Mathematical behavior without a brain?

I have made the argument on more than one occasion that a refreshed look at mathematics may help illuminate the relationship between our experience of the physical and our experience of the thoughtful. Mathematics is a discipline characterized by complex relations among abstract things but, as has been explored from many directions, the action of the brain itself looks mathematical –

*In vision, individual neurons respond to abstractions (particular abstract visual properties contained in an object) like the verticality of edges.

*In navigation, neurons fire in grid-like patterns, internally marking locations.

*In learning, Bayesian probabilities accurately model the development of our intuitive understanding of physical things (like the interplay of  the weight of an object and its size, stability, speed) as well as our expectations of the social behaviors we perceive.

Even more broadly, research led by cognitive and computational neuroscientist Anil Seth supports the idea that all aspects of the brain’s construction of the world are managed through probabilities and inference, where sensory signals are combined with expectations based on prior experience to form the best hypothesis of what’s out there. He defines perception as controlled hallucination, and further argues that this kind of predictive processing can help us understand the nature of consciousness itself, where our sense of self is also generated by the brain’s ‘best guess’ processing. In this light, conscious experience is one of the consequences of the brain’s predictions about sensory signals from within and around the body. In a recent TED talk, Seth says the following:

So our most basic experiences of being a self, of being an embodied organism, are deeply grounded in the biological mechanisms that keep us alive. And when we follow this idea all the way through, we can start to see that all of our conscious experiences, since they all depend on the same mechanisms of predictive perception, all stem from this basic drive to stay alive. We experience the world and ourselves with, through and because of our living bodies.

and later

Finally, our own individual inner universe, our way of being conscious, is just one possible way of being conscious. And even human consciousness generally — it’s just a tiny region in a vast space of possible consciousnesses. Our individual self and worlds are unique to each of us, but they’re all grounded in biological mechanisms shared with many other living creatures.

I don’t think that we have reason to assume that the basic drive is to stay alive. I’m fairly well-convinced that it’s more creative than that. But this also echos the view that was pioneered by biologists Humberto Maturana and Francisco Varela. In their book, The Tree of Knowledge. they describe cognition as “an ongoing bringing forth of a world through the process of living itself.”

These and other studies in cognitive science lend strong support to Yehuda Rav’s argument that at least the bones of mathematics, on which more culturally driven mathematical themes develop, emerge from cognitive processes that have been genetically fixed and driven by natural selection. And the way I see it, questions about these genetically fixed, mathematics-like processes are approached from another direction when brainless creatures seem to demonstrate behaviors that we associate with the presence of consciousness (like learning), or when consciousness is considered very broadly in an evolutionary context.  Studies suggest that the thing we call ‘thought’ exists outside the brain.  Recent New Scientist articles address these issues. Bob Holmes published a piece in May with the title Why be conscious: The improbable origins of our unique mind. Holmes surveyed studies aimed at identifying at least the elements of consciousness that can be found among a diverse set of creatures.

Unlimited associative learning requires an array of brain functions, not only selective attention, but also the ability to combine sensations into one perception, perform compound action patterns and distinguish between self and environment. Scientists have found evidence that this complex learning is surprisingly widespread throughout the animal kingdom. Already, researchers have documented it in almost every vertebrate (except, possibly, lampreys), some arthropods such as insects and crustaceans, a few molluscs including octopuses and, perhaps, some snails. The jury is out on other groups, such as worms, since we don’t have enough evidence to be sure.

…There’s no doubt that human consciousness is special. Whether it is unique in some way or simply richer than that of other animals is still up for debate. However, it is becoming clear that the rudiments of consciousness are all around us.

Then, in July, Erica Tennenhouse contributed an article with the title, Smart but dumb: probing the mysteries of brainless intelligence. Here, findings in various experiments support the idea that “organisms with tiny brains or no brain at all are capable of amazing feats.”

A slime mold, for example, which is neither plant, nor animal, nor even fungus, seemed to learn not to be deterred by compounds like caffeine that were placed strategically in the way of a path toward nutrients. These deterrents were not concentrated enough to harm the slime mold but they were enough to stop them. After several hours, however, they moved through the threat.  And, as time passed, they moved through it more quickly.  After a few days, they almost completely ignored.

Tennenhouse provides a list of organisms, with their neuron count, and one of their feats:

A pea plant, with 0 neurons, when given the option of growing roots in a pot with a steady food supply or one with a “boon-or-bust” supply, will prefer the former but try the latter if they are starved.

Box jellyfish (with 13,000 neurons) “use four of their 24 eyes to peer through the water’s surface at tree canopies, which they use to help them navigate mangrove swamps.”

Bumblebees (1,000,000 neurons) will learn to pull a string to get a sugary treat by watching another bee perform the task.

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.

A 2012 Scientific American article by Ferris Jabr (How Brainless Slime Molds Redefine Intelligence) sums the point up nicely:

In other words, the single-celled brainless amoebae did not grow living branches between pieces of food in a random manner; rather, they behaved like a team of human engineers, growing the most efficient networks possible. Just as engineers design railways to get people from one city to another as quickly as possible, given the terrain—only laying down the building materials that are needed—the slime molds hit upon the most economical routes from one morsel to another, conserving energy. Andrew Adamatzky of the University of the West of England Bristol and other researchers were so impressed with the protists’ behaviors that they have proposed using slime molds to help plan future roadway construction, either with a living protist or a computer program that adopts its decision-making process. Researchers have also simulated real-world geographic constraints like volcanoes and bodies of water by confronting the slime mold with deterrents that it must circumvent, such as bits of salt or beams of light.

And what about time?

Another set of experiments suggests that slime molds navigate time as well as space, using a rudimentary internal clock to anticipate and prepare for future changes in their environments. Tetsu Saigusa of Hokkaido University and his colleagues—including Nakagaki—placed a polycephalum in a kind of groove in an agar plate stored in a warm and moist environment (slime molds thrive in high humidity). The slime mold crawled along the groove. Every 30 minutes, however, the scientists suddenly dropped the temperature and decreased the humidity, subjecting the polycephalum to unfavorably dry conditions. The slime mold instinctively began to crawl more slowly, saving its energy. After a few trials, Saigusa and his colleagues stopped changing the slime mold’s environment, but every 30 minutes the amoeba’s pace slowed anyway. Eventually it stopped slowing down spontaneously. Slime molds did the same thing at intervals of 60 and 90 minutes, although, on average, only about half of the slime molds tested showed spontaneous slowing in the absence of an environmental change.

…Somehow, the slime mold may be keeping track of its own rhythmic pulsing, creating a kind of simple clock that would allow it to anticipate future events.

While none of these reports say so directly, it does begin to look like slime molds have a mathematical way about them.

The plasticity of grids, in our heads and otherwise

Familiar mathematical structure is found in the neural activity that governs how the body orients itself and navigates its environment. Grid cells are neurons, found in areas neighboring the hippocampus, whose individual firings line up in a coordinate-like pattern according to an animal’s movement across the full extent of its environment. Their grid pattern acts much like a coordinate system of locations and produces an abstract spatial framework or cognitive map of the environment. Partnered with place cells that code specific locations, border cells that identify the borders of an environment, and head direction cells that encode the direction we are facing, grid cells generate a series of maps of different scales that tell us where we are. John M. O’Keefe, May-Britt Moser and Edvard I. Moser received Nobel Prizes in 2014 for these groundbreaking observations. In an article for the journal Neuron, Neil Burgess makes the following observation:

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.

More recently, researchers have expanded their results a bit by finding evidence that cells in this area of the brain are not entirely devoted to spatial considerations, but may also accomplish other cognitive tasks with similar action. Studies at Princeton University have investigated rats’ responses to sounds of ever-increasing frequencies on the hunch that the continuum of pitch could be compared to movement along a path.  Neuroscientist Dimitriy Aronov and colleagues monitored the neuronal activity of rats, who were taught to increase the pitch of a tone being played over a loudspeaker, by pressing and then releasing a lever when the tone reached some predetermined frequency range. Cells that functioned as place cells or grid cells also fired in sequences as the rats moved through a progression of frequencies, and these were analogous to the sequences produced by their movement across the space of their environment. The significance of these findings for researchers is the light shed on what these firing patterns may have to do with memory and learning. In a Science Daily report, David Tank, one of the paper’s authors, says the following:

The findings suggest that there are common mechanisms in the hippocampal-entorhinal system that can represent diverse sorts of tasks, said Tank, who is also director of the Simons Collaboration on the Global Brain. “The implication from our work is that these brain areas don’t represent location specifically, but rather they can represent other relevant features of the animal’s experience. When those features vary in a continuous way, sequences of neural activation are produced,” Tank said.

Perhaps this is a bit of a leap but it seems to me that these findings also address the notion of a continuum, and might suggest that the generality these mathematics-like cognitive mechanisms possess resembles the kind of generality that purely mathematical ideas eventually come to exhibit.  One could say that the body solved its fundamental need to navigate, to move and know where it is going, with a system that our thoughtful minds took centuries to put together. Our more willful efforts toward broadening our navigational sense perhaps begin with the ancient development of longitude and latitude. This kind of idea becomes more abstract and universal with the Cartesian coordinate system we learn in high school. And this is finally made most abstract with the vector space, which fully generalizes the notion of a coordinate. One could argue that the productive generalities of the brain’s coding mechanisms, or patterns of neuronal firing, foreshadow the generality and efficiency of the mathematical structures that they resemble.

In mathematics, the use of the notion of a coordinate was extended from being the geometric location of a point, to representing any kind of ‘variable’ or ‘parameter’ in countless physical problems. This happens when we make the transition from seeing the ordered pair of numbers (x, y) as the distance and direction from zero on the horizontal axis, and the distance and direction from zero on the vertical axis, to seeing it as an example of an n-tuple that represents any ordered list of numbers. The list can have some finite number of coordinates or it can be infinitely long. It is used to produce the tremendously useful notion of a vector space, which can be defined on that familiar cross of the x axis with the y axis, but has much broader meaning. The number of variables in the list defines the dimension of the space.

Vector spaces are used to address countless physical problems.  The structure produced by our intuitive ideas of space, location and direction are effective in countless non-spatial systems. In its most general terms, a vector space is a collection of objects, that can be added together or multiplied by a number, and that satisfies a set of axioms about their arithmetic. The extent to which this generalization can be made allows the prolific application of these ideas. The collection of objects can be a collection of functions. In physics, the ‘state’ of a physical system is called a ‘state space,’ but a state space is a vector space. Vector spaces are used extensively in physics and in quantum mechanics in particular.

The broadening of a mathematical idea comes with consistently thorough and precise explorations of the meaning of concepts.  And meaning comes from the relations among them, producing thought-filled weaves of ever increasing complexity.  The generalizations of spatial ideas to non-spatial systems, or spatial metaphors that can clarify non-spatial notions, always captures my attention.

The Mind of Arithmetic

I like finding what I believe are productive generalizations, or some sameness across apparently diverse events.  Often the subject of Mathematics Rising are things like the observation that we seem to search for words in patterns that resemble the trajectories of a rat foraging for food, or the possibility that natural language can be described using a graphic language designed to describe quantum mechanical interactions, or the idea that natural selection could look like a physics principle, or even the observation that a mathematical pattern is used in the social, political, and spiritual organization of  a village in Southern Zambia.  It may be that mathematics can help bridge diverse experiences because it ignores the linguistic barriers that distinguish them.

Perhaps working in the other direction, Professor of Philosophy Roy T. Cook nicely argues, in The illegitimate open-mindedness of arithmetic, that arithmetic is open-minded (a fairly human attitude) by mathematical necessity.  This essay appeared on the Oxford University Press blog with whom Mathematics Rising has recently become partnered.

Where does the mind begin?

The slow and steady march toward a more and more precise definition of what we mean by information inevitably begins with Claude Shannon. In 1948 Shannon published The Mathematical Theory of Communication in Bell Labs’ technical journal. Shannon found that transmitted messages could be encoded with just two bursts of voltage – an on burst and an off burst, or 0 and 1 – immediately improving the integrity of transmissions.  But, of even greater significance, this binary code made it possible to measure the information in a message. The mathematical expression of information has led, I suppose inevitably, to the sense that information is some thing, part (if not all) of our reality, rather than just the human experience of learning. Information is now perceived as acting in its various forms in physics as well as cognitive science. Like mathematics, it seems to be the thing that exists inside of us and outside of us. And, I would argue that every refinement of what we mean by information opens the door to significantly altered views of reality, mind, and consciousness.

Constructor Theory, the current work of author and theoretical physicist David Deutsch, drives the point home in its very premise. Deutsch explains that when speaking, for example, information starts as electrochemical signals in the brain that get converted into signals in the nerves, that then become sound waves, that perhaps become the vibrations of a microphone and electricity, and so on…..The only thing unchanged through this series of transformations is the information itself. Constructor Theory is designed to get at what Deutsch calls the “substrate independence of information.”  Biological information like DNA and what he calls the explanatory information produced by human brains, are aligned in Deutsch’s paradigm. Biological information is distinguished from explanatory information by its limits, not by what it is.

Cosmologist Max Tegmark once remarked:

I think that consciousness is the way information feels when being processed in certain complex ways.

In the same lecture, Tegmark redefines ‘observation,’ to be more akin to interaction, which leads to the idea that an observer can be a particle of light as well as a human being. And Tegmark rightly argues that only if we escape the duality that separates the mind from everything else can we find a deeper understanding of quantum mechanics, the emergence of the classical world, or even what measurement actually is.

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.

IIT was not developed with panpsychism in mind. However, in line with the central intuitions of panpsychism, IIT treats consciousness as an intrinsic, fundamental property of reality.  IIT also implies that consciousness is graded, that it is likely widespread among animals, and that it can be found in small amounts even in certain simple systems.

In his book Probably Approximately Correct computer scientist Leslie Valiant brought computational learning to bear on evolution and life in general. And Richard Watson of the University of Southampton, UK added a new observation last year. He argued that genes do not work independently. They work in concert creating networks of connections that are forged through past evolution (since natural selection will reward gene associations that increase fitness). What Watson observed was that the making of connections among genes in evolution parallels the making of neural networks, or networks of associations, built in the human brain for problem solving. In this way, large-scale evolutionary processes look like cognitive processes.  Watson and his colleagues have been able to go as far as create a learning model demonstrating that a gene network makes use of a kind of generalization when grappling with a problem under the pressure of natural selection. Watson’s work was reported on in a New Scientist article by Kate Douglas with the provocative title Nature’s brain: A radical new view of evolution.

I’ve collected these topics here because these developing and novel theories are grounded in the ways one might perceive the behavior of information.  An unexpected question about where a spider might be gathering and storing information is subject of discussion in a recent paper published in Animal Cognition. The paper, from biologists Hilton F. Japyassu and Kevin Laland, was reported on in a Quanta Magazine article this month from Joshua Sokol.

When the spider was confronted with a problem to solve that it might not have seen before, how did it figure out what to do? “Where is this information?”…Is it in her head, or does this information emerge during the interaction with the altered web?

In February, Japyassú and Kevin Laland, an evolutionary biologist at the University of Saint Andrews, proposed a bold answer to the question. They argued in a review paper, published in the journal Animal Cognition, that a spider’s web is at least an adjustable part of its sensory apparatus, and at most an extension of the spider’s cognitive system.

This kind of thinking suggests the notion of  extended cognition (the view that the mind extends beyond the body to include parts of the environment in which the organism is embedded), and embodied cognition (the view that many features of cognition function as parts of the entire body). It seems that infant spiders (who are a thousand times smaller than adult spiders) are able to build webs that are as geometrically precise as the ones built by adult spiders. Japyassu’s work addressed this surprise by suggesting the possibility that spiders can somehow outsource information processing to their webs. But if the web is part of the spider’s cognitive system, then there should be some interplay between the web and the spider’s cognitive state. Experimenters do find evidence of such interplay. For example, if one section of a web is more effective, a spider may enlarge that section in the future. The idea that the spider is making an informed and objective judgment would be the alternative to the idea that there is an intimate connection between the web and the insects’ cognitive state.

But how these things are interpreted does rely on how one defines cognition. Is cognition acquiring, manipulating and storing information, or must it involve interpreting information in some abstract and more familiarly meaningful way?  My own sense is that many discussions addressing information suggest a continuum of actions involving information, and what we mean by information will continue to be the key to unlocking some new conceptual consistency in our sciences.

Number, insight, and the Riemann Hypothesis

The Riemann Hypothesis came to my attention again recently. More specifically I read a bit about the possibility that quantum mechanical measurements may provide a proof of a centuries-old hypothesis and one of mathematics’ most famous enigmas.

Within mathematics itself, without any reference to its physical meaning, the Riemann Hypothesis highlights the kind of surprises that the very notion of number itself has produced.  Riemann saw a connection between a particular function of a complex variable, and the distribution of prime numbers (numbers evenly divisible by only themselves and 1).  These numbers, like 2, 3, 5, 7, 11, 13, … are found among the infinite set of natural numbers. The function somehow associates information about the prime numbers (which emerge from one of the simplest arithmetic ideas) with a function that describes a relation among complex numbers.  Complex numbers are represented by the sum of a real number and an imaginary number.  The imaginary part of the number is a multiple of the imaginary unit i, defined as the square root of -1.  The function is a zeta function, in particular it is the Riemann zeta function.  It is an infinite sum (first introduced as a function of real variables by Leonhard Euler in the 18th century) whose terms are given by


Riemann extended Euler’s idea by letting n be a complex number. With the tools of complex analysis, he found a relationship between the zeros of the zeta function, ( the n values that produced zeros) and the distribution of prime numbers. It then became possible to use these zeros to count prime numbers and to get information about their positions with respect to the natural numbers as well as to each other. Riemann discovered a formula for calculating the number of primes below a given number. It only works, however, when the real part of those complex numbers that produce zeros is 1/2. While it continues to look like the real part of all of these zeros is 1/2, there is no proof that this generalization is true. Improvements in computing programs have nonetheless made it possible to determine that it is true for at least the first 10 trillion zeros.

I should add that the Riemann zeta function fits into an intellectual weave that includes a number of other functions – ones that express related ideas or that produce specified values. This kind of close investigation of ideas, that culminates in so much mathematics, is fundamentally fueled by little more than what functions can reveal about numbers and vice versa. There are so many intriguing things about the multitude of unexpected results produced by these purely abstract investigations. Yet the nature of this wholly symbolic exploration rarely shows up in discussions of science in general, or of cognition and epistemology in particular. When we dig further into the world of numbers and their relations, (and find so much!) what are we actually looking at? Whether we’re looking at our ourselves or at an ideal world that is inaccessible to our senses, it is equally remarkable that we can see it at all.  I feel strongly that mathematics is particularly fertile ground for planting epistemological questions.

The Riemann zeta function is known for being able to answer questions about the spacing or distribution of prime numbers. And prime numbers are important to the creation of new encryption algorithms that rely on them. But the zeta function has also found application in the physical side of our experience – in things like quantum statistical mechanics and nuclear physics.

The natural fit that the Riemann zeta function enjoys in physics is even more surprising. On the School of Mathematics website at the University of Bristol is this:

The University of Bristol has been at the forefront of showing that there are striking similarities between the Riemann zeros and the quantum energy levels of classically chaotic systems.

From a conference in 1996 in Seattle, aimed at fostering collaboration between physicists and number theorists, came early evidence of correlation between the arrangement of the Riemann zeroes and the energy levels of quantum chaotic systems. If this were true it would prove the Riemann hypothesis.

This is a reversal of what usually happens. Mathematical discoveries usually contribute to physical discoveries and not the other way around. But observations in physics have moved so far outside the range of the senses that physicists necessarily rely more and more heavily on mathematical structure. The disciplines, are inevitably more and more entangled.

Natalie Wolchover wrote about these efforts in a recent Quanta Magazine article.

As mathematicians have attacked the hypothesis from every angle, the problem has also migrated to physics. Since the 1940s, intriguing hints have arisen of a connection between the zeros of the zeta function and quantum mechanics. For instance, researchers found that the spacing of the zeros exhibits the same statistical pattern as the spectra of atomic energy levels. In 1999, the mathematical physicists Michael Berry and Jonathan Keating, building on an earlier conjecture of David Hilbert and George Pólya, conjectured that there exists a quantum system (that is, a system with a position and a momentum that are related by Heisenberg’s uncertainty principle) whose energy levels exactly correspond to the nontrivial zeros of the Riemann zeta function…

If such a quantum system existed, this would automatically imply the Riemann hypothesis.

Wolchover explains that physicists who have their eye on the prize are exploring quantum systems described by matrices whose eigenvalues correspond to the system’s energy levels.   A recent paper in Physical Review Letters, authored by Carl Bender of Washington University in St. Louis, Dorje Brody of Brunel University London and Markus Müller of the University of Western Ontario proposed a candidate system.

If physicists do someday nail down the quantum interpretation of the zeros of the zeta function,… this could provide an even more precise handle on the prime numbers than Riemann’s formula does, since matrix eigenvalues follow very well-understood statistical distributions. It would have other implications as well…

A quantum system that models the distribution of primes might provide a simple model of chaos.  This narrative is a nice story about the insightful nature of mathematics  – about the way it can see in.

The geometry of everything

The idea that geometry in Gothic architecture was used to structure ideas, rather than the edifice itself, has come up before here at Mathematics Rising. But I would like to focus a bit more on this today because it illustrates something about mathematics, and mathematics’ potential, that the modern proliferation of information may be obscuring. Toward this end, I’ll begin with a quote from a paper by architect and architectural historian Nelly Shafik Ramzy published in 2015. The paper’s title is The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry.

The medieval geometry of Euclid had nothing to do with the geometry that is taught in schools today; no knowledge of mathematics or theoretical geometry of any kind was required for the construction process of medieval edifices. Using only a compass and a straight-edge, Gothic masons created myriad lace-like designs, making stone hang in the air and glass seem to chant. In a similar manner, although they did not know the recently discovered principles of Fractal geometry, Gothic artists created a style that was based on the geometry of Nature, which contains a myriad of fractal patterns.

..the paper assumes that the Gothic cathedral, with its unlimited scale, yet very detailed structure, was an externalization of a dual language that was meant to address human cognition through its details, while addressing the eye of the Divine through the overall structure, using what was thought to be the divine language of the Universe.

One of the more intriguing aspects of Ramzy’s analysis is the consideration that the brain is optimized to process fractals, suggesting that fractals are more compatible with human cognitive systems. For Ramzy, this possibility may account for the fact that Gothic artists intuitively produced fractal forms despite the absence of any scientific or mathematical basis for understanding them.

Ramzy does a fairly detailed analysis of the geometric features of a number of cathedrals at various locations.

The geometrically defined proportions of the human body, for example, produced by Roman architect and engineer Marcus Vitruvius Pollio in the 1st century BC, and explored again by Leonardo da Vinci in the 15th century, are displayed in the floor plans of Florence Cathedral, as well as Reims Cathedral and Milan Cathedral. These same proportions are also found in the facades of Notre-Dame of Laon, Notre-Dame of Paris, and Amiens Cathedrals. Natural spirals that reflect the Fibonacci series can be seen in patterns that are found in San Marco, Venice, the windows of Chartres Cathedral, as well as those in San Francesco d’Assissi in Palermo, and in the carving on the pulpit of Strasburg Cathedral.

Vesica Pisces was the name given to the figure created by the intersection of two circles with the same radius in such a way that the center of each circle is on the perimeter of the other.  A figure thus created has some interesting geometric properties, and was often used as a proportioning system in Gothic architecture. It can be found in the working lines of pointed arches, the plans of Beauvais Cathedral and Glastonbury Cathedral, and the facade of Amiens Cathedral. Fractal patterns are also seen in windows of Amiens, Milan, Chartres Cathedrals and Sainte-Chapelle, Paris. And the shapes of Gothic vaults are shown to reflect variations of fractal trees in Wales Cathedral, the Church of Hieronymite Monastery, Portugal, Frauenkirche, Munich and Gloucester Cathedral.

These are just a few of the observations contained in Ramzy’s paper.

And according to Ramzy:

Euclidian applications and fractal applications, geometry aimed at reproducing forms and patterns that are present in Nature, were considered to be the underpinning language of the Universe. Medieval theologians believed that God spoke through these forms and it is through such forms that they should appeal to him, thus Nature became the principal book that made the Absolute Truth visible. So, even when they applied the abstract Euclidian geometry, the Golden Mean and the proportional roots, which they found in the proportions of living forms, governed their works.

Geometric principles and mathematical ratios “were thought to be the dominant ratios of the Universe.” The viewer of these cathedrals was meant to participate in the metaphysics that was contained in its geometry.

There is an interesting interplay here of cosmological, metaphysical, and human attributes.   Human body proportions are found in cathedral floor plans, for example, while God is seen as the architect of a universe whose language is grounded in mathematics.   We are expected to read the universe through the cathedral.  Ramzy’s reference to cognition in his discussion of fractal patterns is contained here:

Fractal Cosmology relates to the usage or appearance of fractals in the study of the cosmos.  Almost anywhere one looks in the universe; there are fractals or fractal-like structures. Scientists claimed that even the human brain is optimized to process fractals, and in this sense, perception of fractals could be considered as more compatible with human cognitive system and more in tune with its functioning than Euclidian geometry. This is sometimes explained by referring to the fractal characteristics of the brain tissues, and therefore it is sometimes claimed that Euclidean shapes are at variance with some of the mathematical preferences of human brains.  These theories might actually explain how Gothic artists intuitively produced fractal forms, even though they did not have the scientific basis to understand them.

It may be that these kinds of observations can help us break the categories that are now habitual in our thinking.  These Gothic edifices blend our experiences of God, nature, and ourselves using mathematics.  I would argue that this creates the appearance that mathematics is a kind of collective cognition.  Perhaps we can find useful bridges, that connect the thoughtful and the material, if we focus on these kinds of blends, rather than on the individual disciplines into which they have evolved. Finding a way to more precisely address the relationship between our universes of ideas, and the material world we find around us, will be critical to deepening our insights across disciplinary boundaries.

The shape of things


Both Quanta Magazine and New Scientist reported on some renewed interest in an old idea.  It was an approach to particle physics, proposed by theoretical physicist Geoffrey Chew in the 1960s, that ignored questions about which particles were most elementary and put a major portion of the weight of discovery on mathematics. Chew expected that information about the strong interaction could be derived from looking at what happens when particles of any sort collide. And he proposed S-matrix theory as a substitute for quantum field theory. S-matrix theory contained no notion of space and time. These were replaced by the abstract mathematical properties of the S-matrix, which had been developed by Werner Heisenberg in 1943 as a principle of particle interactions.

New research, with a similarly democratic approach to matter, is concerned with mathematically modeling phase transitions – those moments when matter undergoes a significant transformation.  The hope is that what is learned about phase transitions could tell us quite a lot about the fundamental nature of all matter.  As New Scientist author, Gabriel Popkin, tells us:

Whether it’s the collective properties of electrons that make a material magnetic or superconducting, or the complex interactions by which everyday matter acquires mass, a host of currently intractable problems might all follow the same mathematical rules. Cracking this code could help us on the way to everything from more efficient transport and electronics to a new, shinier, quantum theory of gravity.

Toward this end, in 1944, Norwegian physicist Lars Onsager solved the problem of modeling material that loses magnetism when heated above a certain temperature. While his was a 2-dimensional model, it has none-the-less been used to simulate the flipping of various physical states from the spread of an infectious disease to neuron signaling in the brain. It’s referred to as the Ising model, named for Ernst Ising, who first investigated the idea in his PhD thesis but without success.

In the 1960s, Russian theorist Alexander Polyakov began studying how fundamental particle interactions might undergo phase transitions, motivated by the fact that the 2D Ising model, and the equations that describe the behavior of elementary particles, shared certain symmetries. And so he worked backwards from the symmetries to the equations.

Popkin explains:

Polyakov’s approach was certainly a radical one. Rather than start out with a sense of what the equations describing the particle system should look like, Polyakov first described its overall symmetries and other properties required for his model to make mathematical sense. Then, he worked backwards to the equations. The more symmetries he could describe, the more he could constrain how the underlying equations should look.

Polyakov’s technique is now known as the bootstrap method, characterized by its ability to pull itself up by its own bootstraps and generate knowledge from only a few general properties. “You get something out of nothing,” says Komargodski. Polyakov and his colleagues soon managed to bootstrap their way to replicating Onsager’s achievement with the 2D Ising model – but try as they might, they still couldn’t crack the 3D version. “People just thought there was no hope,” says David Poland, a physicist at Yale University. Frustrated, Polyakov moved on to other things, and bootstrap research went dormant.

This is part of the old idea.  Bootstrapping, as a strategy, is attributed to Geoffrey Chew who, in the 1960’s, argued that the laws of nature could be deduced entirely from the internal demand that they be self-consistent. In Quanta, Natalie Wolchover explains:

Chew’s approach, known as the bootstrap philosophy, the bootstrap method, or simply “the bootstrap,” came without an operating manual. The point was to apply whatever general principles and consistency conditions were at hand to infer what the properties of particles (and therefore all of nature) simply had to be. An early triumph in which Chew’s students used the bootstrap to predict the mass of the rho meson — a particle made of pions that are held together by exchanging rho mesons — won many converts.

The effort gained greater traction again in 2008 when physicist Slava Rychkov and colleagues at CERN decided to use these methods to build a physics theory that didn’t have a Higgs particle. This turned out not to be necessary (I suppose), but the work was productive none-the-less in the development of bootstrapping techniques.

The symmetries of physical systems at critical points are transformations that, when applied, leave the system unchanged. Particularly important are scaling symmetries, where zooming in or out doesn’t change what you see, and conformal symmetries where the shapes of things are preserved under transformations. The key to Polykov’s work was to realize that different materials, at critical points, have symmetries in common.   These bootstrappers are exploring a mathematical theory space, and they seem to be finding that the set of all quantum field theories forms a unique mathematical structure.

What’s most interesting about all of this is that these physicists are investigating the geometry of a ‘theory space,” where theories live, and where the features of theories can be examined. Nima Arkani-Hamed, Professor of physics at the Institute for Advanced Study has suggested that the space they are investigating could have a polyhedral structure with interesting theories living at the corners. It was also suggested that the polyhedral might encompass the amplituhedron – a geometric object discovered in 2013 that encodes, in its volume, the probabilities of different particle collision outcomes.

Wolchover wrote about the amplituhedron in 2013.

The revelation that particle interactions, the most basic events in nature, may be consequences of geometry significantly advances a decades-long effort to reformulate quantum field theory, the body of laws describing elementary particles and their interactions. Interactions that were previously calculated with mathematical formulas thousands of terms long can now be described by computing the volume of the corresponding jewel-like “amplituhedron,” which yields an equivalent one-term expression.

The decades-long effort is the one to which Chew also contributed.  The discovery of the amplituhedron began when some mathematical tricks were employed to calculate the scattering amplitudes of known particle interactions, and theorists Stephen Parke and Tomasz Taylor found a one term expression that could do the work of hundreds of Feynman diagrams that would translate into thousands of mathematical terms. It took about 30 years for the patterns being identified in these simplified expressions to be recognized as the volume of a new mathematical object, now named the amplituhedron. Nima Arkani-Hamed and Jaroslav Trinka published results in 2014.

Again from Wolchover:

Beyond making calculations easier or possibly leading the way to quantum gravity, the discovery of the amplituhedron could cause an even more profound shift, Arkani-Hamed said. That is, giving up space and time as fundamental constituents of nature and figuring out how the Big Bang and cosmological evolution of the universe arose out of pure geometry.

Whatever the future of these ideas, there is something inspiring about watching the mind’s eye find clarifying geometric objects in a sea of algebraic difficulty.  The relationship between mathematics and physics, or mathematics and material for that matter, is a consistently beautiful, captivating, and enigmatic puzzle.

Life in the inanimate

Not too long ago I wrote about entropy, and what has come to be known as Maxwell’s demon – a hypothetical creature, invented in 1871 by James Clark Maxwell.  The creature was the product of a thought experiment meant to explore the possibility of violating the second law of thermodynamics using information to impede entropy (otherwise known as the gradual but inevitable decline of everything into disorder).  The only reason the demon succeeds in stopping the gradual decline of order is that it has information that can be used to rearrange the behavior of molecules, information that we cannot acquire from out perspective.  That post was concerned with the surprising reality of such a creature, as physicists have now demonstrated that it could be made physical, even mechanical, in the form of an information heat engine or in the action of light beams.

I read about the demon again today in a Quanta Magazine article, How Life (and Death) Spring Fr0m Disorder.  Much of the focus of this article concerns understanding evolution from a computational point of view.  But author Philip Ball describes Maxwell’s creature and how it impedes entropy, since it is this action against entropy that is the key to this new and interesting approach to biology, and to evolution in particular.

Once we regard living things as agents performing a computation — collecting and storing information about an unpredictable environment — capacities and considerations such as replication, adaptation, agency, purpose and meaning can be understood as arising not from evolutionary improvisation, but as inevitable corollaries of physical laws. In other words, there appears to be a kind of physics of things doing stuff, and evolving to do stuff. Meaning and intention — thought to be the defining characteristics of living systems — may then emerge naturally through the laws of thermodynamics and statistical mechanics.

In 1944, Erwin Schrödinger approached this idea by suggesting that living organisms feed on what he called negative entropy.  And this is exactly what this new research is investigating – namely the possibility that organisms behave in a way that keeps them out of equilibrium, by exacting work from the environment with which they are correlated, and this is done by using information that they share with that environment (as the demon does).  Without using this information, entropy, or the second law of thermodynamics, would govern the gradual decline of the organism into disorder and it would die.  Schrödinger’s hunch went so far as to propose that organisms achieve this negative entropy by collecting and storing information.  Although he didn’t know how, he imagined that they somehow encoded the information and passed it on to future generations.  But converting information from one form to another is not cost free.  Memory storage is finite and erasing information to gather new information will cause the dissipation of energy. Managing the cost becomes one of the functions of evolution.

According to David Wolpert, a mathematician and physicist at the Santa Fe Institute who convened the recent workshop, and his colleague Artemy Kolchinsky, the key point is that well-adapted organisms are correlated with that environment. If a bacterium swims dependably toward the left or the right when there is a food source in that direction, it is better adapted, and will flourish more, than one that swims in random directions and so only finds the food by chance. A correlation between the state of the organism and that of its environment implies that they have information in common. Wolpert and Kolchinsky say that it’s this information that helps the organism stay out of equilibrium — because, like Maxwell’s demon, it can then tailor its behavior to extract work from fluctuations in its surroundings. If it did not acquire this information, the organism would gradually revert to equilibrium: It would die.

Looked at this way, life can be considered as a computation that aims to optimize the storage and use of meaningful information. And life turns out to be extremely good at it.

This correlation between an organism and its environment is reminiscent of the structural coupling introduced by biologist H.R. Maturana which he characterizes in this way: “The relation between a living system and the medium in which it exists is a structural one in which living system and medium change together congruently as long as they remain in recurrent interactions.”

And these ideas do not dismiss the notion of natural selection.   Natural selection is just seen as largely concerned with minimizing the cost of computation.  The implications of this perspective are compelling.  Jeremy England at the Massachusetts Institute of Technology has applied this notion of adaptation to complex, nonliving systems as well.

Complex systems tend to settle into these well-adapted states with surprising ease, said England: “Thermally fluctuating matter often gets spontaneously beaten into shapes that are good at absorbing work from the time-varying environment.”

Working from the perspective of a general physical principle –

If replication is present, then natural selection becomes the route by which systems acquire the ability to absorb work — Schrödinger’s negative entropy — from the environment. Self-replication is, in fact, an especially good mechanism for stabilizing complex systems, and so it’s no surprise that this is what biology uses. But in the nonliving world where replication doesn’t usually happen, the well-adapted dissipative structures tend to be ones that are highly organized, like sand ripples and dunes crystallizing from the random dance of windblown sand. Looked at this way, Darwinian evolution can be regarded as a specific instance of a more general physical principle governing nonequilibrium systems.

This is an interdisciplinary effort that brings to mind a paper by Virginia Chaitin which I discussed in another post. The kind of interdisciplinary work that Chaitin is describing, involves the adoption of a new conceptual framework, borrowing the very way that understanding is defined within a particular discipline, as well as the way it is explored and the way it is expressed in that discipline.   Here we have the confluence of thermodynamics and Darwinian evolution made possible with the mathematical study of information.  And I would caution readers of these ideas not to assume that the direction taken by this research reduces life to the physical laws of interactions.  It may look that way at first glance.  But I would suggest that the direction these ideas are taking is more likely to lead to a broader definition of life.  In fact there was a moment when I thought I heard the echo of Leibniz’s monads.

You’d expect natural selection to favor organisms that use energy efficiently. But even individual biomolecular devices like the pumps and motors in our cells should, in some important way, learn from the past to anticipate the future. To acquire their remarkable efficiency, Still said, these devices must “implicitly construct concise representations of the world they have encountered so far, enabling them to anticipate what’s to come.”

It’s not possible to do any justice to the nature of the fundamental, living, yet non-material substance that Leibniz called monads, but I can, at the very least, point to a a few things about them.  Monads exist as varying states of perception (though not necessarily conscious perceptions).   And perceptions in this sense can be thought of as representations or expressions of the world or, perhaps, as information. He describes a heirarchy of functionality among them.  Ones mind, for example, has clearer perceptions (or representations) than those contained in the monads that make up other parts of the body.  But, being a more dominant monad, ones mind contains ‘the reasons’ for what happens in the rest of the body.  And here’s the idea that came to mind in the context of this article. An individual organ contains ‘the reasons’ for what happens in its cells, and a cell contains ‘the reasons’ for what happens in its organelles.  The cell has its own perceptions or representations.  I don’t have a way to precisely define ‘the reasons,’ but like the information-driven states of nonequilibrium being considered by physicists, biologists, and mathematicians, this view of things spreads life out.





The difference between dreaming and being awake

I could go on for quite some time about the difference between dreaming and being awake.  I could see myself picking carefully through every thought I have ever had about the significance of dreams, and I know I would end up with a proliferation of questions, rather than a clarification of anything.  But I think that this is how it should be.  We understand so little about our awareness, our consciousness, and what cognitive processes actually produce for us.  All of this comes to mind, at the moment, because I just read an article in Plus Magazine based on a press conference given by Andrew Wiles at the Heidelberg Laureate Forum in September 2016.  Wiles, of course, is famous for having proved Fermat’s Last Theorem in 1995.  The article highlights some of Wiles’ thoughts about what it’s like to do mathematics, and what it feels like when he’s doing it.  When asked about whether he could feel when a mathematical investigation was headed in the right direction, or when things were beginning to ‘harmonize,’ he said

Yes, absolutely. When you get it, it’s like the difference between dreaming and being awake.

If I had the opportunity, I would ask him to explain this a bit because the relationship between dream sensations and waking sensations has always been interesting to me.  The relationship between language and brain imagery, for example, is intriguing.   I once had a dream in which someone very close to me looked transparent.  I could see through him, and I actually said those words in the dream.  I would learn, in due time, that this person was not entirely who he appeared to be.  But what Wiles seems to be addressing is the clarity and the certainty of being awake, of opening ones eyes, in contrast to the sometimes enigmatic narrative of a dream.  This is what it feels like when you begin to find an idea.

Wiles also had a refreshingly simple response to a question about whether mathematics is invented or discovered:

To tell you the truth, I don’t think I know a mathematician who doesn’t think that it’s discovered. So we’re all on one side, I think. In some sense perhaps the proofs are created because they’re more fallible and there are many options, but certainly in terms of the actual things we find we just think of it as discovered.

I’m not sure if the next question in the article was meant as a challenge to what Wiles believes about mathematical discovery, but it seems posed to suggest that the belief held by mathematicians that they are discovering things is a necessary illusion, something they need to believe in order to do the work they’re doing.  And to this possibility Wiles says,

I wouldn’t like to say it’s modesty but somehow you find this thing and suddenly you see the beauty of this landscape and you just feel it’s been there all along. You don’t feel it wasn’t there before you saw it, it’s like your eyes are opened and you see it. (emphasis added)

And this is the key I think, “it’s like your eyes are opened and you see it.”    Cognitive neuroscientists involved in understanding vision have described the physical things we see as ‘inventions’ of the visual brain.  This is because what we see is pieced together from the visual attributes of objects we perceive (shape, color, movement, etc.), attributes processed by particular cells, together with what looks like the computation of probabilities based on previous visual experience.  I believe that questions about how the brain organizes sensation, and questions about what it is that the mathematician explores, are undoubtedly related.   Trying to describe the sensation of ‘looking’ in mathematics (as opposed to the formal reasoning that is finally written down) Wiles says this:

…it’s extremely creative. We’re coming up with some completely unexpected patterns, either in our reasoning or in the results. Yes, to communicate it to others we have to make it very formal and very logical. But we don’t create it that way, we don’t think that way. We’re not automatons. We have developed a kind of feel for how it should fit together and we’re trying to feel, “Well, this is important, I haven’t used this, I want to try and think of some new way of interpreting this so that I can put it into the equation,” and so on.

I think it’s important to note that Wiles is telling us that the research mathematician will come up with some completely unexpected patterns in either their reasoning or their results.   The unexpected patterns in the results are what everyone gets to see.    But that one would find unexpected patterns in ones reasoning is particularly interesting.  And clearly the reasoning and the results are intimately tied.

Like the sound that is produced from the numbers associated with the marks on a page of music,  there is the perceived layer of mathematics about which mathematicians are passionate.   And this is the thing about which it is very difficult to speak.  Yet the power of what this perceived layer is may only be hinted at by the proliferation of applications of mathematical ideas in every area of our lives.

Best wishes for the New Year!

Points of view and a curve


Roger Antonsen came to my attention with a TED talk recorded in 2015 that was posted in November.  Characterized by the statement, “Math is the hidden secret to understanding the world,” it piqued my curiosity. Antonsen is an associate professor in the Department of Informatics at the University of Oslo. Informatics has been defined as the science of information and computer information systems but its broad reach appears to be related to the proliferation of ideas in computer science, physics, and biology that spring from information-based theories. The American Medical Informatics Association (AMIA) describes the science of informatics as:

…inherently interdisciplinary, drawing on (and contributing to) a large number of other component fields, including computer science, decision science, information science, management science, cognitive science, and organizational theory.

Antonsen describes himself as a logician, mathematician, and computer scientist, with research interests in proof theory, complexity theory, automata, combinatorics and the philosophy of mathematics. His talk, however, was focused on how mathematics reflects the essence of understanding, where mathematics is defined as the science of patterns, and the essence of understanding is defined as the ability to change one’s perspective. In this context, pattern is taken to be connected structure or observed regularity. But Antonsen highlights the important fact that mathematics assigns a language to these patterns. In mathematics, patterns are captured in a symbolic language and equivalences show us the relationship between two points of view.  Equalities, Antonsen explains, show us ‘different perspectives’ on the same thing.

In his exploration of the many ways to represent concepts, Antonsen’s talk brought questions to mind that I think are important and intriguing. For example. what is our relationship to these patterns, some of which are ubiquitous? How is it that mathematics finds them? What causes them to emerge in the purely abstract, introspective world of the mathematician? Using numbers, graphics, codes, and animated computer graphics, he demonstrated, for example, the many representations of 4/3. And after one of those demonstrations, he received unexpected applause. The animation showed two circles with equal radii, and a point rotating clockwise on each of the circles but at different rates – one moved exactly 4/3 times as fast as the other. The circles lined up along a diagonal. The rotating point on each circumference was then connected to a line whose endpoint was another dot. The movement of this third dot looked like it was just dancing around until we were shown that it was tracing a pleasing pattern. The audience was clearly pleased with this visual surprise. (FYI, this particular demonstration happens about eight and a half minutes in) Antonsen didn’t expect the applause, and added quickly that what he had shown was not new, that it was known. He explains in his footnotes:

This is called a Lissajous curve and can be created in many different ways, for example with a harmonograph.

These curves emerge from periodicity, like the curves for sine and cosine functions and their related unit circle expressions. In fact it is the difference in the periods described by the rotation of the point around each of Antonsen’s circles (a period being one full rotation) that produced the curve that so pleased his audience.

I believe that Antonsen wanted to make the point that because mathematics brings a language to all of the patterns that emerge from sensation, and because it is driven by the directive that there is always value in finding new points of view, mathematics is a kind of beacon for understanding, everything. About this I wholeheartedly agree. I have always found comfort in the hopefulness associated with finding another point of view, and the powerful presence of this drive in mathematics may be the root of what captivates me about it.  Mathematics makes very clear that there is no limit to the possibilities for creative and careful thought.

But I also think that the way Antonsen’s audience enjoyed a very mathematical thing deserves some comment. They didn’t see the mathematics, but they saw one of the things that the mathematics is about – a shape, a pattern, that emerges from relationship.  And their impulse was to applaud. This tells us something about what we are not accomplishing in most of our math classes.