress of the Cuban missile crisis in October of 1962, he has said that he was inspired to write the lyrics in question when he saw the hopeful smiles of two babies in strollers, in friendly exchange on a street in Manhattan.

I’m not arguing that my observation of the lyrics defend any particular religious perspective. I want more to express the fact that I can’t help but notice that the song sits comfortably within world views once considered by a 17th century polymath, known for his development of the calculus, and by 20th century biologists whose work redefines life as well as our experience of reality! And there is value in taking note of unintended science-like perspectives in religious images. Even the notion of The Word in Christian literature, translated from the Greek logos, is replete with fundamental views of reality in Ancient Greek philosophy. For the Stoics logos was reason both in the individual and in the cosmos. It was nature as well as God.

Religion and science have a common ancestor and may have a shared destiny.

]]>In late August, paleontologists reported finding the fossil of a flattened turtle shell that “was possibly trodden on” by a dinosaur, whose footprints spanned the rock layer directly above. The [...]]]>

In late August, paleontologists reported finding the fossil of a flattened turtle shell that “was possibly trodden on” by a dinosaur, whose footprints spanned the rock layer directly above. The rare discovery of correlated fossils potentially traces two bygone species to the same time and place.

Cosmologist Nima Arkani-Hamed makes the connection:

Paleontologists infer the existence of dinosaurs to give a rational accounting of strange patterns of bones…We look at patterns in space today, and we infer a cosmological history in order to explain them.

I doubt my 12-year old son has ever thought that the existence of dinosaurs is inferred. For him, the facts are clear. The dinosaurs are just not here anymore. But Arkani-Hamed’s observation caused a few things to go through my mind quickly. First I thought, this is cool – corresponding a tactic in paleontology to one in physics. And then, I realized what very little thought I have given to how we have come to know so much about creatures whose lives occurred completely outside the range of our experience. We have fully life-like images of them, and treat their existence as an unquestionably known quantity. Thinking about the labor it took to transform fossil discoveries into these convincing images highlighted the need, as I see it, to make the labor of science as apparent to non-science audiences as the results of that labor have been. The creativity involved in all of our inquiries is as important to see as the outcomes of those inquiries.

As a species, it seems that we are very good at piecing things together. Some facet of our reasoning and cognitive skills is always on the hunt for patterns with which our intellect or our imagination will then build countless structures – from the brain’s production of visual images created by the flow of visual data it receives, to the patterns in our experience that facilitate our day-to-day navigation of our earthbound lives, to the patterns in the sky that hint at things that are far beyond our experience, and the purely reasoned patterns of science and mathematics. We use these structures to capture, or harness, things like the detail of astronomical events billions of light years away, or the character of particles of matter that we cannot see, or species of animals that we can never meet. The reach or breadth of these reasoned structures likely rivals the extent of the universe itself or, at least our universe. I would argue that it is useful to reflect on how our now deep scientific knowledge is built on pattern and inference because, in the end, it is the imagination that has built them. By this I do not mean to discredit the facts. Rather, I mean to elevate what we think of the imagination and of abstract thought in general.

Wolchover’s article describes how Arkani-Hamed and colleagues have worked on schemes that use spatial patterns among astronomical objects to understand the origins of the universe. (Based on the paper, The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities). Physicists have considered simple correlated pairs of objects for some time.

The simplest explanation for the correlations traces them to pairs of quantum particles that fluctuated into existence as space exponentially expanded at the start of the Big Bang. Pairs of particles that arose early on subsequently moved the farthest apart, yielding pairs of objects far away from each other in the sky today. Particle pairs that arose later separated less and now form closer-together pairs of objects. Like fossils, the pairwise correlations seen throughout the sky encode the passage of time—in this case, the very beginning of time.

But cosmologists are also considering the possibility that rare quantum fluctuations involving three, four or more particles may have also occurred in the birth of the universe. These would create other arrangements, like triangular arrangements of galaxies, or objects forming quadrilaterals, or pentagons. Telescopes have not yet identified such arrangements, but finding them could significantly enhance physicists’ understanding of the earliest moments of the universe.

Wolchover’s article describe physicists’ attempts to access these moments.

Cosmology’s fossil hunters look for the signals by taking a map of the cosmos and moving a triangle-shaped template all over it. For each position and orientation of the template, they measure the cosmos’s density at the three corners and multiply the numbers together. If the answer differs from the average cosmic density cubed, this is a three-point correlation. After measuring the strength of three-point correlations for that particular template throughout the sky, they then repeat the process with triangle templates of other sizes and relative side lengths, and with quadrilateral templates and so on. The variation in strength of the cosmological correlations as a function of the different shapes and sizes is called the “correlation function,” and it encodes rich information about the particle dynamics during the birth of the universe.

This is pretty ambitious. In the end, Arkani-Hamed and colleagues found a way to simplify things. They borrowed a design from particle physicists who found shortcuts to analyzing particle interactions using what’s called the bootstrap.

The physicists employed a strategy known as the bootstrap, a term derived from the phrase “pick yourself up by your own bootstraps” (instead of pushing off of the ground). The approach infers the laws of nature by considering only the mathematical logic and self-consistency of the laws themselves, instead of building on empirical evidence. Using the bootstrap philosophy, the researchers derived and solved a concise mathematical equation that dictates the possible patterns of correlations in the sky that result from different primordial ingredients.

Arkani-Hamed chose to use the geometry of “de Sitter space,” to investigate various correlated objects because the geometry of this space looks like the geometry of the expanding universe. De Sitter space is a 4-dimensional sphere-like space with 10 symmetries.

Whereas in the usual approach, you would start with a description of inflatons and other particles that might have existed; specify how they might move, interact, and morph into one another; and try to work out the spatial pattern that might have frozen into the universe as a result, Arkani-Hamed and Maldacena translated the 10 symmetries of de Sitter space into a concise differential equation dictating the final answer.

It is significant that there is no time variable in this analysis. Time emerges within the geometry. Yet it predicts cosmological patterns that provide information about the rise and evolution of quantum particles at the beginning of time. This suggests that time, itself, is an emergent property that has its origins in spatial correlations.

It should be clear that confidence in the geometric calculations is coming from how they square (no pun intended) with empirical measurements that we do have.

By leveraging symmetries, logical principles, and consistency conditions, they could often determine the final answer without ever working through the complicated particle dynamics. The results hinted that the usual picture of particle physics, in which particles move and interact in space and time, might not be the deepest description of what is happening. A major clue came in 2013, when Arkani-Hamed and his student Jaroslav Trnka discovered that the outcomes of certain particle collisions follow very simply from the volume of a geometric shape called the amplituhedron.

I wrote about this discovery in March.

Arkani-Hamed suspects that the bootstrapped equation that he and his collaborators derived may be related to a geometric object, along the lines of the amplituhedron, that encodes the correlations produced during the universe’s birth even more simply and elegantly. What seems clear already is that the new version of the story will not include the variable known as time.

An important aspect of the issues being discussed is the replacement of time-oriented functional analyses with time-less geometric ones. As I see it, this raises questions broader than how the structure of the universe itself is mathematical. This work highlights the relationships between physical things, abstract or ideal objects, and the constraints of logic. It says as much about us, and what we do, as it says about the origins of the universe or **what we say that time is**. I’ll stress, as I often do, that these issues are relevant to people, not just to science. This shift from one kind of organization of concepts (dynamic change) to another (geometric relationships) should encourage us to consider where these conceptual structures are emerging from and how are they connecting us to our reality.

I’m convinced that paying more attention to how we participate in building our reality will clarify quite a lot.

]]>The mathematical notions of infinity and continuity guide a great number of Leibniz’s observations. But Smolin makes a particular reference to Leibniz’s metaphysical account of the whole of reality, his *Monadology*. It would seem unlikely that a modern physicist would choose this path, but I would argue only because the path is under appreciated. Here’s a little of how the reasoning goes:

As Leibniz saw it, there are no discontinuous changes in nature. The observed absence of abrupt change suggested to him that all matter, regardless of how small, had some elasticity. Since elasticity requires parts, a truly singular thing, with no parts, would not be elastic. That would mean that all material objects, no matter how small, would have to be compounds or amalgams of some sort. If not, they could produce abrupt change. Now anything simple and indivisible, is necessarily without extension, or dimension, like a mathematical point. In other words, it wouldn’t take up any space. Leibniz was convinced that this non-material fundamental substance had to exist. If it didn’t, then everything would be an aggregate of substances. And every aggregate would also be an aggregate, allowing for the endless divisibility of everything, making it impossible to identify anything. According to Leibniz, the universe of extended matter is a consequence of the interaction of simple non-material substances known as monads, or simply the relations among these monads.

But it is not the non-material nature of a monad that Smolin keys on. It is more Leibniz’s conviction that that there is no fundamental space within which the elements of the universe exist, together with the fact that *it is relations among the actions of fundamental unities that produce the universe we experience.* Here’s what Smolin says:

I first read Leibniz at the instigation of Julian Barbour, when I was just out of graduate school. First I read the correspondence between Leibniz and Samuel Clarke, who was a follower of Newton, in which Leibniz criticized Newton’s notion of absolute space and absolute time and argued that observables in physics should be relational. They should describe the relations of one system with another, resulting from their interaction. Later I read the Monadology.

I read it as a sketch for how to make a background- independent theory of physics.I do look at my copy from time to time. There is a beautiful quote in there, where Leibniz says, “Just as the same city viewed from different directions appears entirely different … there are, as it were, just as many different universes, which are, nevertheless, only perspectives on a single one, corresponding to the different points of view of each monad.” That, to me, evokes why these ideas are very suitable, not just in physics but for a whole range of things from social policy and postmodernism to art to what it feels like to be an individual in a diverse society. But that’s another discussion! (Emphasis added)

The key seems to be in what the interviewer refers to as Smolin’s slogan: “The first principle of cosmology must be: There is nothing outside the universe.” Smolin agrees with Leibniz that space, rather than being some thing within which bodies are located and move, it is a system of relations holding between things or, in his terms, ‘an order of situations.’ Space is created by the arrangement of matter, as the family tree is created by the arrangement of ones ancestors (a comparison Leibniz, himself, made). Space comes into existence only when the coexistent parts of the universe come into existence. It seems that Smolin also finds value in Leibniz’s portrayal of the individual monad as something that represents the universe *from one of all possible points of view.*

Leibniz described monads as *complete* in the sense that they cannot be changed by anything outside of themselves nor can they influence each other. It is an inner, pre-established solidarity that defines their relationship to each other. Their completeness requires, however, that they hold within themselves, perhaps as potentialities, all of the properties they will exhibit in the future, as well as some trace of all of the properties that they exhibited in the past. This brings timelessness to the fundamental level of our reality, to which Leibniz also attributes a preexisting harmony. The monad’s singularity also requires that they each, somehow, mirror or reflect the entire universe and every other monad.

It may not be the nature of the monad that has Smolin’s attention. But he has chosen to work on a theory about processes rather than things, the “causal relations among things that happen, not the inherent properties of things that are.”

The fundamental ingredient is what we call an “event.” Events are things that happen at a single place and time; at each event there’s some momentum, energy, charge or other various physical quantity that’s measurable. The event has relations with the rest of the universe, and that set of relations constitutes its “view” of the universe. Rather than describing an isolated system in terms of things that are measured from the outside, we’re taking the universe as constituted of relations among events. The idea is to try to reformulate physics in terms of these views from the inside, what it looks like from inside the universe.

There are so many reasons that I am intrigued by Smolin’s choice. It’s beautifully imaginative. But I’ve always been reassured by Leibniz’s view of things – an unexpected amalgam of rigorous formal reasoning, the conceptual possibilities of mathematics, what was known in physics, and the way that God was understood – all brought to bear in an effort to comprehend everything. Leibniz characterizes space and time as *beings of reason*; they are abstractions, or idealizations (like the geometric continuum) and, as such, are found to be continuous, homogenous, and infinitely divisible. Leibniz was intent on avoiding the blunder of a mind/body duality. His monadology is a unique synthesis of things that sound like biological notions, along with physical observations, and mathematical abstractions. Smolin’s choice to explore Leibniz’s map of the world with the observations of modern physics sounds very promising.