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.

… according to several mathematical theorems, nothing can be localized in the way that the traditional concept of a particle implies

…Fields, too, are not what they appear to be. Modern quantum theories long ago did away with electric and magnetic fields as concrete structures and replaced them with a hard-to-interpret mathematical abstraction

…The deeper physicists dive into reality, the more reality seems to evaporate.

And, he asks:

…What differentiates physical from mathematical objects or a simulation from the original system? Both involve the same sets of relations, so there seems to be nothing to tell them apart.

One can argue that it is physics’ increasing reliance on mathematics that causes reality to evaporate the way that Musser describes. He does discuss some of the ideas that physicists use to reconcile their mathematics with their reality. One of these is a perspective called Qbism, which is an interpretation of quantum mechanical theory that acknowledges and addresses the role played by the scientist in the development of theory. Also from Musser:

Immanuel Kant argued that the structure of our minds conditions what we perceive. In that tradition, physicist Markus Müller of the Institute for Quantum Optics and Quantum Information in Vienna and cognitive scientist Donald Hoffman of the University of California, Irvine, among others, have argued that we perceive the world as divided into objects situated within space and time, not necessarily because it has this structure but be cause that is the only way we could perceive it. The reality we experience looks the way that it does because of the nature of the perceiving agent.

In the same Part 1 is a piece, written by mathematician Kelsey Houston-Edwards, that addresses the creation vs. discovery arguments about mathematics which is, essentially, the question of whether or not mathematics exists, in some way, independently of human experience. She suggests a useful image:

This all seems to me a bit like improv theater. Mathematicians invent a setting with a handful of characters, or objects, as well as a few rules of interaction, and watch how the plot unfolds. The actors rapidly develop surprising personalities and relationships, entirely independent of the ones mathematicians intended. Regardless of who directs the play, however, the denouement is always the same. Even in a chaotic system, where the endings can vary wildly, the same initial conditions will always lead to the same end point. It is this inevitability that gives the discipline of math such notable cohesion. Hidden in the wings are difficult questions about the fundamental nature of mathematical objects and the acquisition of mathematical knowledge.

I like this image because I find it consistent with what does seem to happen in the research done by mathematicians. But it also suggests a focus for the questions we have about the fundamental nature of mathematical objects, that focus being the significance and nature of the interaction of the thoughts we put forward.

The last piece of this triad is an article by cognitive and computational neuroscientist Anil K.Seth. Seth’s work also proposes that our experience is not really an indication of how things really are, but more what our bodies make of the things that are. His central idea is that perception is a process of active interpretation, that tries to predict the source of signals that originate both outside and within the body.

The central idea of predictive perception is that the brain is attempting to figure out what is out there in the world (or in here, in the body) by continually making and updating best guesses about the causes of its sensory inputs. It forms these best guesses by combining prior expectations or “beliefs” about the world, together with incoming sensory data, in a way that takes into account how reliable the sensory signals are.

For Seth the contents of our perceived worlds are what he calls *controlled hallucinations*, the brains best guesses about the unknowable causes of the sensory signals it receives.

What I find interesting about this discussion of truth is that no one is looking directly at what mathematics is doing, or what mathematics might have to say about the relationship between the brain and the world in which it is embedded through the body. Mathematics has the peculiar character of existing in both the perceiver and the perceived. And maybe this isn’t really peculiar. But there is a reason why mathematics is always crucial to correcting the deceptions present in our experience (as it has done with general relativity and quantum mechanics). In physics mathematics does the heavy lifting of defining the data, giving it meaning, finding the patterns, for example, in what we see of particles, and fields, and their interactions, and everything else. And in cognitive science we now see the mathematical nature of the brain processes that construct our reality. But what I hope to see in mathematics is not just about science. I’m convinced that mathematics can help us see how thought and physical reality are not only related, or interacting, but are somehow the same stuff. I suspect that the physical world is full of thoughts, and ideas are as physical as flowers. But I don’t think we’re clear yet on what physicality really is. Mathematics may be the thing that cracks open the stubborn duality in our experience that is obscuring our view.

Another recent article, unrelated to this truth discussion, added a point to my collection of data about the profoundness of what mathematics seems to tell us about ourselves. This past February, Quanta magazine reprinted an article from Wired.com about possible breakthroughs in understanding how the brain creates our sense of time and memory. The brain processes that create memory have been difficult to identify. For neuroscientists Marc Howard and Karthik Shankar, memory is a display of sensory information in much the same way that a visual image is a display of visual information. But neurons do not directly measure time the way that some neurons measure wavelength or brightness, or even verticality. So Howard and Shankar looked for a way (i.e. equations) to describe how the brain might encode time *indirectly.*

…it’s fairly straightforward to represent a tableau of visual information, like light intensity or brightness, as functions of certain variables, like wavelength, because dedicated receptors in our eyes directly measure those qualities in what we see. The brain has no such receptors for time. “Color or shape perception, that’s much more obvious,” said Masamichi Hayashi, a cognitive neuroscientist at Osaka University in Japan. “But time is such an elusive property.” To encode that, the brain has to do something less direct.

It now looks like the way the brain accomplishes this resembles a fairly familiar strategy in mathematics called the Laplace Transform. The Laplace transform translates difficult equations into less difficult ones by replacing the somewhat complex operation of differentiation with the very familiar operation of multiplication. It’s a mapping that changes time and space relations, described by derivatives, into simpler algebraic relations. Once the algebraic relation is understood, there are mechanisms for translating these solutions back into solutions of the original differential equations.

When Howard heard about Tsao’s results, which were presented at a conference in 2017 and published in

Naturelast August, he was ecstatic: The different rates of decay Tsao had observed in the neural activity were exactly what his theory had predicted should happen in the brain’s intermediate representation of experience. “It looked like a Laplace transform of time,” Howard said — the piece of his and Shankar’s model that had been missing from empirical work.

This model for understanding the neurological components of time, developed by Howard and Shankar, began with just the mathematics. It was a purely theoretical model. But the possibility was demonstrated in the lab work of another neuroscientist, Albert Tsao, working independently of Howard and Shankar. Tsao found, in rats, that firing frequencies for certain neurons increased at the beginning of an event (like releasing a rat into a maze to find food) and diminished over the course of the event. At the start of another trial the firing increased again and diminished again in such a way that each trial could be identified by this pattern of enlivened and diminishing activity of brain activity.

As neuroscientist Max Shapiro sees it:

It’s this coding by parsing episodes that, to me, makes a very neat explanation for the way we see time. We’re processing things that happen in sequences, and what happens in those sequences can determine the subjective estimate for how much time passes.

What I think is important here is that the strategy we developed with the Laplace transform is a strategy the body also employs. This happens all the time, but this seems like a particularly unexpected and intimate instance of it. Mathematics, I expect, is pure structure that exists on the edge of everything that we are and all that there is.

]]>What struck me about all of these interviews was how traditionally everyone seemed to address the question from their individual philosophical, theological, or psychological perspectives. George Lakoff points to the way the brain creates metaphors that will produce the sense that there is a separation between the experiencing subject and the self. Warren Brown suggests that the notion of the soul might be better understood with the word person, a word which seems to apply to both the physical and idea-driven aspects to human experience. Charles Tart points out that individuals have seen the part of themselves that is not physical in out of body experiences and near death experiences. Deepak Chopra reminds us that Schrodinger once said that consciousness is a singular that has no plural. And he makes useful corrections to some of our habits of thought. He suggests, for example, that the personal aspect of consciousness is like a wave in the ocean, a pattern of movement. It’s real, but it disappears. He also suggests that our minds have produced evidence for a multiverse, so why are we so suspect of the notion of eternity. Bottom line for Chopra, the ultimate truth is consciousness, and we cannot fully express it. In Part One of the series on the soul, philosopher and parapsychologist Stephen Braude made clear that he is not just an anti-physicalist, but he is anti-mechanist. “There is one kind of stuff,” he says, that can be looked at through any number of conceptual grids. He levels the playing field. Every set of descriptive terms leaves something out. No description of nature can be complete.

I’ve spent a significant amount of time looking for philosophical shifts in physics and biology, as well as unexpected developments in, or applications of mathematics. I’ve tried to identify innovative efforts in these areas because they often lend support to to my own ideas about the nature and value of mathematics. So many novel approaches to biology and physics have important implications for how we might think about mind, consciousness, spirit and the soul. Yet none of these were relevant to the Closer to Truth inquiry. When Plato was invoked, there was no acknowledgement that his eternal world of ideals is tied to our empirical study of our surroundings through mathematics – an observation that warrants some thought.

Listening to the interviews also made me more aware that a fairly provocative and well-known idea has still had only modest impact on how we see ourselves and our world. In 1987 biologists Humberto Maturana and Francisco Varela formalized a new approach to biology and to cognition in particular. It is a perspective defined by the notion of autopoiesis, the self-creating nature of life itself, and the more generalized notion of cognition that this perspective brings about. The key to their strategy is to begin with the understanding that our experience is tied to our individual structure in a binding way. From their point of view, what we experience is due more to our own structure than to what exists around us. Maturana and Varela make their case in the book, *The Tree of Knowledge: The Biological Roots of Human Understanding*. From their book:

The experience of anything out there is validated in a special way by the human structure, which makes possible “the thing” that arises in the description.

The seeds of these ideas appear in *The Neurophysiology of Cognition*, an article published by Maturana in 1969. In that article he raises an unexpected question. Does cognition just transcribe, for us, the truth of the world around us, or is it a biological phenomenon whose nature we do not actually understand? For most of us, our own immediate sense of what we seem to know, feels like the simple gathering of information from the world around us. We believe that the information that we gather is out there and independent of us. If we take this as our starting point, Maturana argues, then questions about cognition will be mostly concerned with how it works and how to use it. But Maturana takes a step back from this. For him, cognition itself is the unknown. As Maturana sees it, the question we should be asking is, “What kind of biological phenomenon is the phenomenon of cognition?” What is it doing? This is broader than even a question about how the mind is related to the brain. If this question is our starting point, the nature of a reality that is independent of us becomes fairly difficult to discern. We are fully and dynamically embedded in our reality. Since we find mathematics in brain processes themselves, this sets the stage for the possibility that mathematics, itself, is something we participate in rather than something we produce. The effect of this embededness would challenge us to be more rigorous in all of our inquiries, whether physical or metaphysical because we are never fully independent of what we see.

Some of the potential in the notion of autopoiesis appears in the work of Karl Friston, a neuroscientist who is known for his contributions to neuroimaging technology but who, more recently, is receiving a lot of attention because of a theoretical framework he has proposed to describe all living systems. His idea, called the free energy principle, is already enjoying multi-disciplinary application. The free energy principle doesn’t build directly on autopoiesis, but it shares some of its most fundamental concepts.

In particular in a video produced by Serious Science, after a brief account of the main points of the free energy principle, Friston concludes that we are all in the game of garnering information that maximizes the evidence of our own existence. And so, he adds, brain structure speaks exactly to the causal structure of the world we inhabit.

The circularity that these perspectives should have a significant effect on our ideas about mind, consciousness, and soul. I’m not suggesting that they make the inquiry meaningless. On the contrary, they open up the inquiry and require that we be more careful and more creative. They make it more difficult and, I expect, more interesting.

]]>Mathematical billiards describe the motion of a mass point in a domain with elastic reflections from the boundary. Billiards is not a single mathematical theory… it is rather a mathematician’s playground where various methods and approaches are tested and honed. Billiards is indeed a very popular subject…

In her public lecture Rom-Kedar started at the beginning. She described the familiar motion of billiard balls as they hit the sides of a billiards table. Once set in motion, a billiard ball will move along a straight line with a constant speed until it hits the side of the table. Its path, after it hits the side, is subject to a familiar law about the reflection of light, specifically, that the angle of incidence equals the angle of reflection. The billiard ball obeys the same law. If it happens to hit the other side head-on it will return to the first side along the same path.

Rom-Kedar then asked the first scientific question: What would happen if the ball just kept moving, traveling in straight lines, as it hit side after side, for an infinitely long time, each time obeying that law of reflection? Will the ball eventually traverse every point on the table? As it turns out, the answer to this question is yes, for some of these events, and no for others. When the ball’s initial move away from the side of the table begins at an angle whose measure has a rational relationship to the dimensions of the table, a periodic orbit gets locked in, and the periodic repetition of paths will never allow the ball to cover all the points on the table. But when the relation between that angle and the dimensions of the table is irrational, the paths are ergodic, i.e. they impinge on all of the points of the given surface or table. These ideal billiards have no mass and hence there is no friction. But in every other way, their behavior is the same as the ordinary billiard ball. I would suggest that there is already something interesting about the correspondence between the rationality of a geometric measure and the action of the ball. Why would there be such a correspondence? It’s like seeing something about numbers through the back of a mirror.

As it turns out, periodic behavior is fairly rare. Ergodic behavior is far more common, There’s a nice narrative about various approaches to this specialization in a 2014 Plus Magazine article by Marianne Freiberger.

In the 1980s mathematicians proved that for the vast majority of initial directions the trajectory will be much wilder: not only will it not retrace its steps, but it will eventually explore the whole of the table, getting arbitrarily close to every point on it. What is more, a typical trajectory will visit each part of the table in equal measure: if you take two regions of the table whose areas are equal, then the trajectory will spend an equal amount of time in both. This behaviour is a consequence of billiards being ergodic. By “vast majority” mathematicians mean that if you pick a direction at random, it will almost certainly behave in this ergodic way.

The absence of a pattern in ergodic behavior makes it very difficult to predict where the ball, or point, might be after some specified amount of time. A computer program could run all the paths fast enough to see what happens, but in true chaotic fashion a very slight change in the direction of the initial trajectory will dramatically change the ball’s later positions. But, as Freiberger explains, because many dynamic physical systems are ergodic, it does give us a handle on something other than the position of a particular point over time. Rather than being able to trace the path of of point.

you can accurately predict what proportion of its time it spends in a certain region of the table. If it’s a gas you are looking at, then you might not be able to say exactly where its many constituent molecules are at any given moment, but you can predict things like its temperature or pressure. So, as chaotic systems go, ergodicity is actually a good thing.

As always happens, mathematicians hunt for all of the generalities associated with all of the imagined, ideal possibilities. And changing the shape of the table introduces a lot of them. Instead of a rectangle, the table could be triangular, hexagonal or L-shaped. It could be round or elliptical. Rom-Kedar said that with these variations, questions about what will happen become “more delicate.” There are many more periodic trajectories in curved figures like circles and ellipses. Most of these events do not explore all of the table.

It is remarkable that billiards models effectively address many phenomena in the physical sciences, that are already described by alternative mathematical models, as well as open questions in mathematics, even number theory. Physicists use it as a close approximation of particle forces and movement. They are relevant to any systems exhibiting chaotic behavior. And, to be clear, billiard models are not restricted to objects on the plane. Billiard models have been developed on various surfaces, including Riemann surfaces.

There is something beautiful about all of this. An observation, of a very specific and pretty limited physical event (a billiard ball on a table) inspires the thoughtful exploration of imagined ideals, that involve infinite times, and are not limited by physicality. These abstractions are a product of looking through the physical consideration to the endless possibilities captured by ideals. Then these thorough investigations of purely idealized possibilities become a way to look at a surprising number of unrelated physical (and mathematical) phenomena. How does the human intellect manage this? And what motivates us to do things like this? It’s beautiful and fascinating.

]]>The authors argue, convincingly, that while individuals diagnosed with dyslexia may have difficulty with the symbolic representation of words, they seem to excel at 3-dimensional spatial reasoning. I thought of my daughter, whose dyslexia was not diagnosed till her freshman year in high school, but who, even as a 5-year old, seemed to have a remarkable ability to know which way to go when we were driving. She could locate herself pretty easily.

The authors describe the coordinate-like action of grid cells in order to point to one of the neurological components of how we all negotiate 3-dimensional space. I wrote about the action of grid cells in a 2014 post, to suggest that cognitive processes themselves have a mathematical nature. There I made this observation:

Spatial relations are translated into what look like purely temporal ones (the timing of neuron firing). The non-sensory system then stores a coded representation of a sensory one. Here again we see, not the mathematical modeling of brain processes but more their mathematical nature.

Drs. Brock and Fernette Eide cite studies done, from various perspectives, which suggest that the presence of strong spatial reasoning in a dyslexic individual is not developed in order to compensate for verbal difficulties but, rather, it is a strength with which dyslexic individuals are born. As a result, many such individuals have chosen careers in areas that include art, architecture, building, engineering, and computer graphics.

I found one of their observations particularly interesting because it contradicted the perfectly reasonable expectation that strong spatial reasoning skills are accompanied by vivid, mental visual images. But, as it is with mathematics, so it is with the brain. Specifically, it seems that it is possible to separate *knowledge of space* from *spatial images* in an individual’s experience. The authors describe a particular case-study where the individual involved lost his ability to create clear visual mental images, but his spatial reasoning abilities were unaffected. In other words, he could manage spatial relations without visualizing them.

MX was a retired building surveyor living in Scotland who’d always enjoyed a remarkable vivid and lifelike visual imagery system, or “mind’s eye.” Unfortunately, four days after undergoing a cardiac procedure MX awoke to discover that though his vision was normal, when he closed his eyes he could no longer voluntarily call to mind any visual image at all.

MX was tested using a whole series of spatial reasoning and visual memory tasks. As a control, a group of high-visualizing architects performed the same tasks. Surprisingly, it was found that although MX could no longer create any mental visual images while performing these tasks, he scored just as well as the architects did. As he performed the tasks, MX’s brain was also scanned with fMRI technology. In contrast to the architects, who heavily activated the visual centers of their brains while solving these tasks, MX used none of his brain’s visual processing regions.

These studies suggested that while MX had lost his ability to

perceive visual imageswhen engaging in spatial reasoning, he could stillaccess spatial informationfrom his spatial database and apply it to Material reasoning tasks with no detectable loss of skill.

It is common place in mathematics to separate spatial information from the visual images with which they can be associated. Analytic or coordinate geometry, for example, studies geometric figures (or figures in space) using their algebraic representations (i.e. only numbers). So there we have the numerical approach to figures and the visual one. A discipline like abstract algebra creates other kinds of spaces and objects by abstracting away not just the particular numbers (like the variables we learn about in high school algebra do) but by abstracting away the previous meaning of things like addition, for example. The plus sign comes to stand for anything that obeys the same rules that addition obeys, like a + b = b + a and a + 0 = a. But a, b and 0 are not necessarily numbers. The point is that mathematics manages to keep finding other places where information exists. Mathematics explores structure as fully and deeply as possible.

The relationship between vision and structure that MX’s experience brought me back to also reminded me of the 2002 AMS article called The World of Blind Mathematicians. The article is full of interesting and unexpected observations of the accomplished blind geometer, Bernard Morin. This is just one of them:

]]>…blind people often have an affinity for the imaginative, Platonic realm of mathematics. For example, Morin remarked that sighted students are usually taught in such a way that, when they think about two intersecting planes, they see the planes as two-dimensional pictures drawn on a sheet of paper. “For them, the geometry is these pictures,” he said. “They have no idea of the planes existing in their natural space.

Physics theoretician Nima Arkani-Hamed, at the Institute for Advanced Study at Princeton, has recently suggested that maybe space and time are not what we think they are. In a recent interview with Natalie Wolchover, for a New Yorker article, he expressed renewed interest in a point made by Richard Feynman in 1964. [1] Feynman took note of the fact that when considering physical systems, it is very often the case that different explanations will produce equally good predictions of events. For example, when predicting the movements of two objects that are gravitationally attracted to each other, three different approaches will produce the same correct prediction. These are, Newton’s law of gravity (that says that objects exert a pull on each other); Einstein’s spacetime (that describes how space bends around massive objects), and the mathematics of what is known as the principle of least action which holds that moving objects follow the path that uses the least energy and is accomplished in the least time. The fact that it is possible to predict physical behavior with more than one mathematical idea could suggest that physics research is actually testing the mathematics more than the physical world. And so Arkani-Hamed has proposed that the universe is actually the answer to a mathematical question we have not yet discovered. From Wolchover’s article:

“The miraculous shape-shifting property of the laws is the single most amazing thing I know about them,” he told me, this past fall. It “must be a huge clue to the nature of the ultimate truth.”

Arkani-Hamed was encouraged in this when he and his colleagues succeeded in predicting the outcomes of subatomic particle interactions using only the volume of their newly discovered, and purely abstract geometric object called the amplituhedron.[2] These volume calculations are made without any reference to physical space or time. If we don’t need space and time to calculate particle interactions then, perhaps, the space and time in which we seem to live is not a fundamental aspect of our reality. Arkani-Hamed is considering that space and time emerge from some deeper structure. If the volume of the amplituhedron encodes the outcomes of particle collisions, maybe mathematical principles, like those that define the intersections of lines and planes, for example is the real fundamental thing. Arkani-Hamed has also found that celestial patterns, that describe the history of the universe, can be represented as geometric volumes. Like Max Tegmark, author of Our Mathematical Universe, Arkani-Hamed is inspired by the possibility that ultimately, it will be a “spectacular mathematical structure,” out of which the past, present, and future of everything emerge.[3] Tegmark’s argument goes more like this:

Remember that two mathematical structures are equivalent if you can pair up their entities in a way that preserves all relations. If you can thus pair up every entity in our external physical reality with a corresponding one in a mathematical structure…then our external physical reality meets the definition of being a mathematical structure – indeed, the same mathematical structure.

Given these kinds of considerations, I would argue that addressing the question of whether or not mathematics exists independently of us is far more complicated than we ever thought. And the possibility that space and time emerge from something more fundamental (like a spectacular mathematical structure) is one of the more extreme attempts to consolidate the physical with the abstract. Again from Wolchover’s article:

“The ascension to the tenth level of intellectual heaven,” he told me, “would be if we find the question to which the universe is the answer, and the nature of that question in and of itself explains why it was possible to describe it in so many different ways.”

[1] Natalie Wolchover, New Yorker Magazine, February 10, 2019

[2] Nima Arkani-Hamedm, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, Jaroslav Trnka, Scattering Amplitudes and the Positive Grassmannian, arXiv:1212.5605

[3] Natalie Wolchover, Visions of Future Physics, Quanta Magazine, September 22, 2015

]]>Building a convincing account of autopoiesis is a book-length enterprise. And the free energy principle is a mathematically complex idea. So I didn’t do justice to either of them in that post. But they are both important to a sense I’ve had, for a number of years now, that mathematics does have a biological nature. Today, I want to make the argument again, but a little differently. I’ll begin with some references to studies that have caught my attention because they concern *mathematical behavior* in insects. A very recent Science article reported on a study that suggests that bees are capable of responding to symbolic representations of addition and subtraction operations. It was a small study (just 14 bees), but the bees were trained to associate addition with the color blue and subtraction with the color yellow

Over the course of 100 appetitive-aversive (reward-punishment) reinforced choices, honeybees were trained to add or subtract one element based on the color of a sample stimulus.

The bees were placed at the entrance of a Y-shaped maze, where they were shown several shapes in either yellow or blue. If the shapes were blue, bees got a reward for choosing a set of objects at the end of the maze that was equal to the first number of shapes plus one. If the first set was yellow, they got a reward for choosing the set of shapes at the end of the maze that was equal to the first number shapes minus one. The alternative or incorrect choice could have more than one shape added, or it could have a smaller number of shapes than the initial set. The bees made the right choice 63% to 72% of the time, much better than random choices would allow.

The full study can be found here. In their introduction they explain why these bees are worth looking at.

Honeybees are a model for insect cognition and vision. Bees have demonstrated the ability to learn a number of rules and concepts to solve problems such as “left/right,“ above/below”, “same/different,” and “larger/smaller.” Honeybees have also shown some capacity for counting and number discrimination when trained using an appetitive (reward- only) differential conditioning framework. Recent advances in training protocols reveal that bees perform significantly better on perceptually difficult tasks when trained with an appetitive-aversive (reward-punishment) differential conditioning framework. This improved learning capacity is linked to attention in bees, and attention is a key aspect of advanced numerosity and spatial processing abilities in the human brain. Using this conditioning protocol, honeybees were recently shown to acquire the numerical rules of “greater than” and “less than” and subsequently apply these rules to demonstrate an understanding that an empty set, zero, lies at the lower end of the numerical continuum.

I understand being impressed with the fact that bees can acquire this kind of discriminating ability. But not so clear is why they are structurally capable of such things. I would argue that it’s because their living relies on the presence of structure that allows these kinds of responses. This is the level that interests me.

Here are a few clips from past posts::

Ants were seen to communicate some kind of numerical information about the location of food.

In the described experiments scouting ants actively manipulated with quantities, as they had to transfer to foragers in a laboratory nest the information about which branch of a ‘counting maze’ they had to go to in order to obtain syrup…

The likely explanation of the results concerning ants’ ability to find the ‘right’ branch is that they can evaluate the number of the branch in the sequence of branches in the maze and transmit this information to each other. Presumably, a scout could pass messages not about the number of the branch but about the distance to it or about the number of steps and so on. What is important is that even if ants operate with distance or with the number of steps, this shows that they are able to use quantitative values and pass on exact information about them.

Zhanna Reznikova and Boris Ryabko, 2011. Numerical competence in animals, with an insight from ants.Behaviour, Volume 148, Number 4, pp. 405-434, 2011

*The abstract of a paper published in Nature in 2001 includes this:*

…honeybees can interpolate visual information, exhibit associative recall, categorize visual information, and learn contextual information. Here we show that honeybees can form ‘sameness’ and ‘difference’ concepts. They learn to solve ‘delayed matching-to-sample’ tasks, in which they are required to respond to a matching stimulus, and ‘delayed non-matching-to-sample’ tasks, in which they are required to respond to a different stimulus; they can also transfer the learned rules to new stimuli of the same or a different sensory modality. Thus, not only can bees learn specific objects and their physical parameters, but they can also master abstract inter-relationships, such as sameness and difference.

**Mathematical behavior without a brain?**

But here’s something interesting about the slime mold – the abstract of a paper published in Nature in September of 2000 reads:

The plasmodium of the slime mould Physarum polycephalum is a large amoeba-like cell consisting of a dendritic network of tube-like structures (pseudopodia). It changes its shape as it crawls over a plain agar gel and, if food is placed at two different points, it will put out pseudopodia that connect the two food sources. Here we show that this simple organism has the ability to find

the minimum-length solution between two points in a labyrinth.(emphasis added)

And here’s another strategy used by researchers that was reported by Tim Wogan in 2010 in Science.

They placed oat flakes (a slime mold favorite) on agar plates in a pattern that mimicked the locations of cities around Tokyo and impregnated the plates with P. polycephalum at the point representing Tokyo itself. They then watched the slime mold grow for 26 hours, creating tendrils that interconnected the food supplies.

Different plates exhibited a range of solutions, but the visual similarity to the Tokyo rail system was striking in many of them… Where the slime mold had chosen a different solution, its alternative was just as efficient.

Autopoietic systems are ones which, through their interactions and transformations, continuously produce or realize the network of processes that defines them. They continuously create themselves. Maturana and Varela proposed that every living system is autopoietic, from individual cells, to the nested autopoietic systems in organs, organisms, and even social organizations. In my last post I connected this interpretation of life to Karl Friston’s free energy principle. But it was pretty sketchy, so I would like to fill it in a little here.

I find it important that the free energy principle has the same circular model of living processes as autopoiesis. But for Friston, the key to a system’s continuously regenerating itself relies on how it manages to maximize expectations and minimize surprise. Minimizing surprise is essentially the same as maintaining a low entropy state, which is synonymous with maintaining ones structure. (The mathematics of entropy in information theory, is easily applied to entropy in thermodynamics.) And so minimizing surprise is the same as minimizing entropy. But the thing that holds it all together, the thing that can formalize the analysis, is the use of Baysian inferences or statistical models because this is a way to quantify uncertainty. With all of this, living systems maintain themselves by keeping themselves within a set of expectations (through sensory information, statistical inferencing, and their own action), If they stray too far from having those expectations met (like a fish out of water), they will no longer exist.

When I consider the different ways that mathematics is present in bees, ants, and slime molds from the perspective of autopoiesis or free energy, mathematics looks like its right in the middle of all the action – in the thick of the organism’s living. It will show up in the interactions and transformations that contribute to life itself (where life is the maintenance of the structural and functional integrity of oneself) because it is part of the regular flow of its living. According to the free energy principle, living systems live by either adjusting their expectations to match the flow of sensations, or adjusting the flow of sensations to match their expectations. It must be true that mathematics is as much a part of this as any biological process.

]]>From my perspective, the notion of structural coupling which developed out of this framework, has the potential to contribute something important to a philosophy of mathematics. Two or more unities are structurally coupled when they enter into a relatedness that accomplishes their autopoiesis by virtue of ‘a history of recurrent interactions’ that leads to their ‘structural congruence.’ Also true is that every autopoietic system is closed, meaning that it lives only with respect to itself. Whether interactions happen within the internal components of a system, or with the medium in which the system exists, the system is only involved in its own continuous regeneration. The view of cognition proposed by Maturana requires that the nervous is just such a closed, autopoietic system, which also functions as a component of the organism that contains it.

Mathematician Yehuda Rav used these ideas to propose a philosophy of mathematics (which I referenced in a 2012 post). In an essay with the title *Philosophical Problems of Mathematics in the Light of Evolutionary Epistemology*, Rav writes:

Thus, Maturna (1980, p. 13) writes: “Living systems are cognitive systems, and living as a process is a process of cognition”. What I wish to stress here is that there is a continuum of cognitive mechanisms, from molecular cognition to cognitive acts of organisms, and that some of these fittings have become genetically fixed and are transmitted from generation to generation. Cognition is not a passive act on the part of an organism, but a dynamic process realized in and through action.

When we form a representation for possible action, the nervous system apparently treats this representation as if it were a sensory input, hence processes it by the same logico-operational schemes as when dealing with an environmental situation. From a different perspective, Maturana and Varela (1980, p. 131) express it this way: “all states of the nervous system are internal states, and the nervous system cannot make a distinction in its process of transformations between its internally and externally generated changes.”

Thus, the logical schemes in hypothetical representations are the same as the logical schemes in coordination of actions, schemes which have been tested through eons of evolution and which by now are genetically fixed.

As it is a fundamental property of the nervous system to function through recursive loops, any hypothetical representation which we form is dealt with by the same ‘logic’ of coordination as in dealing with real life situations. Starting from the elementary logico-mathematical schemes, a hierarchy is established. Under the impetus of socio-cultural factors, new mathematical concepts are progressively introduced, and each new layer fuses with the previous layers. In structuring new layers, the same cognitive mechanisms operate with respect to the previous layers as they operate with respect to an environmental input. …..The sense of reality which one experiences in dealing with mathematical concepts stems in part from the fact that in all our hypothetical reasonings, the object of our reasoning is treated by the nervous system by means of cognitive mechanisms which have evolved through interactions with external reality.

Mathematics is a singularly rich cognition pool of mankind from which schemes can be drawn for formulating theories which deal with phenomena which lie outside the range of daily experience, and hence for which ordinary language is inadequate.

Rav is imagining the development of mathematics as a feature of human cognition. But the perspective proposed by Maturana includes a theory of language. For Maturana, language is not a thing, and the essence of what we call language is not in the words or the grammar. Language happens as we live in the units that our coupling defines – through living systems, interlocked by structural congruences, that build unities. We are languaging beings the way we are breathing beings.

My experience with mathematics has suggested to me that, like words and grammar, the symbolic representation of mathematics is secondary to what mathematics is. Mathematics also seems to happen. And Maturana’s emphasis on autopoiesis and structural coupling has suggested to me that mathematics, like language, happens through the recursive coordination of behaviors. But perhaps unlike language, the relational dynamics that bring it about are somehow fed by the more fundamental structures in the physical world (both living and non-living), to which we are coupled, rather than by the features of the day-to-day experience that we share.

Conceptually, the view of biology proposed by Maturana is significantly different from main stream thinking in the biological sciences. One of the most important differences is the way living systems are each bounded by their individual autopoietic processes and, at the same time, nested within each other, infinitely extending living possibilities. In my opinion, this particular aspect of their thinking is the most promising in the sense that it is this aspect of their thinking that has the greatest potential to produce something new.

A recent article in Wired about the work of Karl Friston suggested to me that I might be right. Friston, a neuroscientist who has made important contributions to neuroimaging technology, is the author of an idea called the free energy principle. Free energy is the difference between the states a living system expects to be in, and the states that its sensors determine it to be in. Another way of saying it is that when free energy is minimized, surprise is minimized. For Friston, a biological system (Maturana’s unity) that resists disorder and dissolution (is autopoietic) will adhere to the free energy principle – “whether it’s a protozoan or a pro basketball team.”

Friston’s unities are separated by what are called Markov blankets.

Markov is the eponym of a concept called a Markov blanket, which in machine learning is essentially a shield that separates one set of variables from others in a layered, hierarchical system. The psychologist Christopher Frith—who has an h-index on par with Friston’s—once described a Markov blanket as “a cognitive version of a cell membrane, shielding states inside the blanket from states outside.

In Friston’s mind, the universe is made up of Markov blankets inside of Markov blankets. Each of us has a Markov blanket that keeps us apart from what is not us. And within us are blankets separating organs, which contain blankets separating cells, which contain blankets separating their organelles. The blankets define how biological things exist over time and behave distinctly from one another. Without them, we’re just hot gas dissipating into the ether.

The free energy principle is mathematical, and grounded in physics, Bayesian statistics, and biology. It involves action, or the living system’s response to surprise, in addition to the systems predictive abilities. This is one reason the theory has far-reaching potential for application. The audience that the free energy principle attracts is consistently expanding.

]]>Today I looked back at a post that I wrote in 2010 with the title, *Archetypes, Image Schemas, Numbers and the Season*. The subject of the post is a chapter from the book, *Recasting Reality: Wolfgang Pauli’s Philosophical Ideas and Contemporary Science*. The chapter was written by cognitive scientist Raphael Nuñez who uses Pauli’s collaboration with Jung to address Pauli’s philosophy of mathematics. Jung understood ‘number’ in terms of archetypes, primitive mental images that are part of our collective unconscious. But Nuñez seems most interested in addressing Platonism. Pauli’s interest in Jung doesn’t address Platonism directly but it is nonetheless implied in many of the things he says. As a cognitive scientist, Nuñez rejects Platonism. Despite the complexity of mathematical abstractions, he argues that the discipline is heavily driven by human experience. While his observations of Pauli’s interest in Jung’s psychology are nicely laid out, and some parallels to his own theory are highlighted, Pauli’s ideas don’t really contribute to the non-mystical position that Nuñez has staked out.

But today I looked at the entire text to which Nuñez contributed, and could see that there are a number of things in Pauli’s view that address my own preoccupation with the nature of mathematics, and more deeply than does the question of whether mathematics exists independent of human experience or not. Pauli was preoccupied with reconciling opposites, finding unity, making things whole, and was strongly motivated to think about the problem of how scientific knowledge, and what he called redemptive knowledge, are related to each other. I find it fairly plausible that mathematics could help with this since it exists in the world of ideal images as well as the world of physical measurement and logical reasoning.

Today I found a really nice essay by physicist Hans von Baeyer with the title Wolfgang Pauli’s Journey Inward. It tells a more intimate story of Pauli’s ardent search for what’s true, and is well worth the read.

In time Pauli came to feel that the irrational component of his personality, represented by the black, female yin, was every bit as significant as its rational counterpart. Pauli called it his shadow and struggled to come to terms with it. What he yearned for was a harmonious balance of yin and yang, of female and male elements, of the irrational and the rational, of soul and body, of religion and science

During his lifetime, Pauli’s fervent quest for spiritual wholeness was unknown to the public and ignored by his colleagues. Today, with the debate between science and religion once more at high tide, Pauli’s visionary pursuit speaks to us with renewed relevance.

I particularly enjoyed von Baeyer’s description of the famous Exclusion Principle for which Pauli received a Nobel Prize in 1945. It went like this:

The fundamental question had been why the six electrons in the carbon atom, say, don’t all carry the same amount of energy — “why their quantum numbers don’t have identical values… it should be expected that the electrons would all seek the same lowest possible energy configuration, the way water seeks the lowest level, and crowd into it.” If this rule applied to electrons in atoms, there would be very little difference between, say, carbon with its six and nitrogen with its seven electrons. There would be no chemistry.

Pauli answered the question by decree: the electrons in an atom, he claimed, don’t have the same quantum numbers because they can’t. If one electron is labeled with, say, the four quantum numbers (5, 2, 3, 0) the next electron you add must carry a different label, say (5, 2, 3, 1) or perhaps (6, 2, 3, 0). He proposed no new force between electrons, no mechanism, not even logic to support this injunction. It was simply a rule, imperious in its peremptoriness, and unlike anything else in the entire sweep of modern physics. Electrons avoid each other’s private quantum numbers for no reason other than, as one physicist put it, “for fear of Pauli.” …With the invention of the fourth quantum number and the exclusion principle Pauli opened the way for the systematic construction of Mendeleev’s entire periodic table.

What struck me from reading von Baeyer’s account was the depth of Pauli’s concern. And the boldness of his Exclusion Principle somehow makes him seem particularly trustworthy. The reconciliation he sought was not one that just allowed for the accepted coexistence of different concerns, but rather one that changed both of them to accommodate something new. As von Baeyer points out, physics has become more and more dominated by “the manipulation of symbols that facilitate thinking but bear only an indirect relationship to observable facts.” Pauli seemed to expect that symbol was the link between the rational and the irrational. This would easily support my hunch. He seemed to expect that science be able to deal with the soul, where the soul in turn informs science.

Eventually, he hoped, science and religion, which he believed with Einstein to have common roots, will again be one single endeavor, with a common language, common symbols, and a common purpose.

This is what I expect. And, at the moment, mathematics seems to be my most trustworthy guide. It lives on the boundary that we think we see between pure thought and material, between mind and matter. Pauli’s conviction is particularly reassuring.

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